Principles of Finance - Exeterpeople.exeter.ac.uk/wl203/BEAM010/Materials/Lecture 3...3 October 19, 2004 Principles of Finance - Lecture 3 5 Lecture 1 recap (2) • If we have a sample

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Principles of Finance - Lecture 3 1October 19, 2004

Principles of Finance

Grzegorz Trojanowski

Lecture 3:Combining assets into portfolios


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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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


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

3


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


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


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


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)

5



• 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)


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 −+−+=

6


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


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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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



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

8


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


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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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


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

=

10


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


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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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


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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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


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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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


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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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



• 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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Feasible setFeasible set


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

16


Minimum variance set and GMV portfolio (2)

GMV Portfolio


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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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

+−−+=∂∂


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

−+−

=

18


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


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


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’


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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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


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


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


Benefits of diversification (4)US example:

22


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.

Documents

Principles of Finance - Exeterpeople.exeter.ac.uk/wl203/BEAM010/Materials/Lecture 3...3 October 19, 2004 Principles of Finance - Lecture 3 5 Lecture 1 recap (2) • If we have a sample