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Risk-Adjusted Performance 303
performance of portfolios and their managers. RAP also allows us to identify
the “best” portfolio, the portfolio that has the highest return for any level of risk.
This is in contrast to the conventional method of evaluating the performance of
portfolios using total return, which can be highly misleading.
RAP provides a simple answer to the question, “do returns adequately compensate
us for the risk that we bear?” Ranking portfolios by RAP yields the same
results as ranking portfolios by the Sharpe ratio, except that RAP yields a score
expressed in basis points, which is a much easier measure for the average investor
to understand.
Finally, the recognition that the risk of a portfolio can be readily altered provides
the basis for an important corollary of the RAP approach: In the pursuit of
superior performance, investors should separate decisions as to which portfolio
to hold from decisions as to how much risk to bear. The portfolio to hold is the
portfolio with the highest RAP, because it will yield the highest return for any
level of risk. Risk can then be tailored to individual preferences through leverage.
We note that this renders leverage a key tool for risk management in the
pursuit of optimal investment performance.
Appendix: Relative Measures of Risk-Adjusted Performance
One may prefer a measure of risk-adjusted performance that is relative to the performance
of the unmanaged market. An obvious measure is therefore RAP relative
to the total return of the unmanaged market portfolio:
RAP * ()= i 100( r()
i r M )
(12.A-1)
which shows by what percent the risk-adjusted performance of portfolio i exceeds
that of the market.
Similarly, the same measure expressed in excess return is RAPA relative to the
excess return of the market:
RAPA * ()= i 100( e()
i e M )
(12.A-2)
Both indexes will equal 100 for a portfolio doing just as well as the market
and exceed (or fall short) of 100 for superior (or inferior) performance. We note
that the two indexes will tend to produce similar, but not necessarily identical
rankings of portfolios; in fact, only RAPA* will rank portfolios in the same
optimal way as RAP. We therefore conclude that RAPA* is the best relative
measure of risk-adjusted performance. We also note that equation (12.A-2) can
be expressed as S i /S M , where S M is the Sharpe ratio for the market (which can
also be thought of as the market price for a unit of risk).