56 Frequently Asked Questions In Quantitative Finance
we have
μ=α+βE[RM]=α+βμM.
Similarly the risk inis measured by
σ=
√
√
√
√
∑N
i= 1
∑N
j= 1
WiWjβiβjσM^2 +
∑N
i= 1
Wi^2 e^2 i.
Note that if the weights are all about the same,N−^1 ,then
the final terms inside the square root are alsoO(N−^1 ).
Thus this expression is, to leading order asN→∞,
σ=
∣
∣
∣
∣
∣
∑N
i= 1
Wiβi
∣
∣
∣
∣
∣
σM=|β|σM.
Observe that the contribution from the uncorrelated
s to the portfolio vanishes as we increase the num-
ber of assets in the portfolio; this is the risk associated
with the diversifiable risk. The remaining risk, which is
correlated with the index, is the undiversifiable system-
atic risk.
Multi-index versions of CAPM can be constructed. Each
index being representative of some important financial
or economic variable.
The parameters alpha and beta are also commonly
referred to in the hedge-fund world. Performance reports
for trading strategies will often quote the alpha and beta
of the strategy. A good strategy will have a high, pos-
itive alpha with a beta close to zero. With beta being
small you would expect performance to be unrelated
to the market as a whole and with large, positive alpha
you would expect good returns whichever way the mar-
ket was moving. Small beta also means that a strategy
should be a valuable addition to a portfolio because of
its beneficial diversification.
Sharpe shared the 1990 Nobel Prize in Economics with
Harry Markowitz and Merton Miller.
