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Risk-Adjusted Performance 295
Nonetheless, it should be recognized that RAP(i) and S i provide very different
measures of risk-adjusted performance. RAP gives an answer in basis points that
is readily understandable by non-experts, while Sharpe produces a ratio that is
difficult for the average investor to interpret.
Jensen–Treynor Measures
The Jensen and Treynor formulation measures risk-adjusted performance by comparing
the excess returns of portfolio i with those of a market portfolio that is
matched to the risk of portfolio i. The risk-matching is accomplished by multiplying
e M by the ratio of b i to the b of the market (which is the same as b i since
the b of the market is one).
We see two problems with this approach. First, using the ratio of the betas is
fundamentally different from using the ratios of the sigmas. To the extent that the
returns of portfolio i are less than perfectly correlated with the returns of the
market benchmark, b i may differ appreciably from s i /s M . In short, RAP reflects
the proposition that in considering which portfolio to select for investing a major
portion of wealth, what matters in judging its performance is total risk, not just
systematic risk.
Moreover, these measures account for risk by adjusting the market portfolio to
match the risk of portfolio i, instead of adjusting portfolio i to match the risk
of the market, as in RAP. While these approaches are closely related, the
Jensen–Treynor measure is less useful in that it may produce misleading rankings.
The portfolio with the highest risk-adjusted return by the criteria of Jensen’s
alpha or the Treynor ratio will not necessarily be the portfolio capable of achieving
the highest return for any level of risk. This is in contrast to RAP and S, which
unambiguously identify the optimal portfolio for any desired risk level.
Graphical Representation of RAP
Figure 12.1 illustrates RAP in graphic form. Measuring standard deviation on the
x-axis and return on the y-axis, any portfolio i can be represented as a point P i ,
with coordinates (s i , r i ). The point P M represents the market portfolio, with standard
deviation s M and total return r M . Similarly, P 0 represents a portfolio of riskless
fixed-income securities with sigma s 0 of zero and return r f . Drawing a straight
line from point P 0 through any portfolio point P i gives us the “leverage opportunity
line,” or l i , for that portfolio.
For any given level of sigma, the vertical distance between l i and the x-axis
represents the total return of portfolio i (similarly, the distance between l i and the
horizontal line through r f represents the corresponding excess return). Levering