Annualizing Quarterly Returns for Means and Covariance
Summary
The document asks how to estimate annual mean returns and a four-asset covariance matrix from quarterly observations. It states the correct compounding relationship: an annual return is the product of the four quarterly gross returns minus one. The proposed shortcut instead raises each individual quarter’s gross return to the fourth power, then computes statistics on those transformed observations.
The question identifies a key aggregation issue: the shortcut does not combine the actual four quarters belonging to each year, so it cannot directly represent observed annual returns or their covariance. The supplied table contains quarterly returns for four assets, but the document provides no worked calculation or answer. Any annual estimate from this sample depends on grouping consecutive quarters into years and on the statistical assumptions used; a short sample may also make annual covariance estimates uncertain.
Key ideas
- An annual simple return compounds the four quarterly gross returns observed in that year.
- Raising each quarter’s gross return to the fourth power does not produce the realized annual return for that year.
- Annual return vectors should be formed by compounding aligned quarters across all assets before calculating annual means and covariance.
- The supplied quarterly table is a data example, but the document gives no computed estimates or worked solution.
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Full text
# Finding mean vector and covariance matrix for annual returns given quarterly returns
# Finding mean vector and covariance matrix for annual returns given quarterly returns
I am currently trying to calculate a vector for the mean annual returns of 4 different asset classes along with their 4x4 covariance matrix in excel. However, I am having problems since the data I have been supplied is quarterly returns (as calculated from the relevant stock indicies). I know that quarterly and annual returns are linked by the following equation $$r_{Annual,x}=\prod_{q=1}^{4}(1+r_{q,x})-1$$ Where $r_{i}$ is the return for the $i^{th}$ quarter and $x$ denotes that this is the $x^{th}$ year. What I have done is calculate $(1+r_{i,j})^4-1$ where $r_{i,j}$ is the $j^{th}$ quarterly return observation for the $i^{th}$ asset class (I have 4 asset class).
This results in 4 new vectors where each quarterly return has been converted to an effective annual return. I then apply the mean and covariance functions to these converted values to get the mean vector and covariance matrix.
However I get the feeling this is wrong. Any feedback would be greatly appreciated and if you could provide a worked example with the following (quarterly return) data simulated in excel that would be great.
```
Asset 1 Asset 2 Asset 3 Asset 4
1.98% 2.99% 1.91% 3.24%
-1.22% -1.87% 1.18% 2.35%
5.00% 6.10% 1.08% 4.46%
2.45% -0.10% 1.75% 3.06%
5.70% 3.70% 1.47% 2.10%
8.29% 1.96% 1.51% 0.62%
-3.36% -1.25% 1.11% 2.72%
-2.01% 0.21% 1.49% 0.90%
-4.99% 2.44% 1.48% 2.02%
4.67% -0.29% 1.48% 1.51%
0.86% 2.92% 1.54% 2.38%
7.52% -0.75% 1.43% 3.36%
-1.24% 3.11% 1.20% 1.93%
-4.25% 1.41% 1.35% 0.84%
3.89% -0.25% 0.86% 1.46%
0.67% 4.13% 1.45% 2.30%
3.93% 1.53% 1.43% 1.35%
-5.00% -0.63% 1.50% 3.35%
3.47% 2.99% 1.50% 3.06%
7.76% 3.61% 0.98% 3.79%
-8.26% 1.03% 1.18% 2.90%
5.45% 1.57% 1.05% 3.38%
-2.65% 2.25% 1.44% 1.45%
0.76% 7.50% 1.52% 1.79%
4.55% -2.72% 1.31% 1.82%
1.32% 8.70% 1.36% 1.24%
-3.02% -1.43% 1.52% 3.92%
2.05% 1.20% 1.94% 2.50%
7.37% 0.29% 1.64% 2.73%
-0.66% 2.36% 1.75% 2.68%
-6.86% -1.40% 1.35% 2.62%
13.55% 6.03% 1.30% 1.33%
-2.23% 5.26% 1.44% 1.48%
-4.26% 0.45% 1.61% 2.93%
6.75% 1.70% 1.08% 1.26%
-0.84% -2.16% 0.89% 2.45%
-0.66% 4.03% 1.66% 2.98%
0.15% 1.94% 1.21% 3.13%
-4.74% 0.26% 1.44% 2.35%
1.29% -0.76% 0.96% 1.52%
13.22% 0.37% 1.74% 4.11%
-2.37% 0.41% 1.25% 2.71%
3.16% 1.27% 1.22% 2.54%
1.56% 4.54% 0.87% 3.55%
0.97% 2.18% 1.52% 1.75%
3.32% 2.41% 1.64% 2.12%
-1.80% -2.07% 1.12% 0.77%
7.16% 1.87% 1.61% 3.68%
10.65% 4.89% 1.25% 2.47%
5.47% 4.26% 1.33% 1.94%
0.31% 5.53% 1.32% 3.89%
3.59% 4.55% 1.46% 2.27%
7.24% 1.50% 1.67% 1.32%
3.06% 0.87% 1.78% 3.43%
7.43% 3.92% 1.58% 3.05%
11.29% 2.17% 1.47% 2.76%
11.85% 3.64% 1.59% 1.57%
1.68% -1.25% 1.48% 2.37%
9.93% -0.53% 1.95% 1.76%
-9.09% -3.53% 1.32% 3.30%
-3.09% 3.00% 1.41% 2.86%
2.99% 1.49% 1.34% 1.97%
0.28% 3.81% 1.30% 2.27%
9.23% 6.06% 1.17% 2.44%
6.74% 7.60% 1.52% 1.66%
0.48% 5.72% 1.61% 1.25%
-2.28% 0.96% 1.30% 2.69%
4.16% -1.94% 1.68% 2.95%
7.21% 4.77% 1.59% 1.40%
-3.97% -0.84% 1.56% 2.11%
2.48% 1.46% 1.22% 3.60%
5.89% 1.65% 1.83% 1.62%
-2.73% 4.73% 1.37% 1.99%
9.12% -0.52% 1.29% 2.89%
1.29% 3.89% 1.35% 2.47%
4.18% -1.76% 1.58% 3.85%
-9.78% 2.66% 1.09% 1.63%
10.73% -1.80% 1.57% 3.11%
3.37% -0.03% 1.67% 1.57%
11.17% 5.92% 1.26% 2.04%
7.13% 3.02% 1.79% 2.29%
13.12% -0.75% 1.25% 2.72%
-2.73% 1.45% 1.04% 3.61%
-8.38% -1.19% 1.16% 2.15%
4.63% 0.53% 1.33% 3.00%
1.93% 4.88% 1.39% 3.65%
-6.75% 0.74% 0.95% 2.40%
5.43% 0.75% 1.33% 2.76%
-4.68% 3.02% 0.90% 2.38%
4.99% 4.99% 1.66% 0.84%
-0.44% 3.68% 1.51% 1.84%
13.18% 3.82% 1.69% 2.88%
8.05% 2.02% 0.79% 2.50%
-3.39% 1.93% 1.15% 2.65%
21.79% 3.30% 1.36% 2.12%
-5.96% 6.49% 1.58% 3.43%
1.43% -1.17% 0.89% 0.57%
10.21% 4.70% 1.27% 1.69%
-13.53% 7.99% 1.31% 2.62%
-5.25% -0.46% 1.08% 3.75%
```Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.