# Half annual covariance $\mathrm{Cov}(X,Y)_{\text{6 months}}$ from annual covariance?

I have annual returns and standard deviations for two funds, $r_{a}$, $r_{b}$, $\mathrm{SD}_{a}$ and $\mathrm{SD}_{b}$ but I do not have individual data, just the annual data. The annual correlation between the prices of the funds is 0.7. If I had the individual data I could use

$\mathrm{Cov}(A,B) =\sum_i (\bar{r}_{a} - r_{i a}) (\bar{r}_{b} - r_{i b}) \>,$

but now I am lost without the data. Are there some approximations to get the half annual data from the annual data or some formula to do it directly?

The Wikipedia entry for covariance is good for learning the rules for manipulating and calculating covariance. The Wikipedia entry on variance is good for that special case.

In your particular case, you can solve your problem fairly easily by using the rule (in the properties section of the covariance article)

$$\operatorname{Cov}(aX+bY, cW+dV) = ac\,\operatorname{Cov}(X,W)+ad\,\operatorname{Cov}(X,V)+bc\,\operatorname{Cov}(Y,W)+bd\,\operatorname{Cov}(Y,V)\$$

If you think of annual returns as the sum of two semi-annual returns, and assume that the semi-annual returns are not correlated across periods and the correlations for the two periods are constant. I think it should turn out that you can simply divide the covariance in half.

• the thing that bogles me is whether the assumption about independence of year halves is valid. If not, $Cov(Year)=Cov(A/2+B/2)=Cov(A/2)+Cov(B/2)+\rho_{A/2, B/2}w_{A/2}w_{B/2}$, where $A/2$ and $B/2$ stand for different half years and $\rho_{A/2, B/2}=\sigma_{A/2}\sigma_{B/2}w_{A/2}w_{B/2}$ where $w$ stands for the proportions. – hhh May 16 '11 at 18:34
• It would be surprising if the two year halves were non-negligibly correlated since asset returns are martingales. – John Salvatier May 16 '11 at 21:03
• @hhh dividing the covariance like this is incorrect. Think in terms of scatterplots: behind each year's paired data is a set of about 250 pairs of daily returns. The question is trying to recover the scatter of those daily returns from observing the scatter of these averaged-out points. There is no fixed relationship, or even useful bounds available, unless one makes strong assumptions. In particular, dividing by two--even assuming no temporal correlation--will not generally produce the correct answer. – whuber May 16 '11 at 22:00
• I think it is better to say that asset returns can be modeled by stochastic processes such as martingales and markov chains. Sometimes, it is useful to analyse things with path-depended models but then again you cannot beat state-depended models in some point-wise analysis. Sorry I am a bit lost what you are referring to with this comment. Can you write it mathematically to your answer? – hhh May 17 '11 at 13:46
• My notion was that assuming independence of halves and equality of the covariance of the halves was probably 'good enough' for your situation since your question implies you think that semi-annual covariance from annual covariance is a reasonable thing to try to do. If these assumptions are not good enough for your purposes, then there's probably little you can do short of getting more data. Is more clear? – John Salvatier May 17 '11 at 18:47