LECTURE 5 :LECTURE 5 :

PORTFOLIO THEORYPORTFOLIO THEORY

(Asset Pricing and Portfolio (Asset Pricing and Portfolio Theory)Theory)

ContentsContents

Principal of diversification Principal of diversification Introduction to portfolio theory (the Introduction to portfolio theory (the

Markowitz approach) – mean-variance Markowitz approach) – mean-variance approachapproach– Combining risky assets – the efficient frontier Combining risky assets – the efficient frontier – Combining (a bundle of) risky assets and the risk Combining (a bundle of) risky assets and the risk

free rate – transformation line free rate – transformation line – Capital market line (best transformation line)Capital market line (best transformation line)– Security market lineSecurity market line

Alternative (mathematical) way to obtain the Alternative (mathematical) way to obtain the MV results MV results – Two fund theorem Two fund theorem – One fund theoremOne fund theorem

IntroductionIntroduction

How should we divide our wealth ? – say How should we divide our wealth ? – say £100£100

Two questions : Two questions : Between different risky assets (Between different risky assets (’s > 0)’s > 0) Adding the risk free rate (Adding the risk free rate ( = 0) = 0)

Principle of insurance is based on concept Principle of insurance is based on concept of ‘diversification’ of ‘diversification’

pooling of uncorrelated events pooling of uncorrelated events

insurance premium relative small proportion insurance premium relative small proportion of the value of the items (i.e. cars, building)of the value of the items (i.e. cars, building)

Assumption : Mean-Assumption : Mean-Variance ModelVariance Model Investors : Investors :

prefer a higher expected return to lower returnsprefer a higher expected return to lower returns ERERAA ≥ ER ≥ ERBB

Dislike riskDislike risk var(Rvar(RAA) ≤ var(R) ≤ var(RBB) or SD(R) or SD(RAA) ≤ SD(R) ≤ SD(RBB) )

Covariance and correlation : Covariance and correlation :

Cov(RCov(RAA, R, RBB) = ) = SD(RSD(RAA) SD(R) SD(RBB) )

Portfolio : Expected Portfolio : Expected Return and VarianceReturn and VarianceFormulas (2 asset case) : Formulas (2 asset case) :

Expected portfolio return : Expected portfolio return : ERERpp = w = wAA ER ERAA + w + wbb ER ERBB Variance of portfolio return : Variance of portfolio return :

var(Rvar(Rpp) = w) = wAA2 2 var(Rvar(RAA) + w) + wBB

22 var(R var(RBB) + 2w) + 2wAAwwBBCov(RCov(RAA,R,RBB) )

Matrix notation : Matrix notation :

Expected portfolio return : Expected portfolio return : ERERpp = w’ER = w’ERii

Variance of portfolio return : Variance of portfolio return : var(Rvar(Rpp) = w’) = w’ww

where where w is (nx1) vector of weights w is (nx1) vector of weights

ERERii is (nx1) vector of expected returns of individual is (nx1) vector of expected returns of individual assetsassets

is (nxn) variance covariance matrixis (nxn) variance covariance matrix

Minimum Variance Minimum Variance ‘Efficient’ Portfolio‘Efficient’ Portfolio 2 asset case : w2 asset case : wAA + w + wBB = 1 or w = 1 or wBB = 1 – w = 1 – wAA

var(Rvar(Rpp) = w) = wAA22 AA

22 + w + wBB22 BB

22 + 2w + 2wAA w wBB AABB

var(Rvar(Rpp) = w) = wAA22 AA

22 + (1-w + (1-wAA))22 BB22 + 2w + 2wAA (1-w (1-wAA) ) AABB

To minimise the portfolio variance : To minimise the portfolio variance : Differentiating with respect to wDifferentiating with respect to wAA

∂∂pp22/∂w/∂wAA = 2w = 2wAAAA

22 – 2(1-w – 2(1-wAA))BB22 + 2(1-2w + 2(1-2wAA))AABB = 0 = 0

Solving the equation : Solving the equation :

wwAA = [= [BB22 – – AABB] / [] / [AA

22 + + BB22 – 2 – 2AABB] ]

= (= (BB22 – – ABAB) / () / (AA

22 + + BB22 – 2 – 2ABAB) )

Power of Power of DiversificationDiversification As the number of assets (n) in the As the number of assets (n) in the

portfolio increases, the SD (total portfolio increases, the SD (total riskiness) fallsriskiness) falls

Assumption : Assumption : – All assets have the same variance : All assets have the same variance : ii

22 = = 22

– All assets have the same covariance : All assets have the same covariance : ijij = = 22

– Invest equally in each asset (i.e. 1/n) Invest equally in each asset (i.e. 1/n)

Power of Power of Diversification (Cont.)Diversification (Cont.) General formula for calculating the General formula for calculating the

portfolio variance portfolio variance 22

pp = = wwii22 ii

22 + + w wiiwwjj ijij

Formula with assumptions imposedFormula with assumptions imposed22

pp = (1/n) = (1/n) 22 + ((n-1)/n) + ((n-1)/n) 22

If n is large (1/n) is small and ((n-1)/n) is close If n is large (1/n) is small and ((n-1)/n) is close to 1. to 1.

