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

300 Miscellanea
that an increase in leverage increases the risk and return of a portfolio in the same
proportion. Measures of downside risk that meet this standard include semivariance,
or average deviation, below the mean (assuming that the distribution
of returns is roughly symmetrical around the mean).
Moreover, the fundamental RAP approach of risk-matching is applicable
to any measure of risk, provided that it satisfies two more general conditions:
1) leverage changes the risk and reward of portfolios in the same direction,
and 2) leverage does not change the ranking of portfolios at any level of
risk. With such an alternative measure of risk, the RAP approach would still
call for levering or unlevering a portfolio until its risk matches the risk of
the market, and then measuring the return of that risk-equivalent portfolio. Again,
this approach will identify the portfolio with the highest return for any level
of risk.
We identify “probability of loss” as an interesting measure of downside risk
that meets these criteria. If returns are approximately normally distributed, then
the probability of a negative return (loss) can be inferred from the ratio of the
mean to the standard deviation.
Arithmetic versus Geometric Returns
Compliance with the Association for Investment Management and Research
(AIMR) performance presentation standards requires the use of geometric rather
than arithmetic average returns when reporting investment results. This has
become the standard for the investment industry.
The arithmetic mean is always higher than the geometric mean, and the difference
between the two grows larger, the greater the variance in returns. The two
averages measure different quantities. The geometric mean measures the return
of an investment that grows in each period at precisely the rate of the return of
the portfolio (before personal taxes). The arithmetic mean measures the return of
an investment that is held constant at the initial level. Both geometric and arithmetic
measures can be considered equally meaningful (and some would argue
that both should be reported).
Deviating from the standards for computing total returns, but in keeping with
the common practice for calculating risk-adjusted performance, and the Sharpe
ratio in particular, we use arithmetic returns in our calculations for this article.
Analyzing the dispersion in returns using geometric figures involves significantly
more complex calculations. The standard deviation associated with the geometric
mean return is approximately the standard deviation of the log of (1 + r). More
important, the effect of leverage on geometric returns is less clear-cut than that
on arithmetic returns and cannot be represented simply.