Capital Asset Pricing Model

Summaries

Equally weighted portfolio risk:

Variance = average_variance / n + (n-1)/n * average_covariance

when n is larger, variance = average_covariance

*** average_covariance is calculated by dividing (n-1)*n

Or variance = average_variance *((1-p)/n + p), where p is average correlation

1)      minimum is average correlation * average variance

2)      larger the p, the few stocks to reach 110% of the minimum

Capital Allocation Line:

1. Inclusion of risk free asset making the risk-return line a straight line
2. Choose the one tangent to the efficient frontier to get CAL

R_p = R_rf + (R_T – Rrf) * sigma_p/sigma_T

Intercept = risk-free rate

Slope = Sharpe ratio (reward to risk ratio) of the optimal portfolio (the tangency portfolio) = (R_T – R_rf)/sigma_T

*** This is also the maximum portfolio return-to-risk ratio available for a give risk free rate

*** If borrowing is not available, the efficient frontier is a straight line (before the tangent) + a curve (after the tangent, the original efficient frontier without risk-free asset)

Capital Market Line (CML)

Assume all investors have homogeneous expectations (one mean, variance and co-variance of all market assets)

So CML is a special case of CAL, where all investors’ CAL merge and tangency portfolio becomes the market portfolio.

Capital Asset Pricing Model (CAPM)

Assumptions:

1)      Investors only know and all have the same expectation on the mean, variance and co-variance of risky assets

2)      There is no fiction in the market (no transaction cost, tax)

3)      Unlimited borrowing of risk-free asset and short selling

4)      All assets are competitive and marketable

Therefore,

1)      All investors have exactly the same portfolio

2)      Beta reflects only the market risk (systematic)

If 2) is violated, less risk-averse investors may hold different risky portfolio from the more risk-averse. The investment will also be below the efficient frontier.

Security Market Line (SML) is the graph of CAPM

E(Ri) = R_rf + Beta_i *(E(Rm)-E_rf)

Market risk premium = E(Rm) – E_rf = compensation for additional unit of systematic risk

Slope is the market risk premium. NOT the beta

*** Although similar to CML, this is for valuating an individual risky asset!

Beta_i = cov(i,m)/var_m = p(i,m) sigma_i/sigma_m