Hence : Hence : 22pp 22

Portfolio risk is ‘covariance risk’. Portfolio risk is ‘covariance risk’.

Random Selection of Random Selection of StocksStocks

Sta

nd

ard

devia

tion

No. of shares in portfolio

Diversifiable / idiosyncratic risk

Market / non-diversifiable risk

20 400 1 2 ...

C

Example : 2 Risky Example : 2 Risky AssetsAssets

Equity 1Equity 1 Equity 2Equity 2

Mean Mean 8.75%8.75% 21.25%21.25%

SDSD 10.83%10.83% 19.80%19.80%

CorrelationCorrelation -0.9549-0.9549

Covariance Covariance -204.688-204.688

Example : Portfolio Example : Portfolio Risk and ReturnRisk and Return

Share of Share of wealth in wealth in

PortfolioPortfolio

ww11 ww22 ERERpp pp

11 11 00 8.75%8.75% 10.83%10.83%

22 0.750.75 0.250.25 11.88%11.88% 3.70%3.70%

33 0.50.5 0.50.5 15%15% 5%5%

44 00 11 21.25%21.25% 19.80%19.80%

Example : Efficient Frontier

0

5

10

15

20

25

0 5 10 15 20 25

Standard deviation

Exp

ec

ted

re

turn

(%

)

1, 0

0, 1

0.5, 0.5

0.75, 0.25

Efficient and Inefficient Efficient and Inefficient PortfoliosPortfolios

ERp

p

x

x

x

x

x

x

xx

xx

xx

x

x

xxB

A

C

P1

P1

x

L

Up* = 10%

p** = 9%

p** p*

Risk Reduction Risk Reduction Through Through DiversificationDiversification

Std. dev.

Exp

ecte

d re

turn

X

Y

= 0.5

= 0

= -0.5

= -1

B A

Z

C

= +1

Introducing Borrowing Introducing Borrowing and Lending : Risk Free and Lending : Risk Free AssetAsset Stage 2 of the investment process : Stage 2 of the investment process :

– You are now allowed to borrow and lend at the You are now allowed to borrow and lend at the risk free rate r while still investing in any risk free rate r while still investing in any SINGLE ‘risky bundle’ on the efficient frontier. SINGLE ‘risky bundle’ on the efficient frontier.

– For each SINGLE risky bundle, this gives a new For each SINGLE risky bundle, this gives a new set of risk return combination known as the set of risk return combination known as the ‘transformation line’. ‘transformation line’.

– Rather remarkably the risk-return combination Rather remarkably the risk-return combination you are faced with is a straight line (for each you are faced with is a straight line (for each single risky bundle) - transformation line. single risky bundle) - transformation line.

– You can be anywhere you like on this line. You can be anywhere you like on this line.

Example : 1 ‘Bundle’ of Example : 1 ‘Bundle’ of Risky Assets + Risk Free Risky Assets + Risk Free RateRate

ReturnsReturns

T-Bill T-Bill (safe)(safe)

Equity Equity (Risky)(Risky)

MeanMean 10%10% 22.5%22.5%

SDSD 0%0% 24.87%24.87%

‘‘Portfolio’ of Risky Portfolio’ of Risky Assets and the Risk Assets and the Risk Free AssetFree Asset Expected return Expected return

ERERNN = (1 – x)r = (1 – x)rff + xER + xERpp

Riskiness Riskiness 22

NN = x = x2222pp or or = x = xpp

where where x = proportion invested in the portfolio of risky assetsx = proportion invested in the portfolio of risky assetsERERpp = expected return on the portfolio containing only risky = expected return on the portfolio containing only risky assetsassetspp = standard deviation of the portfolio of risky assets = standard deviation of the portfolio of risky assetsERERNN = expected return of new portfolio (including the risk free = expected return of new portfolio (including the risk free asset) asset) NN = standard deviation of new portfolio = standard deviation of new portfolio

Example : New Example : New Portfolio With Risk Portfolio With Risk Free AssetFree Asset

