Finance Lecture Notes

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25721

INVESTMENT MANAGEMENT
SESSION 2, 2012

Lecture 5: The Capital Asset Pricing Model

Last Week
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Index models
Systematic and idiosyncratic risks  Calculating covariance 



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

Today
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 

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
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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
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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
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  



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
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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
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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
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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
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 

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)
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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
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 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 
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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
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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

...
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