1
Principles of Finance - Lecture 3 1October 19, 2004
Principles of Finance
Grzegorz Trojanowski
Lecture 3:Combining assets into portfolios
Principles of Finance - Lecture 3 2October 19, 2004
Lecture 3 material
• Required reading:Elton et al., Chapters 4, 5
• Supplementary reading:Luenberger, Chapter 6Sharpe et al., Chapter 6Alexander et al., Chapters 7-8
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Principles of Finance - Lecture 3 3October 19, 2004
Lecture 3: Checklist
• By the end of this lecture you should:Understand the concepts of a portfolio, portfolio weights, and short sellingKnow how to compute the variance-covariance matrix for a set of assetsKnow how to compute the return, expected return, and the variance of returns for a portfolio of assetsBe familiar with the notion of the global minimum variance portfolio
Principles of Finance - Lecture 3 4October 19, 2004
Lecture 1 recap (1)
• Risky assets involve cash flows which are uncertain
• We can therefore think of the return on an asset as a random variable drawn from a probability distribution
• When we consider an asset in isolation, we use the marginal distribution, which is characterised by the expected return and the variance of returns
• When we consider several assets together, we use the joint distribution, which is characterised also by the covariance of returns
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Principles of Finance - Lecture 3 5October 19, 2004
Lecture 1 recap (2)• If we have a sample of return data we can
estimate the expected return, variance and covariance using the following formulae:
Expected return
Variance
Covariance
• We now consider what happens when we combine several stocks into a portfolio
∑==
T
ttrT
r1
1
∑ −==
T
tt rr
T 1
22 )(1σ
∑ −−==
T
tBtBAtABA rrrr
T 1,,, ))((1σ
Principles of Finance - Lecture 3 6October 19, 2004
Definition of a portfolio (1)
• Consider n individual assets
• Consider investing a fraction, wi, of your wealth in asset i
• This investment represents a portfolio of the n assets, and the fractions wi are the portfolio weights
• Portfolio weights must satisfy the property that
• Note, however, that portfolio weights are not restricted to be positive
• Nor are they restricted to be less than one
11
=∑=
n
iiw
4
Principles of Finance - Lecture 3 7October 19, 2004
Definition of a portfolio (2)• A positive weight implies a positive quantity of the asset
in the portfolio, or in other words, a long position in that asset
• A negative weight implies a ‘negative’ quantity of the asset in a portfolio, or in other words a short position in that asset
• A weight that is greater than one implies that the investor has invested more than his wealth in that asset, or in other words, he has a super-long position that asset
• If an investor short sells an asset, he is selling an asset that he does not own; it is accomplished by borrowing the asset from one investor and selling it to another
Principles of Finance - Lecture 3 8October 19, 2004
Example: Two-asset portfolio (1)
Suppose that we have two stocks whose prices we have recorded over 12 months, and that we compute continuously compounded returns of each asset
6.24%87.87-1.27%37.121215-1.39%82.56-7.82%37.5911141.13%83.710.08%40.6510130.03%82.776.95%40.61912
-6.88%82.753.63%37.89811-5.27%88.65-0.15%36.547104.76%93.44-2.92%36.5969
-0.75%89.103.86%37.67581.44%89.780.93%36.2547
12.76%88.505.46%35.91366.34%77.903.25%34.0025
15.29%73.114.77%32.911462.7531.9803
ReturnPriceReturnPriceMonth2Stock ‘B’Stock ‘A’1
EDCBA
=LN(B14/B13)
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Principles of Finance - Lecture 3 9October 19, 2004
Example: Two-asset portfolio (2)
• The mean return, volatility, standard deviation of the two stocks, and their covariance and correlation, are computed as
0.2636Correlation23
0.0007Covariance22
6.35%3.96%Monthly std. dev.21
0.00400.0016Monthly variance20
2.81%1.40%Monthly mean19
Stock ‘B’Stock ‘A’18
CBA
=STDEVP(C4:C15)
=AVERAGE(C14:C15)=VARP(C14:C15)
=COVAR(C14:C15,E4:E15)
=CORREL(C14:C15,E4:E15)
Principles of Finance - Lecture 3 10October 19, 2004
Two asset portfolio• Suppose we combine two individual assets (‘A’ and
‘B’) into a portfolio with weight wA in asset ‘A’ and weight wB = (1 – wA) in asset ’B’
