# Mixing Max Drawdown and Sharpe Ratio in a single utility function : is there a standard approach?

We know that 2 strategies can give the same Sharpe Ratio, but with different Maximum Drawdown. I computed myself these 2 strategies having the same cumulative return and SR, but with considerably different Max Drawdown to highlight this :

I am currently optimising my strategy parameters with either one of these 2 measures (SR and MDD), but the loss function needs to output one single number (a final utility function). How can I "mix" these 2 informations using input utility that is for example :

"I want the better Sharpe Ratio but with Max Drawdown not exceeding 20%"

Is there a standard approach or measure that can mix these 2 informations ? i.e., both the risk-adjusted moment based measure (Sharpe Ratio, or other ratio accounting for higher order moments) and the measure that takes into account the order in which the returns occur (MDD, or Ulcer Index)

EDIT : I have an idea: maybe we could compute an average of the different Sharpe Ratio that would give daily returns dist., 2-days returns dist., 3 days returns dist. etc. This "Sharpe Ratio average" would take into account the order in which the return occur over time because, even if the chart above gives the same SR for daily returns, the standard deviation of 3-months return is much lower for Strategy A than for strategy B. This would lead to an "Average Sharpe Ratio" that is in favor of Strat. A. Is this intuition a common practice that I don't know ?

EDIT 2: ACF of biased strategy (B) shows significant autocorrelation for several lags, while ACF of A shows 0 lag autocorrelated:

• Does the maximum draw-down in your case refer to daily maximum draw-down? Because the overall maximum draw-down of A is obviously less than that of B.
– Hans
May 11 at 12:27
• It’s the maximum drawdown computed from daily equity value. The red line (strat B) has a much lower drawdown than the black line (strat A) here May 11 at 12:49

Why not use the Sortino Ratio instead of the Sharpe Ratio? This only uses downside deviation in its calculation and thus directly includes the idea of drawdown only in your loss function.

In your given example, the black return line would have a higher Sortino Ratio value than that of the red return line, so you could directly optimise for this ratio.

Re: "But here is the thing: the daily returns for the red line (both positive and negative) are exactly the same as for the black line"

Well, yes, maybe in this example that is true, but I believe this is an unrealistic example. In my opinion the returns streams from two, different and unrelated sets of trading rules will not produce identical returns distributions. It is far more likely that the distributions will be different but the summary statistics will be indistinguishable.

By way of example I present the following stylised chartwhich somewhat follows the OP's chart with regard to beginning and ending values.

Strategy A (Black line) is constructed from two different Gaussian distributions, one for positive returns (mean = 40, std = 1) and the other for negative returns (mean = -0.25, std = 0.25) and sorted to produce a highly desirable "stair stepping" equity curve with minimal drawdowns.

Strategy B (Red line) is another set of Gaussian returns with mean and standard deviation equal to that of the combined returns of strategy A and sorted so that all negative returns occur first for a large drawdown, followed by an all positive returns drawup.

The summary statistics for these are:

A_mean_return = 11.323
A_std_return = 18.236
A_downside_deviation_return = 0.2205
Sharpe_A = 0.6209
Sortino_A = 51.354
B_mean_return = 12.024
B_std_return = 19.134
B_downside_deviation_return = 10.714
Sharpe_B = 0.6284
Sortino_B = 1.1223


The Sharpe Ratios are not exactly the same due to the nature of the random generation of the returns, but are similar enough to be statistically indistinguishable. However, the Sortino Ratio does clearly distinguish strategy A as being the more desirable.

If, by some fluke, your different systems produce identical Sharpe and Sortino Ratios then you are running into the sort of problem that is illustrated by Anscombe's Quartet whereby you will have to resort to "graphical" methods. To my mind, the simplest way would be a gain-to-pain ratio calculated thus:

Total_Gain / Max_Drawdown

For the chart above these values are

A_gain_pain_ratio = 149.81
B_gain_pain_ratio = 3.7397


which, obviously, also shows that strategy A is the better one without needing classical statistical measures to tell us this fact.

