Equal-Weighted Long-Short Returns and Portfolio Compounding
Summary
The document examines how to calculate daily performance for a long-short portfolio with equal capital allocated to each side. It defines each asset’s simple daily return from consecutive closing prices, then combines the long return and short return by subtracting the latter and dividing by two. The author tests this formula against a small set of made-up prices and questions whether it remains meaningful as the positions’ market values diverge over time.
The post further asks whether these daily figures can support a Sharpe ratio and whether compounding them produces a cumulative return that maps back to net asset value. It contrasts the equal-weighted return series with dollar changes in a specific share-based position. No answer or resolution is included, so the example illustrates a portfolio-weighting and return-denominator question rather than validating the book’s method. The calculation’s interpretation depends on the rebalancing and capital conventions, which the post leaves open.
Key ideas
- The document defines simple daily asset returns using consecutive closing prices.
- It combines long and short returns by subtracting the short return and dividing by two.
- The author questions whether this measure remains representative as the sides’ market values diverge.
- The post asks whether the resulting series supports Sharpe ratio calculation and compounding but gives no answer.
- Portfolio return interpretation depends on capital and rebalancing conventions that are not resolved here.
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Full text
# Daily returns over many days of two-stock portfolio - averaging
# Daily returns over many days of two-stock portfolio - averaging
I'm reading Ernest Chan's book “Quantitative Trading: How to Build Your Own Algorithmic Trading Business 2E”. In example 3.4 of chapter 3, he defines:
- The daily return of a stock on day $d$ as $DailyRet(d)=\frac{Close(d)-Close(d-1)}{Close(d-1)}$, for example if stock L had close of 100 on day 0 and 110 on day 1, then the daily return on day 1 was $\frac{110-100}{100}=0.1$
- The net daily return of a long-short that starts with equal capital on each side (market neutral) on day $d$ is defined as: $NetDaily_{LS}(d)=\frac{DailyRet_L(d) - DailyRet_S(d)}{2}$, so assuming long instrument (he uses IGE) on the first day (i.e. $d=1$) had $DailyRet_L(1)=0.5$ and short instrument (he uses SPY) had $DailyRet_S(1)=-0.2$ we would have a $NetDaily_{LS}(1)=\frac{0.5 - (-0.2)}{2}=\frac{0.7}{2}=0.35$
He uses this to populate data for multiple days from historical IGE/SPY data and calculate the Sharpe ratio which I am suspicious of.
I've made up some numbers for close prices and here is his example for 3 days of made-up close prices:
| Stock L | | Index S | | Net DailyRet (avg) |
| Close | $DailyRet_L$ | Close | $DailyRet_S$ | $NetDaily_{LS}$ |
| 50 | | 100 | | |
| 75 | 0.5 | 80 | -0.2 | $\frac{0.5-(-0.2)}{2} = \frac{0.7}{2} = 0.35$ |
| 75 | 0 | 60 | -0.25 | $\frac{0-(-0.25)}{2} = \frac{0.25}{2} = 0.125$ |
| 60 | -0.2 | 75 | 0.25 | $\frac{-0.2-0.25}{2} = \frac{-0.45}{2} = -0.225$ |
> My problem is with his use of $NetDaily_{LS}(d)=\frac{DailyRet_L(d) - DailyRet_S(d)}{2}$ formula. This assumes equal weights for long/short sides which may work for day 1, but as soon as the two sides deviate the weights no longer apply. Therefore the numbers in the "Net DailyRet (avg)" column above seem meaningless to me. How can he then go on to use them for calculating the Sharpe ratio?
For example, let's assume you buy a 2:1 ratio at the close of day 0, by shorting 1 share of S (1x\$100) and going long 2 shares of L (2x\$50) for a market neutral position. The \$-amount returns using the close prices above would be:
| Long Capital | Short Capital | NetDailyRet x (Long-Short) |
| 100 | -100 | n/a |
| 150 | -80 | $0.35\times(100-(-100)) = 0.35\times200 = 70$ |
| 150 | -60 | $0.125\times(150-(-80)) = 0.125\times230 = 28.75$ |
| 120 | -75 | $-0.225\times(150-(-60)) = -0.225\times210 = -47.25$ |
As you can see, the net daily of $0.35$ works when the capital is 50-50, but the net daily of $0.125$ when applied to a capital of $230$ the next day, gives a net move of $28.75$ which is wrong.
In the example in the book, the author has no issue with using this net daily to calculate the Sharpe ratio, but I am reluctanct to accept this as "correct".
> Is anyone familiar with the book and can you explain why this definition is correct and can be used to yield a correct Sharpe ratio? Furthermore how can one use the value in the $NetDaily_{LS}$?
For example, for (2) in my mind the answer is "$NetDaily_{LS}(d)$ is the return you can expect on day $d$ if you had an equally balanced portfolio of L and S". In practice this number is only useful for days when that happens (day 0 in this example) and useless for cases where the long/short capital is not equal (all other days in this example).
He even goes on to specify the compound cumulative return using this $NetDaily_{LS}$ as:
$$ CompCumRet(d) = \left(1+NetDaily_{LS}(d)\right) \times \left(1+CompCumRet(d-1)\right) - 1 $$
I would argue that any compound cumulative return definition should have the characteristic that you can take its value for any given day $d$, multiply it with the initial capital on day 0 and get the net asset value for that day $d$. This does not hold for the book's definitions of $CompCumRet(d)$ and $NetDaily_{LS}(d)$.
I would really appreciate if someone can confirm or debunk the definitions used in this book.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.