Ulcer Index
An Alternative Approach to the Measurement of Investment
Risk & Risk-Adjusted Performance
by Peter G. Martin
pgm@seanet.com
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What
is the Ulcer Index?
Ulcer Index (UI) is a method for measuring investment risk
that addresses the real concerns of investors, unlike the widely used standard
deviation of return (SD). It is a measure of the depth and duration of
drawdowns in prices from earlier highs.
Using UI instead of SD can lead to very different conclusions
about investment risk and risk-adjusted return, especially when evaluating
strategies that seek to avoid major declines in portfolio value (market
timing, dynamic asset allocation, hedge funds, etc.).
The Ulcer Index was originally developed by the author
of this page in 1987. Since then, it has been widely recognized by the
investment community. The Index was first described in The Investor's
Guide to Fidelity Funds: Winning Strategies for Mutual Fund Investors,
by Peter Martin & Byron McCann. Originally
published by John Wiley & Sons in 1989, this out-of-print book can be downloaded
from http://www.tangotools.com/ui/igff.htm.
Note: There have been instances of the term "Ulcer Index" being
used for risk measures that do not strictly follow the details described
here. This document explains the correct use of the concept.
What's
Wrong with Standard Deviation of Return?
Standard deviation is a statistical measure of the variability
or unpredictability of an investment's return. It suffers from a number
of serious drawbacks:
-
Both upward and downward changes in value add to the calculated
SD. Real investors associate risk only with the downside. Rising prices
create profits, not risk.
-
The calculated value of SD is not affected by the sequences
in which gains and losses occur. Thus, SD does not recognize the strings
of losses that result in significant drawdowns in value. The three hypothetical
investments in the chart below have the same annualized return and the
same SD, but no rational investor would consider them as having the same
risk.
-
When SD is used to measure the risk of a market timing strategy,
it will tell you roughly how often you were out of the market, but nothing
about whether you were out at the right times. SD doesn't tell you if your
strategy reduced risk by avoiding market downturns.
-
The calculated value of SD depends on the time period used.
For most investments, the SD of annual return is roughly 7.2 times the
SD of weekly return (7.2 is the square root of 52 weeks per year). Since
the time period is often unstated, this creates an opportunity for misunderstandings.
As a result of these weaknesses, SD does not reward an investment
strategy for avoiding market downturns. Using Ulcer Index as a risk measure
avoids all of these problems.
What
About Other Risk Measures?
Other established risk measures have weaknesses too. For
example:
-
Some are based on the single worst event over a time period,
which by definition has no statistical significance (e.g. Worst Trade and
Maximum Drawdown).
-
Some are based on absolute rather than percentage price changes,
which distorts results over periods with strong price trends (e.g. Average
Maximum Retracement).
-
Some cannot be used to compare investment alternatives (e.g.
Percentage Losing Trades cannot be used to compare a market timing strategy
with a buy-and-hold approach, because the latter has no trades).
-
Others share some or all of the problems of standard deviation
(e.g. Beta).
What
Does Ulcer Index Measure?
Ulcer Index measures the depth and duration of
percentage drawdowns
in price from earlier highs. Technically, it is the square root of the
mean of the squared percentage drops in value. The greater a drawdown in
value, and the longer it takes to recover to earlier highs, the higher
the UI. The squaring effect penalizes large drawdowns proportionately more
than small drawdowns (just as it does in the SD calculation).
In effect, UI measures the "severity"
of drawdowns, as represented by the dark regions in the charts below:
Drawdowns in value: S&P 500 index with dividends reinvested
Drawdowns in value: Fidelity Select Precious Metals & Minerals
fund
The algorithm for computing UI is simple, and can be seen
in the pseudo-code fragment below:
SumSq = 0
MaxValue = 0
for T = 1 to NumOfPeriods do
if Value[T] > MaxValue then MaxValue
= Value[T]
else SumSq = SumSq + sqr(100 * ((Value[T]
/ MaxValue) - 1))
UI = sqrt(SumSq / NumOfPeriods)
Unlike SD, the calculated value of UI is essentially the
same regardless of the time interval per data point. Weekly price data
is a good compromise, but daily data can be used as well. As the interval
is extended beyond a week, there is an increasing danger of missing significant
intra-period retracement-and-recovery events. The use of quarterly or longer
intervals is strongly discouraged for this reason.
An Excel spreadsheet showing how to calculate the
Ulcer Index is available at http://www.tangotools.com/ui/UlcerIndex.xls
Measuring
Investment Performance
A popular method for measuring investment "performance" is
to divide the excess return of an investment by its risk. (Excess return
is total return minus the return offered by risk-free investments). This
calculation provides a single number that accounts for both return and
risk. It reports the additional return achieved per unit of risk assumed.
Traditionally the Sharpe Ratio is used, where risk is again represented
by the standard deviation of return:
Sharpe Ratio = (Total return - Risk-free
return) / SD
Just as SD is a poor risk measure, so is this formula a poor
performance measure. This problem is solved by simply replacing SD with
UI. This new performance measure has been dubbed the
"Martin Ratio" or "Ulcer Performance Index" (UPI).
Martin Ratio = (Total return - Risk-free
return) / UI
In either case, compounded annual returns should be used
for consistency. These figures should include reinvestment of dividends
and other distributions; and should be net of all recurring fees, transaction
costs and trading slippage.
When plotting investments on a risk vs return chart, UI
can be used instead of SD for the horizontal (risk) axis.
If a line is drawn between the points representing the
risk-free return and a risky investment, the slope of the line is equal
to the Martin Ratio. As with the Sharpe Ratio, if an investment lies above the line joining
the risk-free return with the S&P 500, the investment is "beating the
market" on a risk-adjusted basis.
Market
Timing Example
The table below shows the results achieved with both UI and
SD. We compared two strategies over the period 1940-1997: buy-and-hold
the S&P 500 index, and timing the index with a simple indicator. Results
include reinvestment of dividends in both cases.
|
Buy-and-Hold |
Timing System |
% Change |
| Annualized Total Return (%/yr) |
12.59 |
14.79 |
17.5 |
| Ulcer Index (%) |
8.85 |
5.14 |
-41.9 |
| UI Performance (Martin Ratio) |
0.92 |
2.01 |
118.9 |
| Standard Deviation (%/yr) |
16.10 |
13.18 |
-18.1 |
| SD Performance (Sharpe Ratio) |
0.51 |
0.78 |
52.9 |
For the timing system, annualized total return is increased
by a modest 2.2 percentage points. SD reports risk 18% lower and performance
(Sharpe Ratio) 53% higher. UI reports risk 42% lower and performance (Martin
Ratio) 119% higher. Thus, UI places a much higher value on the market timing
system.
The timing system offers a modest 2% increase in annual
return, but it more than doubles the risk-adjusted return calculated with
UI.
Other experimental work has shown that many popular market
timing systems have little value when SD is used to measure risk, but significant
value when UI is used instead. This arises largely because SD fails to
recognize the success of timing systems in avoiding major market downturns.
Caveat
Emptor
With any method for computing risk and performance, it is
important to use data covering as long a time period as possible. In particular,
the time period should include both bull and bear markets for the investments
of interest. Needless to say, when comparing multiple investments, the
same time period must be used in each case.
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(C) Copyright 1987-2004 by Peter G. Martin.
All rights reserved.