money, banking & finance lecture 3 risk, return and portfolio theory
TRANSCRIPT
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
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212211
22112122
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2222111
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)()(
,
,)()(
)()(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.