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Weighing up the alternatives Print E-mail

December 18, 2006             

Jeffrey Lins, Executive Director of Quantitative Analysis at Saxo Bank in Copenhagen, evaluates the pitfalls of using maximum drawdown-based performance measures, and points to recent advances that may help to expand opportunities for investment strategies.

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 trade-off 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 peak-to-valley drawdown for the trading program" and "the annual ...rate-of-return 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 decision-making 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.

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 time-invariant 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 artificially-excluded 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?

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 Magdon-Ismail, 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 high-frequency 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 Value-at-Risk based portfolio re-balancing, 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.

Magdon-Ismail, M., A. Atiya. 2004. "Maximum drawdown," Risk, Vol. 17, No. 10, 99-102.
Magdon-Ismail M, A Atiya, A Pratap and Y Abu-Mostafa, 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 1-day volatility to h-day volatility: scaling by root-h is worse than you think." Wharton Financial Institutions Center. Working Paper no. 97-34.

Vecer, J. "Trading maximum drawdown and options on maximum drawdown", Frankfurt Math Finance Workshop Paper. March 27–28, 2006.

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