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Introduction to InvestmentsFINAN 3050
Week 7:
Managing Bond Portfolios (Chp. 10.1-10.3)
Slide 2Week 7
Michael HallingUniversity of Utah
YTM vs. Spot Rates
� Definition of Yield to Maturity (YTM): the yield (constant across time) that gives the observed price; denoted by r (without any subscript)
� Definition of spot rates: the spot rate st is the rate of interest for money held from the present time (0) until time t (measured in years); YTM of zero-
coupon bonds
( ) ( )∑= +
++
=T
tTt
tB
r
FV
r
CP
1 11
( ) ( )∑= +
++
=T
tT
T
t
t
tB
s
FV
s
CP
1 11
Slide 3Week 7
Michael HallingUniversity of Utah
YTM vs. Spot Rates: Example
� Consider a 3-year bond with a coupon rate (annual) of 7%.� Determine the price of the bond if the YTM is 5.48%:
� Determine the price of the bond if the 1-year spot rate s1=5%, the 2-year spot rate s2=5.2% and the 3-year spot rate s3=5.5%
( ) ( ) ( )11.104
055.01
107
052.01
7
05.01
7321=
++
++
+
( ) ( ) ( )11.104
0548.01
107
0548.01
7
0548.01
7321=
++
++
+
Slide 4Week 7
Michael HallingUniversity of Utah
Extracting Spot Rates from Bond Prices
� Assume the following information:
─ There is a 1-year zero-coupon bond trading for $98.─ There is a 2-year, 4%-coupon (annual) bond trading for $102.
� Determine the 1-year and 2-year spot rate (i.e., s1 and s2).
Slide 5Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Bond Pricing Relationships (1)
� Inverse relationship between price and yield (in the following we refer to the
yield to maturity)� An increase in a bond’s yield to maturity results in a smaller price decline than
the price gain associated with a decrease in yield � relationship between price an yield is non-linear (convex)
� The price of long-term bonds are more sensitive to yield changes than prices of
short-term bonds� Price sensitivity is inversely related to a bond’s coupon rate: higher coupon rate
means lower price sensitivity� Price sensitivity is inversely related to the yield to maturity at which the bond is
currently selling
Slide 6Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Bond Pricing Relationships (2)
Slide 7Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration (1)
� A measure of the effective maturity of a bond
� The weighted average of the times until each payment is received, with the
weights proportional to the present value of the payment
� Duration is shorter than maturity for all bonds except zero coupon bonds
� Duration is equal to maturity for zero coupon bonds
� A measure of a bond’s sensitivity to interest rate risk: higher duration means higher sensitivity to interest rate risk
Slide 8Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration (2)
Slide 9Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration (3)
� Calculation:
─D…Duration─ T…Maturity of the bond
─ r...yield to maturity
─Ct…Coupon in period t in $─ FV…Face Value in $
( ) ( )PriceBond
1
PriceBond
1
1
TT
t
tt
rFV
Tr
C
tD+
×++
×=∑=
Slide 10Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration - Example
8% Bond Time years Payment PV of CF (10%) Weight C1 X C4 1 80 72.727 .0765 .0765 2 80 66.116 .0690 .1392 Sum
3
1080 811.420 950.263
.8539 1.0000
2.5617 2.7774
� Calculation of the duration for a 3-year 8% coupon bond. The YTM is 10%.
Slide 11Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration - Exercise
� Calculate the price and the duration for a 3-year zero-coupon bond. The YTM
is 10%.
� Calculate the price and the duration for a 3-year 8% coupon bond. The YTM is 9%.
� Calculate the price and the duration for a 3-year zero-coupon bond. The YTM
is 9%.
Slide 12Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration Price Relationship
� The duration of a bond can be used to approximate its price change if interest rates change.
� Calculation:
─∆P…Change in bond price (i.e., New Price minus Old Price)─∆P/P…Relative change in bond price
─∆r…Change in YTM
( )r
r
D
P
P
DurationModified
∆×+
−=∆
3211 ( )
Prr
DP
DurationModified
×∆×+
−=∆321
1
Slide 13Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration Price Relationship - Example
� Consider a bond with maturity of 30 years, coupon rate of 8% (paid annually)
and a YTM of 9%. Its price is $897.26 and its duration is 11.37 years.
� Question: What will happen to the bond price if the bond’s yield to maturity
increases to 9.1%?
� Calculate the change in YTM: ∆r=9.1%-9%=0.1%
� Change in price equals -$9.36
( )36.9$26.897$
09.01
37.11−=××
+−=∆
0 . 0 0 10 . 0 0 10 . 0 0 10 . 0 0 1P
Slide 14Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration Price Relationship - Exercise
� Consider a 3-year bond with a coupon rate of 8% and a duration of 2.7774. Thebond has a price of $95.026 and a YTM of 10%.
─Consider an increase in YTM by one basis point (i.e., by 0.001 or 0.1%) to 10.1%
• Calculate the resulting percentage change in the bond price
• Approximate the percentage change in the bond price using the bond’s duration
─Repeat the exercise from before for the following situations:• A decrease in YTM by one basis point
• An increase in YTM by 1%
• A decrease in YTM by 1%
� How precise is the approximation using the duration equation in each case?
Slide 15Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Some simple duration rules
� Rule 1: The duration of a zero-coupon bond equals its maturity.
� Rule 2: Holding time to maturity and YTM constant, a bond’s duration is higher when the coupon rate is lower.
� Rule 3: Holding the coupon rate constant, a bond’s duration increases with
time to maturity.
Slide 16Week 7
Michael HallingUniversity of Utah
Application of Duration: Passive Bond Management
� Passive managers take bond prices as fairly set and seek to control only the risk
of their fixed-income portfolios.� Immunization:
─ strategies that investors use to shield their fixed-income portfolios from
exposure to interest rate fluctuations─ especially important for banks, insurance companies and pension funds
that have assets and liabilities, i.e., fixed-income products with different characteristics and different sensitivities to interest rate on both sides of
the balance sheets (inflows and outflows) ─ in this case, immunization means to match the duration of assets and liabilities
Slide 17Week 7
Michael HallingUniversity of Utah
Application of Duration: Immunization – Example (1)
� Consider the following problem:
─ an insurance company faces an obligation (liability) of $19,487 in seven years; the market interest rate equals 10% p.a.
─How can you immunize this liability using a combination of 3-year zero
coupon bonds and a 15-year bond with coupon rate of 2.7%?
� Step 1: calculate the duration of the liability� Step 2: calculate the duration of the asset portfolio
� Step 3: find the combination of bonds on the asset side such that the duration of liabilities matches the duration of the assets
� Step 4: Fully fund the obligation
Slide 18Week 7
Michael HallingUniversity of Utah
Application of Duration: Immunization – Example (2)
� Step 1: calculate the duration of the liability─Duration of the liability equals 7.
� Step 2: calculate the duration of the asset portfolio
─Duration of 3-year zero-coupon equals 3 years─Duration of coupon-paying bond equals 11 years
─ Portfolio duration is the weighted average of duration of each component asset
─w…fraction of the portfolio invested in the zero
─Asset duration = w × 3years + (1-w) × 11years
Slide 19Week 7
Michael HallingUniversity of Utah
Application of Duration: Immunization – Example (3)
� Step 3: find the combination of bonds on the asset side such that the duration of
liabilities matches the duration of the assets─Asset duration = Liability Duration
─w × 3years + (1-w) × 11years = 7 years─w=50%
� Step 4: Fully fund the obligation─ Present value of the liabilities equals $10,000 = 19,478/(1+0.1)7
─We have to invest $10,000 into the asset side• $5,000 into the 3-year zero-coupon bond (we buy 66.5 of these bonds with a face
value of $100 each at a price of $75.13 each)
• $5,000 into the coupon bond (we buy 112.4 of these bonds with a face value of
$100 each at a price of $44.48 each)
Slide 20Week 7
Michael HallingUniversity of Utah
Application of Duration: Immunization – Example (4)
� Check if immunization works: consider a drop in interest rates to 9% on the next day (i.e., there are still 7 years left until liability matures)
─ Present value of liability: $10,660.06 = $19,478/(1+0.09)7
─ Present value of asset portfolio: • Zero-coupon bond: P (FV of 100) = $77.2
• Coupon-paying bond: P (FV of 100) = $49.2
• Portfolio Value = $77.2×66.5 + $49.2×112.4 = $10,671.89
─Gap of -$10,660.06+$10,671.89=$11.4
� This strategy ensures that the average duration of assets and liabilities is the
same � if interest rates change in the short run, both – liabilities and assets –change to the same extent and the assets will be sufficient to cover the liabilities
Slide 21Week 7
Michael HallingUniversity of Utah
Application of Duration: Immunization – Example (5)
� Check if immunization works: consider a drop in interest rates to 9% in one year (i.e., there are only 6 years left until liability matures)
─ Present value of liability: $11,619.5 = $19,478/(1+0.09)6
─ Present value of asset portfolio: • Zero-coupon bond: P (FV of 100; 2 years left until mat.) = $84.2
• Coupon-paying bond: P (FV of 100; 14 years left until mat.) = $50.95
• Portfolio Value = $84.2×66.5 + $50.95×112.4 = $11,328.83
─Gap of -$11,619.5+$11,328.83=-$290.63
� This strategy requires rebalancing, as over one year
─ durations of liabilities and assets change─ present values of liabilities and assets change
Slide 22Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Convexity (1)
� Remember the non-linear
(convex) relationship
between yields and bond
prices.
� Approximating bond price
changes by duration
assumes a linear relationship
� Convexity considers
curvature.
� Do investor like or dislike convexity?
Slide 23Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Convexity (2)
� Remember the approximation of bond price changes using duration (it is linear in r∆):
� The same relationship looks as follows if convexity is considered:
( )r
r
D
P
P
DurationModified
∆×+
−=∆
3211
( )( )2
2
1
1rCXr
r
D
P
P
DurationModified
∆××+∆×+
−=∆
321
Slide 24Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Calculating Convexity
� In the spirit of the duration equation – similar structure.
� Example: convexity for a 3-year 8% coupon bond. The YTM is 10%.
( )( ) ( ) ( ) ( )
+
×+×++
×+×+
= ∑= PriceBond
11
PriceBond
11
1
1
1
2
tT
t
tt
rFV
TTr
C
ttr
CX
8% Bond Time years Payment PV of CF (10%) Weight t × (t+1) × C4 1 80 72.727 .0765 .153 2 80 66.116 .0690 .4176 Sum
3
1080 811.420 950.263
.8539 1.0000
10.2468
10.82 ×××× 1/1.12=8.94
Slide 25Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Convexity - Exercise
� Consider a 3-year bond with a coupon rate of 8%, a duration of 2.7774 and a convexity of 8.94. The bond has a price of $950.263 and a YTM of 10%.
─Consider an increase in YTM by one basis point (i.e., by 0.001 or 0.1%) to 10.1%
• Approximate the percentage change in the bond price using the bond’s duration
AND convexity
─Repeat the exercise from before for the following situations:• A decrease in YTM by one basis point
• An increase in YTM by 1%
• A decrease in YTM by 1%
� Compare your results to the ones from the exercise on slide 14 (approximation
using only duration).
Slide 26Week 7
Michael HallingUniversity of Utah
Interest Rate Risk: Duration Only and Duration-Convexity Approximations – Summary of Results
Change
New
Yield Old Price New Price
Relative
Change
Duration
Appr.
Change
Rel. Dev.
from
Approx.
D-CX
Appr.
Chan.
Rel. Dev.
from
Approx.
0.001 0.101 95.026 94.787 -0.00252 -0.003 -0.18% -0.003 0.00%
-0.001 0.099 95.026 95.267 0.002529 0.003 0.18% 0.003 0.00%
0.01 0.11 95.026 92.669 -0.02481 -0.025 -1.78% -0.025 0.02%
-0.01 0.09 95.026 97.469 0.025702 0.025 1.76% 0.026 0.02%
� This table summarizes results related to the exercises on slide 14 and slide 25.� (Comment: it would be a good exercise to replicate all the results in detail)
Hm
k 1a
1. Y
ou purchased a share of stock for $20. One year later you received $1 as dividend and sold
the share for $24. Your holding-period return w
as __________. 2.
An investor invests 40%
of his wealth in a risky asset w
ith an expected rate of return of 15%
and a variance of 4% and 60%
in a treasury bill that pays 6%. H
er portfolio's expected rate of return and standard deviation are __________ and __________
3. C
onsider the following tw
o investment alternatives. First, a risky portfolio that pays 15%
rate of return w
ith a probability of 60% or 5%
with a probability of 40%
. Second, a treasury bill that pays 6%
. The risk premium
on the risky investment is __________.
4. Y
ou have $500,000 available to invest. The risk-free rate as well as your borrow
ing rate is 8%
. The return on the risky portfolio is 16%. If you w
ish to earn a 22% return, you should
__________. 5.
Risk that can be elim
inated through diversification is called ______ risk 6.
Asset A
has an expected return of 20% and a standard deviation of 25%
. The risk free rate is 10%
. What is the rew
ard-to-variability ratio? 7.
A portfolio is com
posed of two stocks, A
and B. Stock A
has a standard deviation of return of 25%
while stock B
has a standard deviation of return of 5%. Stock A
comprises 20%
of the portfolio w
hile stock B com
prises 80% of the portfolio. If the variance of return on the
portfolio is .0050, the correlation coefficient between the returns on A
and B is __________.
8. The standard deviation of return on investm
ent A is .10 w
hile the standard deviation of return on investm
ent B is .05. If the covariance of returns on A
and B is .0030, the correlation
coefficient between the returns on A
and B is __________.
9. A
portfolio is composed of tw
o stocks, A and B
. Stock A has a standard deviation of return
of 5% w
hile stock B has a standard deviation of return of 15%
. The correlation coefficient betw
een the returns on A and B
is .5. Stock A com
prises 40% of the portfolio w
hile stock B
comprises 60%
of the portfolio. The variance of return on the portfolio is __________. 10. Stocks A
, B, C
and D have betas of 1.5, 0.4, 0.9 and 1.7 respectively. W
hat is the beta of an equally w
eighted portfolio of A, B
and C?
11. Consider the C
APM
. The risk-free rate is 6% and the expected return on the m
arket is 18%.
What is the expected return on a stock w
ith a beta of 1.3? 12. C
onsider the CA
PM. The risk-free rate is 5%
and the expected return on the market is 15%
. W
hat is the beta on a stock with an expected return of 12%
? 13. C
onsider the CA
PM. The expected return on the m
arket is 18%. The expected return on a
stock with a beta of 1.2 is 20%
. What is the risk-free rate?
14. Consider the single factor A
PT. Portfolio A has a beta of 1.3 and an expected return of 21%
. Portfolio B
has a beta of 0.7 and an expected return of 17%. The risk-free rate of return is
8%. If you w
anted to take advantage of an arbitrage opportunity, you should take a short position in portfolio __________ and a long position in portfolio __________.
15. Consider the m
ulti-factor APT w
ith two factors. Portfolio A
has a beta of 0. 5 on factor 1 and a beta of 1.25 on factor 2. The risk prem
iums on the factors 1 and 2 portfolios are 1%
and 7%
respectively. The risk-free rate of return is 7%. The expected return on portfolio A
is __________ if no arbitrage opportunities exist.
16. Security X has an expected rate of return of 13%
and a beta of 1.15. The risk-free rate is 5%
and the market expected rate of return is 15%
. According to the capital asset pricing m
odel, security X
is __________. 17. Y
ou invest $600 in security A w
ith a beta of 1.5 and $400 in security B w
ith a beta of .90. The beta of this form
ed portfolio is __________. 18. Security A
has an expected rate of return of 12% and a beta of 1.10. The m
arket expected rate of return is 8%
and the risk-free rate is 5%. The alpha of the stock is __________.
19. Use the follow
ing to answer questions a-d:
SML
beta
return
15%
10%
5%
01 2
a. W
hat is the expected return on the market?
b. What is the beta for a portfolio w
ith an expected return of 15%
c. W
hat is the expected return for a portfolio with a beta of 0.5?
d. W
hat is the alpha of a portfolio with a beta of 2 and actual return of 15%
? 20. A
ccording to the CA
PM, w
hat is the market risk prem
ium given an expected return on a
security of 13.6%, a stock beta of 1.2, and a risk free interest rate of 4.0%
? 21. U
sing the index model, the alpha of a stock is 4.0%
, the beta if 0.9 and the market return is
10%. W
hat is the residual given an actual return of 15%?
22. The risk premium
for exposure to exchange rates is 5% and the firm
has a beta relative to exchanges rates of 0.4. The risk prem
ium for exposure to the consum
er price index is -6%
and the firm has a beta relative to the C
PI of 0.8. If the risk free rate is 3.0%, w
hat is the expected return on this stock?
23. The small firm
in January effect is strongest ________. 24. Evidence suggests that there m
ay be _______ mom
entum and ________ reversal patterns in
stock price behavior. 25. In a study of investm
ent behavior of men and w
omen, B
arber and Odean find ___
26. Psychological regret theory says ____. 27. M
ental accounting is a form of fram
ing which is consistent w
ith ___ . Solutions 1. 25 percent
2. (
)(
)(
)(
).0800
.04.4
.0960.06
.6.15
.4r
E.5
p p=
=σ
=+
=
3. (
)(
)[
].05
.06-
.05.4
.15.6
=+
=Prem
ium
Risk
4.
()
375,0001-
1.75500,000
751
0816
0822
==
=− −
=
Borrowing
..
..
.y
5. unique, firm
-specific, diversifiable 6.
.40
7. C
orr=.225 C
orr)
05)(.
25)(.
8)(.
2(.2
)05
(.)8
(.)
25(.
)2(.
0050.
22
22
++
=8.
.60(.05)]
.0030/[.10n
Correlatio
==
9.
)5)(.
15)(.
05)(.
6)(.
4(.2
)15
(.)6
(.)
05(.
)4(.
22
22
2+
+=
σ0103
.2=
σ10. .93 11. 21.6%
12. .7
13. 8%
14. short A, long B
; hint: compute the factor’s exp return for each stock
15. 1625.
)07
(.25.1
)01
(.5.
07.
)(
=+
+=
A rE
=16.25%
16. 16.5%
.05)-
1.15(.15.05
benorm
ally
would
)(
=+
x rE
=> overpriced
17. 26.1
)90
(.000,1 400
)5.1
(000,1 600
=+
=p
β
18. 3.7%
19. a. 10% b. ? c. 7.5%
d. ? 20. 8%
21. R
esidual = 15 – (4 + 0.9x10) = 2%
22. 0.2%
23. early in the month
24. short-run, long run 25. that m
en trade more actively than w
omen
26. contends that people have more regret (blam
e themselves m
ore) when a decision that turned
out badly was m
ore unconventional, is consistent with the firm
size anomaly, and is
consistent with the book-to-m
arket anomaly
27. taking a very risky position with one investm
ent account and a very conservative position w
ith another investment account, investors' irrational preference for stocks w
ith high cash dividends, a tendency to ride losing stock positions for too long
Hom
ework 1b – From
BK
M
Ch. 5:
9, 10, 11, C
h. 6: 6, 7, 18, 19
Ch. 7:
2, 3, 4, 6, 7, 28 C
h. 8: 6, 16, 24.
Solutions to book problems from
the Sol Manual:
CH
. 5. 9.
E(rX ) = [0.2 × (–20%)] + [0.5 × 18%
] + [0.3 × 50%)] = 20%
E(rY ) = [0.2 × (–15%
)] + [0.5 × 20%] + [0.3 × 10%
)] = 10%
10. σ
X2 = [0.2 × (–20 – 20) 2] + [0.5 × (18 – 20) 2] + [0.3 × (50 – 20) 2] = 592
σ
X = 24.33%
σY
2 = [0.2 × (–15 – 10) 2] + [0.5 × (20 – 10) 2] + [0.3 × (10 – 10) 2] = 175
σY = 13.23%
11.
E(r) = (0.9 × 20%) + (0.1 × 10%
) = 19%
CH
. 6. Think of your C
APM
project when solving problem
s 6 and 7, we w
ill compute m
eans and st deviation for a variety of portfolios, here w
ith diff proportions in stocks and bonds, they will form
our frontier. Then w
e’ll use some extra form
ulas (that apply only in the case of two-asset portfolios) to help
us identify the min variance portfolio and the tangent one, but the overall objective is to
construct the frontier, and the tangent Capital M
arket Line. 6.
The parameters of the opportunity set are:
E(rS ) = 15%, E(rB ) = 9%
, σB
S = 32%, σ
BB = 23%
, ρ = 0.15, rf = 5.5%
-------------- Y
ou don’t need to know the cov or m
in-var weight calculations but here they are:
From the standard deviations and the correlation coefficient w
e generate the covariance matrix [note
that Cov(rS , rB ) = ρσ
B
S σB
B]:
Bonds
Stocks B
onds529.0
110.4 Stocks
110.4 1024.0
The min-var portf proportions are:
)r,
r(C
ov2
)r,
r(C
ov)S
(w
BS
2B2S
BS
2BM
in−
σ+
σ−
σ=
3142.0
)4.
1102(
5291024
4.110
529=
×−
+−
=
w
Min (B
) = 0.6858
This calc will help us add an extra point to the table below
: 31.42% stocks/68.58%
bond portfolio
-----------------
The mean and standard deviation of the m
inimum
variance portfolio are:
E(rM
in ) = (0.3142 × 15%) + (0.6858 × 9%
) = 10.89%
[
]2 1
BS
BS
2B2B
2S2S
Min
)r,
r(C
ovw
w2w
w+
σ+
σ=
σ
= [(0.3142
2 × 1024) + (0.68582 × 529) + (2 × 0.3142 × 0.6858 × 110.4)] 1/2 = 19.94%
%
in stocks %
in bonds E
xp. return Std dev.
00.00
100.00 9.00
23.00
20.00 80.00
10.20 20.37
31.42
68.58 10.89
19.94 M
inimum
variance 40.00
60.00 11.40
20.18
60.00 40.00
12.60 22.50
70.75
29.25 13.25
24.57 T
angency portfolio 80.00
20.00 13.80
26.68
100.00 00.00
15.00 32.00
7.