My strategy A does indeed have drawdowns and the Octave/MATLAB code given below should enable you to replicate the above and see for yourself.

pkg load statistics ;

## Create Strategy A returns
x_d = normrnd( -0.25 , 0.25 , 250 , 1 ) ; ## drawdown distribution
x_u = normrnd( 40 , 1 , 100 , 1 ) ; ## drawup distribution
A = [ x_d(1:100) ; x_u(1:50) ; x_d(101:200) ; x_u(51:75) ;  x_d(201:250) ; x_u(75:100) ] ; ## distributions combined
A_equity_value = cumsum( [ 10000 ; A ] ) ;

A_mean_return = mean( A ) ;
A_std_return = std( A ) ;
A_downside_deviation_return = std( x_d ) ;
Sharpe_A = A_mean_return / A_std_return ;
Sortino_A = A_mean_return / A_downside_deviation_return ;

## Create Strategy B returns
B = normrnd( A_mean_return , A_std_return , 350 , 1 ) ;
B = sort( B ) ;
B_equity_value = cumsum( [ 10000 ; B ] ) ;
B_ix = find( B < 0 ) ;
B_mean_return = mean( B ) ;
B_std_return = std( B ) ;
B_downside_deviation_return = std( B( B_ix ) ) ;
Sharpe_B = B_mean_return / B_std_return ;
Sortino_B = B_mean_return / B_downside_deviation_return ;

## Gain pain ratio
A_gain_pain_ratio = ( A_equity_value( end ) - A_equity_value( 1 ) ) / max( cummax( A_equity_value ) - A_equity_value ) ;
B_gain_pain_ratio = ( B_equity_value( end ) - B_equity_value( 1 ) ) / ( B_equity_value( 1 ) - min( B_equity_value ) ) ;

if ( ishandle( 1 ) )
clf( 1 ) ;
endif
figure( 1 ) ; h1 = axes( 'position' , [ 0.02 , 0.02 , 0.97 , 0.95 ] ) ;
plot( A_equity_value , 'k' , 'linewidth' , 2 , B_equity_value , 'r' , 'linewidth' , 2 ) ;
title( "Comparison of 2 Strategies' Equity Values Over Time with 'Similar' Moments and Sharpe Ratios" , "fontsize" , 15 ) ;
legend( 'Strategy A Equity Value' , 'Stategy B Equity Value' , 'location' , 'northwest' , 'fontsize' , 15 ) ;


Are you sure that Strat A and B equity over time come from the same return distribution?

No, they do not, but that is the point I'm trying to make and the code makes this explicit. You can have different distributions of returns but the summary statistics of these different distributions can be (almost) identical or indistinguishable from each other, making said summary statistics completely uninformative with respect to choosing between the underlying trading systems.

If you don't like Return / MDD you could try something like Return / Average size of all individual DDs. The denominator in this expression can be adjusted in many ways, e.g. average plus 1 or 2 x standard deviation of DDs.

• But here is the thing: the daily returns for the red line (both positive and negative) are exactly the same as for the black line. They only occur in a different order (in the red line the negative returns tend to appear earlier). So both the standard devoation and the downside deviation are the same I believe. May 11 at 12:06
• Exact ! Sortino or Sharpe ratio are moments-based risk adjusted measure, they only take into consideration the distribution of return without considering the order in which these returns occur. Here I confirm that Sortino or Sharpe ratio give the same result for strategies A and B. When using leverage trading you definitely want to avoid having consecutive negative return, to avoid liquidation. That’s why there is some kind of trade off to do between having a good Sharpe ratio and do not exceed a drawdown that would give a certain prob to reach a margin call May 11 at 12:43
• A doubt that I have. I am willing to believe that past standard deviations (and to a lesser extent means) tell us something about the $\sigma$ and $\mu$ we will experience in the future. Using them for optimization may make some sense. But isn't the exact path taken (and the drawdown) in the backtest period a random outcome that will have no relevance to the future? Do there really exist red securities that can be expected to have higher drawdown than black securities with same vol? May 11 at 13:58
• Actually this is not really securities prices displayed there, but more the equity value across time for a certain strategy (i.e., equity initial + realized PnL + unrealized PnL). The strategy implies dozens of pairs of securities, on which we regularly enter and exit positions at certain prices estimated by an underlying model. So here, the case is more complicated than having a weighted portfolio of assets. However, your point is interesting and I didn't take into consideration that the estimator of the drawdown may not be converging toward a "true value" that we can expect in the future.. May 11 at 17:13
• like $\sigma$ or $\mu$ could be (at the condition of having stationarity). I just believe intuitively that certain parameters that trigger the entry and exit signals have an effect in some way (that I black-box) over the max drawdown. I can have a look at the kurtosis of the return distribution as well : a high kurt indicating the presence of shocks is likely to cause a max drawdown. But what I always miss by looking at the return dist. is the "consecutiveness" of the negative returns. If low negative returns occur 1000 times in a row, I end up with the same big drawdown as with a big shock May 11 at 17:20