Share of Share of wealth inwealth in

PortfolioPortfolio

(1-x)(1-x) xx ERERNN NN

11 11 00 10%10% 0%0%

22 0.50.5 0.50.5 16.25%16.25% 12.44%12.44%

33 00 11 22.5%22.5% 24.87%24.87%

44 -0.5-0.5 1.51.5 28.75%28.75% 37.31%37.31%

Example : Example : Transformation LineTransformation Line

All lending

No borrowing/no lending

0.5 lending + 0.5 in 1 risky bundle

-0.5 borrowing + 1.5 in 1 risky bundle

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40

Standard deviation (Risk)

Transformation Line Transformation Line

Expected Return, N

X Standard Deviation, N

XL

Q

Z

Borrowing/leverage

Lending

r

all wealth in risky asset

all wealth in risk-free asset

The CML – Best The CML – Best Transformation LineTransformation Line

ERp

p

Transformation line 1

Transformation line 2

Transformation line 3 – best possible one

rf

Portfolio A

Portfolio M

Expected return CML

Std. dev., i20

Market Portfolio

rf

Risk Premium / Equity Premium(ERm – rf)

The Capital Market The Capital Market Line (CML)Line (CML)

Expected return SML

Beta, i0.5 1 1.2

Market Portfolio

The larger is i, the larger is ERi

rf

Risk Premium / Equity Premium(ERi – rf)

The Security Market The Security Market Line (SML)Line (SML)

Risk Adjusted Rate of Risk Adjusted Rate of Return MeasuresReturn Measures Sharpe Ratio : Sharpe Ratio : SRSRii = (ER = (ERii – r – rff) / ) / ii

Treynor Ratio : Treynor Ratio : TRTRii = (ER = (ERii – r – rff) / ) / ii

Jensen’s alpha : Jensen’s alpha : (ER(ERii – r – rff))tt = = ii + + ii(ER(ERmm – r – rff))tt + + itit

Objective : Objective : Maximise Sharpe ratio (or Treynor ratio, or Maximise Sharpe ratio (or Treynor ratio, or Jensen’s alpha)Jensen’s alpha)

Portfolio ChoicePortfolio Choice

ER

Capital Market Line

r

A

K

M

ERm

ERm - r

IB

IA Y

Z’

m

Q

L

Math ApproachMath Approach

Solving Markowitz Solving Markowitz Using Lagrange Using Lagrange MultipliersMultipliers Problem : min ½(Problem : min ½(wwiiwwjjijij))

Subject to Subject to

wwiiERERii = k (constant) = k (constant)

wwii = 1 = 1

Lagrange multiplier Lagrange multiplier and and L = ½ L = ½ wwiiwwjjijij – – ((wwiiERERii – k) – – k) – ((wwii – 1) – 1)

Solving Markowitz Using Solving Markowitz Using Lagrange Multiplier Lagrange Multiplier (Cont.)(Cont.) Differentiating L with respect to the Differentiating L with respect to the

weights (i.e. wweights (i.e. w11 and w and w22) and setting ) and setting the equation equal to zero the equation equal to zero

For 2 variable case For 2 variable case 11

22ww11 + + 1212ww22 – – kk11 – – = 0 = 0

11ww11 + + 2222ww22 – – kk22 – – = 0 = 0

The two equations can now be solved for The two equations can now be solved for the two unknowns the two unknowns and and . .

Together with the constraints we can now Together with the constraints we can now solve for the weights. solve for the weights.

The Two-Fund The Two-Fund TheoremTheorem Suppose we have two sets of weight : wSuppose we have two sets of weight : w11 and w and w2 2

(obtained from solving the Lagrangian), then (obtained from solving the Lagrangian), then

ww11 + (1- + (1-)w)w22

for -∞< for -∞< < ∞ are also solutions and map out < ∞ are also solutions and map out the whole efficient frontierthe whole efficient frontier

Two fund theorem : Two fund theorem : If there are two efficient portfolios, then any other If there are two efficient portfolios, then any other efficient portfolio can be constructed using those two. efficient portfolio can be constructed using those two.

One Fund TheoremOne Fund Theorem

With risk free lending and borrowing is With risk free lending and borrowing is introduced, the efficient set consists of a introduced, the efficient set consists of a single line. single line.

One fund theorem : One fund theorem : There is a single fund M of risky assets, so There is a single fund M of risky assets, so that any efficient portfolio can be that any efficient portfolio can be constructed as a combination of this fund constructed as a combination of this fund and the risk free rate. and the risk free rate.

Mean = Mean = rrff + (1- + (1-))SD = SD = rfrf + (1- + (1-))

References References

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial (2004) ‘Quantitative Financial Economics’, Chapter 5 Economics’, Chapter 5

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and (2001) ‘Investments : Spot and Derivatives Markets’, Chapters 10 Derivatives Markets’, Chapters 10 and 18and 18

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