Investment opportunity set
for stocks and bonds
min var
B
SC
AL
0 2 4 6 8
10 12 14 16 18
010
2030
Standard Deviation (%
)
40
The graph approximates the points:
E(r)
σ M
inimum
Variance Portfolio
10.89%
19.94%
Tangency Portfolio 13.25%
24.57%
18.
The expected rate of return on the stock will change by beta tim
es the unanticipated change in the m
arket return: 1.2 × (8% – 10%
) = – 2.4%
Therefore, the expected rate of return on the stock should be revised to: 12% – 2.4%
= 9.6%
19. a.
The risk of the diversified portfolio consists primarily of system
atic risk. Beta m
easures systematic
risk, which is the slope of the security characteristic line (SC
L). The two figures depict the stocks'
SCLs. Stock B
's SCL is steeper, and hence Stock B
's systematic risk is greater. The slope of the SC
L, and hence the system
atic risk, of Stock A is low
er. Thus, for this investor, stock B is the riskiest.
b.
The undiversified investor is exposed primarily to firm
-specific risk. Stock A has higher firm
-specific risk because the deviations of the observations from
the SCL are larger for Stock A
than for Stock B
. Deviations are m
easured by the vertical distance of each observation from the SC
L. Stock A
is therefore riskiest to this investor. C
h. 7 2.
a. E(rX ) = 5%
+ 0.8(14% – 5%
) = 12.2%
"X = 14%
– 12.2% = 1.8%
E(rY ) = 5% + 1.5(17%
– 5%) = 18.5%
"Y = 17%
– 18.5% = –1.5%
b. i. For an investor w
ho wants to add this stock to a w
ell-diversified equity portfolio, Kay should
recomm
end Stock X because of its positive alpha, w
hile Stock Y has a negative alpha. In
graphical terms, Stock X
’s expected return/risk profile plots above the SML, w
hile Stock Y’s
profile plots below the SM
L. Also, depending on the individual risk preferences of K
ay’s clients, Stock X
’s lower beta m
ay have a beneficial impact on overall portfolio risk.
ii. For an investor who w
ants to hold this stock as a single-stock portfolio, Kay should
recomm
end Stock Y, because it has higher forecasted return and low
er standard deviation than Stock X
. Stock Y’s Sharpe ratio is: (0.17 – 0.05)/0.25 = 0.48
Stock X’s Sharpe ratio is only: (0.14 – 0.05)/0.36 = 0.25
The market index has an even m
ore attractive Sharpe ratio: (0.14 – 0.05)/0.15 = 0.60
How
ever, given the choice between Stock X
and Y, Y
is superior. When a stock is held in
isolation, standard deviation is the relevant risk measure. For assets held in isolation, beta as a
measure of risk is irrelevant. A
lthough holding a single asset in isolation is not typically a recom
mended investm
ent strategy, some investors m
ay hold what is essentially a single-asset
portfolio (e.g., the stock of their employer com
pany). For such investors, the relevance of standard deviation versus beta is an im
portant issue. 3.
E(rP ) = rf + β[E(rM ) – rf ]
20% = 5%
+ β(15% – 5%
) ⇒ β = 15/10 = 1.5
4. If the beta of the security doubles, then so w
ill its risk premium
. The current risk premium
for the stock is: (13%
- 7%) = 6%
, so the new risk prem
ium w
ould be 12%, and the new
discount rate for the security w
ould be: 12% + 7%
= 19%
If the stock pays a constant dividend in perpetuity, then w
e know from
the original data that the dividend (D
) must satisfy the equation for a perpetuity:
Price = D
ividend/Discount rate
40 = D/0.13 ⇒
D = 40 × 0.13 = $5.20
At the new
discount rate of 19%, the stock w
ould be worth: $5.20/0.19 = $27.37
The increase in stock risk has lowered the value of the stock by 31.58%
. 6.
a. False. β = 0 im
plies E(r) = rf , not zero. b. False. Investors require a risk prem
ium for bearing system
atic (i.e., undiversifiable) risk. c.
False. You should invest 0.75 of your portfolio in the m
arket portfolio, and the remainder in T-
bills. Then: βP = (0.75 × 1) + (0.25 × 0) = 0.75
7. a.
The beta is the sensitivity of the stock's return to the market return. C
all the aggressive stock A and the defensive stock D
. Then beta is the change in the stock return per unit change in the m
arket return. We com
pute each stock's beta by calculating the difference in its return across the tw
o scenarios divided by the difference in market return.
00
.220
532
2A
=− −
=β
70
.020
514
5.3
D=
− −=
β
b. With the tw
o scenarios equal likely, the expected rate of return is an average of the two possible
outcomes: E(rA ) = 0.5 × (2%
+ 32%) = 17%
E(rB ) = 0.5 × (3.5% + 14%
) = 8.75%
B
c. The SM
L is determined by the follow
ing: T-bill rate = 8% w
ith a beta equal to zero, beta for the m
arket is 1.0, and the expected rate of return for the market is:
0.5 × (20% + 5%
) = 12.5%
E(r)
β
8%
12.5%
1.0 2.0 A
SML
M
.7
αD
D
The equation for the security market line is:
E(r) = 8% + β(12.5%
– 8%)
d.
The aggressive stock has a fair expected rate of return of:
E(rA ) = 8% + 2.0(12.5%
– 8%) = 17%
The security analyst’s estimate of the expected rate of return is also 17%
. Thus the alpha for the aggressive stock is zero. Sim
ilarly, the required return for the defensive stock is:
E(rD ) = 8%
+ 0.7(12.5% – 8%
) = 11.15%
The security analyst’s estim
ate of the expected return for D is only 8.75%
, and hence:
α
D = actual expected return – required return predicted by CA
PM
= 8.75%
– 11.15% = –2.4%
The points for each stock are plotted on the graph above.
e. The hurdle rate is determ
ined by the project beta (i.e., 0.7), not by the firm’s beta. The
correct discount rate is therefore 11.15%, the fair rate of return on stock D
. 28. Equation 7.11 applies here:
E(rP ) = rf + βP1 [E(r1 ) − rf ] + β
P2 [E(r2 ) – rf ] W
e need to find the risk premium
for these two factors: γ1 = [E(r1 ) − rf ] and γ2 = [E(r2 ) − rf ]
To find these values, we solve the follow
ing two equations w
ith two unknow
ns:
40% = 7%
+ 1.8γ1 + 2.1γ2
10% = 7%
+ 2.0γ1 + (−0.5)γ2The solutions are: γ1 = 4.47%
and γ2 = 11.86%
Thus, the expected return-beta relationship is: E(rP ) = 7% + 4.47β
P1 + 11.86βP2
Ch. 8.
6. d.
16. a.
The grandson is recomm
ending taking advantage of (i) the small firm
anomaly and (ii) the January
anomaly. In fact, this seem
s to be one anomaly: the sm
all-firm-in-January anom
aly.
b. (i) C
oncentration of one’s portfolio in stocks having very similar attributes m
ay expose the portfolio to m
ore risk than is desirable. The strategy limits the potential for diversification.
(ii) Even if the study results are correct as described, each such study covers a specific tim
e period. There is no assurance that future tim
e periods would yield sim
ilar results.
(iii) A
fter the results of the studies became publicly know
n, investment decisions m
ight nullify these relationships. If these firm
s in fact offered investment bargains, their prices m
ay be bid up to reflect the now
-known opportunity.
24. i.
Mental accounting is best illustrated by Statem
ent #3. Sampson’s requirem
ent that his income
needs be met via interest incom
e and stock dividends is an example of m
ental accounting. M
ental accounting holds that investors segregate funds into mental accounts (e.g., dividends and
capital gains), maintain a set of separate m
ental accounts, and do not combine outcom
es; a loss in one account is treated separately from
a loss in another account. Mental accounting leads to
an investor preference for dividends over capital gains and to an inability or failure to consider total return.
ii. Overconfidence (illusion of control) is best illustrated by Statem
ent #6. Sampson’s desire to select
investments that are inconsistent w
ith his overall strategy indicates overconfidence. O
verconfident individuals often exhibit risk-seeking behavior. People are also more confident
in the validity of their conclusions than is justified by their success rate. Causes of
overconfidence include the illusion of control, self-enhancement tendencies, insensitivity to
predictive accuracy, and misconceptions of chance processes.
iii.
Reference dependence is best illustrated by Statem
ent #5. Sampson’s desire to retain poor
performing investm
ents and to take quick profits on successful investments suggests reference
dependence. Reference dependence holds that investm
ent decisions are critically dependent on the decision-m
aker’s reference point. In this case, the reference point is the original purchase price. A
lternatives are evaluated not in terms of final outcom
es but rather in terms of gains and
losses relative to this reference point. Thus, preferences are susceptible to manipulation sim
ply by changing the reference point.
Extra Questions(no solutions):
4. The Fama-French (1996) m
odel has three factors what are they?
6. What are the day of the w
eek effect, the January effect and the small-size firm
effect? 7. Provide at least one exam
ple for each of the “mental errors” (forecasting errors, overconfidence,
conservatism, representativeness) and “behavioral biases” (fram
ing, mental accounting, regret, loss
aversion)
Hm
k 2 C
h. 3: 2, 3, 4, 6, 8, 12, 13, 14, 20, 21 C
h. 4: 13 C
h. 12: 3, 5, 8, 18, 19 C
h. 17: 6, 11abcd C
h. 3. 2.
a. In principle, potential losses are unbounded, grow
ing directly with increases in the price of
IBM
.
b. If the stop-buy order can be filled at $128, the m
aximum
possible loss per share is $8. If the price of IB
M shares go above $128, then the stop-buy order w
ould be executed, limiting the
losses from the short sale.
3. a.
The stock is purchased for: 300 × $40 = $12,000 The am
ount borrowed is $4,000. Therefore, the investor put up equity, or m
argin, of $8,000. b. If the share price falls to $30, then the value of the stock falls to $9,000. B
y the end of the year, the am
ount of the loan owed to the broker grow
s to:
$4,000 × 1.08 = $4,320
Therefore, the rem
aining margin in the investor’s account is:
$9,000 − $4,320 = $4,680
The percentage m
argin is now: $4,680/$9,000 = 0.52 = 52%
Therefore, the investor will not receive a m
argin call.
c. The rate of return on the investm
ent over the year is:
(Ending equity in the account − Initial equity)/Initial equity
= ($4,680 − $8,000)/$8,000 = −0.415 = −41.5%
4. a.
The initial margin w
as: 0.50 × 1,000 × $40 = $20,000 A
s a result of the increase in the stock price Old Econom
y Traders loses:
$10 × 1,000 = $10,000
Therefore, margin decreases by $10,000. M
oreover, Old Econom
y Traders must pay the
dividend of $2 per share to the lender of the shares, so that the margin in the account
decreases by an additional $2,000. Therefore, the remaining m
argin is:
$20,000 – $10,000 – $2,000 = $8,000
b.
The percentage margin is: $8,000/$50,000 = 0.16 = 16%
So there w
ill be a margin call.
c.
The equity in the account decreased from $20,000 to $8,000 in one year, for a rate of return of:
(−$12,000/$20,000) = −0.60 = −60%
6. a.
The buy order will be filled at the best lim
it-sell order price: $50.25
b. The next m
arket buy order will be filled at the next-best lim
it-sell order price: $51.50
c.
You w
ould want to increase your inventory. There is considerable buying dem
and at prices just below
$50, indicating that downside risk is lim
ited. In contrast, limit sell orders are sparse,
indicating that a moderate buy order could result in a substantial price increase.
8. a.
Initial margin is 50%
of $5,000 or $2,500.
b. Total assets are $7,500 ($5,000 from
the sale of the stock and $2,500 put up for margin).
Liabilities are 100P. Therefore, net worth is ($7,500 – 100P). A
margin call w
ill be issued w
hen:
P
100P
100500,7
$−
= 0.30 ⇒ w
hen P = $57.69 or higher
12. The broker is instructed to attempt to sell your M
arriott stock as soon as the Marriott stock trades
at a bid price of $38 or less. Here, the broker w
ill attempt to execute, but m
ay not be able to sell at $38, since the bid price is now
$37.85. The price at which you sell m
ay be more or less than $38
because the stop-loss becomes a m
arket order to sell at current market prices.
13. a.
55.50
b. 55.25
c.
The trade will not be executed because the bid price is low
er than the price specified in the limit
sell order.
d. The trade w
ill not be executed because the asked price is greater than the price specified in the lim
it buy order. 14.
a. In an exchange m
arket, there can be price improvem
ent in the two m
arket orders. Brokers for
each of the market orders (i.e., the buy and the sell orders) can agree to execute a trade inside the
quoted spread. For example, they can trade at $55.37, thus im
proving the price for both custom
ers by $0.12 or $0.13 relative to the quoted bid and asked prices. The buyer gets the stock for $0.13 less than the quoted asked price, and the seller receives $0.12 m
ore for the stock than the quoted bid price.
b.
Whereas the lim
it order to buy at $55.37 would not be executed in a dealer m
arket (since the asked price is $55.50), it could be executed in an exchange m
arket. A broker for another
customer w
ith an order to sell at market w
ould view the lim
it buy order as the best bid price; the tw
o brokers could agree to the trade and bring it to the specialist, who w
ould then execute the trade.
20. (d)
The broker will sell, at current m
arket price, after the first transaction at $55 or less. 21.
(b) C
h.4. 13.
Start of year NA
V = $20
D
ividends per share = $0.20
End of year N
AV
is based on the 8% price gain, less the 1%
12b-1 fee:
End of year N
AV
= $20 × 1.08 × (1 – 0.01) = $21.384
R
ate of return = 20
$20.0
$20
$384.
21$
+−
= 0.0792 = 7.92%
Ch. 12
3. a.
gk D
P1
0−
=
g16
.02$
50$
−=
⇒
%12
12.0
50$
2$16.0
g=
=−
=
b.
18.
18$
05.0
16.0
2$g
k DP
10
=−
=−
=
The price falls in response to the m
ore pessimistic forecast of dividend grow
th. The forecast for current earnings, how
ever, is unchanged. Therefore, the P/E ratio decreases. The lower P/E
ratio is evidence of the diminished optim
ism concerning the firm
's growth prospects.
5. a.
g = RO
E × b = 0.20 × 0.30 = 0.06 = 6.0%
D
1 = $2(1 – b) = $2(1 – 0.30) = $1.40
33.
23$
06.0
12.0
40.1
$g
k DP
10
=−
=−
=
P/E = $23.33/$2 = 11.67
b.
PVG
O = P0 –
k E0= $23.33 –
66.6
$12
.0 00.2
$=
c.
g = RO
E × b = 0.20 × 0.20 = = 0.04 = 4.0%
D
1 = $2(1 – b) = $2(1 – 0.20) = $1.60
00
.20
$04
.012
.060
.1$
gk D
P1
0=
−=
−=
P/E = $20/$2 = 10.0
PV
GO
= P0 – k E
0= $20.00 –
12.0 00.2
$= $3.33
8. a.
k = rf + β (kM – rf ) = 6%
+ 1.25(14% – 6%
) = 16%
g = (2/3) × 9%
= 6%
D
1 = E0 × (1 + g) × (1 – b) = $3 × 1.06 × (1/3) = $1.06
60
.10
$06
.016
.006
.1$
gk D
P1
0=
−=
−=
b.
Leading P0 /E1 = $10.60/$3.18 = 3.33
Trailing P0 /E0 = $10.60/$3.00 = 3.53
c.
PVG
O = P0 –
k E0
= $10.60 – 15
.8$
16.0 3$
−=
The low
P/E ratios and negative PVG
O are due to a poor R
OE (9%
) that is less than the market
capitalization rate (16%).
d. Now
, you revise the following:
b = 1/3 g = 1/3 × 0.09 = 0.03 = 3.0% D
1 = E0 × 1.03 × (2/3) = $2.06
85.
15$
03.0
16.0
06.2
$g
k DV
10
=−
=−
=
V0 increases because the firm
pays out more earnings instead of reinvesting earnings at a
poor RO
E. This information is not yet know
n to the rest of the market.
18. a.
The value of a share of Rio N
ational equity using the Gordon grow
th model and the
capital asset pricing model is $22.40, as show
n below.
Calculate the required rate of return using the capital asset pricing m
odel:
k = rf + β (kM – rf ) = 4%
+ 1.8(9% – 4%
) = 13%
Calculate the share value using the G
ordon growth m
odel:
40
.22
$12
.013
.0)
12.0
1(20
.0$
gk
g)(1
DP
o0
=−
+×
=−
+×
=
b.
The sustainable growth rate of R
io National is 9.97%
, calculated as follows:
g = b × RO
E = Earnings Retention R
ate × ROE = (1 – Payout R
atio) × ROE =
%97.9
0997.0
35.
270$
16.
30$
16.
30$
20.3
$1
Equity
Beginning
Income
Net
Income
Net ividends
D1
==
×⎟⎠ ⎞⎜⎝ ⎛
−=
×⎟⎠ ⎞⎜⎝ ⎛
−
19. a. To obtain free cash flow
to equity (FCFE), the tw
o adjustments that Shaar should m
ake to cash flow
from operations (C
FO) are:
1. Subtract investment in fixed capital: C
FO does not take into account the investing
activities in long-term assets, particularly plant and equipm
ent. The cash flows
corresponding to those necessary expenditures are not available to equity holders and therefore should be subtracted from
CFO
to obtain FCFE.
2. Add net borrow
ing: CFO
does not take into account the amount of capital supplied to the
firm by lenders (e.g., bondholders). The new
borrowings, net of debt repaym
ent, are cash flow
s available to equity holders and should be added to CFO
to obtain FCFE.
b.
Note 1: R
io National had $75 m
illion in capital expenditures during the year. Adjustm
ent: negative $75 million
The cash flows required for those capital expenditures (–$75 m
illion) are no longer available to the equity holders and should be subtracted from
net income to obtain FC
FE.
Note 2: A
piece of equipment that w
as originally purchased for $10 million w
as sold for $7 m
illion at year-end, when it had a net book value of $3 m
illion. Equipment sales are unusual for
Rio N
ational. Adjustm
ent: positive $3 million
In calculating FCFE, only cash flow
investments in fixed capital should be considered. The $7
million sale price of equipm
ent is a cash inflow now
available to equity holders and should be added to net incom
e. How
ever, the gain over book value that was realized w
hen selling the equipm
ent ($4 million) is already included in net incom
e. Because the total sale is cash, not just
the gain, the $3 million net book value m
ust be added to net income. Therefore, the adjustm
ent calculation is: $7 m
illion in cash received – $4 million of gain recorded in net incom
e = $3 m
illion additional cash received that must be added to net incom
e to obtain FCFE.
Note 3: The decrease in long-term
debt represents an unscheduled principal repayment; there
was no new
borrowing during the year.
Adjustment: negative $5 m
illion The unscheduled debt repaym
ent cash flow (–$5 m
illion) is an amount no longer available
to equity holders and should be subtracted from net incom
e to determine FC
FE.
Note 4: O
n 1 January 2002, the company received cash from
issuing 400,000 shares of com
mon equity at a price of $25.00 per share.
No adjustm
ent Transactions betw
een the firm and its shareholders do not affect FC
FE. To calculate FC
FE, therefore, no adjustment to net incom
e is required with respect to the issuance of
new shares.
Note 5: A
new appraisal during the year increased the estim
ated market value of land
held for investment by $2 m
illion, which w
as not recognized in 2002 income.
No adjustm
ent The increased m
arket value of the land did not generate any cash flow and w
as not reflected in net incom
e. To calculate FCFE, therefore, no adjustm
ent to net income is required.
c.
Free cash flow to equity (FC
FE) is calculated as follows:
FCFE = N
I + NC
C – FC
INV
– WC
INV
+ Net B
orrowing
where
NC
C = non-cash charges
FCIN
V = investm
ent in fixed capital W
CIN
V = investm
ent in working capital
M
illion $ Explanation
NI =
$30.16 From
Exhibit 18B
NC
C =
+$67.17 $71.17 (depreciation and am
ortization from Exhibit 18B
)
– $4.00* (gain on sale from N
ote 2) FC
INV
= –$68.00
$75.00 (capital expenditures from N
ote 1) – $7.00* (cash on sale from
Note 2)
WC
INV
= –$24.00
–$3.00 (increase in accounts receivable from Exhibit 18A
+ –$20.00 (increase in inventory from
Exhibit 18A) +
–$1.00 (decrease in accounts payable from Exhibit 18A
) N
et Borrow
ing = +(–$5.00)
–$5.00 (decrease in long-term debt from
Exhibit 18A)
FCFE =
$0.33
*Supplemental N
ote 2 in Exhibit 18C affects both N
CC
and FCIN
V.
Ch. 17.
6. a.
Actual: (0.70 × 2.0%
) + (0.20 × 1.0%) + (0.10 × 0.5%
) = 1.65%
B
ogey: (0.60 × 2.5%) + (0.30 × 1.2%
) + (0.10 × 0.5%) = 1.91%
U
nderperformance = 1.91%
– 1.65% = 0.26%
b. Security Selection:
Market
Portfolio Perform
ance Index
Performance
Excess Perform
ance M
anager’s Portfolio W
eight C
ontribution
Equity 2.0%
2.5%
-0.5%
0.70
-0.35%
Bonds
1.0%
1.2%
-0.2%
0.20 -0.04%
C
ash 0.5%
0.5%
0.0%
0.10
0.00%
Contribution of security selection:
-0.39%
c.
Asset Allocation:
Market
Actual W
eight B
enchmark
Weight
Excess Weight
Index Return
Minus Bogey
Contribution
Equity 0.70
0.60 0.10
0.59%
0.059%
Bonds
0.20 0.30
-0.10 -0.71%
0.071%
C
ash 0.10
0.10 0.00
-1.41%
0.000%
Contribution of asset allocation:
0.130%
Summ
ary
Security selection -0.39%
A
sset allocation 0.13%
Excess perform
ance -0.26%
11.
a. The spreadsheet below
displays the monthly returns and excess returns for the V
anguard U
.S. Grow
th Fund, the Vanguard U
.S. Value Fund and the S&
P 500. Note that the
inception date for the Vanguard U
.S. Value Fund w
as 6/29/2000.
