money, banking & finance lecture 3 risk, return and portfolio theory

Money, Banking & FinanceLecture 3

Risk, Return and Portfolio Theory

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 portfolio

choice based on the mean and variance of portfolio returns.

Overview• Investors choose a set of risky assets (stocks)

plus a risk-free asset.

• The risk-free asset is a term deposit or government Treasury bill.

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

• Investors like return but dislike risk (risk averse).

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.

Investor Utility function

0)(

)(/

0)(

0;0)(

),(

2

1

21

21

U

U

d

RdE

RdE

dU

d

dU

d

dUU

RdE

dUUdU

UU

URE

U

REUU

p

p

pp

pp

pp

pp

Preference Function

E(Rp) Expected return

σp Risk

U0

U2

Expected return

)()( 2211

2211

1

RERERE

RRR

RR

p

p

n

iiip

Risk

212,12122

22

21

21

2

21

212,1

212211

22112122

22

21

21

2222111

22

2

)()(

,

,)()(

)()(2

)()()(

p

ppp

RVarRVar

RRCov

RRCovRERRERE

RERRERE

RERRERERERE

Return and risk

• How do return and risk vary relative to each other as the investor alters the proportion of each of the assets in the portfolio?

• Assume that returns, risk and the covariance are fixed and simply vary the weights in the portfolio.

• Let E(R1)=8.75% and E(R2)=21.25• Let w1=0.75 and w2=0.25• E(Rp)=.75x8.75+.25x21.25=11.88• σ1=10.83, σ2=19.80, ρ1,2=-.9549

Portfolio Risk

• σ2p=(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

• σp=√13.7=3.7

• Calculate risk and return for different weights

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%

Locus of risk-return points

Expected return

Risk=standard deviation

(0,1)

(.5,.5)

(.75,.25)

(1,0)

Risk – return locus

• Can see that the locus of risk and returns vary according to the proportions of the equity held in the portfolio.

• The proportion (0.75,0.25) is the lowest risk point with 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 correlation coefficient and is called the efficient frontier

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)/2

covariances - becomes intractable• A short-cut is the Single Index Model (SIM) where each

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

• For ‘n’ assets the efficient frontier defines a ‘bundle’ of risky assets.

‘n’ asset case

n

i

n

jijjijip

n

iiip RERE

1 1

2

1

How is the efficient frontier derived?

• The shape of the efficient frontier will depend on the correlation between the asset returns of the two assets.

• If the correlation is ρ = +1 then the portfolio risk is the weighted average of the risk of the portfolio components.

• If the correlation is ρ = -1 then the portfolio risk can be diversified away to zero

• When ρ < +1 then not all the total risk of each investment is non-diversifiable. Some of it can be diversified away

Correlation of +1

21

221

2122

221

2

2,1

212,122

221

2

)1())1((

)1(2)1(

1

)1(2)1(

p

p

Correlation of -1

21

2

221

21

221

2122

221

2

0

0)1(

min

)1(

)1(2)1(

p

p

p

risk

Check

0

1

21

21

21

21

221

21

21

2

p

p

Correlation < +1

21 )1( p

Efficient frontier

Ρ = +1

Ρ = -1

-1 < Ρ < +1

E(Rp)

σp

The general case – applied to two assets

]2[

)(

)2(

2)(

02)1(

042)1(22

)1(2)1(

)1(2)1(

212,122

21

12,122

212,122212,1

22

21

212,1212,122

22

21

212,1212,122

21

212,1212,122

21

2

212,122

221

22

212,122

221

2

d

d p

p

p

Efficient Frontier

X

Y

E(Rp)

σp

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

rate of interest Rf.• The rate Rf is known with certainty and has

zero variance and therefore no covariance with the portfolio.

• Such a rate could be a short-term government bill or commercial bank deposit.

One bundle of risky assets

• Take one bundle of risky assets and allow the investor to lend 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 that relates the return on the portfolio with one risk-free asset and risk.

Transformation line

Np

Np

fnNfNfp

Nfp RRRE

)1(

)1(

)1()1(

)1(

222

22222

Linear Opportunity set

• Let the risk-free rate Rf = 10% and the return on the bundle of assets RN = 22.5%.

• The standard deviation of the returns on the bundle σN = 24.87%.

• The weights on the risky bundle and the risk-free asset can be varied to produce a range of new portfolio returns.

Portfolio Risk and Return

State T-bill Equity E(Rp) σp

(1-φ) φ

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%

Transformation line

• The transformation line describes the linear risk-return relationship for any portfolio consisting of a combination of investment in one safe asset and one ‘bundle’ of risky assets.

• At every point on a given transformation line the investor holds the risky assets in the same fixed proportions of the risky portfolio ωi.

Transformation line

Rf

No lending all investment in bundle

E(Rp)

σp

All lending

0.5 lending + 0.5 in risky bundle

-0.5 borrowing + 1.5 in risky bundle

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 of efficient portfolios.

• The single risky asset is replaced by a risky portfolio.

• We can find a dominant portfolio with the riskless asset that will be superior to all other combinations.

Combining risk-free and risky portfolios

A

B

C

Rf

E(Rp)

σp

Borrowing and Lending

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

• The risk-free rate of interest Rf represents the rate on Treasury Bills or some other risk-free asset.

• The efficiency boundary is redefined to include borrowing.

Borrowing and lending frontier

E(Rp)

σp

Rf

A

B

C

Combined borrowing and lending at different rates of

interest• The investor can borrow at the rate of

interest Rb

• Lend at the rate of interest Rf

• The borrowing rate is greater than the risk-free rate. Rb > Rf

• Preferences determine the proportions of lending or borrowing,

Combining borrowing and lending

E(Rp)

σp

Rb

A

B

C

D

Rf

P

Q

Separation Principle

• Investor makes 2 separate decisions• Given knowledge of expected returns, variances

and covariances the investor determines the efficient frontier. The point M is located with reference to Rf.

• The investor determines the combination of the risky portfolio and the safe asset (lending) or a leveraged portfolio (borrowing).

Market portfolio and risk reduction

Portfolio risk

Diversifiable risk

Non-diversifiable risk

Number of securities

20

Summary

• We have examine the theory of portfolio diversification

• We have seen how the efficient frontier is constructed.

• We have seen that portfolio diversification reduces risk to the non-diversifiable component.