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Money, Banking & Finance

Lecture 3

Risk, Return and Portfolio Theory

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Aims

Explain the principles of portfolio diversification

Demonstrate the construction of the efficient

frontier Show the trade-off between risk and return

Derive the Capital Market Line (CML)

Show the calculation of the optimal portfoliochoice based on the mean and variance ofportfolio returns.

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Overview

Investors choose a set of risky assets(stocks) plus a risk-free asset.

The risk-free asset is a term deposit orgovernment Treasury bill.

Investors can borrow or lend as much asthey like at the risk-free rate of interest.

Investors like return but dislike risk (riskaverse).

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Preferences of Expected return

and risk We have seen how expected return is defined in Lecture 2. The investor faces a number of stocks with different

expected returns and differ from each other in terms of

risk. The expected return on the portfolio is the weighted mean

return of all stocks. First moment.

Risk is measured in terms of the variance of returns or

standard deviation. Second moment. Investor preferences are in terms of the first and second

moments of the distribution of returns.

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Investor Utility function

0)(

)(/

0

)(

0;0)(

),(

2

1

21

21

U

U

d

RdE

RdE

dU

d

dUd

dUU

RdE

dUUdU

U

U

URE

U

REUU

p

p

pp

pp

pp

pp

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Preference Function

E(Rp)Expected return

pRisk

U0

U2

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Expected return

)()( 22112211

1

RERERE

RRR

RR

p

p

n

iiip

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Risk

212,121

2

2

2

2

2

1

2

1

2

21

212,1

212211

221121

2

2

2

2

2

1

2

1

2

222111

22

2

)()(

,

,)()(

)()(2

)()()(

p

ppp

RVarRVar

RRCov

RRCovRERRERE

RERRERE

RERRERERERE

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Return and risk

How do return and risk vary relative to each otheras the investor alters the proportion of each of theassets in the portfolio?

Assume that returns, risk and the covariance arefixed and simply vary the weights in the portfolio.

LetE(R1)=8.75% andE(R2)=21.25

Let w1=0.75 and w2=0.25 E(Rp)=.75x8.75+.25x21.25=11.88 1=10.83, 2=19.80, 1,2=-.9549

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Portfolio Risk

2p=(0.75)2x(10.83)2+(0.25)2x(19.80)2+2x(0.75)x(0.25)x(-0.95)x(10.83)x(19.80)

=13.7 p=13.7=3.7

Calculate risk and return for differentweights

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Portfolio risk and return

Equity 1 Equity 2 E(Rp) Risk

State w1 w2

1 1 0 8.75% 10.83%

2 0.75 0.25 11.88% 3.70%

3 0.5 0.5 15% 5%

4 0 1 21.25 19.8%

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Locus of risk-return points

Expected

return

Risk=standard

deviation

(0,1)

(.5,.5)

(.75,.25)

(1,0)

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Riskreturn locus

Can see that the locus of risk and returns varyaccording to the proportions of the equity held inthe portfolio.

The proportion (0.75,0.25) is the lowest risk pointwith highest return.

The other points are either higher risk and higher

return or low return and high risk. The locus of points vary with the correlationcoefficient and is called the eff icient frontier

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Choice of weights

How does the portfolio manager choose the weights? That will depend on preferences of the investor. What happens if the number of assets grows to a large

number.

If n is the number of assets then will need n(n-1)/2covariances - becomes intractable

A short-cut is the Single Index Model (SIM) where each

asset return is assumed to vary only with the return of thewhole market (FTSE100, DJ, etc).

For n assets the efficient frontier defines a bundle ofrisky assets.

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n asset case

n

i

n

j ijjijip

n

iiip RERE

1 1

2

1

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How is the efficient frontier

derived? The shape of the efficient frontier will depend on

the correlation between the asset returns of the twoassets.

If the correlation is = +1 then the portfolio risk isthe weighted average of the risk of the portfoliocomponents.

