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INVESTMENT MANAGEMENT

SESSION 2, 2012

Lecture 5: The Capital Asset Pricing Model

Last Week

2

Index models

Systematic and idiosyncratic risks Calculating covariance

Case study

Calculating systematic and idiosyncratic risks Investment strategies Required return Reward-to-risk ratio

Today

3

Asset pricing models: what and why The Capital Asset Pricing Model (CAPM) Assumptions The claim Implications The economic mechanism The reality check Applications Extensions

Asset Pricing

4

Central issue: what is the “fair” or “required” return of a risky asset?

Sarah Wolfe of BMC Macro economy:

Why do we care?

Efficient capital allocation for growth Bubbles and crashes

Social welfare: Pension investments Firms’ cost of capital Performance evaluation:

Fund managers, trading strategies

Equilibrium Asset Pricing

5

E(Ri), i, ρij, i,j = 1,…N

Markowitz Portfolio Optimization

The minimum variance set (MVS) and the optimal risky portfolio

Capital Asset Pricing Model

The CAPM Assumptions

6

All investors are price takers, meanvariance optimisers, and have identical information and holding periods. All assets are marketable and divisible. The market portfolio includes ALL assets There is a single risk-free rate at which one can borrow or lend any amount No market imperfections (no taxes, short selling restriction, transaction costs, etc)

CAPM Conclusion

7

The market portfolio is the tangency portfolio on the efficient frontier: Investors: Same model (Markowitz) + same input parameters = same tangency portfolio (The Separation Property) Equilibrium: supply = demand Total shares issued by firms (market portfolio) = aggregate holdings of all investors (tangency portfolio)

CAPM Implications

8

The optimal portfolio for all investors is a combination of the risk-free asset and the market portfolio.

Growth of index funds

The market risk premium is proportional to its total risk: E(rM)-rf = Aσ , A is the average degree of risk aversion. The risk premium on individual assets is proportional to the market risk premium: E(ri)-rf = i[E(rM)-rf ] where i=Cov(ri,rM)/σ .

The Market Portfolio

9

The market portfolio has N assets:

E(rM)-rf=w1[E(r1)-rf ]+…+wi[E(ri)-rf ]+…+wN[E(rN)-rf ] σ ∑ ∑ w wjCov(ri,rj) =∑ w Cov(ri,∑ wr) =w1Cov(r1,rM)+…+ wiCov(ri,rM)+…+ wNCov(rN,rM)

Stock i’s contribution to the market risk premium is wi[E(ri)-rf ] and its contribution to the market risk is wiCov(ri,rM). The marginal reward-to-risk ratio for stock i is ,

.

Reward to Risk

10

For the market portfolio, the ratio is In equilibrium, this ratio is the same for all stocks and the market portfolio: for all i.

.

If not, what happens? (Case study)

The CAPM equation E(ri)-rf = i[E(rM)-rf ] with i=Cov(ri,rM)/ .

Security Market Line (SML)

11

In [E(r), ] space: E(ri) = rf + [E(rM)-rf ]i

E(r) M A rf rf βA βM=1

β

CML P MVS

E(r)

SML P M A

βP

as a Measure of Risk

12

is the scaled (by the variance of the market portfolio) covariance of the asset with the market portfolio. Covariance is about the timing of payoffs.

Good and bad returns relative to the market.

It is the contribution of a stock to the market/systematic risk. CAPM = SIM = SCL Page 312, #10-#16.

Limitations of

13

Risk is measured against one particular portfolio, the market portfolio. There are other risks, e.g. labor income risk. Static one-period measure What determines ? How does change over time?

A measure of the second moment

Risk measures based on skewness and kurtosis

Portfolio P

14

Portfolio is the weighted average of individual stock’s n Cov(rP ,rM ) P 2 M

n

Cov( wr,rM ) i i

i 1 2 M

...