Appropriate method for calculating negative returns on a trading strategy?

Question

I have a cumulative profit/loss time series below for a trading strategy, what is the appropriate way to calculate the returns in percentage for such a series?

My issue is the appropriate calculation for when the strategy becomes negative and going forward. I can't take log of a negative number and I am not sure if the arithmetic calculation is appropriate. I would appreciate any suggestions. I am using matlab. Thanks

 1.0e+003 *

    5.2735
    4.3922
    3.1878
    3.6250
    3.6982
    5.3774
    5.5748
    5.0108
    1.0355
   -2.4639
   -4.6589
   -4.2990
   -3.8678
   -3.1051
   -4.5356
   -4.8130
   -8.9671
   -9.0438
   -7.2986
   -7.1849

Answer 1

So those are cumulative pnl figures and you are interested in the percent changes in pnl from one data point to the next? Don't use log returns, simply generate the percent changes through r(t)/r(t-1)-1.

4.3922/5.2735-1 = -16.71% (in your example time series I made the assumption that the time series is in ascending order. Given your description of the above time series data points, the change of your pnl from absolute 5.2735 -> 4.3922 constitutes a decrease by 16.71% of generated pnl. (I am not sure why you wanna get to those numbers this way but this is how you described it, nonetheless).

Using arithmetic returns is completely fine, in fact its preferable over log returns in some instances. I would consider return calculation for return attribution purposes such situation. I know I will probably be down voted by someone for this comment but I am happy to defend this statement if being asked.

Answer 2

For me, I would calculate daily returns for such a series by backing out the daily PnL and dividing by some volatility number.

lets define your cumsum as "c_pnl":

daily_pnl   = c_pnl - [0; c_pnl(1:length(c_pnl-1)]
max_draw    = max(cummax(c_pnl) - c_pnl)
pct_returns = daily_pnl / max_draw # in terms of drawdown

Don't you have capital already in the assumptions of your backtest? Why not just use capital? log((capital + pnl)/capital).

Do you trade through time with constant capital or are you compounding on PnL?

edit: jlowin points out margin as another good way. Margin is probably better because capital as it incorporates some level of risk of the strategy and you can compare pnl /margin more easily.

Answer 3

This is a very interesting problem.

I disagree with the use of any form of percent returns -- conventional or logarithmic -- simply because they are nonsensical for negative equity values (I am assuming this price series is from some sort of margined position?). You are forced to choose between the mathematically "correct" returns for those which are "correct" from the point of view of the account holder.

Consider the final two results: a negative equity balance of -7.30 followed by an improvement to a negative balance of -7.18. Conventionally, that's a -1.64% move (-7.18 / -7.30 - 1), even though of course we can see that the account balance is moving positively (in the account holder's favor).

Is it "right"? Sure, mathematically -7.18 is -1.64% from -7.30. But if you try to calculate any statistics off these returns, the winning and losing days will be reversed -- so it's not right from a practicality standpoint and will ruin your perception of the account's performance (like average return, trend, risk).

It is correct, however, if we consider that it is the return experienced by whomever is "in the money" with regard to this margin account at that time. When the account balance is positive, the account holder is (presumably) experiencing gains. When the balance is negative, the margin lender is experiencing gains. Percent returns calculated conventionally are a reflection of their respective positions times the sign of their cumulative profit.

Logarithmic returns suffer from exactly the same problem -- moving toward zero is always considered a "negative" return, disregarding the fact that there is a negative equity balance.

However, for the sake of answering part of your question, you can in fact calculate logarithmic returns for your negative price series, with the exception of the one point where it crosses zero. Note that log(x) - log(y) = log(x/y). Therefore, instead of differencing the log of two numbers, just take the logarithm of their ratio. The example I gave above becomes log(-7.18 / -7.30) = -1.65%, which is very close to the arithmetic result.

At the crossing point, this method will fail, and you may have to fall back on an arithmetic calculation (with all the caveats above).

The best way to do this is to use a margin balance or risk capital position. Add that to the price series to de-lever it and avoid negative balances. Then either method of calculating returns will work as normal.

Answer 4

I have a cumulative profit/loss time series below for a trading strategy, what is the appropriate way to calculate the returns in percentage for such a series?

Correct me if I am wrong, but it seems to me there is no such thing as return for P&L. Your series contains net income which itself is a return. You could calculate the relative change for your series, but it has no meaning when you switch from profit to loss and vice versa. Here is the relevant discussion with the link to WSJ not providing percent change for net income in such scenario.

Answer 5

One workaround for this is to add to the cumulative profit and loss the initial equity and transform the points gained in cash.

Here is what i usually do. I trade forex.

dollarPerPip = contracts * 100; % Here i calculate how much a pip worth in dollar, this value is true for EURUSD forex
cpnl = cumsum(pnl);
cpnl = (cpnl * dollarperPip) % Here transform pips in cash
cpnl = cpnl + initialEquity; % add the initialEquity
maxdd = maxdrawdown(cpnl); % calculate the max drawdown

If a strategy go under the required margin to trade, in my opinion, its not worth to calculate the max drawdown, so i discard.

I hope this helps.