Monthly R
ates of Return: M
ay 2000 - April 2005
G
rowth
Value
Excess R
eturns M
onth Fund
Fund T-B
ills S&
P500 G
rowth Fund
Value Fund S&
P500 M
ay-00 -5.22
N/A
0.50-2.19
-5.72N
/A -2.69
Jun-00 9.09
N/A
0.492.39
8.61N
/A 1.91
Jul-00 -2.35
N/A
0.51-1.63
-2.86N
/A -2.15
Aug-00 10.01
8.77 0.52
6.079.48
8.24 5.55
Sep-00 -5.96
0.09 0.52
-5.35-6.48
-0.42 -5.86
Oct-00
-5.19 1.59
0.52-0.49
-5.721.07
-1.02N
ov-00 -18.67
-3.87 0.53
-8.01-19.20
-4.40 -8.54
Dec-00
-3.12 6.23
0.500.41
-3.615.74
-0.09
Jan-01 3.88
0.27 0.44
3.463.44
-0.17 3.02
Feb-01 -21.65
-0.45 0.42
-9.23-22.07
-0.87 -9.65
Mar-01
-15.24 -1.99
0.38-6.42
-15.61-2.37
-6.80Apr-01
15.16 4.34
0.337.68
14.834.00
7.35M
ay-01 -0.83
2.30 0.31
0.51-1.14
1.99 0.20
Jun-01 -5.94
-0.17 0.30
-2.50-6.24
-0.47 -2.80
Jul-01 -3.46
1.13 0.30
-1.07-3.75
0.83 -1.37
Aug-01 -9.00
-3.68 0.29
-6.41-9.28
-3.97 -6.70
Sep-01 -12.42
-8.36 0.22
-8.17-12.64
-8.58 -8.40
Oct-01
7.83 -0.68
0.181.81
7.64-0.86
1.63N
ov-01 10.95
8.40 0.16
7.5210.79
8.24 7.36
Dec-01
0.11 1.80
0.140.76
-0.041.66
0.61Jan-02
-5.25 1.68
0.14-1.56
-5.391.54
-1.70Feb-02
-7.18 -0.87
0.15-2.08
-7.33-1.02
-2.22M
ar-02 4.87
4.30 0.15
3.674.72
4.15 3.52
Apr-02 -10.22
-1.35 0.15
-6.14-10.37
-1.49 -6.29
May-02
-2.65 -0.43
0.15-0.91
-2.80-0.57
-1.05Jun-02
-9.30 -7.54
0.14-7.25
-9.45-7.68
-7.39Jul-02
-7.84 -9.18
0.14-7.90
-7.98-9.32
-8.04Aug-02
1.75 1.63
0.140.49
1.611.50
0.35Sep-02
-11.02 -
0.14-11.00
-11.15-10.98
-11.14O
ct-02 9.75
5.74 0.13
8.649.61
5.61 8.51
Nov-02
4.48 5.64
0.105.71
4.385.54
5.60D
ec-02 -8.19
-5.04 0.10
-6.03-8.29
-5.14 -6.13
Jan-03 -2.42
-2.87 0.10
-2.74-2.52
-2.97 -2.84
Feb-03 -0.94
-3.06 0.10
-1.70-1.04
-3.16 -1.80
Mar-03
3.19 -0.79
0.100.84
3.10-0.89
0.74Apr-03
6.77 8.41
0.108.10
6.688.31
8.01M
ay-03 2.90
7.34 0.09
5.092.81
7.25 5.00
Jun-03 0.68
2.05 0.08
1.130.61
1.97 1.05
Jul-03 3.63
0.57 0.08
1.623.55
0.50 1.55
Aug-03 1.53
2.19 0.08
1.791.45
2.11 1.71
Sep-03 -1.08
-0.74 0.08
-1.19-1.16
-0.82 -1.27
Oct-03
5.52 6.57
0.085.50
5.446.49
5.42N
ov-03 0.76
2.29 0.08
0.710.68
2.21 0.63
Dec-03
3.28 4.04
0.085.08
3.203.97
5.00Jan-04
3.04 2.89
0.081.73
2.972.82
1.65Feb-04
-0.06 1.53
0.081.22
-0.141.45
1.14M
ar-04 -1.03
-0.55 0.08
-1.64-1.11
-0.63 -1.72
Apr-04 -2.47
-2.63 0.08
-1.68-2.55
-2.71 -1.76
May-04
2.53 0.16
0.091.21
2.440.08
1.12Jun-04
1.43 2.53
0.111.80
1.322.42
1.69Jul-04
-7.49 -1.91
0.11-3.43
-7.60-2.02
-3.54Aug-04
-0.76 0.97
0.130.23
-0.890.85
0.10Sep-04
2.51 1.53
0.140.94
2.371.39
0.80O
ct-04 1.43
0.16 0.15
1.401.28
0.01 1.25
Nov-04
4.69 5.30
0.183.86
4.525.12
3.68D
ec-04 3.59
3.23 0.19
3.253.40
3.04 3.06
Jan-05 -3.46
-2.33 0.20
-2.53-3.66
-2.52 -2.73
Feb-05 0.77
3.65 0.22
1.890.55
3.43 1.68
Mar-05
-2.86 -2.37
0.23-1.91
-3.09-2.60
-2.15Apr-05
-2.42 -2.65
0.24-3.65
-2.65-2.88
-3.88S
td.Dev.
7.03 4.21
0.154.51
b. The standard deviations for the U
.S Grow
th Fund and the U.S. V
alue Fund are 7.03% and
4.21%, respectively, as show
n in the Excel spreadsheet above.
c. The betas for the U
.S. Grow
th Fund and the U.S. V
alue Fund are 1.432 and 0.799, respectively, as show
n in the Excel spreadsheets below:
GR
OW
TH FU
ND
SUM
MA
RY O
UTPU
T OF EXC
EL REG
RESSIO
N
Regression Statistics
Multiple R
0.919596
R S
quare 0.845657
Adj. R
Square
0.842996
Standard E
rror 2.799924
Observations
60.000000
AN
OV
A
df S
S
MS
F
Significance F
Regression
12491.311496
2491.31317.78651
3.31E-25
Residual
58454.695412
7.83958
Total 59
2946.006908
Coefficients
Std E
rror t S
tat P
-value
Intercept -0.632155
0.36381611-1.7376
0.0875945
S&
P 500
1.4319940.08032921
17.82663.308E
-25
VALU
E FUN
D
SUM
MA
RY O
UTPU
T OF EXC
EL REG
RESSIO
N
Regression Statistics
Multiple R
0.879497
R S
quare 0.773514
Adj. R
Square
0.769396
Standard E
rror 2.021425
Observations
57
AN
OV
A
df
SS
M
S
F S
ignificance F R
egression 1
767.547925767.548
187.84101 2.22E
-19R
esidual 55
224.7386534.08616
Total
56992.286579
Coefficients
Std E
rror t S
tat P
-value
Intercept 0.811917
0.26926013.01536
0.0038794
S&
P500
0.7992230.05831399
13.70552.223E
-19
d.
The formulas for the three m
easures are:
Sharpe:
p
fpσ
rr
−
Treynor:
p
fpβ
rr
−
Jensen:
{})r
r(β
rr
αf
Mp
fp
p−
+−
=
The values for the three m
easures are computed as follow
s:
U
.S. Grow
th Fund U
.S. Value Fund
Sharpe (─
1.16 ─ 0.21)/7.03 = ─
0.1945 (0.61 ─
0.19)/4.21 = 0.0999
Treynor (─
1.16 ─ 0.21)/1.432= ─
0.9551 (0.61 ─
0.19)/0.799 = 0.5184
Jensen ─
1.16 ─ {0.21 + 1.432 × (─
0.31 ─ 0.21)} = ─
0.6322
0.61 ─ {0.19 + 0.799 × (─
0.30 ─ 0.19)} =
0.8181
Note that, in the above calculations, rf and rM are calculated over different tim
e periods for the G
rowth Fund and the V
alue Fund because of the 6/29/2000 inception date of the Value Fund.
Also, there are som
e discrepancies due to rounding.
Hm
k 3 1. U
nderwriting is one of the services provided by w
hom?
2. Under firm
comm
itment underw
riting, who assum
es the full risk that the shares cannot be sold to the public at the stipulated offering price?
3. What is an EC
N?
4. The bid-ask spread exists because of the need for dealers to cover expenses and make a m
odest profit. True/False
5. Consider the follow
ing limit order book of a specialist. The last trade in the stock occurred at a price of
$40. If a market buy order for 100 shares com
es in, at what price w
ill it be filled?
Limit B
uy Orders
Limit Sell O
rdersPrice
SharesPrice
Shares$39.75
100$40.25
100$39.50
100$40.50
100
6. Assum
e you purchased 200 shares of XY
Z comm
on stock on margin at $80 per share from
your broker. If the initial m
argin is 60%, the am
ount you borrowed from
the broker is how m
uch? 7. Y
ou sold short 200 shares of comm
on stock at $50 per share. The initial margin is 60%
. Your initial
investment w
as how m
uch? 8. Y
ou short-sell 200 shares of Tuckerton Trading Co., now
selling for $50 per share. What is your
maxim
um possible loss? W
hat is your maxim
um possible gain?
9. You purchased 100 shares of A
BC
comm
on stock on margin at $50 per share. A
ssume the initial m
argin is 50%
and the maintenance m
argin is 30%. B
elow w
hat stock price would you get a m
argin call? A
ssume the stock pays no dividend and ignore interest on m
argin. 10. B
orrowing a security from
your broker in order to sell it, with the intention of repurchasing it later
when the price is low
er, is called what?
12. The margin requirem
ent on a stock purchase is 15%. Y
ou fully use the margin allow
ed to purchase 100 shares of M
SFT at $35. If the price drops to $32, what is your percentage loss?
13. What are the difference s betw
een active and passively managed m
utual funds? 15. U
nder SEC rules, the m
anagers of certain funds are allowed to deduct charges for advertising,
brokerage comm
issions, and other sales expenses, directly from the fund assets rather than billing
investors. These fees are known as…
..? 19. The Stone H
arbor Fund is a closed-end investment com
pany with a portfolio currently w
orth $300 m
illion. It has liabilities of $5 million and 9 m
illion shares outstanding. If the fund sells for $30 a share, w
hat is its premium
or discount as a percent of NA
V?
21. What is a R
EIT? 22. W
hat is market tim
ing? What funds are likely to be subject to it?
23. You purchased X
YZ stock at $50 per share. The stock is currently selling at $65. Y
our gains could be protected by placing a
A
) lim
it-buy order B) lim
it-sell order C) m
arket order D) stop-loss order
24. The market capitalization rate on the stock of A
berdeen Wholesale C
ompany is 10%
. Its expected RO
E is 12%
and its expected EPS is $5.00. If the firm's plow
-back ratio is 40%, its P/E ratio w
ill be __________.
25. Gagliardi W
ay Corporation has an expected R
OE of 15%
. Its dividend growth rate w
ill be __________ if it follow
s a policy of paying 30% of earning in the form
of dividends. 26. R
ose Hill Trading C
ompany is expected to have EPS in the upcom
ing year of $6.00. The expected RO
E is 18.0%
. An appropriate required return on the stock is 14%
. If the firm has a plow
back ratio of 60%
, its growth rate of dividends should be __________.
27. Rose H
ill Trading Com
pany is expected to have EPS in the upcoming year of $6.00. The expected R
OE
is 18.0%. A
n appropriate required return on the stock is 14%. If the firm
has a plowback ratio of
70%, its intrinsic value should be __________.
28. Cache C
reek Com
pany is expected to pay a dividend of $3.36 in the upcoming year. D
ividends are expected to grow
at 8% per year. The riskfree rate of return is 4%
and the expected return on the m
arket portfolio is 14%. Investors use the C
APM
to compute the m
arket capitalization rate, and the constant grow
th DD
M to determ
ine the value of the stock. The stock's current price is $84.00. U
sing the constant growth D
DM
, the market capitalization rate is __________.
29. Grott and Perrin, Inc. has expected earnings of $3 per share for next year. The firm
's RO
E is 20% and
its earnings retention ratio is 70%. If the firm
's market capitalization rate is 15%
, what is the
present value of its growth opportunities?
30. Annie's D
onut Shops, Inc. has expected earnings of $3.00 per share for next year. The firm's R
OE is
18% and its earnings retention ratio is 60%
. If the firm's m
arket capitalization rate is 12%, w
hat is the value of the firm
excluding any growth opportunities?
32. Cache C
reek Manufacturing C
ompany is expected to pay a dividend of $4.20 in the upcom
ing year. D
ividends are expected to grow at the rate of 8%
per year. The riskfree rate of return is 4%
and the expected return on the market portfolio is 14%
. Investors use the CA
PM to com
pute the m
arket capitalization rate on the stock, and the constant growth D
DM
to determine the
intrinsic value of the stock. The stock is trading in the market today at $84.00. U
sing the constant grow
th DD
M and the C
APM
, the beta of the stock is __________. 33. W
estsyde Tool Com
pany is expected to pay a dividend of $2.00 in the upcoming year. The risk-free
rate of return is 6% and the expected return on the m
arket portfolio is 12%. A
nalysts expect the price of W
estsyde Tool Com
pany shares to be $29 a year from now
. The beta of W
estsyde Tool Com
pany's stock is 1.20. Using a one-period valuation m
odel, the intrinsic value of W
estsyde Tool Com
pany stock today is 34. C
aribou Gold M
ining Corporation is expected to pay a dividend of $6 in the upcom
ing year. Dividends
are expected to decline at the rate of 3% per year. The risk-free rate of return is 5%
and the expected return on the m
arket portfolio is 13%. The stock of C
aribou Gold M
ining Corporation
has a beta of -0.50. Using the constant grow
th DD
M, the intrinsic value of the stock is
__________. 35. The Free cash flow
to the firm is $300m
in perpetuity, the cost of equity equals 14% and the W
AC
C is
10%. If the m
arket value of the debt is $1.0 billion, what is the value of the equity using the free
cash flow valuation approach?
36. The free cash flow to the firm
is reported as $405 million. The interest expense to the firm
is $76 m
illion. If the tax rate is 35% and the net debt of the firm
increased by $50, what is the free cash
flow to the equity holders of the firm
? 37. The free cash flow
to the firm is reported as $205 m
illion. The interest expense to the firm is $22
million. If the tax rate is 35%
and the net debt of the firm increased by $25, w
hat is the market
value of the firm if the FC
FE grows at 2%
and the cost of equity is 11%?
U
se the following to answ
er questions 38-40: In a particular year, Lost H
ope Mutual Fund earned a return of 2%
by making the follow
ing investments in
asset classes:
38. The total excess return on the m
anaged portfolio was __________.
39. The contribution of asset allocation across m
arkets to the total excess return was __________.
40. The contribution of security selection w
ithin asset classes to the total excess return was __________.
41. A portfolio generates an annual return of 13%
, a beta of 0.7 and a standard deviation of 17%. The
market index return is 14%
and a standard deviation of 21%. W
hat is the Treynor measure of the
portfolio if the risk free rate is 5%?
42. A
portfolio generates an annual return of 17%, a beta of 1.2 and a standard deviation of 19%
. The m
arket index return is 12% and a standard deviation of 16%
. What is the Treynor m
easure of the portfolio if the risk free rate is 4%
? 43. A
portfolio generates an annual return of 13%, a beta of 0.7 and a standard deviation of 17%
. The m
arket index return is 14% and a standard deviation of 21%
. What is the Sharpe m
easure of the portfolio if the risk free rate is 5%
? 44. A
portfolio generates an annual return of 17%, a beta of 1.2 and a standard deviation of 19%
. The m
arket index return is 12% and a standard deviation of 16%
. What is the Sharpe m
easure of the portfolio if the risk free rate is 4%
?
45. A portfolio generates an annual return of 13%
, a beta of 0.7 and a standard deviation of 17%. The
market index return is 14%
and a standard deviation of 21%. W
hat is the Jensen measure of the
portfolio if the risk free rate is 5%?
46. A portfolio generates an annual return of 17%
, a beta of 1.2 and a standard deviation of 19%. The
market index return is 12%
and a standard deviation of 16%. W
hat is the Jensen measure of the
portfolio if the risk free rate is 4%?
47. The portfolio that contains the benchm
ark asset allocation against which a m
anager will be m
easured is often called the _____________.
Solutions 1. ib
2. u 3. el com
net
4. t
5. 40.25 or less 6.
7.
400,6
)60
.1)(
80(
200B
orrowing
=−
=000
,6)
60)(.
50(
200Investm
ent=
=
8. Ans: There is no upper lim
it to the price of a share of stock, therefore no upper limit the price you w
ill have to pay to replace the 200 shares of Tuckerton. Tuckerton could go bankrupt w
ith a share price of $0. Y
ou could keep the entire proceeds from the short sale.
Maxim
um gain
= proceeds – m
inimum
possible replacement cost
=
200 ( $50 ) – 200 ( $0 )
=
$10,000
9. A
ns:71.
3530
.1
50.
150
=− −
=⎟⎠ ⎞
⎜⎝ ⎛P
or 2500=.7x100xX
10. sh s
12. Ans: M
argin = 35 x 100 x .15 = 525 Stock value at start = 100 x 35 = 3500 Stock value at end = 100 x 32 = 3200 D
ecrease in value = 300
Percentage loss = 300 / 525 = 57%
15. 12b-1 19. 8.5% d 23. B
24. 11.54 25. 10.5 26. 10.8%
27. $128.57 28. 12%
29. $70 30. $25 32. 0.9 33. $27.39 34. $150 35. 2bn 36. 406m
37. 2,397 38.
Excess R
eturn = .0200 – .0750 = –.0550 39.
(.10 – .50)(.1000) + (.90 – .50)(.0500) = –.0200
40.
(.1100 – .1000).10 + (.0100 – .05).90 = –.0350 41. 0.1143
42. 0.1083 43. 0.4706 44. Sh=0.6842 45. 0.017
46. 0.034 47. B
ogey
Ch 9 – Problem
s: 4, 9, 10, 11, 13, 14, 15, 18, 19, 24, 28, 31, 32, 33, 34, 39bcd L
egend: PVIF(n, r) = 1 / (1+r) n
PVA
F(n, r) = 1/r – 1/[r(1+r) n] FVA
F(n,r)= [(1+r) n-1]/r 4.
The bond price will be low
er. As tim
e passes, the bond price, which is now
above par value, w
ill approach par. 9.
a. The bond pays $50 every six m
onths. Current price:
[$50 × PV
AF(4%
, 6)] + [$1000 × PVIF(4%
, 6)] = $1,052.42
A
ssuming the m
arket interest rate remains 8%
semi=4%
per half year, price six m
onths from now
:
[$50 × PV
AF(4%
, 5)] + [$1000 × PVIF(4%
, 5)] = $1,044.52
b. R
ate of return =
months
six per
%00.4
0400.0
42.
052,1
$90.7
$50
$42.
052,1
$)
42.
052,1
$52.
044,1
($50
$=
=−
=−
+=
8% sem
i 10.
a. From
[$40 × PVA
F(x%, 40)] + [$1000 × PV
IF(x%, 40)] = $950 solve for x.
You w
ill find that the yield to maturity per 6 m
onths is x= 4.26%. This
implies a bond equivalent yield to m
aturity of: 4.26% × 2 = 8.52%
semi
Effective annual yield to m
aturity = (1.0426) 2 – 1 = 0.0870 = 8.70%
b.
Since the bond is selling at par, the yield per 6 months is the sam
e as the sem
i-annual coupon, 4%. The bond equivalent yield to m
aturity is 8%.
Effective annual yield to m
aturity = (1.04) 2 – 1 = 0.0816 = 8.16%
c.
Keeping other inputs unchanged except P = 1050, w
e find a bond equivalent yield to m
aturity of 7.52% sem
i, or 3.76% per 6 m
onths.
Effective annual yield to m
aturity = (1.0376) 2 – 1 = 0.0766 = 7.66%
11. Since the bond paym
ents are now m
ade annually instead of semi-annually, the
bond equivalent yield to maturity is the sam
e as the effective annual yield to m
aturity. The equation is [$80 × PVA
F(x%, 20)] + [$1000 × PV
IF(x%, 20)] = P
The resulting yields for the three bonds are:
Bond Price
Bond equivalent yield =
Effective annual yield $ 950
8.53%
$1000 8.00%
$1050
7.51%
9-1
The yields com
puted in this case are lower than the yields calculated w
ith semi-annual
coupon payments. A
ll else equal, bonds with annual paym
ents are less attractive to investors because m
ore time elapses before paym
ents are received. If the bond price is the sam
e with annual paym
ents, then the bond's yield to maturity is low
er. 13.
Rem
ember that the convention is to use sem
i-annual periods:
Price M
aturity (years)
Maturity
(half-years) Per 6 m
onths Y
TM
Bond equivalent sem
i YTM
$400.00
20.00 40.00
2.317%
4.634%
$500.00 20.00
40.00 1.748%
3.496%
$500.00
10.00 20.00
3.526%
7.052%
$376.89 10.00
20.00 5.000%
10.000%
$456.39
10.00 20.00
4.000%
8.000%
$400.00 11.68
23.36 4.000%
8.000%
14.
Zero 8%
coupon10%
coupon a.
Current prices
$463.19$1000
$1134.20 b.
Price one year from now
$500.25$1000
$1124.94
Price increase $ 37.06
$ 0.00 -$ 9.26
C
oupon income
$ 0.00$80.00
$ 100.00
Income
$ 37.06$80.00
$ 90.74
Rate of R
eturn 8.00%
8.00%
8.00%
15. The reported bond price is: 100 2/32 percent of par = $1,000.625 H
owever, 15 days have passed since the last sem
iannual coupon was paid, so
accrued interest equals: $35 × (15/182) = $2.885 The invoice price is the reported price plus accrued interest: $1003.51
18. The solution is obtained using Excel:
A
B
C
D
E 1
5.50%
coupon bond,
2
m
aturing March 15, 2014
3
Formula in C
olumn B
4
Settlem
ent date 2/22/2006
DA
TE(2006,2,22)
5
Maturity date
3/15/2014D
ATE
(2014,3,15)
6 A
nnual coupon rate 0.055
7
Yield to m
aturity 0.0534
8
Redem
ption value (% of face value)
100
9 C
oupon payments per year
2
10
11
12 Flat price (%
of par) 101.03327
PR
ICE
(B4,B
5,B6,B
7,B8,B
9) 13
Days since last coupon
160C
OU
PD
AY
BS
(B4,B
5,2,1)
9-2
14 D
ays in coupon period 181
CO
UP
DA
YS
(B4,B
5,2,1) 15
Accrued interest
2.43094(B
13/B14)*B
6*100/2
16 Invoice price
103.46393B
12+B15
19. The solution is obtained using Excel:
A
B
C
D
E
F G
1
Sem
iannual A
nnual 2
coupons
coupons 3
4 S
ettlement date
2/22/20062/22/2006
5 M
aturity date
3/15/2014
3/15/2014 6
Annual coupon rate
0.0550.055
7 B
ond price
102
102 8
Redem
ption value (% of face value)
100100
9 C
oupon payments per year
2
1 10
11
Yield to m
aturity (decimal)
0.051927
0.051889 12
13
14
Formula in cell E
11: Y
IELD
(E4,E
5,E6,E
7,E8,E
9) 24. a.
