How Leverage Changes Compounded Returns Along a Price Path
Summary
The document examines how to calculate returns for a long-only strategy when leverage is applied to a changing price series. It first compares simple and logarithmic period returns and shows that, when compounded across the stated path, both produce the same cumulative result as measuring the overall unlevered price change. It then applies leverage by multiplying each period’s simple return by two and compounds those leveraged returns.
The example ends with a cumulative return below twice the unlevered return, even though the start and end prices alone imply a doubled gain. This raises the central issue: a leveraged strategy’s result depends on the sequence of gains and losses, because each period’s return applies to the changing account value. The text is a question rather than a full answer, so it does not specify financing costs, margin rules, or liquidation effects. Its figures illustrate path dependence for the particular price sequence, not a universal performance estimate.
Key ideas
- Simple and logarithmic returns can yield the same cumulative return when properly compounded across the example path.
- The example applies leverage by scaling each period’s simple return before compounding.
- The resulting leveraged return differs from simply multiplying the full-period price change by leverage.
- The document illustrates path dependence but does not cover financing costs, margin rules, or liquidation.
Tags
Full text
# How to properly calculate leveraged returns # How to properly calculate leveraged returns Imagine we have a certain price movement, where the price starts at 1000 and ends at 1200, with some fluctuations in the middle. For the sake of the example, imagine it's hourly timestamps, and it's a long only strategy. I went ahead and calculated the simple and log returns, as well as the cumulative returns. As expected, the cumulative returns calculated for both the simple and log returns match. | price | simple_returns | log_returns | cum_sum_simple | cum_sum_log | | 1000 | nan | nan | nan | nan | | 1100 | 0.10000 | 0.09531 | 1.10000 | 1.10000 | | 900 | -0.18182 | -0.20067 | 0.90000 | 0.90000 | | 800 | -0.11111 | -0.11778 | 0.80000 | 0.80000 | | 950 | 0.18750 | 0.17185 | 0.95000 | 0.95000 | | 1100 | 0.15789 | 0.14660 | 1.10000 | 1.10000 | | 1500 | 0.36364 | 0.31015 | 1.50000 | 1.50000 | | 1200 | -0.20000 | -0.22314 | 1.20000 | 1.20000 | This outputs a return of `0.2` (20%), which is expected. `(1200 - 1000) / 1000` is also 20%. Now imagine we introduce leverage into the equation, for example 2. From my current understanding, in order to introduce the leverage into the returns, we have to multiply it by the simple returns, not the log returns. We can then calculate the cumulative sum as before. So it would be: | price | lev_returns | cum_sum_lev | | 1000 | nan | nan | | 1100 | 0.20000 | 1.20000 | | 900 | -0.36364 | 0.76364 | | 800 | -0.22222 | 0.59394 | | 950 | 0.37500 | 0.81667 | | 1100 | 0.31579 | 1.07456 | | 1500 | 0.72727 | 1.85606 | | 1200 | -0.40000 | 1.11364 | This means that the final return would be `0.113`, or 11.3%. This in itself is already a little counter intuitive—as one would expect the gains to be double, so 40%—but this could in theory make sense since losses (with simple returns) are more amplified than the gains. What is really confusing me, is that if we were to calculate the total final return independently of the path the price took to get there—so by taking only the last and first prices—, then: `r = ((1200 - 1000) / 1000) * 2` `r = 0.4` What am I missing here?
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