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CAPITAL ASSET PRICING MODEL

(CAPM)
AND
ARBITRAGE PRICING THEORY
(APT)
OPPORTUNITY SET WITH MANY RISKY ASSETS
AND 1 RISK FREE ASSET
The opportunity set with N risky assets and 1 risk-free asset case can
be easily extended from the N risky only assets case:
• Any point on (and inside) the opportunity set can be combined with the
risk free asset
• The combinations will lie on a straight line. Example: for any 1>a>0
invested in Rf and (1-a) in A, the combination will lie on the straight line
ABRf.
• The efficient set now becomes RfMA
• However, a risk-free asset is equivalent to having the ability to borrow
and lend at the rate Rf.
• The efficient set can be extended (through short-selling the Rf asset to
become a linear set with M at a tangent to curve AMBCD
THE CAPITAL MARKET LINE
Similar to Fisher Separation, we get Two-Fund Separation
• All investors will seek a combination of Rf and M to
maximize their utility.
• Recall: Fisher separation – All investors agreed there was
one optimal production point determined by the market
rate of interest, regardless of their personal preferences.
• Personal preferences only determined present vs future
consumption choice, not the production choice.
• Two-Fund Separation – All investors agree that the two relevant assets to be
holding are the risk free asset and the portfolio corresponding to point M.
• Personal preferences determine only the weights assigned to Rf vs M.
The line RfMC is called the Capital Market Line (CML)
THE CAPITAL MARKET LINE
• Once again, we see the introduction of a capital market
(ability to borrow & lend) improves the aggregate welfare
• Utility for some improves, and no one is worse off
(Pareto optimality)
• In equilibrium, M will be the market portfolio (why?)
• All assets must be held at market clearing prices in
equilibrium i.e. excess demand to what is already held
in the market portfolio must be zero
• Slope of the CML will be (E(Rm) – Rf)/σ(Rm)
• The fact that all investors hold a combination of only 2 assets – the risk free asset
and the market portfolio – is a very useful result
• We don’t have to worry about computing the efficient set for the entire combination of
risky assets (the full curve MB). Just the knowledge of the market portfolio will
suffice.
PORTFOLIO DIVERSIFICATION AND INDIVIDUAL
ASSET RISK
• Recall: Portfolio Variance calculation involves N variance terms and N(N-1)
covariance terms
• Assume an equally weighted portfolio i.e. wi = 1/N
• The portfolio variance can be separated into variance and covariance
terms
• = 12∑ σ + =1 12 ∑ ∑ σ=1
≠
=1
• Suppose the largest individual asset variance is L and the average
covariance is σ�
• Then 1
2
∑ σ ≤=1 /N and as N becomes large the limit approaches 0
• 1
2
∑ ∑ σ

=1
≠
=1 = (N-1)/N * σ� which in the limit approaches σ�
PORTFOLIO DIVERSIFICATION AND INDIVIDUAL
ASSET RISK
• As we increase the size of a portfolio (in terms of number of assets, N), the
covariance terms become relatively more important – and in the limit are
the only contributor to portfolio volatility or risk
• Diversification reduces the impact of the variance terms
• Typically, for a portfolio of stocks the first 10-15 stocks have the largest
impact
Contribution of a single asset to portfolio
risk:
• Evaluate the change in portfolio variance
w.r.t change in asset’s weight ∂()∂ = 2wiσi2 + 2∑ σ=1≠
• Again, suppose wi = 1/N
• In the limit, the marginal contribution
of a single asset to portfolio risk is a
function of its average covariance with
the other assets in the portfolio
• The above insight was crucial to the development of the CAPM
• While variance is a good measure of portfolio risk, covariance is the
right measure of risk for an individual security
THE CAPITAL ASSET PRICING MODEL (CAPM)
Based on similar suppositions we have encountered previously..
Main assumptions of the CAPM are:
1. Investors are risk-averse and maximize expected utilty
2. Asset returns have joint normal distributions (and there is agreement amongst
investors i.e. homogenous beliefs)
3. Existence of a risk free asset and capital markets that allows for borrowing and
lending at this rate
• This also implies frictionless markets. In addition no market imperfections such as
taxes, or regulations such as restrictions on short selling are assumed
Note: Above assumptions imply the existence of a Capital Market Line
and a linear efficient set
THE CAPITAL ASSET PRICING MODEL (CAPM)
We will also accept the claim that the Market Portfolio is an efficient
portfolio in equilibrium (i.e. lies on the efficient set)
• Proof of the market portfolio being efficient is quite involved but can be inferred
from other observations
• As long as investors have homogenous beliefs, there will be one efficient set or
frontier