The bond sells for P=$1,124.72 based on the 3.5% yield to m
aturity:
[$40 × PV
AF(3.5%
, 60)] + [$1000 × PVIF(3.5%
, 60)] = $1,124.72
[n = 60; i = 3.5; FV = 1000; PM
T = 40]
Therefore, yield to call is x=3.368%
per 6 months, 6.736%
semi:
[$40 × PVA
F(x%, 10)] + [$1100 × PV
IF(x%, 10)] = $1,124.72
[n = 10; PV = 1124.72; FV
= 1100; PMT = 40]
b.
If the call price were $1050, w
e would set FV
= 1050 and redo part (a) to find that yield to call is x=2.976%
per 6 months, 5.952%
semi. W
ith a lower call
price, the yield to call is lower.
c.
Yield to call is x=3.031%
per 6 months, 6.062%
semi:
[$40 × PVA
F(x%, 4)] + [$1100 × PV
IF(x%, 4)] = $1,124.72
[n = 4; PV
= 1124.72 ; FV = 1100; PM
T = 40] 28. A
pril 15 is midw
ay through the semi-annual coupon period. Therefore, the invoice
price will be higher than the stated ask price by an am
ount equal to one-half of the sem
iannual coupon. The ask price is 101.125 percent of par, so the invoice price is:
$1,011.25 + (1/2 × $50) = $1,036.25
9-3
31. a. (1)
Current yield = C
oupon/Price = 70/960 = 0.0729 = 7.29%
(2)
YTM
= 3.993% per 6 m
onths or 7.986% sem
i bond equivalent yield
(3) R
ealized compound yield is 4.166%
(per 6 months), or 8.332%
semi
bond equivalent yield. To obtain this value, first calculate the future value of reinvested coupons. There w
ill be six payments of $35 each, reinvested
semiannually at a per period rate of 3%
: 35 x FV
AF(6, 3%
) = $226.39 [PV = 0; PM
T = $35; n = 6; i = 3%]
The bond w
ill be selling at par value of $1,000 in three years, since coupon is forecast to equal yield to m
aturity. Therefore, total proceeds in three years w
ill be $1,226.39. To find realized compound yield on a sem
iannual basis (i.e., for six half-year periods), w
e solve:
$960 × (1 + x
realized ) 6 = $1,226.39 ⇒ x
realized = 4.166% (per 6 m
onths)
b.
Shortcomings of each m
easure:
(1)
Current yield does not account for capital gains or losses on bonds
bought at prices other than par value. It also does not account for reinvestment
income on coupon paym
ents.
(2) Y
ield to maturity assum
es that the bond is held to maturity and that all
coupon income can be reinvested at a rate equal to the yield to m
aturity.
(3) R
ealized compound yield (horizon yield) is affected by the forecast of
reinvestment rates, holding period, and yield of the bond at the end of the investor's
holding period. 32. a.
The yield to maturity of the par bond equals its coupon rate, 8.75%
. All else
equal, the 4% coupon bond w
ould be more attractive because its coupon rate is far
below current m
arket yields, and its price is far below the call price. Therefore, if
yields fall, capital gains on the bond will not be lim
ited by the call price. In contrast, the 8.75%
coupon bond can increase in value to at most $1050, offering a m
aximum
possible gain of only 5%
. The disadvantage of the 8.75% coupon bond in term
s of vulnerability to a call show
s up in its higher promised yield to m
aturity.
b. If an investor expects rates to fall substantially, the 4%
bond offers a greater expected return.
c.
Implicit call protection is offered in the sense that any likely fall in yields
would not be nearly enough to m
ake the firm consider calling the bond. In
this sense, the call feature is almost irrelevant.
33. Market conversion value = value if converted into stock = 20.83 × $28 = $583.24
C
onversion premium
= Bond value – m
arket conversion value
= $775 – $583.24 = $191.76
9-4
34. a. The call provision requires the firm
to offer a higher coupon (or higher prom
ised yield to maturity) on the bond in order to com
pensate the investor for the firm
's option to call back the bond at a specified call price if interest rates fall sufficiently. Investors are w
illing to grant this valuable option to the issuer, but only for a price that reflects the possibility that the bond w
ill be called. That price is the higher prom
ised yield at which they are w
illing to buy the bond.
b.
The call option reduces the expected life of the bond. If interest rates fall substantially so that the likelihood of call increases, investors w
ill treat the bond as if it w
ill "mature" and be paid off at the call date, not at the stated m
aturity date. O
n the other hand if rates rise, the bond must be paid off at the m
aturity date, not later. This asym
metry m
eans that the expected life of the bond will be
less than the stated maturity.
c.
The advantage of a callable bond is the higher coupon (and higher promised
yield to maturity) w
hen the bond is issued. If the bond is never called, then an investor w
ill earn a higher realized compound yield on a callable bond issued at
par than on a non-callable bond issued at par on the same date. The
disadvantage of the callable bond is the risk of call. If rates fall and the bond is called, then the investor receives the call price and w
ill have to reinvest the proceeds at interest rates that are low
er than the yield to maturity at w
hich the bond w
as originally issued. In this event, the firm's savings in interest paym
ents is the investor's loss.
39.
b. (3) The yield on the callable bond m
ust compensate the investor for the risk
of call.
Choice (1) is w
rong because, although the owner of a callable bond receives
principal plus a premium
in the event of a call, the interest rate at which he
can subsequently reinvest will be low
. The low interest rate that m
akes it profitable for the issuer to call the bond m
akes it a bad deal for the bond’s holder.
C
hoice (2) is wrong because a bond is m
ore apt to be called when interest
rates are low. There w
ill be an interest saving for the issuer only if rates are low
.
c. (3)
d.
(2)
9-5
Ch 14 Problem
s: 1, 2, 4, 5, 6, 7, 8, 24 C
h 15 Problems 1, 3, 5, 6, 7, 8, 11, 14, 16, 17
CH
14 1.
c is false. This is the description of the payoff to a put, not a call. 2.
c is the only correct statement.
4.
Cost
Payoff Profit
Call option, X
= 85 3.82
5.00 1.18
Put option, X = 85
0.15 0.00
-0.15 C
all option, X = 90
0.40 0.00
-0.40 Put option, X
= 90 1.80
0.00 -1.80
Call option, X
= 95 0.05
0.00 -0.05
Put option, X = 95
6.30 5.00
-1.30 5.
In terms of dollar returns:
Price of Stock Six M
onths From N
ow
Stock price:$80
$100 $110
$120 A
ll stocks (100 shares) 8,000
10,000 11,000
12,000 A
ll options (1,000 shares)
0
0 10,000
20,000 B
ills + 100 options 9,360
9,360 10,360
11,360
In terms of rate of return, based on a $10,000 investm
ent:
Price of Stock Six Months From
Now
Stock price:
$80 $100
$110 $120
All stocks (100 shares)
-20%
0%
10%
20%
All options (1,000 shares)
-100%
-100%
0%
100%
Bills + 100 options
-6.4%
-6.4%
3.6%
13.6%
All options
All stocks
Bills plus options
ST
100
–100 0
– 6.4
Rate of return (%
)
100
110
6.
a. Purchase a straddle, i.e., both a put and a call on the stock. The total cost of the straddle w
ould be: $10 + $7 = $17
b. Since the straddle costs $17, this is the am
ount by which the stock w
ould have to m
ove in either direction for the profit on either the call or the put to cover the investm
ent cost (not including time value of m
oney considerations). 7.
a. Sell a straddle, i.e., sell a call and a put to realize prem
ium incom
e of:
$4 + $7 = $11
b.
If the stock ends up at $50, both of the options will be w
orthless and your profit w
ill be $11. This is your maxim
um possible profit since, at any other
stock price, you will have to pay off on either the call or the put. The stock
price can move by $11 (your initial revenue from
writing the tw
o at-the-m
oney options) in either direction before your profits become negative.
c.
Buy the call, sell (w
rite) the put, lend the present value of $50. The payoff is as follow
s:
Final Payoff
PositionInitial O
utlayS
T < XS
T > X
Long call C
= 7 0
ST – 50
Short put -P = -4
-(50 – ST )
0 Lending
50/(1 + r) (1/4)50
50 Total
7 – 4 + [50/(1 + r) (1/4)] S
TS
T
The initial outlay equals: (the present value of $50) + $3 In either scenario, you end up w
ith the same payoff as you w
ould if you bought the stock itself.
8. a.
By w
riting covered call options, Jones receives premium
income of $30,000.
If, in January, the price of the stock is less than or equal to $45, he will keep
the stock plus the premium
income. Since the stock w
ill be called away from
him
if its price exceeds $45 per share, the most he can have is:
$450,000 + $30,000 = $480,000
(We are ignoring interest earned on the prem
ium incom
e from w
riting the option over this short tim
e period.) The payoff structure is: Stock price
Portfolio value Less than $45
(10,000 times stock price) + $30,000
Greater than $45
$450,000 + $30,000 = $480,000
This strategy offers some prem
ium incom
e but leaves the investor with
substantial downside risk. A
t the extreme, if the stock price falls to zero,
Jones would be left w
ith only $30,000. This strategy also puts a cap on the final value at $480,000, but this is m
ore than sufficient to purchase the house. b.
By buying put options w
ith a $35 strike price, Jones will be paying $30,000 in
premium
s in order to insure a minim
um level for the final value of his
position. That minim
um value is: ($35 × 10,000) – $30,000 = $320,000
This strategy allows for upside gain, but exposes Jones to the possibility of a
moderate loss equal to the cost of the puts. The payoff structure is:
Stock price Portfolio value
Less than $35 $350,000 – $30,000 = $320,000
Greater than $35
(10,000 times stock price) – $30,000
c.
The net cost of the collar is zero. The value of the portfolio will be as
follows:
Stock price Portfolio value
Less than $35 $350,000
Betw
een $35 and $4510,000 tim
es stock price G
reater than $45 $450,000
If the stock price is less than or equal to $35, then the collar preserves the $350,000 in principal. If the price exceeds $45, then Jones gains up to a cap of $450,000. In betw
een $35 and $45, his proceeds equal 10,000 times the
stock price.
The best strategy in this case is (c) since it satisfies the two requirem
ents of preserving the $350,000 in principal w
hile offering a chance of getting $450,000. Strategy (a) should be ruled out because it leaves Jones exposed to the risk of substantial loss of principal.
O
ur ranking is: (1) c
(2) b (3) a
24. a.
Joe’s strategy
Final Payoff
PositionInitial O
utlayS
T < 1200 S
T > 1200Stock index
1200
ST
ST
Long put (X = 1200)
60
1200 – ST
0 Total
1260
1200 S
TProfit = payoff – 1260
-60
ST – 1260
Sally’s Strategy
Final Payoff
PositionInitial O
utlayS
T < 1170 S
T > 1170Stock index
1200
ST
ST
Long put (X = 1170)
45
1170 – ST
0 Total
1260
1170 S
TProfit = payoff – 1245
-75
ST – 1245
Profit
JoeS
ally
-60-75
11701200
ST
b.
Sally does better when the stock price is high, but w
orse when the stock price is
low. (The break-even point occurs at S = $1185, w
hen both positions provide losses of $60.)
c. Sally’s strategy has greater system
atic risk. Profits are more sensitive to the
value of the stock index. 25.
This strategy is a bear spread. The initial proceeds are: $9 – $3 = $6 The payoff is either negative or zero:
Position S
T < 50 50 < S
T < 60S
T > 60
Long call (X = 60)
0 0
ST – 60
Short call (X = 50)
0 – (S
T – 50) – (S
T – 50)
Total 0
– (ST – 50)
–10
B
reakeven occurs when the payoff offsets the initial proceeds of $6, w
hich occurs at a stock price of S
T = $56.
0 S
T50
60
6-10
- 4P
rofit
Payoff
26.
Buy a share of stock, w
rite a call with X
= 50, write a call w
ith X = 60, and buy a call
with X
= 110.
Position S
T < 50 50 < S
T < 60 60 < S
T < 110 S
T > 110
Buy stock
ST
ST
ST
ST
Short call (X = 50)
0 – (S
T – 50) – (S
T – 50) – (S
T – 50) Short call (X
= 60) 0
0 – (S
T – 60) – (S
T – 60) Long call (X
= 110) 0
0 0
ST – 110
Total S
T 50
110 – ST
0
The investor is making a volatility bet. Profits w
ill be highest when volatility is low
so that the stock price ends up in the interval betw
een $50 and $60. C
H15
1. Put values also increase as the volatility of the underlying stock increases. W
e see this from
the parity relationship as follows:
C = P + S
0 – PV(X
) – PV(D
ividends)
Given a value of S and a risk-free interest rate, if C
increases because of an increase in volatility, so m
ust P in order to keep the parity equation in balance.
Num
erical example:
Suppose you have a put w
ith exercise price 100, and that the stock price can take on one of three values: 90, 100, 110. The payoff to the put for each stock price is:
Stock price 90
100110
Put value 10
00
N
ow suppose the stock price can take on one of three alternate values also centered
around 100, but with less volatility: 95, 100, 105. The payoff to the put for each
stock price is: Stock price
95 100
105Put value
5 0
0
The payoff to the put in the low
volatility example has one-half the expected value
of the payoff in the high volatility example.
3. N
ote that, as the option becomes progressively m
ore in the money, its hedge ratio
increases to a maxim
um of 1.0:
X
Hedge ratio
X
H
edge ratio 115
85/150 = 0.567
50
150/150 = 1.000 100
100/150 = 0.667
25 150/150 = 1.000
75 125/150 = 0.833
10
150/150 = 1.000 5.
a. W
hen S = 130, then P = 0.
When S = 80, then P = 30.
The hedge ratio is: [(P
u – Pd )/(uS
0 – dS0 ) = [(0 – 30)/(130 – 80)] = –3/5
b.
Riskless portfolio
S =80 S = 130
3 shares 240
390 5 puts
150 0
Total 390
390
Present value = $390/1.10 = $354.545 c.
Portfolio cost = 3S + 5P = $300 + 5P = $354.545
Therefore 5P = $54.545 ⇒ P = $54.545/5 = $10.91
6. The hedge ratio for the call is: [(C
u – Cd )/(uS
0 – dS0 )] = (20 – 0)/(130 – 80) = 2/5
Riskless portfolio
S =80 S = 130
2 shares 160
260 Short 5 calls
0 -100
Total 160
160
–5C
+ 200 = $160/1.10 = $145.455 ⇒ C
= $10.91 Put-call parity relationship: P = C
– S0 + PV
(X)
$10.91 = $10.91 + ($110/1.10) – $100 = $10.91
7. d1 = 0.3182
N(d1 ) = 0.6248
d2 = –0.0354
N(d2 ) = 0.4859
X
e –rT = $47.56
C = S
0 N(d
1 ) − Xe –rT N
(d2 ) = $8.13
8. P = $5.69
This value is from
our Black-Scholes spreadsheet, but note that w
e could have derived the value from
put-call parity:
P = C – S
0 + PV(X
) = $8.13 – $50 + $47.56 = $5.69 11.
The call price will decrease by less than $1. The change in the call price w
ould be $1 only if: (i) there w
ere a 100% probability that the call w
ould be exercised;
and (ii) the interest rate were zero.
14. The call option w
ith a high exercise price has a lower hedge ratio. The call option
is less in the money. B
oth d1 and N(d1 ) are low
er when X
is higher. 16.
The call option’s implied volatility has increased. If this w
ere not the case, then the call price w
ould have fallen. 17.
The put option’s implied volatility has increased. If this w
ere not the case, then the put price w
ould have fallen.
Introduction to InvestmentsFINAN 3050
Week 7:
Equity Valuation (Chp. 13.2-13.4)
Slide 2Week 7
Michael HallingUniversity of Utah
Intrinsic Value and Market Price
� Problem: determine the value of equity (i.e., of common shares)
� Market Price (P0,MP): consensus value of all potential traders
� Intrinsic Value (P0,IV):
─ Self assigned value
─ Variety of models are used for estimation
� Trading Signal
─ P0,IV > P0,MP: Buy
─ P0,IV < P0,MP: Sell or Short Sell
─ P0,IV = P0,MP: Hold or Fairly Priced
Slide 3Week 7
Michael HallingUniversity of Utah
Realized vs. Expected Holding Period Return (1)
� Remember our definition of HPR (slide 10 of
week 2):
� Useful to evaluate an investment ex-post (i.e., when it is done and all cash-
flows and prices are observable and known).
� If we think about doing an investment and want to evaluate it ex-ante � we
need to look at the Expected HPR!
─ We know the current price P0.
─ We must estimate the expected
dividend payments and the expected
price at the end of the holding period.
( )( ) ( )
MP
t
i
iMPMPt
tP
DEPPE
rE,0
1
,0, ∑=
+−
=
MP
t
i
iMPMPt
tP
DPP
r,0
1
,0, ∑=
+−
=
Slide 4Week 7
Michael HallingUniversity of Utah
Realized vs. Expected Holding Period Return (2)
� Again, two ways to come up with the expected HPR:
─ Market-Oriented: use the CAPM to estimated the return of a stock
required by the market (we refer to this rate as E(k))
─ Look at the firm, estimate future dividends and estimate the expected
HPR (we refer to this rate as E(r))
� Before, we called the difference between E(r) and E(k) the alpha of a stock.
� Trading Signal
─ E(r) > E(k): Buy (positive alpha case)
─ E(r) < E(k): Sell or Short Sell (negative alpha case)
─ E(r) = E(k): Hold or Fairly Priced (zero alpha case)
� Comparing P0,IV and P0,MP or E(r) and E(k) always gives the same signal!
Slide 5Week 7
Michael HallingUniversity of Utah
For a 1-Year Holding Period (1)
� Consider the definition
of the Expected HPR:
� Assume that we know P0,MP, the required rate of return E(k) and have
estimates for E(P1) and E(D1).
� Goal: determine if we should buy or sell this stock
� Two ways to solve this problem:
1. Assume that E(k)=E(r) (i.e., the expected rate of return equals the
required rate of return) and check whether P0,IV=P0,MP (i.e., the
intrinsic value of the stock equals its market price)
2. Assume that P0,IV=P0,MP (i.e., the intrinsic value of the stock equals its
market price) and check whether E(k)=E(r) (i.e., the expected rate of
return equals the required rate of return)
( ) ( )
MP
MPMP
P
DEPPErE
,0
1,0,1
1)(+−
=
Slide 6Week 7
Michael HallingUniversity of Utah
For a 1-Year Holding Period (2)Solution 1
� Solution 1: we assume that E(k)=E(r)
� Then, we calculate the IV out of the equation for the Expected HPR:
� Then we compare the quoted price P0,MP and the calculated intrinsic value
P0,IV:
─ P0,IV > P0,MP : buy
─ P0,IV < P0,MP : sell
─ P0,IV = P0,MP : fairly priced
( ) ( ))(1
,11
,0kE
PEDEP
MP
IV+
+=
( ) ( )
IV
IVMP
P
DEPPEkE
,0
1,0,1)(
+−=
Slide 7Week 7
Michael HallingUniversity of Utah
For a 1-Year Holding Period (3)Solution 2
� Solution 1: we assume that P0,MP=P0,IV
� The, we calculate the expected rate of return using the equation for the
Expected HPR:
� Then we compare the expected rate of return E(r) and the required rate of
return E(k) (could come from the CAPM):
─ E(r) > E(k): Buy (positive alpha case)
─ E(r) < E(k): Sell or Short Sell (negative alpha case)
─ E(r) = E(k): Hold or Fairly Priced (zero alpha case)
( ) ( )
MP
MPMP
P
DEPPErE
,0
1,0,1)(
+−=
Slide 8Week 7
Michael HallingUniversity of Utah
For a 1-Year Holding Period (4)Example
� You expect the price of IBX stock to be E(P1,MP)=$59.77 per share a year from
now. Its current market price is P0,MP=$50, and you expect it to pay a
dividend one year from now of E(D1)=$2.15 per share. The stock has a beta
of 1.15, the risk-free rate is 6% per year, and the expected rate of return on
the market portfolio is 14% per year. Determine if this stock is a good
investment!
� Solution 1:
─ Determine E(k): E(k) = 6%+1.15×(14%-6%) = 15.2%
─ Determine P0,IV = ($2.15+$59.77)/1.152 = $53.75
─ Compare P0,IV and P0,MP: $53.75 > $50 � BUY!
Slide 9Week 7
Michael HallingUniversity of Utah
For a 1-Year Holding Period (5)Example
� You expect the price of IBX stock to be E(P1,MP)=$59.77 per share a year from
now. Its current market price is P0,MP=$50, and you expect it to pay a
dividend one year from now of E(D1)=$2.15 per share. The stock has a beta
of 1.15, the risk-free rate is 6% per year, and the expected rate of return on
the market portfolio is 14% per year. Determine if this stock is a good
investment! (same problem as on the previous slide)
� Solution 2:
─ Determine E(r): E(r) = ($59.77-$50+$2.15)/$50 = 0.2384
─ Determine E(k): E(k) = 6%+1.15×(14%-6%) = 15.2%
─ Compare E(r) and E(k): 23.84% > 15.2% � BUY!
Slide 10Week 7
Michael HallingUniversity of Utah
For a 1-Year Holding Period (6)Exercise
� Consider again the problem discussed on the previous two slides. What does
the expected price of the stock in one year, i.e. E(P1,MP), has to be to make
the stock be priced fairly?