• The return on such a portfolio is computed as
• The expected return is computed as
• The variance of the portfolio is computed as
BAAAP RwRwR )1( −+=
)()1()()( BAAAP RwRwR Ε−+Ε=Ε
ABAABAAAP wwww σσσσ )1(2)1( 22222 −+−+=
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Principles of Finance - Lecture 3 11October 19, 2004
Example: Two-asset portfolio (3)• Suppose we
combine stocks ‘A’ and ‘B’ into portfolio in equal proportions
• The returns for the portfolio are computed as follows
=B31*$D$29+C31*(1-$D$29)
2.48%6.24%-1.27%1242
-4.60%-1.39%-7.82%1141
0.61%1.13%0.08%1040
3.49%0.03%6.95%936
-1.63%-6.88%3.63%838
-2.71%-5.27%-0.15%737
0.92%4.76%-2.92%636
1.55%-0.75%3.86%535
1.19%1.44%0.93%434
9.11%12.76%5.46%333
4.79%6.34%3.25%232
10.03%15.29%4.77%131
Portf.‘B’‘A’Month30
0.50Proportion of stock ‘A’29
DCBA
Principles of Finance - Lecture 3 12October 19, 2004
Example: Two-asset portfolio (4)• The mean, variance,
and std. deviation of the portfolio returns are computed as follows
4.16%6.35%3.96%Monthly std. dev.45
0.00170.00400.0016Monthly variance44
2.10%2.81%1.40%Monthly mean43
2.48%6.24%-1.27%1242
-4.60%-1.39%-7.82%1141
0.61%1.13%0.08%1040
3.49%0.03%6.95%936
-1.63%-6.88%3.63%838
-2.71%-5.27%-0.15%737
0.92%4.76%-2.92%636
1.55%-0.75%3.86%535
1.19%1.44%0.93%434
9.11%12.76%5.46%333
4.79%6.34%3.25%232
10.03%15.29%4.77%131
Portf.‘B’‘A’Month30
0.50Proportion of stock ‘A’29
DCBA
=AVERAGE(D31:D42)
or
=D29*B43+(1-D29)*C43
=VARP(D31:D42)
or
=D29^2*B44+(1-D29)^2*C44
+2*D29*(1-D29)*B22
=STDEVP(D31:D42)
or
=SQRT(D44)
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Principles of Finance - Lecture 3 13October 19, 2004
Feasible set
• The mean and standard deviation of the portfolio return depend on the proportions in which the two assets are combined
• Calculating the mean and standard deviation of the portfolio return for all possible combinations of the two assets yields the feasible set
Principles of Finance - Lecture 3 14October 19, 2004
Example: Two-asset portfolio (5)
0.0007Covariance22
6.35%3.96%Monthly std. dev.21
0.00400.0016Monthly variance20
2.81%1.40%Monthly mean19
Stock ‘B’Stock ‘A’18
CBA
0.70%5.95%1.539
0.84%5.46%1.438
::::
2.10%4.16%0.529
::::
3.23%8.03%-0.321
3.37%8.62%-0.420
3.51%9.21%-0.519
E[R]σ[r]Weight of ‘A’18
HGF
=F29*$B$19+(1-F29)*$C$19
=SQRT(F21^2*$B$20+(1-F21)^2*$C$20+2*F21*(1-F21)*$B$22)
Computed earlier
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Principles of Finance - Lecture 3 15October 19, 2004
Example: Two-asset portfolio (6)• We can plot the feasible set using Excel’s XY (Scatter) Chart function
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%
Standard deviation
Expe
cted
retu
rn
Stock ‘B’Stock ‘A’
50:50 portfolio
Principles of Finance - Lecture 3 16October 19, 2004
Feasible set for two-asset case• The shape of feasible set depends on the correlation
coefficient between the two assets• Example:
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Principles of Finance - Lecture 3 17October 19, 2004
Multiple asset portfolios (1)
• Consider the more general case with many individual assets (not just two)
• Things are greatly simplified by using matrix notation
• Consider N assets whose returns are given by
• A portfolio is defined as a combination of these N assets
with portfolio weights given by where
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
Nr
rR M
1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
Nw
wW M
1
11
=∑=
N
iiw
Principles of Finance - Lecture 3 18October 19, 2004
Multiple asset portfolios (2)• The expected returns of the N assets are given by
• The return on the portfolio is a weighted average of the
returns on the individual assets
where W T is the transpose of W, namely W T = [w1 … wN]
• The expected return of the portfolio is given by
RWrwrN
iiiP
Τ
=∑ ==
1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Ε
Ε=Ε
)(
)()(
1
Nr
rR M
)()()(1
RWrwrN
iiiP Ε=∑ Ε=Ε Τ
=
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Principles of Finance - Lecture 3 19October 19, 2004
Multiple asset portfolios (3)• The variance of the portfolio return is given by
• If we define the variance-covariance matrix of R as
we can express the portfolio variance as
• The covariance between two portfolios whose portfolio weight vectors are W1 and W2 is given by
