Why Higher Average Returns Can Produce Lower Cumulative Returns

Summary

The document distinguishes arithmetic average returns from cumulative, compounded performance. A portfolio can have a higher average periodic return yet finish with a lower cumulative value, because cumulative performance depends on the sequence and magnitude of returns as they compound. The examples compare return streams with the same arithmetic average but different volatility: larger gains and losses leave a lower terminal value after compounding.

The answer gives an approximate link between annualized geometric and arithmetic returns: geometric return is reduced by roughly half the return variance. It uses this relationship to suggest that the portfolio with the higher average in the question may also be more volatile, which could explain its weaker cumulative path. That explanation is plausible rather than demonstrated from the portfolios’ data; the examples illustrate the mechanism, but a specific comparison requires examining realized volatility, return timing, and compounding assumptions.

Key ideas

• Arithmetic average returns do not determine cumulative compounded performance on their own.
• Larger gains and losses can reduce terminal wealth even when the arithmetic average is unchanged.
• Higher volatility generally lowers geometric returns relative to arithmetic returns.
• Portfolio comparisons should consider compounded returns and volatility as well as average returns.

Tags

# average return Vs cumulative return interpretation


# average return Vs cumulative return interpretation












I am looking for the interpretation which distinguishes between average return and cumulative return. I have two portfolios : the average return of portfolio 2 = 3 10E-4 per day while the average return of portfolio 4 = 4.21 10E-4 .. portfolio 4 outperforms portfolio 2.

However, when I graph the cumulative returns of the two portfolios, it's clear that portfolio 2 exceeds portfolio 4. I would be grateful if you could help me to understand and interpret the following graphic.

## Answer by Helin (score 5, accepted)

https://quant.stackexchange.com/a/35501

Consider these two simple portfolios:

- Portfolio 1 returns -10% in month 1 and 10% in month 2. Average arithmetic return is zero, and cumulative return is $(1-10\%)(1+10\%)=0.99$.

- Portfolio 2 returns -50% in month 2 and 50% in month 2. Average arithmetic return is still zero, but cumulative return is $(1-50\%)(1+50\%)=0.75$, a much lower terminal value!

In general, arithmetic return and geometrically compounded return are linked as follows: $$ \text{annualized geometric return} \approx \text{annualized arithmetic return} - \frac{\sigma^2}{2}, $$ where $\sigma$ is volatility. The more volatile a return stream is, the less cumulative compounding effect you get (all else equal).

In your case, one return stream has higher average arithmetic return than the other, but it likely is more volatile, resulting in less cumulative compounding over time.

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