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

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Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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Page 1: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Money, Banking & FinanceLecture 3

Risk, Return and Portfolio Theory

Page 2: Money, Banking & Finance Lecture 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.

Page 3: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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).

Page 4: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.

Page 5: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 6: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Preference Function

E(Rp) Expected return

σp Risk

U0

U2

Page 7: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Expected return

)()( 2211

2211

1

RERERE

RRR

RR

p

p

n

iiip

Page 8: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 9: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 10: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 11: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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%

Page 12: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Locus of risk-return points

Expected return

Risk=standard deviation

(0,1)

(.5,.5)

(.75,.25)

(1,0)

Page 13: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 14: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.

Page 15: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

‘n’ asset case

n

i

n

jijjijip

n

iiip RERE

1 1

2

1

Page 16: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 17: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 18: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Correlation of -1

21

2

221

21

221

2122

221

2

0

0)1(

min

)1(

)1(2)1(

p

p

p

risk

Page 19: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Check

0

1

21

21

21

21

221

21

21

2

p

p

Page 20: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Correlation < +1

21 )1( p

Page 21: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Efficient frontier

Ρ = +1

Ρ = -1

-1 < Ρ < +1

E(Rp)

σp

Page 22: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 23: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Efficient Frontier

X

Y

E(Rp)

σp

Page 24: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.

Page 25: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.

Page 26: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Transformation line

Np

Np

fnNfNfp

Nfp RRRE

)1(

)1(

)1()1(

)1(

222

22222

Page 27: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.

Page 28: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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%

Page 29: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.

Page 30: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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

Page 31: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.

Page 32: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Combining risk-free and risky portfolios

A

B

C

Rf

E(Rp)

σp

Page 33: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.

Page 34: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Borrowing and lending frontier

E(Rp)

σp

Rf

A

B

C

Page 35: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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,

Page 36: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Combining borrowing and lending

E(Rp)

σp

Rb

A

B

C

D

Rf

P

Q

Page 37: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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).

Page 38: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

Market portfolio and risk reduction

Portfolio risk

Diversifiable risk

Non-diversifiable risk

Number of securities

20

Page 39: Money, Banking & Finance Lecture 3 Risk, Return and Portfolio Theory

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.