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=Ω
221
22221
11221
NNN
N
N
σσσ
σσσσσσ
L
MOMM
L
L
∑∑∑ +∑∑ =∑ +∑∑ ===
>===
≠=== =
N
i
N
ijj
ijji
N
iii
N
i
N
ijj
ijji
N
iii
N
i
N
jijjiP wwwwwwww
1 11
22
1 11
22
1 1
2 2 σσσσσσ
WWP Ω= Τ2σ
2112 WW Ω= Τσ
Principles of Finance - Lecture 3 20October 19, 2004
Matrix functions in Excel• Matrix multiplication, transposition, and inversion can
be performed using the MMULT, TRANSPOSE, and MINVERSE Excel’s functions
• To use these functions:
Highlight the area that you want to use as output (making sure that it is of the right dimensions)
Enter the formula and do not press [Enter] afterwards
Press [Ctrl]+[Shift]+[Enter] instead, while still in the formula editing window
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Principles of Finance - Lecture 3 21October 19, 2004
Multi asset portfolios: Example (1)
• Consider four individual assets with the expected return vector and variance-covariance matrix given below
• Consider two portfolios of these assets (as given below)
0.60.115%0.500.02-0.040.056
0.10.410%0.020.400.060.035
0.10.38%-0.040.060.300.014
0.20.26%0.050.030.010.103
W2W12
Portfolio weightsMean return vectorVariance-covariance matrix1
IHGFEDCBA
Principles of Finance - Lecture 3 22October 19, 2004
Multi asset portfolios: Example (2)
• The mean, variance, standard deviation, covariance, and correlation can be computed as follows
0.4540Correlation14
0.0714Covariance13
12
45.10%34.87%Std. dev.11
0.20340.1216Variance10
12.00%9.10%Mean9
Portfolio 2Portfolio 18
CBA
=MMULT(TRANSPOSE(H3:H6),$F$3:$F$6)
=MMULT(TRANSPOSE(H3:H6),MMULT($A$3:$D$6,H3:H6))
=SQRT(C10)
=MMULT(TRANSPOSE(H3:H6),MMULT(A3:D6,I3:I6))
=B13/(B11*C11)
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Principles of Finance - Lecture 3 23October 19, 2004
Computing covariance matrix
• In practice, the elements of variance-covariance matrix have to be estimated
• Each element of the matrix Ω can be estimated
by
• The most direct approach to computing the variance-covariance matrix is to compute excess returns first
)()(1ˆ1
jjt
T
tiitij RRRR
T−∑ −=
=σ
Principles of Finance - Lecture 3 24October 19, 2004
Computing covariance matrix: Example (1)• Consider the following return data for six US stocks
0.12100.10250.15290.15010.05310.2032Mean12
0.26400.20661.86820.31100.36800.1942198311
0.04560.2243-0.26150.6968-0.14931.0642198210
0.04790.0913-0.7427-0.0275-0.2042-0.026419819
0.36570.2002-0.18940.33500.47510.012419808
0.22540.02150.07930.08980.0158-0.265919797
-0.13460.13720.2751-0.0573-0.04520.166319786
-0.27210.0712-0.0938-0.0490-0.4271-0.203419775
0.07810.12760.58150.25500.36650.732919764
0.35690.02130.22270.37190.24720.708319753
0.2331-0.0758-0.2107-0.4246-0.1154-0.350519742
UKMOHRGEBSAMR1
GFEDCBA
=AVERAGE(G2:G11)
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Principles of Finance - Lecture 3 25October 19, 2004
Computing covariance matrix: Example (2)• The excess return matrix is given by
0.14300.10411.71530.16090.3149-0.0090198325
-0.07540.1218-0.41440.5467-0.20240.8610198224
-0.0731-0.0112-0.8956-0.1776-0.2573-0.2296198123
0.24470.0977-0.34230.18490.4220-0.1908198022
0.1044-0.0810-0.0736-0.0603-0.0373-0.4691197921
-0.25560.03470.1222-0.2074-0.0983-0.0369197820
-0.3931-0.0313-0.2467-0.1991-0.4802-0.4066197719
-0.04290.02510.42860.10490.31340.5297197618
0.2359-0.08120.06980.22180.19410.5051197517
0.1121-0.1783-0.3636-0.5747-0.1685-0.5537197416
UKMOHRGEBSAMR15
GFEDCBA
=G2-G$12
Principles of Finance - Lecture 3 26October 19, 2004
Computing covariance matrix: Example (3)• Finally, the covariance matrix is computed as follows
H
220.0392-0.00150.02740.01480.04060.0059UK21
-0.00150.00830.01930.01940.00890.0208MO200.02740.01930.44350.04430.10280.0493HR190.01480.01940.04430.08670.03550.1077GE180.04060.00890.10280.03550.07900.0375BS170.00590.02080.04930.10770.03750.2060AMR16
UKMOHRGEBSAMR15GFEDCBA
=MMULT(TRANSPOSE(B16:G25),B16:G25)/10
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Principles of Finance - Lecture 3 27October 19, 2004
The feasible set for N assets (1)
• Consider N assets with varying degrees of correlation between their returns
• Each can be plotted on the mean-standard deviation diagram, and each subset of them can be formed into a portfolio, with any set of portfolio weights, some of which may be negative