Slide 11Week 7
Michael HallingUniversity of Utah
A More General Model: Dividend Discount Model (1)Motivation
� Consider the 1-year definition of the intrinsic value:
� What is E(P1,MP)? Assume that the stock sells for its intrinsic value in one year,
i.e. in one year P1,IV=P1,MP. We can estimate P1,IV and plug it into the equation
for P0,IV.
( ) ( ))(1.11
,0kE
PEDEP MP
IV+
+=
( ) ( ) ( ) ( ))(1.22
,1,1kE
PEDEPEPE MPIVMP
+
+==
( )( )
( ) ( )( )( )2
221,0
11 kE
PEDE
kE
DEP MP
IV+
++
+=
Slide 12Week 7
Michael HallingUniversity of Utah
A More General Model: Dividend Discount Model (2)Equation
� If you continue to substitute for expected future market prices (i.e.,
for E(P2,MP), E(P3,MP), etc.) indefinitely you get:
� Description:
─ P0,IV…Intrinsic value of stock at t=0
─ E(k)...required return/discount rate
─ E(Dt)…expected dividend payment in period t
( )( )
( )( )( )
( )( )( )
...111 3
3
2
21,0 +
++
++
+=
kE
DE
kE
DE
kE
DEP IV
Slide 13Week 7
Michael HallingUniversity of Utah
A More General Model: Dividend Discount Model (3)Equation
� How to use this model? � we need more assumptions on future dividends
� Assumption 1: constant dividend � E(D1)=E(D2)=…=E(D)
� Assumption 2: dividend that grows at the constant rate g
)(
)(,0
kE
DEP IV =
( )( )( )gkE
DEP IV
−= 1
,0
( ) ( )gDDE +×= 101
( ) ( )202 1 gDDE +×=
Slide 14Week 7
Michael HallingUniversity of Utah
A More General Model: Dividend Discount Model (4)Example
� High Flyer Industries has just paid its annual dividend of D0=$3 per share.
The dividend is expected to grow at a constant rate of g=8% until infinity.
The beta of High Flyer stock is 1.0, the risk-free rate is 6%, and the market
risk premium is 8%.
� What is the intrinsic value of the stock?
─ E(k)=6%+1.0×8%=14%
─ P0,IV=$54
� What is the intrinsic value if the beta changes to 1.25?
─ E(k)=6%+1.25×8%=16%
─ P0,IV=$40.50
( )( )
54$08.014.0
08.013$,0 =
−
+×=IVP
Slide 15Week 7
Michael HallingUniversity of Utah
A More General Model: Dividend Discount Model (5)Exercise
� IBX’s stock dividend at the end of this year is expected to be E(D1)=$2.15,
and it is expected to grow at g=11.2% per year forever. If the required rate
of return on IBX stock is E(k)=15.2% per year, what is its current intrinsic
value, P0,IV?
� What is IBX next year’s expected intrinsic value, P1,IV?
� If an investor were to buy IBX stock now and sell it after receiving the $2.15
dividend a year from now, what is the expected holding-period return?
Slide 16Week 7
Michael HallingUniversity of Utah
A More General Model: Dividend Discount Model (6)Some More Remarks
� Constant Growth Dividend Discount Model:
─ Only valid if g is less than k
─ What happens if g is larger than k? Why can this not be the case?
� Implications of the constant growth DDM: a stock’s value will be greater
─ The larger the expected dividend per share
─ The lower the market capitalization rate, k
─ The higher the expected growth rate of dividends, g
Slide 17Week 7
Michael HallingUniversity of Utah
Multistage Growth Dividend Discount Models (1)
� Consider a firm that quickly grows in its early years and then, once it matures,
yields a dividend with constant growth until infinity.
─ Dividends in the early years: $.32, $.41, $.50, $.60
─ Constant growth of dividends thereafter: g=9.375%
─ The market capitalization rate (i.e., E(k)) equals 13.1%.
� The current intrinsic value is
( )4
,4
432,0131.1131.1
60$.
131.1
50$.
131.1
41$.
131.1
32$. MP
IV
PEP ++++=
Slide 18Week 7
Michael HallingUniversity of Utah
Multistage Growth Dividend Discount Models (2)
� Determine E(P4,MP): assume that the stock trades for its intrinsic value in four
years; thus we must calculate the intrinsic value of a constant growth
dividend discount model in four years
� Plug E(P4,MP) back into our equation for P0,IV:
( ) ( ) ( )( )
62.17$09375.0131.0
09375.0160$.,4,4 =
−
+×== IVMP PEPE
08.12$131.1
62.17$
131.1
60$.
131.1
50$.
131.1
41$.
131.1
32$.4432,0 =++++=IVP
Slide 19Week 7
Michael HallingUniversity of Utah
Multistage Growth Dividend Discount Models (3)Exercise
� Consider the firm of the previous example. Now assume that its beta equals
1.0 (the risk-free rate is 5% and the market risk premium is 6%).
─ Calculate the new discount rate.
─ Determine the current intrinsic value of this company.
Slide 20Week 7
Michael HallingUniversity of Utah
Determination of Dividend Growth Rate (1)
� Problem: how to come up with an estimate of dividend growth rate g
� Solution: look at the firm’s earnings (rather profits) and investment strategy;
basically a firm has two choices to spend its profits
1. Reinvest them into the firm
2. Pay them out as dividends
3. A combination of (1) and (2)
Slide 21Week 7
Michael HallingUniversity of Utah
Determination of Dividend Growth Rate (2)Equation
� The dividend growth rate depends on the reinvestment rate and on the return
generated on these reinvestments.
� Description:
─ b…earnings reinvestment rate, earnings retention rate
─ ROE…return on equity; rate of return earned on reinvestments
─ g...dividend growth rate
bROEg ×=
Slide 22Week 7
Michael HallingUniversity of Utah
Determination of Dividend Growth Rate (3)Example
� Consider a firm with $100 million of assets (all equity-financed) and 3 million
shares outstanding. The Return on Equity (ROE) equals 15%. Earnings
account for $15 million or $5 per share.
─ Management decides to reinvest b=60%. i.e., 0.6×15 = $9 million
─ The capital stock of the firm increases by $9 million, i.e., 9% (given the
initial capital of $100 million).
─ Endowed with 9% more capital the firm earns 9% more income per
year and dividends grow by 9% per year.
%960.0%15 =×=×= bROEg
Slide 23Week 7
Michael HallingUniversity of Utah
Determination of Dividend Growth Rate (4)Different Retention Rates
� What is the influence of different retention rates on the firm value? If two
firms are identical but one reinvests 60% and the other one reinvests 0%
which one is going to have a higher firm value?
� Answer: this depends on the profitability of the reinvestment, i.e., compare
the ROE to the required rate of return as implied by the CAPM.
� Example: Consider a firm with $100 million of assets (all equity-financed)
and 3 million shares outstanding. The Return on Equity (ROE) equals 15%.
The risk-adjusted discount rate implied by the CAPM is k=12.5%. Earnings
account for $15 million or $5 per share. Management decides to reinvest
b=60%.
Slide 24Week 7
Michael HallingUniversity of Utah
Determination of Dividend Growth Rate (5)Example
� From slide 22 we know that if b=60% then g=9%.
� The value of the firm in this case is:
� If the firm does not reinvest, there is no dividend growth and the value is:
( ) ( )14.57$
09.0125.0
2$1,0 =
−=
−=
gk
DP IV
40$125.0
5$10 ===
k
DP
( ) 2$6.015$1 =−×=D
Slide 25Week 7
Michael HallingUniversity of Utah
Determination of Dividend Growth Rate (6)Exercise
� Consider the firm from before. Now assume that the ROE equals 12.5%. Does
an earnings retention ratio (i.e., b or reinvestment ratio) of 60% increase firm
value relative to the case of 0% earnings retention ratio? Explain your result.
� Consider the firm from before. Now assume that the ROE equals 10%. Does
an earnings retention ratio (i.e., b or reinvestment ratio) of 60% increase firm
value relative to the case of 0% earnings retention ratio? Explain your result.
Fin
ance 3
050
Hom
ework
#1
Sprin
g 2
007
Consid
er the fo
llow
ing 3
investm
ents:
Investm
ent
E(R
)F
A.1
9.2
158
Cov(a,b
) = .0
014562
B.1
0.0
632
Cov(a,c) =
-.0055285
C.1
2.1
404
Cov(b
,c) = .0
034125
1.
On g
raph p
aper, p
lot th
e three in
vestm
ents. U
se F o
n th
e x-ax
is, and E
(R) o
n th
e y-ax
is.
2.
Calcu
late the p
ortfo
lio E
(R) an
d F
for each
of th
e follo
win
g co
mbin
ations:
10%
A - 9
0%
B10%
A - 9
0%
C10%
B - 9
0%
C
50%
A - 5
0%
B50%
A - 5
0%
C50%
B - 5
0%
C
90%
A - 1
0%
B90%
A - 1
0%
C90%
B - 1
0%
C
Plo
t each p
oin
t on th
e grap
h
3.
Fin
d th
e min
imum
varian
ce portfo
lio p
roportio
ns fo
r each tw
o asset p
ortfo
lio,
(A,B
); (A,C
); (B,C
). The m
inim
um
varian
ce portfo
lio can
be fo
und b
y th
e form
ula
Where
W1
= p
roportio
n o
f the p
ortfo
lio in
investm
ent 1
, and
(1
- W1 )=
pro
portio
n o
f the p
ortfo
lio in
investm
ent 2
Calcu
late the E
(R) an
d F
for each
min
imum
varian
ce portfo
lio. P
lot each
varian
ce- min
imizin
g
portfo
lio o
n th
e grap
h.
Draw
a curv
e show
ing th
e portfo
lio p
ossib
ilities for each
of th
e three p
airs
(A,B
); (A,C
); (B,C
). Use a d
ifferent co
lor fo
r each cu
rve.
4.
Suppose in
vesto
rs could
only
invest in
A,B
,C o
r som
e com
bin
ation o
f any tw
o (b
ut n
ot all
three).
a)Id
entify
on y
our g
raph a seg
men
t where all in
vesto
rs would
wan
t to select a p
ortfo
lio.
b)
Mig
ht an
y in
vesto
r wan
t to in
vest in
A alo
ne? W
hat ab
out B
? What ab
out C
?
c)W
ould
investo
rs wan
t to in
vest in
any co
mbin
ation o
f A&
B?, A
&C
?, B&
C?
Fin
ance 3
050
Hom
ework
#2
S
prin
g 2
007
I.V
isit the w
ebsite h
ttp://fin
ance.y
ahoo.co
m an
d g
ather th
e follo
win
g in
form
ation fo
r Pfizer,
Inc, (tick
er sym
bol: P
FE
):
1.
The clo
sing p
rice for P
FE
’s com
mon sto
ck o
n T
uesd
ay. 4
/10/0
7
2.
PFE
’s curren
t annual d
ivid
end
3.
PFE
’s EP
S fo
r last yea
r
4.
The cu
rrent (as o
f 4/1
0/0
7) an
alysts’ co
nsen
sus estim
ate of P
FE
’s EP
S g
row
th fo
r the
nex
t five y
ears (per an
num
)
II.A
ssum
ing th
e follo
win
g in
form
ation:
E(r
m) =
12.5
.0%
Rf =
5.2
5%
PFE
’s beta =
0.7
0*
III.C
alculate th
e follo
win
g:
a.B
ased o
n p
art II, what rate o
f return
should
PFE
’s shareh
old
ers require o
n th
eir investm
ent?
b.
If PFE
retains 6
2%
of its earn
ings, an
d can
earn a retu
rn o
f 10%
on rein
vested
funds, at w
hat
rate can d
ivid
ends g
row
indefin
itely?
c.B
ased o
n y
our an
swer to
(a) and in
form
ation o
btain
ed in
part I, if P
FE
’s div
iden
d (P
art I,
item 2
) is expected
to g
row
indefin
itely at th
e rate found in
part (b
), what is a sh
are of P
FE
stock
worth
today
? (A
ssum
e that th
e div
iden
d y
ou fo
und in
part I is D
0 )
d.
By h
ow
much
is PFE
stock
overp
riced o
r underp
riced w
hen
com
parin
g y
our an
swer to
part
(c) with
the cu
rrent p
rice (part I, item
1)?
e.W
hat is P
FE
’s div
iden
d y
ield?
f.If P
FE
’s earnin
gs g
row
at the estim
ated g
row
th rate fo
r 5 y
ears (item I, p
art 5), w
hat w
ill EP
S
be at th
e end o
f 2011 (b
egin
with
the E
PS
for ‘0
6 [p
art 1, item
3])?
g.
Usin
g y
our an
swer to
(f), and assu
min
g th
at PFE
’s (trailing) P
/E ratio
is 11 at th
e end o
f
2011, w
hat sh
ould
a share o
f stock
sell for at th
at time?
h.
Assu
me th
at div
iden
ds will n
ot grow
for th
e nex
t 5 y
ears. If you b
uy a sh
are of P
FE
stock
at the cu
rrent p
rice (part I, item
1), receiv
e the cu
rrent d
ivid
end (p
art I, item 2
) for each
of
the n
ext 5
years, th
en sell th
e stock
for th
e price fo
und in
(g), w
hat w
ill be y
our d
ollar-
weig
hted
averag
e annual rate o
f return
(Calcu
late IRR
usin
g P
0 , D1 -D
5 and P
5 )?
*Ig
nore an
y actu
al beta fo
r PFE
that y
ou m
ay fin
d in
yah
oo o
r any o
ther so
urce.
Finance 3
050
Midterm
#1
Part I - P
roblem Solving
1.a.
Suppose y
ou sell sh
ort 3
00 sh
ares of JN
J at $55. Y
ou put in
$8,250 of y
our o
wn m
oney
for m
argin purposes. If th
e price o
f JNJ falls to
$47, h
ow m
uch do you gain
or lo
se as a
percen
tage of your original investm
ent?
b.
If the p
rice of JN
J falls to $45 an
d a d
ividend of $
1/sh
are is paid
, what is th
e % m
argin in
your acco
unt?
2.
Municip
al bond A pays in
terest at the rate o
f 3.5%, w
hile C
orporate b
ond B pays in
terest
at the rate o
f 5.50%. If L
ynette p
ays tax
at a marg
inal rate o
f 36%, w
hich
bond sh
ould
she ch
oose to
invest in
? Support y
our an
swer w
ith fig
ures.
3.
Suppose y
ou buy 400 sh
ares of C
SCO at $
25.00 usin
g $5,000 of y
our o
wn m
oney an
d
borro
wing th
e other $
5,000 on m
argin. If th
e main
tenance m
argin m
ust b
e 25%, h
ow lo
w
can th
e price o
f CSCO go befo
re a marg
in call is issu
ed? (Ig
nore in
terest)
Midterm
#2
1.
If inflatio
n in
a certain eco
nomy is ex
pected
to be 1
50%, w
hat n
ominal rate o
f interest
must b
e charg
ed to
earn a real rate o
f return of 8
%?
2Douglas earn
s the fo
llowing rates o
ver fo
ur y
ears of in
vestin
g: r
1 = .2
2, r
2 = .1
2, r
3 = -.0
8,
r4 =
.09. W
hat is th
e geometric av
erage o
f these retu
rns?
3.
Suppose y
ou m
anage a p
ortfo
lio with
E(r) =
16% an
d F = 30% T
he risk
-free rate is 2%.
a.If a clien
t of y
ours ch
ooses to
invest 6
5% of h
er wealth
in your p
ortfo
lio an
d th
e rest in t-
Bills, w
hat w
ould be th
e E(r) an
d F of h
er complete p
ortfo
lio?
b.
Ignorin
g part (a) ab
ove, if y
our clien
t wanted
to ach
ieve an
E(r) o
f 10% on her allo
cation
portfo
lio, w
hat p
roportio
n of h
er wealth
would sh
e need
to in
vest in
your p
ortfo
lio?
Final E
xam
1.
A sto
ck is ex
pected
to pay a d
ividend of $
1.48/sh
are at the en
d of each
of th
e next th
ree
years. A
t that tim
e, the sto
ck is ex
pected
to have y
ear-end earn
ings o
f $5.95 an
d a P
/E ratio
of
19. If in
vesto
rs require a retu
rn of 1
1% on th
is stock, h
ow m
uch sh
ould a sh
are be w
orth
today?
2.
Fairv
iew In
dustrial P
roducts In
c. Is expected
to have y
ear-end earn
ings (E
1 )of
$7.85/sh
are. Fairv
iew in
tends to
plowback
60% of earn
ings o
n which
a return (R
OE) o
f 12% is
expected
. What is th
e PVGO fo
r Fairv
iew if sh
areholders req
uire a 1
0% retu
rn on th
eir
investm
ent?
3.
BMC eq
uipment in
c. has ju
st paid
a dividend (D
0 ) of $
3.28. D
ividends are ex
pected
to
grow at a rate o
f 5.2% in
defin
itely. If B
MC is co
rrectly (fairly
) priced
at $31.95, w
hat is th
e
mark
et capitalizatio
n rate (k
) for th
is stock?
4.
Suppose y
ou buy one p
ut o
ptio
n co
ntract (1
00 sh
ares) on W
hitb
ury co
. with
an ex
ercise
price o
f $40 an
d six
months to
expiratio
n. T
he cu
rrent p
rice of th
e stock is $
44.35 an
d
the p
remium on th
e optio
n is $
.68. If th
e price o
f the sto
ck is $
38.75 on th
e expiratio
n
date o
f the o
ptio
n, w
hat is y
our p
rofit (o
r loss) in
dollars an
d as a p
ercent o
f your o
riginal
investm
ent (all p
rices given are p
er share)?
5.
Find th
e valu
e of a call o
ptio
n with
the fo
llowing ch
aracteristics:
Today’s p
rice of th
e underly
ing sto
ck (S
0 ):$50
Price o
f the sto
ck in
one y
ear (S1 ):
$65 or $
37
Exercise p
rice of th
e call optio
n (X
):$45
Risk
-free interest rate (r
f ):3.5%
1st H
omew
ork Assignm
ent
Due on
September 16
th before class
Important inform
ation (read carefully): a.
The first page of the homew
ork that you turn in must be the standardized C
over Page (i.e., the first page of this docum
ent). Clearly indicate on your cover sheet
which exercises you solved and are ready to present in class (these are the
exercises which are graded).
b. M
ake a copy of your solutions before you turn them in.
c. Show
your calculations. You m
ight get (partial) credit for wrong solutions if
your way of approaching the question m
akes sense. No credit w
ill be given to correct solutions that are not accom
panied by the appropriate calculations. Problem
1 A
T-bond expires in 4 years (assume that the bond w
as just issued). It has a coupon rate of 8%
, a face value of $1000 and pays coupons annually. The asked price is quoted as 102:10 w
hile the bid price is 101:20. a.
What is the asked price expressed in dollar term
s? b.
What is the bid price expressed in dollar term
s? c.
If you hold the bond until maturity, w
hat is going to be your holding period return? W
hat is going to be the annualized holding period return? d.
What is the internal rate of return if you invest into the bond?
Solution Problem 1
a. The asked price is P(asked)=(102+10/32)%
of $1000=$1023.125 b.
The bid price is (101+20/32)% of $1000 = $1016.25
c. Price at m
aturity = 1000; intermediate C
Fs are 4*80=320; price at which you
bought the bond is the ask price: HPR
=(1000+320)/1023.125 – 1=0.29; the annualized H
PR is (1+0.29)^0.25-1 = 0.0658
d. The bond pays 4 paym
ents of $80 every year in addition to the par of $1000 in the end of the fourth year. The price of the bond is your answ
er to part a., $1023.125. Y
ou need to find the interest rate that this cash flow stream
provides (N=4, PV
= -1023.125, FV
=1000, PMT
=80, P/YR
=1, I/YR
=?). The answ
er is 0.0731. A
lternatively, you can specify the CFs explicitly: C
Fj=-1023.125, CFj=80,
CFj=80, C
Fj=80, CFj=1080, IR
R/Y
R=???. T
he answer is 0.0731.
- 1 -
Problem 2
a. Y
ou own $2847. The interest rate is 6.43%
p.a. and compounding happens daily
(assume that there are 250 business days per year). W
hat is your future value after 3 years?
b. N
ow again you ow
n $2847 and you want to invest them
for 20 years. What m
ust be the interest rate such that you end up w
ith exactly $5000 after 20 years if com
pounding happens yearly? c.
Again you start out w
ith $2847. The interest rate is 4.65% p.a.. H
ow long do you
have to
invest your
money
in order
to get
$3500 if
there is
monthly
compounding?
Solution Problem 2
a. $2847*(1+6.43%
/250)^(250*3)=$3452.65 (N=250*3=750; PV
=-2847; I/Y
R=6.43 (enter like this); P/Y
R=250; FV
=?) b.
I/YR
=2.856% (N
=20; P/YR
=1; FV=5000; I/Y
R=?) or 2847*(1+r)^20=5000;
r=(5000/2847)^(1/20)-1=0.02856 c.
N=53.4 (I/Y
R=4.65; FV
=3500; P/YR
=12; N=?); 2847*(1+0.0465/12)^N
=3500; N
=ln(3500/2847)/ln(1+0.0465/12)=53.4 months
Problem 3 to 4: The follow
ing information is relevant for solving Problem
3 to 4. C
onsider the following price and dividend history for stock X
YZ:
Year
Beginning-of-Y
ear PriceD
ividend Paid at Year-End
2002 $100
$4 2003
$110 $4
2004 $90
$4 2005
$95 $4
An investor buys three shares of X
YZ at the beginning of 2002, buys another tw
o shares at the beginning of 2003, sells one share at the beginning of 2004, and sells all four rem
aining shares at the beginning of 2005. Problem
3: a.
What are the arithm
etic and geometric average 1-year holding period returns of
this investment?
b. W
hat is the 2-years holding period return in 2004 (i.e., the 2-years holding period return in the period 2002 to 2004)?
c. W
hat is the 3-years holding period return of this investment in 2005?
Solution Problem 3
- 2 -
(A) The tim
e-weighted average returns are based on year-by-year rates of return.
