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Introduction of Risk and Return
Text: Chapter 9
Introduction to Risk and Return
Common stocks 13.0% 9.2% 20.3%
Small-company stocks 17.7 13.9 33.9
Long-termcorporate bonds 6.1 2.3 8.7
Long-termgovernment bonds 5.6 1.8 9.2
Intermediate-termgovernment bonds 5.4 1.6 5.7
U.S. Treasury bills 3.8 3.2
Inflation 3.2 4.5
Risk premium Arithmetic (relative to U.S. Standard
Series mean Treasury bills) deviation
-90% 90%0%
The Value of an Investment of $1 in 1900
$1
$10
$100
$1,000
$10,000
$100,000
Start of Year
Doll
ars
Common Stock
US Govt Bonds
T-Bills
15,578
14761
20
04
The Value of an Investment of $1 in 1900
Real returns
$1
$10
$100
$1,000
Start of Year
Do
lla
rs
Equities
Bonds
Bills
719
6.81
2.80
20
04
Rates of Return 1926-1997
Source: Ibbotson Associates
-60
-40
-20
0
20
40
60
26 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Common Stocks
Long T-Bonds
T-Bills
Year
Per
cent
age
Ret
urn
Measuring Risk
1 1 24
12 1113
1013
3 20123456789
10111213
-50
to -
40
-40
to -
30
-30
to -
20
-20
to -
10
-10
to 0
0 to
10
10 t
o 20
20 t
o 30
30 t
o 40
40 t
o 50
50 t
o 60
Return %
# of Years
Histogram of Annual Stock Market ReturnsHistogram of Annual Stock Market Returns
Average Market Risk Premia (by country)
4.3 4.7 5.1 5.3 5.8 5.9 5.9 6.3 6.4 6.67.6 8.1 8.2 8.6 9.3
10 10.7
0123456789
1011
Den
ma
rk
Bel
giu
m
Sw
itze
rla
nd
Sp
ain
Ca
na
da
Irel
an
d
Ger
ma
ny
UK
Av
era
ge
Net
her
lan
ds
US
A
Sw
eden
So
uth
Afr
ica
Au
stra
lia
Fra
nce
Ja
pa
n
Ita
ly
Risk premium, %
Country
Variance and Standard Deviation
VAR (ri~) = E [ ri
~ - E(ri~) ]2 = ri
2 ,
COV( r1~, r2
~ ) = E [(r1~ - E(r1
~)) (r2~ - E(r2
~))]
where ri~ is actual return governed by probability distribution
EX: The return of asset i next period is ether .2 with prob. 60% or -.1 with prob. 40%
E(ri~) = .6*.2 + .4*(-.1) = .08
Var(ri~) = .6*(.2-.08)2 + .4*(-.1-.08)2 = .0216
Return and Variance of Two Assets
Calculating Portfolio risks of two stocks
E(r~) Weight SD(r~)Stock A .15 .6 18.6Stock B .21 .4 28
E(rp~) = x1*E(r1
~) + x2*E(r2~), where x1 + x2 = 1
E(rp~) = .6*.15 + .4*.21 = .174
What about variance? x1*12 + x22
2?
No!!!
Covariance of a Portfolio of Two Assets
p2 = E [ rp
~ - E(rp~) ]2
= E [ x1r1~ + x2r2
~ - x1*E(r1~) - x2*E(r2
~) ]2
= E[ x1* (r1~ - E(r1
~)) + x2* (r2~-E(r2
~)) ]2
= E[ x12(r1
~-E(r1~))2 + x2
2(r2~-E(r2
~))2 +
x1x2(r1~-E(r1
~))(r2~-E(r2
~)) + x1x2(r1~-E(r1
~))(r2~-E(r2
~))]
= x121
2 + x222
2 + 2x1x2COV(r1~, r2
~)
Define COV(r1~, r2
~) = E[(r1~-E(r1
~)) (r2~-E(r2
~))] = 12
Correlation Coefficient
To get rid of the unit, we define Correlation coefficient
12 = COV(r1~, r2
~) / 12 , where -1<= <= 1
Thus, p2 = x1
212 + x2
222 + 2x1x21212
If 12 = 1, then p = X11 + X22
If 12 < 1, then p < X11 + X22
Stock 1 Stock 2
Stock 1 x121
2 x1x2COV(r1~, r2
~)
Stock 2 x1x2COV(r1~, r2
~) x222
2
Portfolio Risk
Example
Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The expected return on your portfolio is:
%12)1540(.)1060(. ReturnExpected
Portfolio Risk
2222
22
211221
2112212221
21
)3.27()40(.σx3.272.181
60.40.σσρxxCola-Coca
3.272.181
60.40.σσρxx)2.18()60(.σxMobil-Exxon
Cola-CocaMobil-Exxon
Example
Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.
Portfolio Risk
% 18.3 9.333 DeviationStandard
9.33318.2x27.3)2(.40x.60x
]x(27.3)[(.40)
]x(18.2)[(.60) Variance Portfolio22
22
The Composition of Portfolio Variance
Two risky assets Three assets Four assets N risky assets
Variance of a Diversified Portfolio
What is the variance of portfolio if the number of stock increases?
General Formula: a portfolio with equally weighted N stocks
Portfolio variance:
= N (1/N)2 * average var. + (N2-N)(1/N)2 * average cov.
= 1/N * average var. + (1-1/N) * average cov.
As N increases, the variance of each individual stock becomes less
important, and the average covariance becomes dominant.
How Does Diversification Reduce Risks?
The central message: total risk can be decomposed into two parts: systematic and unsystematic risks.
Therefore diversification can only eliminate unique risks (or unsystematic risks, diversifyable risks), can not eliminate market risk (systematic risks, undiversificable risk)
What is unsystematic risks? RD program, new product introduction, labor relations,
personal changes, lawsuits.
The risk of a well-diversified portfolio depends on the market risk of the securities included in the portfolio.
Measuring Risk
0
5 10 15
Number of Securities
Po
rtfo
lio s
tan
da
rd d
ev
iati
on
Market risk
Uniquerisk
How Individual Securities Affect Portfolio Risk?
A B
Row 1 A . .62*18.62 .6*.4*.2*18.6*28 Row 2 B .6*.4*.2*18.6*28 .42*282
Row 1 = .6 * [.6*18.62 + .4*.2*18.6*28] = .6 * 249
Row 2 = .4 * [.6*.2*18.6*28 + .4*282] = .4 * 376
Total = 300
The contribution of stock A to portfolio risk is WEIGHT * COVARIANCE WITH ALL THE SECURITIES IN THE PORTFOLIO (INCLUDING ITSELF)
The risk of a stock not only depend on its own risks, but also its contribution to the risk of whole portfolio.
Stock’s Beta
If the portfolio is the market portfolio, then we have the formal definition of Beta
Beta - Sensitivity of a stock’s return to the return on the market portfolio.
= Cov (ri~, rm
~) / Var(rm~)
= i,mi m/ m2
= i,m [i/ m]
Conclusions
Markets risk accounts for most of the risk of a well-diversified portfolio.
The beta of an individual security measure its sensitivity to market movement.
A nice property of Beta:
p = Xii, where Xi is the weight of market value of asset I
Does corporate diversification add value for shareholders?