Monotonicity of the Calmar Ratio and Drawdown
Summary
The document asks whether the Calmar ratio, defined as expected return divided by maximum drawdown, satisfies monotonicity as a performance measure. It contrasts Calmar with the gain-loss ratio, which the author describes as monotone, and the Sharpe ratio, which is described as failing monotonicity. The discussion considers cumulative return paths ordered pointwise and observes that their maximum drawdowns need not share that ordering, even though their running maxima do.
The author seeks an explicit example showing whether Calmar can rank a pointwise higher return path worse because of its drawdown, and asks how to interpret ex-post comparisons when the drawdowns are fixed. No example or resolution is included. The key conceptual caveat is that the ratio combines a return numerator with a path-dependent drawdown denominator, so ordering returns alone does not determine the ratio’s ordering. The note raises the monotonicity question but does not establish a formal definition or prove a conclusion.
Key ideas
- Calmar divides expected return by maximum drawdown, which depends on the path of cumulative returns.
- Pointwise ordering of return paths does not guarantee the same ordering of their drawdowns.
- The document asks whether Calmar can violate monotonicity and requests an explicit example.
- No example, formal definition, or conclusion is provided.
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Full text
# May Calmar ratio be considered to satisfy monotonicity?
# May Calmar ratio be considered to satisfy monotonicity?
We have the following definitions
$\text{Gain-loss ratio} = \frac{E[X^+]}{|E[X^-]|}=\frac{E[X^+]}{-E[X^-]}=\frac{E[X]}{-E[X^-]}+1$;
where $X$ are the returns, $E[X^+]$ is the expected gain, i.e. $E[X|X\ge 0]\times P[X\ge0]$, $E[X^-]$the expected loss i.e. $E[X|X< 0]\times P[X<0]$; and we have $E[X]=E[X^+]-E[X^-].$
$\text{Calmar ratio} = \frac{E[X]}{MDD}$
where $\text{MDD}$ is the Maximum DrawDown.
"Given a fixed time time horizon [0,T], with T > 0, and given two cumulative return paths X,Y (stochastic processes with time index running over [0,T]) such that $X(\omega) < Y(\omega)$ pointwise (for every outcome $\omega$ in the sample space) , then it easily seen that there is not a unique ordering between their drawdowns, say D(X) and D(Y). Both D(X) < D(Y) or D(X) > D(Y) are compatible with X(ω) < Y(ω) for every ω. In fact, it is only the running maximum max_t X_t which is less than the running maximum max_t Y_t, provided that X(ω) < Y(ω) for every ω."
However, does there exist any exact example?
I think the $X$,$Y$ should be the return series as opposed to the the cumulative return series.
More over less, the higher the better. Sharpe ratio does not satisfy monotonicity due to it violate "More over less". Cherny and Madan (2009) gave an example very clear to show that the Sharpe ratio does not satisfy monotonicity.
The Gain-loss ratio is considered to satisfy monotonicity.
However, may Calmar ratio (E[X]/MDD) be considered to satisfy monotonicity?
Does here exist any ex-post example to demonstrate the gain-loss ratio satisfies monotonicity, but the Calmar ratio does not?
Furthermore, for ex-post performance comparison, D(X) and D(Y) are fixed.
If X(ω) < Y(ω) for every ω,
if D(X) > D(Y), we have Calmar(X) < Calmar(Y),
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A. we can conclude Fund X not better than Fund Y.
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if D(X) < D(Y), we may have two outcomes,
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B. if Calmar(X) < Calmar(Y), we can also conclude Fund X not better than Fund Y.
C. if Calmar(X) > Calmar(Y), may we conclude Fund X is better than Fund Y?
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If for C, we can conclude Fund X is better than Fund Y, Together with A and B, then may we say Calmar ratio is "monotone"?
If the Gain-Loss ratio is considered a monotone measure (ex-post), may Calmar ratio also be considered a monotone measure?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.