Why Daily Returns Do Not Add Up to Monthly Returns

Summary

The document asks whether summing daily value-weighted returns, with dividends reinvested monthly, should match the corresponding monthly return. The reply explains that returns over successive periods compound multiplicatively rather than add directly. Its example pairs a loss with a gain of the same stated percentage: the product of the two growth factors leaves a small net loss, even though their arithmetic sum is zero.

This illustrates why a daily return sum can differ from a return calculated over the full month. To aggregate sequential simple returns, multiply their gross-return factors and subtract one; addition is only an approximation in limited circumstances. The exchange does not discuss the construction of the value-weighted series, dividend timing, or other sources of measurement differences, so compounding explains the basic discrepancy without diagnosing every possible deviation in the user's data.

Key ideas

• Sequential simple returns combine by multiplying their gross-return factors.
• Returns of equal size with opposite signs do not generally cancel over time.
• Adding daily returns can therefore differ from the return measured across the full month.
• The example illustrates compounding but does not address dividend timing or portfolio-calculation details.

Tags

# Should the sum of daily returns be close to monthly returns


# Should the sum of daily returns be close to monthly returns












I am calculating value-weighted returns with monthly dividends reinvested and for some reason when I sum the daily returns some are a little bit off with monthly returns.

Is this normal?

## Answer by Charles Fox (score 1, accepted)

https://quant.stackexchange.com/a/45115

No, you should expect them to be different. For example, consider -10% and +10%. 1.1*0.9 = 0.99 -> -1% return vs 0% if you use addition.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.