how to chain monthly excess returns into annual?

Question

I want to calculate annual excess returns on portfolios using monthly returns for a CAPM (for the assets in the portfolio as well as for the benchmark), in order to have more information on the correlations, more precise betas.

Because the CAPM comes from monthly correlations, I shall calculate excess returns for each month, right? But if I only have year-end snapshots of portfolios, I should chain the monthly excess returns up (compound them) and multiply the initial value with each surprise return? Is this essentially the same as doing the annual calculation? (I suspect an argument about integrating a continuous price process into some return observations anyway.)

I have information on holdings $q_0$ and want to calculate surprise returns on this initial portfolio over the following year, $r^s_{0,12} \cdot q_0$. (Where annual returns are between moment 0 and moment 12.)

For $r^s_{0,12}$, I thought to use $r^s_{0,12} = \left( \prod_{t =1}^{12} R^s_{t,a} \right)-1$, where monthly surprise returns gross come from a monthly CAPM (of log returns): $ R^s_{t,a} = R_{t,a} / R^{exp}_{t,a} $ where $ \log R^{exp}_{t,a} = \left( \hat{\beta_a}(\log(R^m_t)-\log(R^f_t)) \right)$.

I hope the net vs gross returns and divisions or differences of logs are not too confusing.

Full disclosure: This breaks down my longer question into specifics. Please bear with me. From: annual excess returns from CAPM on monthly total returns

Answer 1

Compounding the monthly excess returns won't provide the annual excess return. You need to compute the difference between the annual return of the portfolio and the annual return of the benchmark.

To illustrate this let's look at an example. Consider the following two situations:

1. The benchmark performs well with a $2\%$ return each month;
2. The benchmark performs badly with a $2\%$ loss (a return of $-2\%$) each month.

Suppose that in both situations the portfolio has a monthly excess return of $1\%$.

Then, the compounded monthly excess return is $(1 + 0.01)^{12} - 1 = 12.7\%$.

In situation 1, the benchmark annual return is $(1 + 0.02)^{12} - 1 = 26.8\%$, and the portfolio annual return is $(1 + 0.03)^{12} - 1 = 42.6\%$. So the annual excess return is $15.8\%$, which is different from the compounded monthly excess return of $12.7\%$.

In situation 2, the benchmark annual return is $(1 - 0.02)^{12} - 1 = -21.5\%$, and the portfolio annual return is $(1 - 0.01)^{12} - 1 = -11.4\%$. So the annual excess return is $10.1\%$, which is also different from the "compounded excess return" of $12.7\%$.

This simple example illustrates that just knowing the excess returns of the portfolio is not enought to get the annual excess return. One needs also the monthly/annual return of the benchmark; it's a consequence of the effect of compounding.