# How to calculate holding period return of a long-short strategy?

I have daily close prices of two stocks, A and B. Suppose that we long stock A and short stock B. Assume that we do the long-short every day and hold that portfolio for some days. How to calculate each day's holding period return of this long-short strategy?

My idea is that we first calculate the holding period return for the long position and the short position seperately, then we subtract each day's long position holding period return by the short position holding period return. Is that right?

To calculate long position's hodling period return, I use $$R_A = \frac{\text{Final Price_A} - \text{Initial Price_A}}{\text{Initial Price_A}}$$

To calculate short position's holding period return, I use $$R_B = \frac{ \text{Initial Price_B} - \text{Final Price_B}}{\text{Initial Price_B}}$$

Then, to find day $$k$$'s holding period return, I use $$R_A - R_B$$

The generic way to compute such returns is to multiply the asset returns over a specific period with the weights at the start of that period. For a short position, the weight is negative. If you do not rebalance over the k days, you should simply multiply each asset's k-day return with the initial weight, and then sum those weighted returns. The advantage of this generic approach is that it can easily handle rebalancing, i.e. changing weights.

To provide an example: Suppose you have a fixed notional amount of money to invest; 100 euros, say. You invest 100 euros (100% of your capital) in stock A, i.e. you buy 100/share_price_of_A units of A. You sell short 100 euros (-100% of 100 euros) of stock B, i.e. you sell 100/share_price_of_B units of B.

Suppose that over the holding period of k days, stock A rises by 20% and stock B by 10%.

So what is your return? 100% * 20% + -100% * 10% = 1*0.2 + -1*0.1 = 0.1 = 10% (Or in monetary units, 10 euros on the 100 euros notional amount.)

In your description, you define the return for a short position different from (as the negative of) the return of a long position. In that case should indeed add those returns, not subtract them. The advantage of working with explicit weights is that the single-asset return computation does not depend on your position.

If you wish to work with shares as opposed to weights, you'll need to make some assumption regarding a notional; there is no generally-established "correct" way to compute returns in this case. You would then compute success in units of currency, and then divide by the chosen notional. (Or skip returns altogether and measure your success in units of currency only; futures traders often do that when the notional is not obvious.)

• For each stock, I would only go long or short on one share at a time. So, it seems like my calculations of the holding period return for the long and short positions are correct, but I need to sum these two returns to get the portfolio's holding period. Do I understand your answer correctly? Commented Feb 14 at 15:45
• I do not understand what you mean by "-100% in B". Do you mean if I have a billion, then I long a billion in A and short a billion in B? Please let me clarify. My idea is that if I see a signal, then I will long one share of A and short one share of B, then hold these for some days. At the end, I want to calculate this portfolio's holding period return. Commented Feb 15 at 7:54
• I use $[(A1 - A0)/A0] - [(B0 - B1)/B0]$ in my question, but I am actually unsure whether the minus sign between the two squre brackets should be a $+$ sign. Since $(B0 - B1)/B0$ is the holding period return of our short position, subtracting this from the long's holding period return, $[(A1 - A0)/A0]$, would not only fail to add to what we have gained but also reduce it, right? Shouldn't we be adding these two holding period returns together to obtain the total holding period return? Commented Feb 15 at 8:00
• Why do we need to multiply the weights $wA$ and $wB$? Would you mind using an example to illustrate it? Thank you! Commented Feb 15 at 8:07
• buying/selling one share: So your signal might tell you to buy one share of Microsoft (ca. 400 USD) and sell one share of Intel (ca. 40 USD)? Commented Feb 15 at 17:26