If the correlation is = -1 then the portfolio risk

can be diversified away to zero When < +1 then not all the total risk of eachinvestment is non-diversifiable. Some of it can bediversified away

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Correlation of +1

21221

21

2

2

22

1

2

2,1

212,1

2

2

22

1

2

)1())1((

)1(2)1(

1

)1(2)1(

p

p

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Correlation of -1

21

2

221

21

2

21

21

2

2

22

1

2

0

0)1(

min

)1(

)1(2)1(

p

p

p

risk

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Correlation < +1

21 )1( p

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Efficient frontier

= +1

= -1

-1 < < +1

E(Rp)

p

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The general caseapplied to

two assets

]2[

)(

)2(

2)(

02)1(

042)1(22

)1(2)1(

)1(2)1(

212,1

2

2

2

1

12,122

212,122212,1

22

21

212,1212,1

2

2

2

2

2

1

212,1212,1

2

2

2

1

212,1212,1

2

2

2

1

2

212,1

2

2

22

1

22

212,1

2

2

22

1

2

dd p

p

p

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Efficient Frontier

X

Y

E(Rp)

p

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Risk-free asset Lets introduce a risk-free asset that pays a

rate of interestRf.

The rateRfis known with certainty and haszero variance and therefore no covariancewith the portfolio.

Such a rate could be a short-termgovernment bill or commercial bankdeposit.

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One bundle of risky assets

Take one bundle of risky assets and allow the investor tolend or borrow at the safe rate of interest. The investor can;

Invest all his wealth in the risky bundle and undertake no

lending or borrowing. Invest less than his total wealth in the single risky bundle

and the rest in the risk-free asset.

Invest more than his total wealth in the risky bundle by

borrowing at the risk-free rate and hold a levered portfolio. These choices are shown by the transformation line thatrelates the return on the portfolio with one risk-free assetand risk.

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Transformation line

Np

Np

fnNfNfp

Nfp RRRE

)1(

)1(

)1()1(

)1(

222

22222

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Linear Opportunity set

Let the risk-free rateRf= 10% and the return on thebundle of assetsRN= 22.5%.

The standard deviation of the returns on the bundleN= 24.87%.

The weights on the risky bundle and the risk-freeasset can be varied to produce a range of new

portfolio returns.

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Portfolio Risk and Return

State T-bill Equity E(Rp) p

(1-)

1 1 0 10% 0%

2 0.5 0.5 16.25% 12.44%

3 0 1 22.5% 24.87%

4 -0.5 1.5 28.75% 37.31%

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Transformation line

The transformation line describes the linear risk-return relationship for any portfolio consisting of a

combination of investment in one safe asset andone bundleof risky assets.

At every point on a given transformation line theinvestor holds the risky assets in the same fixed

proportions of the risky portfolio i.

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Transformation line

Rf

No lending all

investment in

bundle

E(Rp)

p

All lending

0.5 lending + 0.5

in risky bundle

-0.5 borrowing + 1.5

in risky bundle

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A riskless asset and a risky

portfolio An investor faces many bundles of risky assets (eg

from the London Stock Exchange).

The efficient frontier defines the boundary ofefficient portfolios.

The single risky asset is replaced by a riskyportfolio.

We can find a dominant portfolio with the risklessasset that will be superior to all othercombinations.

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Combining risk-free and risky

portfolios

A

BC

Rf

E(Rp)

p

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Borrowing and Lending

The investor can lend or borrow at the risk-free rate of interest rate.

The risk-free rate of interest Rf representsthe rate on Treasury Bills or some other

risk-free asset.

The efficiency boundary is redefined toinclude borrowing.

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Borrowing and lending frontier

E(Rp)

p

Rf

A

B

C

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Combined borrowing and

lending at different rates of

interest The investor can borrow at the rate of

interestRb

Lend at the rate of interestRf The borrowing rate is greater than the risk-

free rate.Rb > Rf

Preferences determine the proportions oflending or borrowing,

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Combining borrowing and

lendingE(Rp)

p

Rb

A

B

C

D

Rf

P

Q

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Separation Principle

Investor makes 2 separate decisions

Given knowledge of expected returns, variances

and covariances the investor determines theefficient frontier. The point M is located with

reference toRf.

The investor determines the combination of therisky portfolio and the safe asset (lending) or aleveraged portfolio (borrowing).

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Market portfolio and risk

reductionPortfolio

risk

Diversifiable

risk

Non-

diversifiable

risk

Number of

securities

20

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Summary

We have examine the theory of portfoliodiversification

We have seen how the efficient frontier isconstructed.

We have seen that portfolio diversification

reduces risk to the non-diversifiablecomponent.