"Life Cycle" Hypothesis of Saving: Aggregate ... - Arabictrader.com

Risk-Adjusted Performance 293
suggests an alternative, slightly simpler measure of risk-adjusted performance,
based exclusively on excess returns:
RAPA()= i e()= i ( s s ) e
M i i
(12.7)
RAP and RAPA are essentially interchangeable in that they rank portfolios
identically, but the reference value for RAPA is the excess return of the market,
e M , frequently referred to as the equity risk premium.
Sigma as a Measure of Risk
Our analysis relies on the fact that the market as a whole is willing to pay a
price—give up expected returns—to avoid variability. A portfolio consisting only
of short-term riskless assets faces no dispersion in returns—it bears essentially
no risk. If one is prepared to bear risk—say, for instance, the risk associated with
holding the market—one can increase the expected return by the market mean
excess return, which on average has been substantial. This excess return is the
premium one can earn for bearing the market risk. Its existence offers anyone the
opportunity to trade off an increment of excess returns for an equal percentage
increment of sigma.
The observation that the market rewards greater uncertainty with higher
expected returns implies that, on average, investors are risk-averse. That is,
investors tend to behave as though incremental gains are less and less valuable
(or produce decreasing incremental satisfation). If this is the case, then an investment
returning $10,000 with certainty—i.e., without dispersion—is preferable to
one that has the same expected value but is dispersed, such as an investment that
is equally likely to return $5,000 or $15,000. It takes more than the prospect of
an extra $5,000 to offset the equal probability of falling $5,000 short. More generally,
dispersion has to be compensated by higher expected return because upside
dispersion is accompanied by more costly downside risk.
For this reason, the view that dispersion is a valid measure of risk is consistent
with the assertion that what really matters to investors is “downside risk.”
While here we have employed the traditional notion of sigma as a measure of
risk, in fact, the RAP methodology is applicable to many alternative definitions
of risk including several measures of downside risk.
RAP as a Tool for Optimal Portfolio Selection
We have said that any portfolio with dispersion s i can be transformed into another
portfolio with different risk, say, s p , by levering it through a choice of d = s p /s i