Allocating Cumulative Portfolio Losses Across Monthly Returns
Summary
The document asks how to attribute the total loss from a sequence of monthly returns to selected months. It points out that returns compound multiplicatively, so simply adding the monthly percentages and assigning a fraction of that sum to the ending loss can give an unsatisfying attribution. It also notes that a proposed approach based on log returns fails when returns have opposite signs.
The answer treats the starting capital as a portfolio value and calculates the dollar loss in each period from the portfolio’s value immediately before that return. The selected months’ losses can then be summed and divided by the total dollar decline, with the remaining periods accounting for the rest. This produces an additive breakdown of the realized change in portfolio value for the specified return sequence. The allocation depends on the order of returns and the chosen initial capital; it is an attribution of dollar changes along that path, not a unique order-independent decomposition of compounded performance.
Key ideas
- Monthly returns combine multiplicatively, so summing return percentages does not directly allocate the final loss.
- Calculate each period’s dollar gain or loss using the portfolio value at the start of that period.
- Summing those dollar changes gives an additive breakdown of the ending portfolio decline.
- The attribution depends on return order and represents one realized path through portfolio value.
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Full text
# Tricky question about returns # Tricky question about returns I have a list of monthly returns. `-10% -20% -70% -30% -15% -60%` The total end return is -94.859%. Because you calculate = 100 x (1+ -10%) x (1+ -20%) x ... Now I want to know how much the two months -70% and -60% contributed to the end return. How can I calculate that? The sum of all returns in the list is -205%. The -70% and -60% entries adds up to -130%. Can I conclude that these both months stand for 63.4% of all losses? Because 130/205=0.634. If this is true, then these both months contributes 0.634 x -94.859% = -60.155% of the total end return. This means the rest of the months contributes (-94.859% - 60.155%) = -34.705% to the total end return. But I dont like this. It feels wrong. I guess the correct answer uses addition of log returns? Do you have a better answer? UPDATE: Alper below gives the answer in the comments. He writes: if you have two returns -10% and -20%, then the total returns would be 72%. Each return gives a contribution like this: -10% gives a contribution of ln(100%-10%)/ln(72%) = 32% and -20% gives a contribution of ln(100%-20%)/ln(72%) = 68%. These numbers add up to 100%. However, if the returns have opposite signs, say -10% and +20%, then this formula does not work. ## Answer by KaiSqDist (score 0) https://quant.stackexchange.com/a/76740 You can treat the initial 100 as the dollar value of a portfolio. As you mentioned, at each point in time, the 100 decreases by 10%, 20% ... 60%. Therefore, the portfolio value after each increase is: | Time | Portfolio Value | % Returns | Dollar Value of Losses | | 0 | 100.0 | - | - | | 1 | 90.00 | -10% | 10.00 | | 2 | 72.00 | -20% | 18.00 | | 3 | 21.60 | -70% | 50.40 | | 4 | 15.12 | -30% | 6.480 | | 5 | 12.85 | -15% | 2.270 | | 6 | 5.140 | -60% | 7.710 | As the the cumulative returns is 94.86, -70% and -60% contributed 58.11/94.86 = 61.26% and the rest contributed 36.75/94.86 = 38.74%. You can see both proportions sum to 61.26% + 38.74% = 100%. Hopefully this helps?
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