December 18,
2006
Jeffrey
Lins, Executive Director of Quantitative Analysis at Saxo Bank in
Copenhagen, evaluates the pitfalls of using maximum drawdownbased
performance measures, and points to recent advances that may help to
expand opportunities for investment strategies.
BACKGROUND
When contemplating managed
asset opportunities, investors are confronted with a dizzying array of
choices from an ever increasing number of performance measures. Each of
these seems to offer a different perspective on what is really a very
slender, straightforward set of considerations: the risks and rewards
of a given investment, and the tradeoff between the two.
From the
perspective of managed asset advisors, such as CTAs, the framing of
these considerations is not always just a matter of offering
information which help match investor preferences to the contours of
managed asset programs. Quite often they are a matter of regulatory
mandate, such as the Commodity Futures Trading Commission's (CFTC) and
National Futures Association's (NFA) performance disclosure
requirements.
Particularly, the CFTC requires CTAs and CPOs to disclose "the worst
peaktovalley drawdown for the trading program" and "the annual
...rateofreturn for the program". These two measures constitute
essentially the denominator and numerator of the Calmar ratio commonly
used by hedge funds for performance reporting.
However, maximum drawdown and the related Calmar ratio in investment
decisionmaking are not free of limitations or spurious application.
Moreover, perhaps more importantly, if investor preference for risk and
reward is expressed in the paradigm, then asset managers should perhaps
look towards mechanisms which allow its commoditisation more directly.
PROBLEM
It should be immediately apparent to any investor, upon some
reflection, that for the typical noisy path of investment returns, the
longer the period of observation, the greater the probability of larger
drawdowns. Therefore the application of the maximum drawdown measure
suffers from a negative bias towards trading histories of increasing
length. Similarly, the Calmar ratio will also be biased, and thus
neither may be considered a timeinvariant metric. Limiting variation
in measurement time horizons for comparison between histories means the
exclusion of past information, and if the assumption is that the past
contains some information which may be useful for investors in their
selection of future opportunities, it may be exactly these
artificiallyexcluded events which would have been of most interest to
them.
If we are able to find a normalised approach for the application of
maximum drawdown and the Calmar ratio across investments of varying
time spans of trading history, what methodology might then be
constructed for protecting, guaranteeing or even offering tradable
contracts on performance as comprehended by these measures?
SOLUTION
Recent developments appearing in academic literature and professional
journals in the field of statistics and financial engineering seem to
be rapidly paving the way towards many answers to the hard questions
and some new innovative opportunities.
Particularly the work of MagdonIsmail, Atiya, et al. (see references)
proposes that the Calmar ratio may be scaled intertemporally as a
function of relationship between an expected maximum drawdown and
statistical moments of the distribution of returns and essentially the
relationship therefore of the Calmar relationship to the Sharpe ratio
via complex integral expansions.
Through this relationship, a normalised scaling factor is derived akin
to the law for scaling the Sharpe ratio by the square root of the
length of time of returns. We point out that this scaling law remains
challenged by the results produced in recent years within the
literature on highfrequency finance, demonstrating some germane
pitfalls of assumptions that financial returns are strictly
independently and identically distributed, or with regard to the tail
quantiles of their distribution. However, in principle, whatever comes
to work for the Sharpe ratio in this context should then be applicable
to the Calmar ratio by this scaling factor.
But
of even more interest may be the appearance in the work of Jan Vecer,
proposing the treatment of maximum drawdown as a contingent claim,
which may then be priced and hedged as a derivative contract. The basis
for this seems to go well beyond capacity of vanilla puts or lookback
options for insuring against drawdowns.
Finally, we propose a look at synthesising not only insurance against
drawdowns, but optionality for both aspects of risk and reward.
Although this is not really meaningfully commoditisable in terms of
risk as expressed as the standard deviation in the Sharpe ratio or
ValueatRisk based portfolio rebalancing, it should become more
readily comprehended as a function of the portfolio constraints based
linearly upon drawdown and expected returns, which may then, as
suggested above, yield tradable contingent claims, even perhaps through
further hybridisation with options protecting against explicit losses
on trading strategies, such as passport options.
REFERENCES
MagdonIsmail, M., A. Atiya. 2004. "Maximum drawdown," Risk, Vol. 17,
No. 10, 99102.
MagdonIsmail M, A Atiya, A Pratap and Y AbuMostafa, 2004. "On the
maximum drawdown of a Brownian motion," Journal of Applied Probability
41(1), March.
Diebold, F. X., A. Hickman, A. Inoue and T. Schuermann, 1997.
"Converting 1day volatility to hday volatility: scaling by rooth is
worse than you think." Wharton Financial Institutions Center. Working
Paper no. 9734.
Vecer,
J. "Trading maximum drawdown and options on maximum drawdown",
Frankfurt Math Finance Workshop Paper. March 27–28, 2006.