Year
Return = (C
apital Gain+D
ividend)/price 2002-2003
(110-100+4)/100=14.00%
2003-2004 (90-110+4)/110=-14.55%
2004-2005
(95-90+4)/90=10.00%
Arithm
etic Mean: (14%
-14.55%+10%
)/3=3.15%
Geom
etric Mean: [(1+0.14)*(1-0.1455)*(1+0.1)]^(1/3)-1=2.33%
(B
) (90-100+8)/100 = -0.02 (C
) (95-100+12)/100 = 0.07 Problem
4: W
hat is the investor’s internal rate of return? (Hint: C
arefully prepare a table with all the
cash flows for the four dates.)
Solution Problem 4
Prepare a table of Cash-Flow
s Tim
e C
ash-Flow
Explanation 0
-300 B
uy three shares at 100 1
-208 B
uy another two shares at 110 plus dividend incom
e on three shares held
2 90+5*4=110
Dividends on 5 shares plus sale of one share at $90
3 4*95+4*4=396
Dividend on four shares, plus sale of four shares at 95
Calculate the internal rate of return: -300+-208/(1+r)^1+110/(1+r)^2+396/(1+r)^3=0
(P/YR
=1; CFj=-300; C
Fj=-208; CFj=110; C
Fj=396; IRR
/YR
=?) IR
R=-0.166%
Problem
5 to 8: The following inform
ation is relevant for solving Problem 5 to 8.
Assum
e that you manage a risky portfolio, called “Y
our Fund”, with an expected rate of
return of 17% and a standard deviation of 27%
. The T-Bill rate (i.e., the risk-free rate) is
7%. The risky portfolio includes the follow
ing investments in the given proportions:
Stock A
27%Stock B
33%Stock C
40% Problem
5:
- 3 -
a. Y
our client chooses to invest 70% of a portfolio in “Y
our Fund” and 30% in the
T-Bill. W
hat is the expected return and standard deviation of your client’s portfolio?
b. W
hat are the investment proportions of your client’s overall portfolio, i.e. your
client’s position in stock A, stock B
, stock C and the T-B
ill? c.
What is the rew
ard-to-variability ratio (S) of “Your Fund” and your client’s
overall portfolio? d.
Draw
the CA
L of “Your Fund” on an expected return/standard deviation diagram
. W
hat is the slope of the CA
L? Show the position of your client on your fund’s
CA
L. Solution Problem
5 a.
E(rp)=0.3*7%+0.7*17%
=14%; SD
p=0.7*27%=18.9%
b.
T-Bills = 30%
; Stock A = 0.7*27%
=18.9%; Stock B
= 0.7*33%=23.1%
; Stock C
= 0.7*40%=28.0%
c.
Your S = (17-7)/27=0.3704; your client’s S = (14-7)/18.0 = 0.3704
d.
E(r)
σ
7
27
14 17P
CA
L ( slope=.3704)%
%
18.9
client
Problem
6: Suppose the sam
e client from the previous problem
decides to invest in “Your Fund” a
proportion y of his total investment budget so that his overall portfolio w
ill have an expected rate of return of 15%
. a.
What is the proportion y?
b. W
hat are your client’s investment proportions in your three stocks and the T-B
ill? c.
What is the standard deviation of the rate of return on your client’s portfolio?
- 4 -
Solution Problem 6
a. E(rp)=(1-y)*7%
+y*17%=15%
; 15%-7%
=y*(-7%+17%
); y=(15-7)/10=0.8 b.
T-Bills = 20%
; Stock A = 0.8*27%
=21.6%; Stock B
= 0.8*33%=26.4%
; Stock C
= 0.8*40%=32.0%
c.
SDp=0.8*27%
=21.6%
Problem 7:
Suppose the same client from
the previous problem decides this tim
e to invest in “Your
Fund” a proportion y of his total investment budget so that the rate of return of his overall
portfolio will have a standard deviation of 20%
. a.
What is the investm
ent proportion y? b.
What is the expected rate of return on your client’s portfolio?
Solution Problem 7
a. 20%
=y*27%; y=20%
/27%; y=0.7407
b. E(rp)=(1-0.7407)*7%
+0.7407*17%=14.407%
Problem
8: Y
ou estimate that a passive portfolio invested to m
imic the S&
P 500 stock index yields an expected rate of return of 13%
with a standard deviation of 25%
. Draw
the CM
L and “Y
our Fund’s” CA
L on an expected return/standard deviation diagram.
a. W
hat is the slope of the CM
L? b.
Characterize in one short paragraph the advantage of “Y
our Fund” over the passive fund.
Solution Problem 8
a. Slope of the C
ML: (13-7)/25=0.24 (scan the graph from
the solution manual)
b. M
y fund allows an investor to achieve a higher expected rate of return for any
given standard deviation than would a passive strategy.
Problem 9 to 10: The follow
ing information is relevant for solving Problem
9 to 10. Im
agine that the returns of the two stocks, A
mgen and C
oca Cola, w
hich caught your attention w
hen you last flipped through the WSJ depend on the overall m
arket conditions in the follow
ing way:
Am
gen would m
ake a return of -20% (i.e. a loss) if m
arkets are bearish, 18%, if they are
normal, and 50%
, if markets are bullish. The corresponding num
bers are -15%, 20%
, and 10%
for Coca C
ola. The probability of Bear M
arkets is 20%, of N
ormal M
arkets is 50%,
and of Bull M
arkets is 30%. This inform
ation is summ
arized in the table below:
- 5 -
B
ear Market
Norm
al Market
Bull M
arket Probability
0.2 0.5
0.3 R
eturn of Am
gen -20%
18%
50%
R
eturn of Coca C
ola -15%
20%
10%
Problem
9 W
hat are the expected returns and standard deviations of the returns of Am
gen and Coca
Cola?
Solution Problem 9: expected return
Expected Return of A
mgen=E
[rA] = (Probability of B
ear Market)*(R
eturn of Am
gen in B
ear Market) + (Probability of N
ormal M
arket)*(Return of A
mgen in N
ormal M
arket) + (Probability of B
ull Market)*(R
eturn of Am
gen in Bull M
arket) = = 0.2 *(-20%
)+0.5*(18%)+0.3*(50%
)=0.2 *(-0.2)+0.5*(0.18)+0.3*(0.50)= 20% = 0.2
Expected Return of C
oca Cola=E
[rC
C]=(Probability of Bear M
arket)*(Return of C
oca C
ola in Bear M
arket) + (Probability of Norm
al Market)*(R
eturn of Coca C
ola in Norm
al M
arket) + (Probability of Bull M
arket)*(Return of C
oca Cola in B
ull Market) =
= 0.2 *(-15%)+0.5*(20%
)+0.3*(10%) = 0.2 *(-0.15)+0.5*(0.20)+0.3*(0.10) = 10%
= 0.1 Solution Problem
9: standard deviations V
ariance[rA] = (Probability of B
ear Market) * ( R
eturn of Am
gen in Bear M
arket -E
[rA]) 2 + (Probability of N
ormal M
arket) * ( Return of A
mgen in N
ormal M
arket -E
[rA]) 2 + (Probability of B
ull Market) * ( R
eturn of Am
gen in Bull M
arket -E[rA
]) 2 = = (0.2) * (-0.2-0.2) 2 + (0.5)*(0.18-0.2) 2+(0.3)*(0.5-0.2) 2
= (0.2)*(-0.4) 2 + (0.5)*(-0.02) 2+(0.3)*(0.3) 2
= 0.0592 (in decimals) or 592 if you used percentages
SD is the square root of the variance: 0.0592^(1/2)= 0.2433 (or 24.33%
) V
ariance[rCC
] = (Probability of Bear M
arket)*( Return of C
oca Cola in B
ear Market -
E[rC
C] ) 2
+ (Probability of Norm
al Market)*( R
eturn of Coca C
ola in Norm
al Market -E
[rCC
]) 2
+ (Probability of Bull M
arket)*( Return of C
oca Cola in B
ull Market -E
[rCC
]) 2
= (0.2)*(-0.15-0.1) 2 + (0.5)*(0.2-0.1) 2+(0.3)*(0.1 – 0.1) 2
= (0.2)*(-0.25) 2 + (0.5)*(0.1) 2+(0.3)*(0) 2
= 0.0175 (in decimals) or 175 if you used percentages
SD is the square root of the variance: 0.0175^(1/2)= 0.1323 (or 13.23%
) Problem
10 A
ssume that of your $10,000 portfolio you invest $9000 in Stock A
mgen and $1000 in
Coca C
ola.
- 6 -
a. C
alculate the returns on your portfolio for every market condition, i.e., B
ear M
arket, Norm
al Market and B
ull Market.
b. W
hat is the expected return on your portfolio? c.
What is the standard deviation of the returns on your new
ly formed portfolio?
Solution Problem 10
The weight on A
mgen in this portfolio is w
(Am
gen) = 9/10 = 0.9. The weight on C
oca C
ola is w(C
oca Cola) = 1/10 = 0.1.
a. The returns in the different m
arket conditions are: i.
Bear M
arket: -19.5%
ii. N
ormal M
arket: 18.2%
iii. B
ull Market: 46%
b.
E[rP]= w(A
mgen)*E[rA
] + w(C
oca Cola)*E[rC
C] = 0.9*0.2+0.1*0.1 = 0.18+0.01
= 0.19 (in decimals); alternatively, you can calculate 0.2*(-19.5) + 0.5*18.2 +
0.3*46 = 19 c.
Variance[rP] = (Probability of B
ear Market)*(R
eturn of Portfolio in Bear M
arket -E
[rP]) 2 + (Probability of Norm
al Market)*( R
eturn of Portfolio in Norm
al M
arket-E[rP]) 2 + (Probability of B
ull Market)*(R
eturn of Portfolio in Bull
Market -E
[rP]) 2 = = (0.2)*(-0.195-0.19) 2 + (0.5)*(0.182-0.19) 2+(0.3)*(0.46 – 0.19) 2 = (0.2)*(-0.3850) 2 + (0.5)*(-0.008) 2+(0.3)*(0.27) 2
= 0.0515 SD
= 0.0515^(1/2)=0.2269
- 7 -
FINA
N 3050
Introduction to Investments
Cover Page
2nd H
omew
ork Assignm
ent
Student name:
Student ID
:
M
ark exercises solved:
Problem 1
Problem
2
Problem 3
Problem
4
Problem
5
Problem 6
Problem
7
Problem 8
Problem
9
Problem 10
C
omm
ents:
- 1 -
2nd H
omew
ork Assignm
ent
Important inform
ation (read carefully): a.
The first page of the homew
ork that you turn in must be the standardized C
over Page (i.e., the first page of this docum
ent). Clearly indicate on your cover sheet
which exercises you solved and are ready to present in class (these are the
exercises which are graded).
b. M
ake a copy of your solutions before you turn them in.
c. Show
your calculations. You m
ight get (partial) credit for wrong solutions if
your way of approaching the question m
akes sense. No credit w
ill be given to correct solutions that are not accom
panied by the appropriate calculations. Problem
1-3:
Abigail G
race has a $900,000 fully diversified portfolio. She subsequently inherits
comm
on stock of company “M
yStock” worth $100,000. H
er financial advisor provided
her with the follow
ing forecasted information:
E
xpected Monthly
Returns
Standard Deviation
of Monthly R
eturns O
riginal Portfolio 0.67%
2.37%
MyStock
1.25%2.95%
The correlation coefficient of “M
yStock” stock returns with the original portfolio returns
is 0.40.
Problem 1:
Assum
ing Grace keeps the “M
yStock” stock, calculate the:
a. Expected m
onthly return of the new portfolio w
hich includes the “Mystock” stock
b. C
ovariance of “MyStock” stock returns w
ith the original portfolio returns.
c. Standard deviation of her new
portfolio which includes the “M
yStock” stock.
Solution Problem 1:
- 2 -
a. (0.9 ×
0.67) + (0.1 × 1.25) = 0.728%
b. C
OV
= 0.4 × 2.37 ×
2.95 = 2.7966
c. SD
= [(0.9^2 × 2.37^2) + (0.1^2 ×
2.95^2) + (2 × 0.9 ×
0.1 × 2.7966)]^0.5 =
2.2673%
Problem 2:
Assum
ing Grace sells the “M
yStock” stock and invests the proceeds in risk-free
government securities yielding 0.42%
per month, calculate the:
a. Expected m
onthly return of the new portfolio w
hich includes the government
securities
b. C
ovariance of the government security returns w
ith the original portfolio returns.
[Hint: think about how
much covariation there can be betw
een a risky and a risk-
free asset?]
c. Standard deviation of her new
portfolio which includes the governm
ent securities.
Solution Problem 2:
a. (0.9 ×
0.67) + (0.1 × 0.42) = 0.645%
b. C
OV
= 0 × 2.37 ×
0 = 0
c. SD
= [(0.9^2 × 2.37^2) + (0.1^2 ×
0) + (2 × 0.9 ×
0.1 × 0)]^0.5 = 2.133%
Problem 3:
Based on conversations w
ith her husband, Grace is considering selling the $100,000 of
“MyStock” stock and acquiring $100,000 of “O
therStock” comm
on stock instead.
“OtherStock” stock has the sam
e expected return and standard deviation as “MyStock”
stock. Her husband com
ments, “It does not m
atter whether you keep all of the “M
yStock”
stock or replace it with $100,000 of “O
therStock” stock.” State whether her husband’s
comm
ent is correct or incorrect. Justify your response.
Solution Problem 3:
The comm
ent is not correct. It also depends on the covariances between each security and
the original portfolio.
- 3 -
Problem 4:
An investor can design a risky portfolio based on tw
o stocks, A and B
. Stock A has an
expected return of 18% and a standard deviation of return of 20%
. Stock B has an
expected return of 14% and a standard deviation of return of 5%
. The correlation
coefficient between the returns of A
and B is 0.5. The risk-free rate of return is 10%
.
The proportion of the optimal risky portfolio that should be invested in stock A
is ?
A. 40%
B. 100%
C. 0%
D. 60%
[Hint: think about the condition that defines the optim
al risky portfolio – slide 4 of week 4
– and check this condition for the four choices.]
Solution Problem 4:
A. 0.4*18 + 0.6*14 = 15.6; V
ariance = 97 SD
= 9.8; S = (15.6-10)/9.8 = 0.57
B. 1.0*18 + 0.0*14 = 18.0; SD
= 20; S = (18-10)/20 = 0.4
C. 0.0*18 + 1.0*14 = 14.0; SD
= 5; S = (14-10)/5 = 0.8
D. 0.6*18 + 0.4*14 = 16.4; V
ariance = 172 SD
= 13.1; S = (16.4-10)/13.1 = 0.5
Problem 5:
Karen K
ay, a portfolio manager at C
ollins Asset M
anagement, is using the C
apital Asset
Pricing Model for m
aking recomm
endations to her clients. Her research departm
ent has
developed the information show
n in the following table:
R
eturn Forecast by R
esearch Dep.
Standard Deviation
Beta
Stock X
14.0%36%
0.8
Stock Y
17.0%25%
1.5
Market Index
14.0%15%
1.0
Risk-Free A
sset 5.0%
- 4 -
Calculate the expected return and alpha for each stock.
Solution Problem 5:
For stock X:
E(r) = 5%
+ 0.8×
(14%-5%
) = 12.2%
Alpha(x) =
14% - 12.2%
= 1.8%
For stock Y:
E(r) = 5%
+ 1.5×
(14%-5%
) = 18.5%
Alpha(y) =
17% - 18.5%
= -1.5%
Problem 6:
Imagine that this tim
e you have two stocks nam
ed AB
C and X
YZ w
hose returns depend
on the state of the world econom
y. The market portfolio constructed of the shares of
10000 firms (including the ones of A
BC
and XY
Z) has a return that also depends on the
state of the world econom
y.
R
ecessionR
egular B
oom
Probability 0.25
0.5 0.25
Return of A
BC
10%
20%
-10%
R
eturn of XY
Z
-20%
25%
10%
Return of M
arket Portfolio 0%
5%
10%
R
eturn of Riskless A
sset 5%
5%
5%
a. W
hat is asset AB
C’s m
arket Beta?
b. W
hat is asset XY
Z’s market B
eta?
c. Im
agine that you invest half of your funds in AB
C, and the other half in X
YZ,
what is the m
arket Beta of this portfolio?
Solution Problem 6:
(a) For asset AB
C:
Expected R
eturn of AB
C: 0.25×
10 + 0.5×
20 + 0.25×
(-10) = 10%
- 5 -
Expected R
eturn of Market: 0.25×
0 + 0.5×
5 + 0.25×
10 = 5%
(b) For asset X
YZ:
Expected R
eturn of XY
Z: 0.25×(-20) +
0.5×25 +
0.25×10 =
10%
Expected R
eturn of Market: 0.25×
0 + 0.5×
5 + 0.25×
10 = 5%
(c) For a portfolio: Beta(p) =
(-2)×0.5 +
3×0.5 =
0.5
Problem 7:
Biopharm
a is a pharmaceutical com
pany. Biopharm
a’s annual stock returns have a CA
PM beta
of 1.25 (i.e. β=1.25). The market portfolio’s return is 13%
, and the risk-free rate is 5%.
a. W
hat is the required expected return for Biopharm
a according to the CA
PM?
b. The firm
has the opportunity to develop a new drug. This project requires an initial
outlay of $400,000 and it will bring expected revenues of $100,000 in each of the next
6 years. The riskiness of this project is the same as the overall riskiness of B
iopharma.
Should the managem
ent team of B
iopharma do the project? O
r should it rather not do
it? Explain your answer.
Solution Problem 7:
(a) Required expected return:
- 6 -
(b) Investment decisions:
The PV
equals 378,448.27 and the Net Present V
alue (NPV
) is, as a consequence,
negative (-400,000 + 378,448.27).
Alternatively you can calculate the project’s IR
R (C
Fj -400,000; then six times the C
Fj
100,000) IR
R = 12.98%
IR
R < 15%
do not undertake the project.
Problem 8:
If the simple C
APM
is valid, is the following situation possible? Explain your answ
er.
PortfolioE
xpected Return
Beta
A
20%1.4
B
25%1.2
Solution Problem 8:
Not possible. Portfolio A
has a higher beta than Portfolio B, but the expected return for
portfolio A is low
er.
Problem 9:
If the simple C
APM
is valid, is the following situation possible? Y
ou also know that the
Market R
isk Premium
(E(rM )-rf ) equals 20%. Explain your answ
er. [Hint: plug everything
you know into the C
APM equation – slide 14 of w
eek 4 – and verify that the implied risk-
free rate makes sense.]
PortfolioE
xpected Return
Beta
A
25%1.5
B
5%0.5
Solution Problem 8:
Not possible. Plug the values into the C
APM
equation and solve for the risk-free rate:
25 = rf + 1.5*20 and 5 = rf + 0.5*20 the risk-free rate w
ould have to be minus 5%
to
make these relationships hold – this does not m
ake sense!
- 7 -
Problem 10:
If the simple C
APM
is valid, is the following situation possible? Explain your answ
er.
[Hint: Think about the “optim
ality” of the Market portfolio (refer to slide 4 and slide 7 of
week 4). Verify that the M
arket portfolio is optimal in that sense.]
Portfolio E
xpected Return
Standard Deviation
Risk-Free
10%0%
M
arket 18%
24%
A
16%12%
Solution Problem
10:
Not possible. The rew
ard-to-variability ratio for Portfolio A is better than that of the
market, w
hich is not possible according to the CA
PM, since the C
APM
predicts that the
market portfolio is the m
ost efficient portfolio.
S(A) = (16-10)/12 = 0.5
S(M) = (18-10)/12 = 0.33
- 8 -
FINA
N 3050
Introduction to Investments
Cover Page
3rd H
omew
ork Assignm
ent
Student name:
Student ID
:
M
ark exercises solved:
Problem 1
Problem
2
Problem 3
Problem
4
Problem
5
Problem 6
Problem
7
Problem 8
Problem
9
Problem 10
Problem
11
Problem 12
C
omm
ents:
- 1 -
3rd H
omew
ork Assignm
ent Problem
1:
Find the price of a coupon bond (par value of $1000) which pays 12%
coupon semi-
annually, matures in 5 years, and has a yield to m
aturity of 10%.
Solution: FV=1000, PM
T=60, i=5% (per 6-m
onth period), n=10→ PV
=-1077.22
Problem 2:
A coupon bond w
hich pays coupon of $100 annually has a par value of $1000, matures in
5 years, and is selling today at a $72 discount from par value (i.e., at $928). Find the yield
to maturity of this bond.
Solution: FV=1000, PM
T=100, n=5, PV= -928 →
i=12%
Problem 3:
Rank the follow
ing bonds in order of descending duration (note that you should not
actually calculate the duration).
Bond
Coupon
Tim
e to Maturity
Yield to M
aturity
A
15%
20 years 10%
B
15%
15 years 10%
C
0%
20 years 10%
D
8%
20 years 10%
E 15%
15 years
15%
Solution:
C:
Highest m
aturity, zero coupon
D:
Highest m
aturity, next-lowest coupon
A:
Highest m
aturity, same coupon as rem
aining bonds
B:
Lower yield to m
aturity than bond E
E: H
ighest coupon, shortest maturity, highest yield of all bonds.
- 2 -
Problem 4:
Consider a zero-coupon bond (face value of $1000) that m
atures in 3 years. The YTM
is
10%. Find the price, the duration and the convexity of this bond.
Solution:
1. Price: 1000/1.1
3 = 751.3148
2. D
uration = 3
3. C
onvexity = 1/1.12×3×(3+1)×1 = 9.92
Problem 5:
Consider a 15%
-coupon bond that matures in 3 years (face value of $1000). The Y
TM is
10%. Find the price, the duration and the convexity of this bond.
Solution:
C
onvexity: 1/1.12×(1×2×0.1213 + 2×3×0.1103 + 3×4×0.7685) = 8.369
Problem 6:
You purchased a 5-year annual interest coupon bond one year ago. Its coupon interest
rate was 6%
and its par value was $1,000. A
t the time you purchased the bond, the yield
to maturity w
as 4% (and the tim
e to maturity w
as 5 years). If you sold the bond after
receiving the first interest payment and the bond's yield to m
aturity had changed to 3%
(and the time to m
aturity is now 4 years), your annual total rate of return on holding the
bond for that year would have been __________.
