SESSION 2, 2012
Lecture 5: The Capital Asset Pricing Model
Systematic and idiosyncratic risks Calculating covariance
Calculating systematic and idiosyncratic risks Investment strategies Required return Reward-to-risk ratio
Asset pricing models: what and why The Capital Asset Pricing Model (CAPM) Assumptions The claim Implications The economic mechanism The reality check Applications Extensions
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
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
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)
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)
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
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
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)
In [E(r), ] space: E(ri) = rf + [E(rM)-rf ]i
E(r) M A rf rf βA βM=1
CML P MVS
SML P M A
as a Measure of Risk
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.
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 is the weighted average of individual stock’s n Cov(rP ,rM ) P 2 M
Cov( wr,rM ) i i
i 1 2 M