• The set of points that contains all possible portfolios made up of different combinations of the N assets is called the feasible set
Principles of Finance - Lecture 3 28October 19, 2004
The feasible set for N assets (2)
• The feasible set has two properties:
It is solid since any portfolio within its bound
can be achieved by the appropriate choice of
assets and portfolio weights
It is convex to the vertical axis
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Principles of Finance - Lecture 3 29October 19, 2004
The feasible set for N assets (3)
Feasible setFeasible set
Principles of Finance - Lecture 3 30October 19, 2004
Minimum variance set and GMV portfolio (1)• The left boundary of the feasible set is known
as the minimum variance set, or the envelope, and comprises the portfolios that have the lowest standard deviation (or variance) for any given expected return
• The portfolio with the lowest standard deviation of all these is known as global minimum variance portfolio or GMV portfolio
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Principles of Finance - Lecture 3 31October 19, 2004
Minimum variance set and GMV portfolio (2)
GMV Portfolio
Principles of Finance - Lecture 3 32October 19, 2004
Minimum variance set and GMV portfolio (3)
• The weights of assets corresponding to the GMV portfolio can be obtained analytically through the application of the following procedure:
Differentiating the expression for the portfolio variance
Setting the result to zero
Solving for the portfolio weights
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Principles of Finance - Lecture 3 33October 19, 2004
Minimum variance set and GMV portfolio: Example (1)
• Recall the two-asset example (Slide 15). The portfolio
variance is given by:
• Differentiation yields:
ABAABAAAP wwww σσσσ )1(2)1( 22222 −+−+=
ABAABBAAAA
P wwww
σσσσσ 42)1(22 222
−+−−=∂∂
ABBABBAAA
P ww
σσσσσσ 22)422( 2222
+−−+=∂∂
Principles of Finance - Lecture 3 34October 19, 2004
Minimum variance set andGMV portfolio: Example (2)
• Then, we can solve for wA and obtain XXXXX
• Sincesolving this equation renders the composition of the GMV portfolio:
02
=∂∂
A
P
wσ
⎩⎨⎧
==
2115.07885.0
B
A
ww
0007.0 and ,0040.0 ,0016.0 22 ≈≈≈ ABBA σσσ
ABBA
ABBAw
σσσσσ
222
2
−+−
=
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Principles of Finance - Lecture 3 35October 19, 2004
Minimum variance set and GMV portfolio: Example (3)
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%
Standard deviation
Expe
cted
retu
rnGMV Portfolio
Principles of Finance - Lecture 3 36October 19, 2004
Minimum variance set and GMV portfolio: Example (4)• The composition of the GMV portfolio can easily be
obtained in Excel using the SOLVER tool
4.16%Std. dev.8
0.0017Variance0.5000W_27
2.10%Return0.5000W_16
GMV featuresGMV weights5
4
0.00400.00072.81%Asset 230.00070.00161.40%Asset 12
Covariance matrixE(R)1
ONMLK Computed earlier
Type in an arbitrary number here(e.g. 0.5): it is just the starting value! =1-L6 =SQRT(O7)
=MMULT(TRANSPOSE
(L6:L7),L2:L3 )
=MMULT
(TRANSPOSE
(L6:L7),MMULT
(N2:O3,L6:L7))
19
Principles of Finance - Lecture 3 37October 19, 2004
Minimum variance set and GMV portfolio: Example (5)• Go to the Tools/Solver option that generates the
following dialogue box
Enter here the address of the cell containing portfolio variance
Enter here the address(es) of the cell(s) to be changed (here the onecontaining w1)
Tick here to select minimisationoption
Confirm by clicking on ‘Solve’
Principles of Finance - Lecture 3 38October 19, 2004
Minimum variance set and GMV portfolio: Example (6)• In the box that emerges, choose ‘Keep Solver Solution’
and confirm by clicking ‘OK’
• The following result will obtain
3.71%Std. dev.8
0.0014Variance0.2115W_27
1.70%Return0.7885W_16
GMV featuresGMV weights5
4
0.00400.00072.81%Asset 230.00070.00161.40%Asset 12
Covariance matrixE(R)1
ONMLK
GMV portfolioweights
GMV portfoliocharacteristics
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Principles of Finance - Lecture 3 39October 19, 2004
Benefits of diversification (1)
• A portfolio of assets can be less risky than any of the assets constituting such a portfolio
• How much risk can be eliminated?
