How sharp is the Sharpe-ratio? - Risk-adjusted Performance
Measures
Carl Bacon, Chairman, StatPro
Any discussion on risk-adjusted performance
measures must start with the grandfather of all risk measures the
Sharpe Ratio or Reward to Variability which divides the excess
return of a portfolio in excess of the risk free rate by its
standard deviation or variability.
Most risk measures are best described graphically, a measure
of
return in the vertical axis and a measure of risk in the horizontal
axis.
Ideally if investors are risk averse they should be looking for
high return
and low variability of return, in other words in the top left-hand
quadrant
of the graph. The Sharpe ratio simply measures the gradient of the
line from the risk free rate (the natural starting point for any
investor) to the combined return and risk of each portfolio, the
steeper the gradient, the higher the Sharpe ratio the better the
combined performance of risk and return.
Funds are ranked in order of preference with the Sharpe ratio but
it is difficult to judge the extent of relative performance. M2;
first proposed by Leah Modigliani and her grandfather Professor
Franco Modigliani (1997) offers an alternative risk-adjusted return
using the Sharpe ratio of the portfolio but calculated at the risk
of the benchmark thus allowing direct comparison.
Investment statistics can either be grouped as Sharpe type
combining risk and return in a ratio, risk adjusted returns such as
M2 or descriptive statistics which are neither good nor bad but
provide information about the pattern of returns of the portfolio
manager. The first moment of a return series is the mean, the
second moment is the variance, the third moment is skewness and the
fourth moment kurtosis. Kurtosis measures the weight of returns in
the tails or the peakedness of a return distribution. Investors
should prefer high average returns, lower variance or standard
deviation, positive skewness and lower kurtosis. The adjusted
Sharpe ratio suggested by Pezier (2006) explicitly rewards positive
skewness and low kurtosis (below 3, the kurtosis of a normal
distribution) in its calculation and thus potentially removes one
of the possible criticisms of the Sharpe ratio.
The regression statistics b (or systematic risk), r (correlation)
and R2 are descriptive statistics. Jensen’s alpha is often
misquoted as the portfolio manager’s excess return above the
benchmark, more accurately it the excess return adjusted for
systematic risk.
Treynor ratio or Reward to Volatility is similar to Sharpe ratio,
the numerator (or vertical axis graphically speaking) is identical
but in the denominator (horizontal axis) instead of total risk we
have systematic risk or volatility as calculated by beta. Although
well known the Treynor ratio is less useful precisely because it
ignores specific risk.
The appraisal ratio first suggested by Treynor & Black (1973)
is similar in concept to the Sharpe ratio but using Jensen’s alpha,
excess return adjusted for systematic risk in the numerator,
divided by specific risk not total risk in the denominator.This
measures the systematic risk adjusted reward for each unit of
specific risk taken.
In the same way that absolute return and absolute risk are combined
in Sharpe ratio excess return and tracking error (the standard
deviation of excess return) are combined in the information ratio,
although given the need of an appropriate benchmark less useful for
hedge funds.
The Sharpe, appraisal, Treynor and information ratios are familiar
measures used by the industry for decades. More recently hedge
funds have encouraged the use of further risk measures designed to
accommodate the risk concerns of different types of investors.
These measures can be categorised as based on normal measures of
risk, regression, higher or lower partial moments, drawdown or
value at risk (VaR).
Predominately hedge fund management styles are designed to be
asymmetric in their return patterns. If successful this leads to
variability of returns on the upside but not on the downside.
Investors are less concerned with variability on the upside but of
course are extremely concerned about variability on the downside.
This leads to an extended family of risk-adjusted measures
reflecting the downside risk tolerances of investors seeking
absolute not relative returns.
Standard deviation and the symmetrical normal distribution are the
foundations of Modern Portfolio Theory. Post-modern Portfolio
Theory recognises that investors prefer upside risk rather than
downside risk and utilises semi-standard deviation.
Downside risk measures the variability of underperformance below a
minimum target rate. The minimum target rate could be the risk free
rate, the benchmark or any other fixed threshold required by the
client. All positive returns are included as zero in the
calculation of downside risk or semi-standard deviation.Downside
potential is simply the average of returns below target, upside
potential the average of returns above target.
In their article “A Universal Performance Measure” (2002) Shadwick
& Keating suggest a gain-loss ratio, Omega (W) that captures
the information in the higher moments of a return distribution
implicitly adjusting for both skewness and kurtosis; dividing
upside potential by downside potential.
A natural extension of the Sharpe and Omega is suggested by Sortino
(1991) which uses downside risk in the denominator. Total risk has
simply been replaced by downside risk, portfolio managers will not
be penalised for upside variability but will be penalised for
variability below the minimum target return.
The upside potential ratio suggested by Sortino, Van de Meer &
Platinga (1999) can also be used to rank portfolio performance and
combines upside potential with downside risk. Even “Prospect
Theory” the fact that investors dislike losses far greater than
they like gains can be built into a Sharpe like measure in the form
of the Prospect ratio.
If value at risk is your preferred measure of risk then, of course,
there is a Sharpe type measure that replaces standard deviation
with VaR in the denominator; called reward to VaR. VaR does not
provide any information about the shape of the tail or the expected
size of loss beyond the confidence level. In this sense it is a
very unsatisfactory risk measure; of more interest is conditional
VaR otherwise know as expected shortfall, mean expected loss, tail
VaR or tail loss which takes into account the shape of the tail.
Historical simulation methods which make no assumptions of
normality are particularly suitable for calculating conditional
VaR. The conditional Sharpe ratio replaces VaR with conditional
VaR.
Perhaps the simplest measure of risk in a return series from an
absolute return investor’s perspective, wishing to avoid losses, is
any continuous losing return period or drawdown. The average
drawdown is the average continuous negative return over an
investment period, three years being a typical period of
measurement for comparison purposes.
The maximum drawdown not to be confused with the largest individual
drawdown is the maximum potential loss over a specific time period,
typically three years. Maximum drawdown represents the maximum loss
an investor can suffer in the fund buying at the highest point and
selling at lowest.The Calmar ratio is a Sharpe type measure that
uses maximum drawdown rather than standard deviation to reflect the
investor’s risk. In the context of hedge fund performance it is
easy to understand why investor’s might prefer the maximum possible
loss from peak to valley as an appropriate measure of risk.
The Sterling ratio replaces the maximum drawdown in
the Calmar ratio with the average largest drawdowns.
Similar measures including the Pain ratio and the Ulcer Performance
ratio incorporate the duration and depth of drawdowns since the
previous high water mark. The range of combined risk and return
measures available for hedge fund investors is almost
limitless.
With so many similar ratios the natural question to ask is “which
is the best measure to use?” In fact Eling & Schuhmacher (2006)
have published an article “Does the Choice of Performance Measure
Influence the Evaluation of Hedge Funds” which concludes that most
of these measures are all highly correlated and do not lead to
significantly different rankings. Both the question and their
article to some degree miss the point, risk like beauty is in the
eye of the beholder, the investor most decide ex-ante which
measures of return and risk best reflect their preferences and
choose the combined ratio which reflects those preferences. One,
and only one, of the above ratios are most likely to reflect the
preferences of the investor. Care should also be taken to ensure
hedge funds are not hiding volatility by using smoothed valuations.
Consistent valuation criteria must by applied each month, although
Global Investment Performance Standards (GIPS) do not require that
specific risk measures are used they do require documented policies
and procedures for valuations consistently applied and are
therefore valuable and a source of comfort for any potential
investor.
The
full 13 pages article is available for download
here.
Back to
Whitepapers