# how to chain monthly excess returns into annual?

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

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.

• @Quantlbex, thanks, but now I am confused. I needed to have monthly returns for a CAPM, otherwise the data is too sparse. But then I cannot annualize those monthly excess returns somehow? I do know monthly returns for both the asset and the benchmark (and the risk-free rate), of course. – László Jul 19 '13 at 11:15
• If you want to compute the monthly excess return for month $i$ expressed on an annual basis, you could simply use $(1+r_i^p)^{12} - (1+r_i^b)^{12}$, where $r_i^p$ denotes the portfolio return for month $i$, and $r_i^b$ denotes the benchmark return for month $i$. – QuantIbex Jul 19 '13 at 11:26
• @Quantlbex, Thanks again. I need to use the surprise returns unexplained by a CAPM month-by-month. Can't I compound those into an annual measure? I think annualization (scaling) is not the real issue here. But I need the correct compounding. You say I need to use expected (CAPM-predicted) returns as well, not just compound the gross unexpected ones? – László Jul 19 '13 at 13:31
• I don't understand what you mean by "the surprise returns unexplained by a CAPM month-by-month". Would you consider editing the question with proper (mathematical) notations to clarify the quantities that are known and what you need? – QuantIbex Jul 19 '13 at 13:59
• @Quantlbex, let me know if this helped. Or put differently, the question is about what do you with beta from monthly returns to calculate annual excess returns? Simply use $r^p-r^f-\hat{\beta} (r^b-r^f)$ for annual returns but monthly beta? The betas are supposed to be independent from the time series resolution they are estimated upon? Maybe I should check that in my time-series statistics notes… – László Jul 20 '13 at 18:52