- 3 -
Solution: 7.57%
1. B
uy price (one year ago; 5 years to maturity; Y
TM 4%
): 1089.04
2. Sell price (4 years to m
aturity; YTM
3%): 1111.51
3. C
apital Gain: 22.47
4. Interest Incom
e: 60
5. H
olding Period Return = (60+22.47)/1089.04 = 0.0757
Problem 7:
Consider the follow
ing $1,000 par value zero-coupon bonds:
C
alculate the expected one-year interest rates one year (from year 1 to 2), tw
o years
(from year 2 to 3), three years (from
year 3 to 4) and four years (from year 4 to 5) from
now.
Solution:
One Y
ear: (1+0.06) 1×(1+f1) = (1+0.07) 2 r = 0.0801
Two Y
ears: (1+0.07) 2×(1+f2) = (1+0.0799) 3 r = 0.10
Three Years: (1+0.0799) 3×(1+f3) = (1+0.0941) 4
r = 0.138
Four Years: (1+0.0941) 4×(1+r) = (1+0.1070) 5
r = 0.1601
Problem 8:
You w
ill be paying $10,000 a year in tuition expenses at the end of the next two years.
Bonds currently yield 8%
.
1. W
hat is the present value and duration of your obligation?
2. If you w
ant to invest only in one zero-coupon bond to imm
unize your obligation,
what does the m
aturity of this zero-coupon bond have to be?
- 4 -
3. V
erify that your answer to (2.) really im
munizes your obligation. For that
purpose, suppose the rates imm
ediately increase to 9%. W
hat happens to your net
position, that is, to the difference between the value of the bond and that of your
tuition obligation? What happens if rates fall to 7%
?
Solution:
1. The present value of the obligation is $17,832.65 and the duration is 1.4808 years.
2. To im
munize the obligation, invest in a zero-coupon bond m
aturing in 1.4808
years. Since the present value of the zero-coupon bond must be $17.832.65, the
face value must be: 17328.65 × 1.08
1.4804 = 19985.26
3. If the interest rate increases to 9%
, the zero-coupon bond would fall in value to
$17590.92. The PV of the tuition obligation w
ould fall to $17591.11, so that the
net position changes by $0.19.
4. If the interest rate decreases to 7%
, the zero-coupon bond would rise in value to
$18079.99. The PV of the tuition obligation w
ould rise to $18080.18, so that the
net position changes by $0.19.
Problem 9:
The risk-free rate of return is 10%, the required rate of return on the m
arket is 15%, and
High-Flyer stock has a beta coefficient of 1.5.
a. If the dividend per share expected during the com
ing year, D1 , is $2.50 and g=5%
,
at what price should a share sell?
b. If the share sells at $25, w
hat must be the m
arket’s expectation of the growth rate
of dividends (keeping all other information the sam
e)?
Solution:
(a) K=10%
+ 1.5×(15%-10%
)=17.5%
Therefore, P0 = $2.50/(0.175-0.05) = $20
(b) $25 = $2.50/(0.175-g) g = 7.5%
- 5 -
Problem 10:
Investment firm
SmartInvest w
ants to value firm SkiU
tah. The following inform
ation is
available for SkiUtah: risk-free rate = 5.0%
, expected market return = 12%
, beta = 1.09.
SmartInvest expects that SkiU
tah’s dividends are going to grow by 15%
in the first 3
years and by 8% thereafter. The current dividend (that has just been paid out) is $2.10.
Find the intrinsic value of SkiUtah using the m
ultistage DD
M and the C
APM
.
Solution:
Market capitalization rate: 0.05 + 1.09*(0.12-0.05) = 0.1263
Next 3-years dividends are: 2.10*1.15 = 2.415; 2.415*1.15 = 2.777; 2.777*1.15 = 3.194
Thereafter there is a constant growth rate of 8%
: 3.194*1.08/(0.1263-0.08) = 74.5
The intrinsic
value is:
2.415/1.1263^1 +
2.777/1.1263^2 +
3.194/1.1263^3 +
74.5/1.1263^3 = 58.71
Problem 11:
MF C
orp. has an RO
E of 16% and a retention rate of 50%
. The market capitalization rate
k equals 12%. If the com
ing year’s earnings are expected to be $2 per share, at what price
will the stock sell? W
hat price do you expect MF shares to sell for in three years?
Solution:
g=RO
E×b=0.16×0.5=0.08
D1 =$2×(1-0.50)=$1.00
P0 =$1.00/(0.12-0.08)=$25
P3 =P
0 ×(1+g) 3=$31.49
Problem 12:
Your prelim
inary analysis of two stocks has yielded the inform
ation set forth below. The
market capitalization rate k for both stock A
and stock B is 10%
per year.
Stock A
Stock B
- 6 -
Expected return on equity, RO
E14%
12%
Estimated earnings per share
$2.00 $1.65
Estimated dividends per share
$1.00 $1.00
Current m
arket price per share $27.00
$25.00
Determ
ine the intrinsic value of each stock and decide in which, if either, of the tw
o
stocks would you choose to invest.
Solution:
Stock A: b=50%
; g = 0.14×0.5 = 7.0%; V
0 = 1/(0.10-0.07) = $33.33
Stock B: b=1-$1/$1.65=0.394; g = 0.12×0.394 = 4.728%
; V0 = 1/(0.10-0.04728)=$18.97
You w
ould choose to invest in Stock A since its intrinsic value exceeds its price. Y
ou
might choose to sell short stock B
.
- 7 -
FINA
N 3050
Introduction to Investments
Cover Page
4th H
omew
ork Assignm
ent
Student name:
Student ID
:
M
ark exercises solved:
Problem 1
Problem
2
Problem 3
Problem
4
Problem
5
Problem 6
Problem
7
Problem 8
Problem
9
Problem 10
C
omm
ents:
- 1 -
4th H
omew
ork Assignm
ent
Important inform
ation (read carefully): a.
The first page of the homew
ork that you turn in must be the standardized C
over Page (i.e., the first page of this docum
ent). Clearly indicate on your cover sheet
which exercises you solved and are ready to present in class (these are the
exercises which are graded).
b. M
ake a copy of your solutions before you turn them in.
c. Show
your calculations. You m
ight get (partial) credit for wrong solutions if
your way of approaching the question m
akes sense. No credit w
ill be given to correct solutions that are not accom
panied by the appropriate calculations. Problem
1:
Suppose you think Wal-M
art stock is going to appreciate substantially in value in the
next year. Say the stock’s current price, S0 , is $100, and the call option expiring in one
year has an exercise price, X, of $100 and is selling at a price, C
, of $10. With $10,000 to
invest, you are considering three alternatives:
a. Invest all $10,000 in the stock, buying 100 shares.
b. Invest all $10,000 in options, buying 1000 options.
c. B
uy 100 options for $1,000 and invest the remaining $9,000 in a m
oney market
fund paying 4% interest annually.
What is your rate of return for each alternative for four stock prices one year from
now?
Price of stock 1 year from
now
$80
$100 $110
$120
a. All Stocks
b. All O
ptions
c. Money m
arket fund + options
Solution:
Price of stock 1 year from
now
$80
$100 $110
$120
- 2 -
a. All Stocks
800010000
1100012000
b. All O
ptions 0
0 10000
20000
c. Money m
arket fund + options9360
9360 10360
11360
Price of stock 1 year from
now
$80
$100 $110
$120
a. All Stocks
-20%
0%
10%
20%
b. All O
ptions -100%
-100%0%
100%
c. Money m
arket fund + options-6.4%
-6.4%
3.6%
13.6%
Problem 2:
The comm
on stock of the P.U.T.T. C
orporation has been trading in a narrow price range
for the past month, and you are convinced it is going to break far out of that range in the
next three months. Y
ou do not know w
hether it will go up or dow
n, however. The current
price of the stock is $100 per share, the price of a three-month call option w
ith an
exercise price of $100 is $10, and a put with the sam
e expiration date and exercise price
costs $7.
Consider buying a straddle. D
raw a diagram
showing the payoff and profit of this
strategy at maturity. H
ow far w
ould the price have to move in either direction for you to
make a profit on your initial investm
ent?
Solution:
The straddle costs $17 stock price has to m
ove by at least $17 in either direction.
Problem 3:
A vertical com
bination is the purchase of a call with exercise price X
2 and a put with
exercise price X1 , w
ith X2 greater than X
1 . Graph the payoff to this strategy (do not graph
the profit, i.e., ignore the costs to establish the positions).
- 3 -
Solution:
See solution graph in the manual on page 14-4
Problem 4:
You w
rite a call option with X
= $50 and buy a call with X
= $60. The options are on the
same stock and have the sam
e maturity date. O
ne of the calls sells for $3; the other sells
for $9. Draw
a graph including the payoff and profit of this strategy at the option maturity
date. What is the break-even point for this strategy?
Solution:
Solution graph is in the manual on page 14-14.
Breakeven occurs w
hen the payoff offsets the initial proceeds of $6, which occurs at a
stock price of $56.
Problem 5:
Joseph Jones, a manager at C
omputer Science, Inc. (C
SI), received 10000 shares of
company stock as part of his com
pensation package. The stock currently sells at $40 a
share. Joseph would like to defer selling the stock until the next tax year. In January,
however, he w
ill need to sell all his holdings to provide for a down paym
ent on his new
house. Joseph is worried about the price risk involved in keeping his shares. A
t current
prices, he would receive $400,000 for the stock. If the value of his stock holdings falls
below $350,000, his ability to com
e up with the necessary dow
n payment w
ould be
jeopardized. On the other hand, if the stock value rises to $450,000 he w
ould be able to
maintain a sm
all cash reserve even after making the dow
n payment. Joseph considers
three investment strategies:
1. Strategy A
is to write January call options on the C
SI shares with a strike price of
$45. These calls are currently selling for $3 each.
2. Strategy B
is to buy January put options on CSI w
ith strike price $35. These
options also sell for $3 each.
3. Strategy C
is to write the January call options and buy the January put options
(this strategy is called a zero-cost collar).
- 4 -
Evaluate each of these strategies with respect to Joseph’s investm
ent goals. What are the
advantages and disadvantages of each? Which w
ould you recomm
end?
Solution:
By w
riting covered call options, Jones receives premium
income of $30,000. If, in
January, the price of the stock is less than or equal to $45, he will keep the stock plus the
premium
income. Since the stock w
ill be called away from
him if its price exceeds $45
per share, the most he can have is:
$450,000 + $30,000 = $480,000
(We are ignoring interest earned on the prem
ium incom
e from w
riting the option over
this short time period.) The payoff structure is:
Stock price Portfolio value
Less than $45 (10,000 tim
es stock price) + $30,000
Greater than $45
$450,000 + $30,000 = $480,000
This strategy offers some prem
ium incom
e but leaves the investor with substantial
downside risk. A
t the extreme, if the stock price falls to zero, Jones w
ould be left with
only $30,000. This strategy also puts a cap on the final value at $480,000, but this is more
than sufficient to purchase the house.
By buying put options w
ith a $35 strike price, Jones will be paying $30,000 in prem
iums
in order to insure a minim
um level for the final value of his position. That m
inimum
value is: ($35 × 10,000) – $30,000 = $320,000
This strategy allows for upside gain, but exposes Jones to the possibility of a m
oderate
loss equal to the cost of the puts. The payoff structure is:
Stock price Portfolio value
Less than $35 $350,000 – $30,000 = $320,000
Greater than $35
(10,000 times stock price) – $30,000
The net cost of the collar is zero. The value of the portfolio will be as follow
s:
Stock price Portfolio value
Less than $35 $350,000
- 5 -
Betw
een $35 and $4510,000 tim
es stock price
Greater than $45
$450,000
If the stock price is less than or equal to $35, then the collar preserves the $350,000 in
principal. If the price exceeds $45, then Jones gains up to a cap of $450,000. In between
$35 and $45, his proceeds equal 10,000 times the stock price.
The best strategy in this case is (c) since it satisfies the tw
o requirements of
preserving the $350,000 in principal while offering a chance of getting $450,000.
Strategy (a) should be ruled out because it leaves Jones exposed to the risk of substantial
loss of principal.
O
ur ranking is: (1) c
(2) b (3) a
Problem 6:
Draw
the payoff diagram for the follow
ing investment strategy using A
BC
share and call
options on AB
C shares:
1. B
uy a share of AB
C
2. W
rite a call on AB
C w
ith X=50
3. W
rite a call on AB
C w
ith X=60
4. B
uy a call on AB
C w
ith X=110
Solution:
Devise a portfolio using only call options and shares of stock w
ith the following
Buy a share of stock, w
rite a call with X
= 50, write a call w
ith X = 60, and buy a call w
ith X
= 110.
Position ST < 50
50 < ST < 60 60 < ST < 110
ST > 110 B
uy stock ST
ST ST
ST Short call (X
= 50) 0
– (ST – 50) – (ST – 50)
– (ST – 50) Short call (X
= 60) 0
0 – (ST – 60)
– (ST – 60) Long call (X
= 110) 0
0 0
ST – 110 Total
ST 50
110 – ST 0
The investor is making a volatility bet. Profits w
ill be highest when volatility is low
so that the stock price ends up in the interval betw
een $50 and Look at the picture in the book at page 513.
- 6 -
Problem 7:
A put option w
ith strike price $60 currently sells for $2. To your amazem
ent, a put on the
same firm
with sam
e expiration date but with a strike price of $62 also sells for $2. H
ow
can you exploit this obvious mispricing using a sim
ple zero-net-investment strategy (i.e.,
a strategy that has zero net investment costs) that gives you a non-negative payoff at
expiration of the options with certainty? A
lso draw the payoff diagram
at expiration in
order to verify that your strategy yields a non-negative payoff for every stock price ST .
Solution:
Buy the X
= 62 put (which should cost m
ore than it does) and write the X
= 60 put. Since the options have the sam
e price, the net outlay is zero. Your proceeds at
maturity m
ay be positive, but cannot be negative.
Position ST < 60
60 < ST < 62ST > 62
Long put (X = 62)
62 – ST 62 – ST
0 Short put (X
= 60) – (60 – ST)
0 0
Total 2
62 – ST 0
0S
T
2
6062
Payoff = P
rofit (because net investment = 0)
Problem
8:
Assum
e the current stock price, S0 , equals $100. In one year the stock price can either go
up to $200 or down to $50. The risk-free interest rate equals 8%
. Determ
ine the price of a
call with strike price, X
, being equal to $75.
- 7 -
Solution:
H=0.833
buy 5 stocks and write 6 options
Risk-free payoff equals: 5×200-6×125 = 250 and 5×50 = 250
5×100-6×C=250/1.08
C=44.75
Problem 9:
Assum
e the current stock price, S0 , equals $100. In one year the stock price can either go
up to $300 or down to $25. The risk-free interest rate equals 8%
. Determ
ine the price of a
call with strike price, X
, being equal to $75.
Solution:
H=0.8181=
buy 9 stocks and write 11 options
Risk-free payoff equals: 9×300-11×225 = 225 and 9×25 = 225
9×100-11×C=225/1.08
C=62.88
Problem 10:
Consider the call options given in Problem
8 and Problem 9. D
etermine the appropriate
prices of put options with a strike price of X
=$75 using the put-call parity relationship.
Solution:
P = C-S
0 +PV(X
)
Given the data in Problem
9: P = 44.75 – 100 + 75/1.08 = 14.19
Given the data in Problem
10: P = 62.88 – 100 + 75/1.08 = 32.32
- 8 -
Introduction to InvestmentsFINAN 3050
Week 3: Asset allocation: risky vs. risk-free assets (Chp. 5.5,5.6)
Asset allocation: optimal risky portfolios (Chp. 6.1,6.2)
Slide 2Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsOverview
� Goal: split investment between safe and risky assets
� Safe asset (risk-free asset): T-bills
� Risky asset: portfolio of stocks
� Issues:
─ Examine the risk-return tradeoff─ Influence of investor’s risk aversion on the allocation
Slide 3Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsEquation: Expected Returns and Standard Deviation
� Notation – Input:
─ rf…risk-free rate─ E(rp)…expected return of the risky asset
─ y...share invested in risky asset (in %); (1-y)…share invested in risk-free asset (in %)
─σrf/p…standard deviation of the risk-free asset (rf) or the risky asset (p)� Notation – Output:
─ E(rc)…expected return of the combined portfolio
─σc…standard deviation of the combined portfolio (c)
( ) ( ) ( ) fpc ryrEyrE ×−+×= 1pc y σσ ×=
Slide 4Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsExample: Expected Returns and Standard Deviation
� Consider the following data:─Risk-free asset:
• Rate of return: rf = 7%
• Standard deviation of return (SD): σrf= 0%─Risky asset:
• Expected rate of return: E(rp) = 15%
• Standard deviation of return (SD): σp = 22%─ y=50%
( ) ( ) 11750.011550.0 =×−+×=crE 112250.0 =×=cσ
Slide 5Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsInvestment Opportunity Set
Slide 6Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsEquation: Slope of the Capital Allocation Line
� The slope S can be calculated in the following way:
� The slope has a nice interpretation: ─ It represents the ratio of the risk premium to the standard deviation
─ It measures the additional return for taking on one more unit of risk─ It is called the “reward-to-variability ratio” or “Sharpe Ratio”
( )
c
fc rrES
σ
−=
Slide 7Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsExercise
� For each of the following combinations determine ─ the expected rate of return,
─ the risk premium, ─ the standard deviation,
─ the reward-to-variability ratio and ─ the point in the graphic on slide 5.
� Case 1: y=1.0
� Case 2: y=0.0
� Case 3: y=0.75
Slide 8Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsExercise
� Assume that you manage a risky portfolio with an expected rate of return of
14% and a standard deviation of 22%. The risk-free rate is 5%. The risky portfolio includes the following investments in the given proportions: 60% in a
U.S. Stock Portfolio and 40% in an Emerging Market Stock Portfolio.
� Suppose a client would like to invest in your risky portfolio a proportion y of
his total investment budget so that his overall portfolio will have an expected rate of return of 10%.
─ What is the proportion y?─ What are your client’s investment proportions in the U.S. Stock
Portfolio, the Emerging Market Stock Portfolio and the T-Bill?
Slide 9Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsInvesting more than 100% into the risky asset (1)
� If we invest 100% into the risky asset, we get to point P on the CAL.
� How can we invest even more in the risky asset?� Answer: by borrowing money at the risk-free rate (leverage) and invest it into
the risky asset
Slide 10Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsInvesting more than 100% into the risky asset (2)
� Example: assume we want to invest 125% into the risky asset
─ Portfolio weights: y = 125% and (1-y) = -25% (i.e., borrowing)
─ Expected return of the combined portfolio:
─ Standard deviation of the combined portfolio:
─Reward-to-variability ratio:
( ) ( ) 17725.111525.1 =×−+×=crE
5.272225.1 =×=cσ
36.05.27
717=
−=S
Slide 11Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsInvesting more than 100% into the risky asset (3)
� Usually investors cannot borrow at the risk-free rate.
� Consider again the example where you borrow 25%. But now you can only borrow at a rate of 9%: rB = 9%
─ Portfolio weights: y = 125% and (1-y) = -25%
─ Expected return of the
combined portfolio:
─ Standard deviation of the combined portfolio:
─Reward-to-variability ratio:
( ) ( ) 5.16925.111525.1 =×−+×=crE
5.272225.1 =×=cσ
27.05.27
95.16=
−=S
Slide 12Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsInvesting more than 100% into the risky asset (4)
Slide 13Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsOptimal allocation between risky asset and risk-free asset
� How to determine the optimal allocation between risky asset and risk-free asset?
─ The investor’s risk aversion determines the “optimal” point on the CAL.
─Greater levels of risk aversion lead to larger proportions of the risk-free
rate.
─ Lower levels of risk aversion lead to larger proportions of the risky asset.
Slide 14Week 3
Michael HallingUniversity of Utah
Allocating Capital between Risky and Risk-Free AssetsCapital Market Line
� Capital Market Line: is the Capital Allocation Line using the market index
portfolio as the risky asset.
� Historical evidence:
Slide 15Week 3
Michael HallingUniversity of Utah
Optimal Risky Portfolios (Chapter 6)Equation: Expected Portfolio Return
� Consider a portfolio of a stock (S) and a risky corporate bond (B)
─ wS…proportion of funds invested in stock─ wB…proportion of funds invested in bond
─ Because we only have two risky assets: wS+wB= 1
─ E(rS)…expected return of stock S─ E(rB)…expected return of bond B
─ E(rP)…expected return of portfolio P
� Expected return of a portfolio P of stock S and bond B:
( ) ( ) ( )BBsSp rEwrEwrE ×+×=
Slide 16Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosExample: Portfolio of Two Risky Assets (1)
� Consider the following scenario analysis:
� What are the expected rates of return?
─ Stock fund:
─ Bond fund:
-4%27%30%Boom
6%13%40%Normal
16%-11%30%Recession
Rate of Return for Bond FundRate of Return for Stock FundProbabilityScenario
( ) ( ) 10273.134.113. =×+×+−×=SrE
( ) ( ) 643.64.163. =−×+×+×=BrE
Slide 17Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosExample: Portfolio of Two Risky Assets (2)
� What are the variances and standard deviations?
─ Stock fund:
─ Bond fund:
( ) ( ) ( )
75.760
60643.664.6163.2222
==
=−−×+−×+−×=
B
B
σ
σ
( ) ( ) ( )
92.146.222
6.22210273.10134.10113.2222
==
=−×+−×+−−×=
S
S
σ
σ
Slide 18Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosExample: Portfolio of Two Risky Assets (3)
� What about the risk and return characteristics of a portfolio with 60% invested
in the stock fund and 40% invested in the bond fund?� First, calculate the portfolio return in each scenario:
─Recession:
─Normal:
─ Boom:
� Second, calculate the expected portfolio return:
( ) 2.0164.0116.0 −=×+−×
2.1064.0136.0 =×+×
( ) 6.1444.0276.0 =−×+×
( ) ( ) 4.86.143.2.104.2.03. =×+×+−×=PrE
Slide 19Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosExample: Portfolio of Two Risky Assets (4)
� Third, calculate the portfolio variance and portfolio standard deviation:
( ) ( ) ( )
92.5
016.354.86.143.04.82.104.04.82.03.02222
=
=−×+−×+−−×=
p
p
σ
σ
Slide 20Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosExample: Portfolio of Two Risky Assets (5)
� Notice that…─ the expected portfolio return equals the average of the expected returns of the two assets
─ the portfolio standard deviation DOES NOT equal the average of the standard deviations of the two assets
─ the portfolio standard deviation is actually less than that of either asset (5.92 < 14.92 & 5.92 < 7.75), i.e., the portfolio of the two assets is less risky than either asset individually (the POWER OF DIVERSIFICATION)
( ) 4.864.106. =×+×=PrE
92.5052.1275.74.92.146. ≠=×+×
Slide 21Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosExample: Portfolio of Two Risky Assets (6)
� Why is this the case?─ Inverse relationship between the performance of the two funds.
