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.

Cumulative returns


Consider these two simple portfolios:

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

  2. 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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