Review of Portfolio Theory and CAPM

(Text reference: Chapters 5, 6, and 7)

� Real vs. nominal returns

� The historical record

� Measuring risk and risk aversion

� Capital allocation line and capital market line

� Market risk and diversifiable risk

� Portfolio selection

� Capital asset pricing model (CAPM)

� Security market line

� Black’s zero β model

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Real vs. nominal returns

� notation: R � nominal rate, r � real rate, and i � inflation rate

� an investment of $1 for 1 year � $1�1 � R � , but in real terms this is

only worth

1 � R1 � i

� 1 � r

� r � R � i1 � i R � i

� real returns are risky, even if nominal returns are certain

– due to uncertain future inflation

– bigger effect for longer horizons

2

� text Table 5.3 provides an illustration of the effects of inflation: 20year zero, price = $103.67, R � 12%

i 0% 4% 6% 8% 10% 12%

r 12.00% 7.69% 5.66% 3.70% 1.82% 0.00%

� example: r � 3% � i � 2% � R � 5 � 06%. Consider a 25 year $1,000

par value bond with a 6% coupon rate (annual coupons).

P � $60 � A25�0506 � $1 � 000

1 � 050625 � $1 � 131 � 69

Suppose i � 2 � 5% � R � 5 � 575%.

P � $60 � A25�05575 � $1 � 000

1 � 0557525 � $1 � 056 � 60 �

so P falls by about 6.6%.

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The historical record� sample period 1957-2001, text Table 5.2:

T-Bill Long Term Bond TSE 300 ∆CPI

Mean 7.2% 9.0% 10.8% 4.4%

Standard deviation 3.7% 10.6% 16.2% 3.3%

Maximum 18.4% 45.8% 44.8% 12.3%

Minimum 2.5% -7.4% -25.9% 0.0%

� the risk premium for stocks was about 3.6%

� these numbers can be very sensitive to the time period; between 1926-1988

risk premium in Canada was around 7.5%� see Table 6.1 for sub-period variation:

Period Average Risk Premium

1957-1969 5.17%

1970-1984 2.55%

1985-2001 3.38%

4

Measuring risk and risk aversion

� to most people, “risk” means the chance of a bad outcome

� is standard deviation an appropriate measure of risk? Yes, if the

probability distribution of returns is symmetric (e.g. normal):

E�r � rE

�r ��� σ E

�r ��� σ

5

� what if the distribution is skewed?

� do investors care about skewness?

� risk aversion:

– people differ, but the average investor certainly appears to dislike

risk

– a useful way to represent this is to think of the investor as having

a utility function of the form:

U � E�r � � 1

2Aσ2

where A is a risk aversion parameter

6

Capital allocation line and capital market line

� allocation between risk free and risky assets

σ

E�r �

Fr f

P

σP

E�rP �

CAL

7

� let y be the fraction of the investor’s wealth invested in P and let C denote a

combination of P and F . Then

E�rC ��� yE

�rP � �

�1 � y � r f

� r f � y�E�rP � � r f �

σ2C � y2σ2

P

� specifying a utility function, we can find the optimal value of y:

maxy

U�y ��� E

�rC ��� 1

2Aσ2

C

� r f � y�E�rP � � r f � � 1

2Ay2σ2

P

∂U∂y

� E�rP � � r f � Ayσ2

P

� y � � E�rP ��� r f

Aσ2P

8

� a passive strategy

– simply pick a broad market index and combine it with risk free

assets

– very cheap (low fees and commissions, infrequent trading, no

security analysis)

– free rider benefit

– can be quite hard to beat (most mutual fund managers

underperform relative to it)

– in this case we call the CAL the capital market line (CML)

9

Market risk and diversifiable risk

� recall the distinction between market risk and unique risk: even

simple strategies such as equally-weighted portfolios will diversify

away unique risk

N

σ2P

1

10

Portfolio selection

� expected return on a portfolio:

E�rP � �

N

∑i � 1

wiE�ri �

� variance of return on a portfolio:

σ2P �

N

∑i � 1

N

∑j � 1

wiw jσi j

�N

∑i � 1

N

∑j � 1

wiw jσiσ jρi j

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σ

E�r �

Two Asset Case

σ

E�r �

N Asset Case

12

� combining the N asset case with a risk free asset:

σ

E�r �

13

� the general sequence of steps:

1. Specify the asset classes to be included in the portfolio of risky

assets

2. Specify expected returns, standard deviations, and covariances

3. Find the efficient frontier of risky assets

4. Find the optimal portfolio of risky assets

5. Allocate funds between it and the risk free asset

� investors in practice will select different risky portfolios because of

different asset classes under consideration and different ways of

specifying means, variances, and covariances

� they also may face different constraints: no borrowing, dividend

yields, divergent borrowing/lending rates

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� utility functions

σ

E�r �

σ

E�r �

σ

E�r �

15

� portfolio selection without a risk free asset

σ

E�r �

16

� portfolio selection with no borrowing

σ

E�r �

17

� portfolio selection with different borrowing and lending rates

σ

E�r �

18

Capital asset pricing model

� assumptions:

1. All investors are price takers

2. All investors have a one period horizon

3. Investors can trade stocks and bonds, borrow/lend at r f

4. No taxes or transactions costs

5. All investors are mean-variance optimizers

6. Homogeneous expectations

� implication: all investors choose the same portfolio of risky assets,

and this must be a value-weighted portfolio of all risky assets (the

“market portfolio” M)

19

σ

E�r �

� note that the “passive strategy” of buying a market index and risk free

borrowing/lending is optimal

20

� assuming a utility function of the form on slide 6, from slide 8 we have that

the proportion of each investor’s wealth allocated to M is:

y � E�rM � � r f

Aσ2M

� on average y � 1, so E�rM ��� r f � Aσ2

M , where A is the average risk aversion

across investors

� the market risk premium E�rM ��� r f � Aσ2

M

� recall that the variance of M is:

σ2M � N

∑i � 1

N

∑j � 1

wiw jσi j

� the contribution of any particular stock B is:

N

∑j � 1

wBw jσB j � wB

N

∑j � 1

w jσB j � wBσBM

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� simple derivation of CAPM relationship:

– consider an investor who puts 100% in M:

E�r � � E

�rM �

σ2 � σ2M

– suppose the investor increases the amount invested in M by some

small percentage δ:

E�r � � �

1 � δ � E�rM � � δr f

σ2 � �1 � δ � 2σ2

M� ∆E

�r � � δ

�E

�rM � � r f �

∆σ2 � �δ2 � 2δ � σ2

M

2δσ2M

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– therefore∆E

�r �

∆σ2 � E�rM � � r f

2σ2M

– repeat the exercise, but have the investor put δ into some stock B

instead of M:

� E�r � � E

�rM � � δE

�rB � � δr f

σ2 � σ2M � δ2σ2

B � 2δσBM

� ∆E�r � � δ

�E

�rB � � r f �

∆σ2 � δ2σ2B � 2δσBM

2δσBM

� ∆E�r �

∆σ2 � E�rB � � r f

2σBM

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– in equilibrium, all marginal prices of risk must be equal:

E�rB � � r f

2σBM� E

�rM � � r f

2σ2M

� E�rB � � r f � �

E�rM � � r f � σBM

σ2M

or E�rB � � r f � βB

�E

�rM � � r f �

� remarks:

1. βM � 1 � βr f � 0

2. CAPM relationship holds for any portfolio P since

E�rP � � ∑k wkE

�rk � and βP � ∑k wkβk

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Security market line

β0 1

E�r �

r f

E�rM �

SML

25

Black’s Zero β Model

� there are many CAPM extensions (non-traded assets, multiple periods, taxes,

heterogeneous expectations, etc.), but we will concentrate on the case where

there is no risk free asset

� consider some properties of the efficient set of risky assets:

(i) Any portfolio constructed by combining efficient portfolios is an efficient

portfolio.

(ii) Every efficient portfolio has a companion portfolio on the inefficient part

of the frontier with which it is uncorrelated.

σ

E�r �

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(iii) The expected return on any asset can be written as a function of

the expected return on any two portfolios on the efficient frontier.

Call these portfolios P and Q. Then:

E�ri � � E

�rQ � � �

E�rP � � E

�rQ � � σiP � σPQ

σ2P � σPQ

� all of this follows from the mathematics of the efficient frontier. If we

make CAPM assumptions (except for there being a risk free asset),

then all investors will derive the same efficient frontier and select

some portfolio of risky assets on it.

� the market portfolio M is just a combination of individual investors’

portfolios and therefore by (i) is also efficient.

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� if we use M and Z�M � in (iii) we obtain:

E�ri � � E

�rZ � M � � ��� E �

rM � � E�rZ � M � ��� σiM � σMZ � M �

σ2M � σMZ � M �

� but M and Z�M � are uncorrelated:

� E�ri � � E

�rZ � M � � � � E �

rM � � E�rZ � M � � � σiM

σ2M

� this means that we simply replace r f in the standard CAPM by

E�rZ � M � �

� this model can also be applied in cases where there is lending but no

borrowing, different borrowing and lending rates, etc.

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