52 Frequently Asked Questions In Quantitative Finance
stocks (choose two fromNwithout replacement, order
unimportant). To Markowitz all investments and all port-
folios should be compared and contrasted via a plot
of expected return versus risk, as measured by stan-
dard deviation. If we writeμAto represent the expected
return from investment or portfolio A (and similarly
forB,C,etc.)andσBfor its standard deviation then
investment/portfolio A is at least as good as B if
μA≥μB and σA≤σB.
The mathematics of risk and return is very simple.
Consider a portfolio,,ofNassets, withWibeing
the fraction of wealth invested in theith asset. The
expected return is then
μ=
∑N
i= 1
Wiμi
and the standard deviation of the return, the risk, is
σ=
√
√
√
√
∑N
i= 1
∑N
j= 1
WiWjρijσiσj,
whereρijis the correlation between theith andjth
investments, withρii=1.
Markowitz showed how to optimize a portfolio by find-
ing theWs giving the portfolio the greatest expected
return for a prescribed level of risk. The curve in the
risk-return space with the largest expected return for
each level of risk is called theefficient frontier.
According to the theory, no one should hold portfolios
that are not on the efficient frontier. Furthermore, if
you introduce a risk-free investment into the universe of
assets, the efficient frontier becomes the tangential line
shown below. This line is called theCapital Market Line