─ In a recession: stocks perform poorly but bonds perform well
─ In a boom: stock perform very well but bonds fall
� How can we measure the tendency of the returns of two assets to vary either in tandem or in opposition to each other:
─Correlation (notation: ρS,B)─Covariance (notation Cov(rS,rB) or σS,B)
Slide 22Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosEquation: Covariance Between Two Risky Assets
� Notation:
─ S…number of different states of the world to take into consideration─ pi…the probability of state i to take place
─ rS,i…return of stock S in state i─ rB,i…return of bond B in state I
─ E(rS)…average return of stock S (across all states of the world)─ E(rB)…average return of bond B (across all states of the world)
� Covariance:
( ) ( )( ) ( )( )BiB
S
i
SiSiBS rErrErprrCov −×−×=∑=
,
1
,,
Slide 23Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosEquation: Correlation Between Two Risky Assets
� Covariance is, however, difficult to interpret � correlation coefficient
� Notation: σS…standard deviation of stock S, σB…standard deviation of bond B
� Correlation coefficient:
� Correlation ranges from values of -1 to +1─ -1 indicates perfect negative correlation: i.e., if the return of stock Sgoes up by +10%, the return of bond B goes down by -10%
─+1 indicates perfect positive correlation─ 0 indicates that the returns on the two assets are unrelated to each other
( )
BS
BSSB
rrCov
σσρ
×=
,
Slide 24Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosEquation: Portfolio Variance
� Notation:
─ ws…weight invested in stock S─ wB…weight invested in bond B (has to be 1-ws)
─σS…standard deviation of stock S, ─σB…standard deviation of bond B
─ ρS,B…correlation between stock S and bond B─ Cov(rS,rB)…covariance between stock S and bond B
� Portfolio variance for two risky assets:
( ) ( )
( )BSBSBBSS
BSBBSSBBSSp
rrCovwwww
wwww
,2
2
2222
,
22222
×××+×+×=
=×××××+×+×=
σσ
ρσσσσσ
Slide 25Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosExample: Portfolio of Two Risky Assets (7)
� Continued from slide 20; data on slide 16.� Calculate the covariance:
� Calculate the correlation:
� Calculate the portfolio standard deviation:
( ) ( ) ( ) ( ) ( ) ( ) ( )114
6410273.6610134.61610113.,
−=
=−−×−×+−×−×+−×−−×=BS rrCov
99.75.792.14
114−=
×
−=SBρ
( )
92.5
016.351144.6.2604.6.2226.222
=
=−×××+×+×=
P
P
σ
σ
Slide 26Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosDiversification – Some Simple Rules
� If the correlation between two assets is equal to +1 (perfect positive correlation) there are no benefits from diversification.
� There are benefits to diversification whenever asset returns are less than perfectly correlated.
� If the correlation between two assets is equal to -1 (perfect negative
correlation) we can reduce the portfolio standard deviation all the way down to zero (maximum benefits from diversification).
Slide 27Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosExercise: The Power of Diversification
� Consider the following input data: E(rB)=6%, E(rS)=10%, σB=12%, σS=25%,
ρBS=0
� Case 1: What is your expected return and standard deviation if you invest 100% into bonds.
� Case 2: What is your expected return and standard deviation if you invest 75% in bonds and only 25% in stocks?
� Case 3: What is your expected return and standard deviation if you invest 87.6% in bonds and only 12.4% in stocks?
� Compare the different portfolios and explain “the power of diversification”!� Compare Case 2 and Case 3: which portfolio would you prefer?
Slide 28Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosThe Opportunity Set
Slide 29Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosThe Mean-Variance Criterion
� Mean-Variance Criterion: how to compare two different portfolios?
─ If two portfolios have the same expected return, we prefer the one with the lower standard deviation.
─ If two portfolios have the same standard deviation, we prefer the one
with the higher expected return.
� Further: a portfolio A is said to dominate portfolio B ifand (portfolio A would lie to the northwest of portfolio B)
� Consider “Portfolio Z” and “Stocks” in the previous graph: which one
dominates?
( ) ( )BA rErE ≥
BA σσ ≤
Slide 30Week 3
Michael HallingUniversity of Utah
Optimal Risky PortfoliosDiversification – Further Exercises
� Consider the following input data: E(rB)=6%, E(rS)=10%, σB=12%, σS=25%� Suppose that your are required to invest 50% of your portfolio in bonds and
50% in stocks.─ If the portfolio standard deviation of your portfolio is 15%, what must be the correlation coefficient between stock and bond returns?
─What is the expected rate of return on your portfolio?─Now suppose that in an alternative scenario the correlation between stock and bond returns is 0.22 but that you are free to choose whatever portfolio proportions you desire. Are you likely to be better or worse off than you were in part (a)?
Introduction to InvestmentsFINAN 3050
Week 4:
Asset allocation: optimal risky portfolios (Chp. 6.3)
Capital Asset Pricing Model (Chp. 7.1-7.2)
Slide 2Week 4
Michael HallingUniversity of Utah
Efficient DiversificationIncluding a Risk-Free Asset – Putting Things Together
� Problem: what is the optimal risky portfolio?
� Consider a risk-free asset (assume, for example, a risk-free rate of 5%) and think about how to draw a Capital Allocation Line.
� Remember that you prefer steep CALs (i.e., the steeper the better).
Slide 3Week 4
Michael HallingUniversity of Utah
Efficient DiversificationExample: Including a Risk-Free Asset – Putting Things Together
� Consider the following information: E(rBOND)=6%, E(rSTOCK)=10%,
σBOND=12%, σSTOCK=25%, rf=5%, ρSTOCK,BOND = 0.2� Evaluate the following “potential” CALs (through portfolio A or B)
─ Portfolio A: wSTOCK=12.94% and wBOND=87.06%
─ Portfolio B: wSTOCK=20% and wBOND=80%
� Sharpe Ratio of CALA: ─ E(rA) = 0.1294×10 + 0.8706×6 = 6.52%
─ 0.12942×252 + 0.87062×122 + 2×0.1294×25×0.8706×12×0.2 = 133.13─ σA = 11.54─ SA = (6.52-5)/11.54 = 0.13
� Exercise:─ What is the Sharpe Ratio of CALB?─ Which CAL (i.e., which risky portfolio) do you prefer?
Slide 4Week 4
Michael HallingUniversity of Utah
Efficient DiversificationFinding the Optimal Risky Portfolio
� Optimal Risky Portfolio � the portfolio that yields the steepest CAL!� Solution: draw a line starting at the risk-free rate that is a tangent to the
investment opportunity set
S = (8.68-5)/17.97 = 0.20
Slide 5Week 4
Michael HallingUniversity of Utah
Efficient DiversificationComplete Portfolio
� Remember the two steps of the investment process:─ Finding the optimal risky portfolio
─Combining the optimal risky portfolio with the risk-free asset (i.e., find the optimal point on the CAL that goes through the optimal risky
portfolio)
� Complete portfolio: depends on the investor’s risk aversion
� Note that all investors hold the same optimal risky portfolio but investors differ in their allocation of funds into this risky portfolio.
Slide 6Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Motivation
� Idea: if all investors follow the concepts we have just developed, what are the implications for financial markets and asset prices?
� To answer this question, we need an equilibrium theory like the CAPM!� The CAPM is a theory but it has important real-world implications for
investors and firms.� Key assumptions underlying the CAPM:
─ Single-period investment horizon; investments are limited to traded
financial assets; no taxes, and transaction costs─ Investors are rational mean-variance optimizers─Homogeneous expectations (everybody has the same information on
expectations and risk)
Slide 7Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Optimal Risky Portfolio
� All investors will hold the same portfolio of risky assets.─Why? Because all investors are identical: same mean-variance analysis,
same universe of securities, identical time horizon, etc. � all investors get the same efficient frontier and the same tangency portfolio
� This tangency portfolio (i.e., optimal risky portfolio) equals the market portfolio and contains all securities.
─Why? Because if the optimal risky portfolio does not include some stock
ABC � no one would buy stock ABC (i.e., no demand) � price of
stock ABC goes down � stock ABC becomes more attractive/cheaper � at some point becomes part of the optimal portfolio
Slide 8Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (1)
� What is the portfolio risk that investors are exposed to when holding the optimal risky portfolio?
� Excursus: decomposing portfolio risk─Component 1: Firm-specific risk (also called unique risk, diversifiable
risk, or nonsystematic risk, or idiosyncratic risk); e.g. firm wins a big deal, firm faces litigation
─Component 2: Market risk (also called systematic, or nondiversifiablerisk); e.g. inflation, employment rates, economic cycle
Slide 9Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (2)
� Diversification (i.e., including many firms in the risky portfolio)
reduces firm-specific risk.
� Diversification does not reduce market risk because all firms are
exposed to market risk.
Total Risk
Slide 10Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (3)
Slide 11Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (4)
� Implications of the CAPM:
─ Investors hold the market portfolio, i.e., investors hold an optimally diversified portfolio.
─All firm-specific risk is diversified away.─ Investors are only exposed to market risk.
� Remember that we used the volatility of returns as our measure of risk
� the volatility of return, however, measures overall risk (firm-specific and market risk).
� In the CAPM, investors are only compensated for market risk � we need an appropriate measure for market risk!
Slide 12Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (5)Equation
� How to measure systematic (i.e., market) risk of a specific stock: look at the covariance of the stock’s return with the market’s return and normalize by the variance of the market’s return
� This measure of systematic risk is called Beta.� Notation:
─ Cov(ri,rM)…Covariance between the returns of stock i and the returns of the market
─ …variance of the returns of the market
( )2
,
M
Mii
rrCov
σβ =
2
Mσ
Slide 13Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Portfolio Risk of the Optimal Risky Portfolio (6)Example
� Problem: Given the following information, calculate the Beta of Stock A
� Solution:─Calculate the covariance between Stock A and the market portfolio
─Calculate the variance of the market portfolio and then the Beta
10%12%70%Boom
-7%-14%30%Recession
Market PortfolioStock AProbabilityScenario
( ) ( ) ( ) ( ) ( ) 82.929.4102.4127.09.472.4143.0, =−×−×+−−×−−×=MA rrCov
( ) ( ) 69.609.4107.09.473.0222 =−×+−−×=Mσ 53.1
69.60
82.92==iβ
Slide 14Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (1)Equation
� Our insight that investors are only compensated for the exposure to systematic
risk has important implications for the expected returns on individual assets.� The rate of return on any asset exceeds the risk-free rate by a risk
premium equal to the asset’s systematic risk measure (its beta) times the risk premium of the market portfolio.
� Notation:─ rf...risk-free rate of return
─ E(ri)…expected rate of return for some stock i
─ βi…Beta of stock i─ E(rM)…expected rate of
return of the market
( ) ( )( )fMifi rrErrE −×+= β
Slide 15Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (2)Example
� Consider the following information for a stock XYZ: E(rM) = 14%, rf = 6%,
βXYZ = 1.2
� What is the expected return of stock XYZ, i.e. E(rXYZ)?
� What happens if your read the Wall Street Journal and a well-known asset manager believes that the stock is going to yield an expected return of 17%?
─According to the CAPM, the stock should yield 15.6%.─ The difference between the 17% and the 15.6% (which is called alpha
or αααα) implies some mispricing of the stock.
( )( ) ( ) 6.156142.16 =−×+=−×+ fMif rrEr β
Slide 16Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (3)Equation: Alpha
� The Alpha of stock A is the difference between some “subjective” expected
rate of return (coming from an expert, a research department, personal beliefs, past evidence) and the expected rate of return according to the CAPM.
─ Positive alpha: stock is expected to earn more than expected according
to the CAPM � the stock is a good investment─Negative alpha: vice versa
� Notation:─ E(ri)…expected rate of return for some stock i according to the CAPM
─ SE(ri)…”subjective” expected rate of return for some stock i
( ) ( ) ( ) ( )( )[ ]fMifiii rrErrSErErSE −×+−=−= βα
Slide 17Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (4)
� Security Market Line (SML):
Graphical representation of the relationship between an asset’s
expected return and Beta
� Slope of the SML: risk premium of the market portfolio
� CML or CAL (in contrast): graphs the relationship between
an asset’s expected return and standard deviation
Slide 18Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (5)Exercise
� Consider the following information:─ Stock XYZ: expected return of 12% by some expert; Beta of 1.0
─ Stock ABC: expected return of 13% by some expert; Beta of 1.5
─ Market: expected return of 11%─ Risk-free rate equals 5%
� What is the expected return according to the CAPM and the alpha of each
stock?� According to the CAPM, which stock is a better buy?
Slide 19Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Individual Securities (6)Exercise
� If the CAPM holds, is the following situation possible?
� Does the given data imply anything about the Betas of Portfolio A and B?
25%40%B
35%30%A
Standard DeviationExpected ReturnPortfolio
Slide 20Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Portfolios (1)Equation
� Same argument as for individual securities: only the systematic risk matters �therefore, we need the portfolio beta!
� Equation: the beta of a portfolio is simply the weighted average of the betas of the stocks in the portfolio where the weights are the appropriate portfolio
weights.
� What is the Beta of the Market Portfolio?
Slide 21Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Investors: Expected Returns on Portfolios (2)Example
� Example: market risk premium = 7.5%; rf=5%; calculate the portfolio beta,
the portfolio expected return and the portfolio risk premium?
� Portfolio Beta = 1.2×0.5 + 0.8×0.3 + 0×0.2 = 0.84� Portfolio Expected Return = 5+0.84×7.5 = 11.3%� Portfolio Risk Premium = 11.3 – 5 = 6.3%
0.20.0Gold
0.30.8Con Edison
0.51.2Microsoft
Portfolio WeightBetaAsset
Slide 22Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Firms: Capital Budgeting (1)
� Consider the following problem: as a manager of a firm you have to decide on
the execution of a risky project
� How should you make this decision?
� Answer 1: you should do the project if it is “profitable”.─ But what does “profitable” mean?─Does an expected internal rate of return of 1% justify execution of the
project? What do you think?
Slide 23Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Implications for Firms: Capital Budgeting (2)
� Answer 2: The project has to yield a return that is acceptable for the investors given the risk of the project.
� The CAPM can be used to obtain this cutoff IRR or “hurdle rate” for the project.
� Example: suppose Silverado Springs Inc. is considering a new spring-water
bottling plant; the expected IRR is 14%; research shows the Beta of similar products is 1.3; the risk-free rate is 4%; market risk premium is estimated to be
8%
─ The hurdle rate of the project should be: 4 + 1.3×8 = 14.4%─ Because the IRR is less than the hurdle rate � the project has a negative
net present value � should not be done.
Slide 24Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)How to take it?
� Logical inconsistency:─ If a passive strategy is costless and efficient, why would anyone follow
and active strategy?─ But if no one does any security analysis, what brings about the efficiency
of the market portfolio?
� The CAPM depends on important assumptions but its popularity and use is some indication that its predictions are reasonable.
Slide 25Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM)Betas of AT&T Over Time (1966-1996)
� Betas are not observed but estimated! Therefore, a “true” Beta does not exist.
0
0.2
0.4
0.6
0.8
1
1.2
Slide 26Week 4
Michael HallingUniversity of Utah
Capital Asset Pricing Model (CAPM) Betas by Industries
1.05Steel
0.60Energy Utilities1.20Shipping
0.75Telephone Utilities1.25Chemicals
0.85Banks1.30Producer Goods
0.90Liquor1.45Consumer Durables
1.00Food1.60Electronics
1.00Agriculture1.80Airlines
BetaIndustryBetaIndustry
Introduction to InvestmentsFINAN 3050
Week 10:
Options Valuation
(15.1, 15.2, Put-Call Parity p. 519 to 521)
Slide 2Week 10
Michael HallingUniversity of Utah
Option Values
� Intrinsic value - profit that could be made if the option was immediately exercised
─Call: stock price minus exercise price
─ Put: exercise price minus stock price
� Time value - the difference between the option price and the
intrinsic value
Slide 3Week 10
Michael HallingUniversity of Utah
Factors Influencing Option Values: Calls
increases
increases
decreases
increases
Effect on value
Asymmetric payoff of options: Longer maturity implies more chances for S to go up
Asymmetric payoff of options: zero payoff if not exercised, ST-X if exercised
payoff of call equals ST-X if call is exercised
payoff of call equals ST-X if call is exercised
Explanation
Time to expiration
Volatility of stock price
Exercise price (X)
Stock price (S)
Factor
� Exercise: prepare a similar table for the determinants of put option values.
Slide 4Week 10
Michael HallingUniversity of Utah
Factors Influencing Option Values: Exercise
� Consider the following information:
─Which put option is written on the stock with the lower price?� Consider the following information:
─Which call option is written on the stock with higher volatility?
50
50
X
0.5
0.5
T
100.25B
100.20A
Price of OptionVolatility of StockPut
55
50
X
0.5
0.5
T
755B
1055A
Price of OptionS0Call
Slide 5Week 10
Michael HallingUniversity of Utah
Binomial Option Pricing (1)
100
200
50
Stock Price
C
75
0
Call Option
Value X = 125
� Two-state (binomial) option pricing
� Assume a stock price can only take two possible values at option expiration:
─ Increase by a factor u=2─Decrease by a factor d=0.5
� Assume a call option with X=$125 and
T=1. Assume further the risk-free rate of interest is 8%.
Slide 6Week 10
Michael HallingUniversity of Utah
Binomial Option Pricing (2)
� Consider the following portfolio: one share of stock and 2 calls written (X =
125)
� What are the payoffs to this portfolio in each state?
100-2××××C
50=200-2××××75
50=50-0
Portfolio
Payoff
Slide 7Week 10
Michael HallingUniversity of Utah
Binomial Option Pricing (3)
� We created a risk-free portfolio with a payout pf $50. What is the value of a
risk-free profit of $50 in one period?
� Note: we can use the risk-free rate to discount ONLY because the payoffs are risk-free, i.e. do not depend on the price of the stock any more)
� $46.30 also has to be the value of the portfolio (the stock plus the two written
calls) today! Products with the same payoffs must have the same value (replication/arbitrage argument). Therefore, the price of the call is:
30.46$08.01
50$=
+
85.26$
30.46$2100$
=
=×−
C
C
Slide 8Week 10
Michael HallingUniversity of Utah
Binomial Option Pricing: Exercise
� Consider the same stock price dynamics as before (i.e., the stock price can go
up to $200 or go down to $50; the current stock price equals $100). The riskfree rate is again 8%.
� You want to determine the price of a call option with X=$125.
� Consider the following portfolio – buy 0.5 shares and write one option:─Determine the payoffs of this portfolio at the end of the period in each of
the two states.─Determine the present value of this payoff.
─Determine the price of the call option.
Slide 9Week 10
Michael HallingUniversity of Utah
Binomial Option Pricing (4)
� How to come up with a portfolio that creates a
risk-free payoff?� Portfolio weights depend on the hedge ratio:
� The hedge ratio equals the ratio of the weight in the
stock and the weight in the option.� For the previous example: twice as much weight in
the option as in the stock
100
200
50
Stock Price
C
75
0
Call Option
Value X = 125
du
du
SS
CCH
−
−=
5.050200
075=
−
−=H
Slide 10Week 10
Michael HallingUniversity of Utah
Binomial Option Pricing: Exercise
� Reconsider the example from before. Now we
change the exercise price of the option to X=$100
� What is the new hedge ratio?
� What does the portfolio with risk-free payoffs look
like, for example?
� Determine the risk-free payoffs?
� Determine the call option price.
100
200
50
Stock Price
C
??
??
Call Option
Value X = 100
Slide 11Week 10
Michael HallingUniversity of Utah
Put-Call Parity Relationship (1)
� Prices of European Puts and European Calls are linked together in an important
relationship – the Put-Call Parity Relationship.� Consider the following portfolio: buy a call option and write a put
option, each with the same exercise price X and the same expiration date T
ST-XST-XTotal
0-(X-ST)Minus payoff of put written
ST-X0Payoff of call held
ST>XST≤X
Slide 12Week 10
Michael HallingUniversity of Utah
Put-Call Parity Relationship (2)
Slide 13Week 10
Michael HallingUniversity of Utah
Put-Call Parity Relationship (3)
� Why is the combined payoff called “Leveraged Equity”?� Consider the following portfolio: buy the stock and take out a loan with face
value X at the risk-free rate
ST-XST-XTotal
-X-XMinus payoff of paying back the loan
STSTPayoff of stock
ST>XST≤X
Slide 14Week 10
Michael HallingUniversity of Utah
Put-Call Parity Relationship (4)
� By the argument of arbitrage/replication the two portfolios – (A) long a call plus short a put, and (B) long the stock plus loan with face value of X – must
have the same price/value today
C - P = S0 - X/(1+rf)T
� Assume S0=$110, call price (T=0.5 years, X=$105) equals $14, put price
(T=0.5 years, X=$105) equals $5, risk-free rate is 5%─Verify that put-call parity holds:
• $14-$5 = $9
• $110-$105/(1+0.05)0.5 = $7.53
• Violation of put-call parity
Slide 15Week 10
Michael HallingUniversity of Utah
Put-Call Parity Relationship (5)
� Example continued: what does the violation of put-call parity imply � a risk-free profit (i.e., arbitrage)
─ Buy the “cheap” portfolio, i.e. the stock plus the loan
─ Sell the “expensive” portfolio, i.e. the call and the short put
Cash-Flow (CF) in 0.5 years
1.47
-5
+14
+102.47
-110
CF at t=0
-105-105Borrow X/(1+rf)T
-(ST-105)0Sell call
00Total
0105- STBuy put
+ ST+STBuy stock
ST≥105ST<105Position