• Assume a very simple diversification scheme (i.e. investing the same amount in each of the Nassets available):
• Recall that
Nwww N
121 ==== K
∑∑∑ +=Ω==
≠==
ΤN
i
N
ijj
ijji
N
iiiP wwwWW
1 11
222 σσσ
Principles of Finance - Lecture 3 40October 19, 2004
Benefits of diversification (2)
Case 1: All assets are independent
• The independence of assets implies that
• Therefore, the formula for portfolio variance
simplifies to
• Let denote the average variance of the stock
in the portfolio. Then
0: =≠∀ ijji σ
22 1iP N
σσ =
2iσ
∑=∑ ⎟⎠⎞
⎜⎝⎛=
==
N
i
iN
iiP NNN 1
2
1
22
2 11 σσσ
21
Principles of Finance - Lecture 3 41October 19, 2004
Benefits of diversification (3)Case 2: Portfolio assets are not independent• This is more realistic case• The formula for portfolio variance is given by
• Replacing summations by averages we get
( ) ijijiijiP NNN
Nσσσσσσ +−=
−+= 222 111
∑∑−
−+∑=
∑∑+∑ ⎟⎠⎞
⎜⎝⎛=
=≠==
=≠==
N
i
N
jj
ijN
i
iP
N
iij
N
jj
N
iiP
NNNN
NN
NNN
1111
22
1111
22
2
)1(11
111
σσσ
σσσ
Principles of Finance - Lecture 3 42October 19, 2004
Benefits of diversification (4)US example:
22
Principles of Finance - Lecture 3 43October 19, 2004
Benefits of diversification (5)UK example:
Principles of Finance Week 4: October 26, 2004
Tutorial problems
Problem 1
• EGBG Chapter 4, Exercise 1, Questions C and D, p. 64-65
Recall that during Week 2 tutorials we have shown that:
• =1R 12%; 2R = 6%; 3R = 14%; 4R = 12%
• 1σ = 2.83%; 2σ = 1.41%; 3σ = 4.24%; 4σ = 3.27%
• The variance/covariance matrix for all pairs of assets is:
1 2 3 4
1 8 −4 12 0
2 −4 2 −6 0
3 12 −6 18 0
4 0 0 0 10.7
• The correlation matrix for all pairs of assets is:
1 2 3 4
1 1 −1 1 0
2 −1 1 −1 0
3 1 −1 1 0
4 0 0 0 1
Problem 2
• EGBG Chapter 4, Exercise 2, Question E, p. 65-66
Recall that during Week 2 tutorials we have shown that:
• %22.1=AR %95.2=BR %92.7=CR
• %92.3=Aσ %8.3=Bσ %78.6=Cσ
• 17.2=ABσ 24.7=ACσ 89.19−=BCσ
• 15.0=ABρ 27.0=ACρ 77.0−=BCρ
Problem 3
• EGBG Chapter 4, Exercises 3-4, p. 66
Problem 4 (optional)
• EGBG Chapter 4, Exercise 6, p. 66
Problem 5
• EGBG Chapter 5, Exercise 1, Question B1, p. 96
Problem 6 (Based on EGBG Chapter 5, Exercises 2-3, p. 96)
• Consider assets analysed in Problem 2 above. Assume short selling is allowed. Find the
composition of the portfolio that has minimum variance for each two-security
combinations possible.