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Time-variation in portfolio performance

1. Time-variation in portfolio performance

Until now, we have taken a static view on portfolio performance. But,

2. Bulls & bears

due to the business cycle, the occurrence of unexpected events, and swings in the market psychology, portfolio performance tends to be anything but constant over time. It is dynamic. In terms of market direction, there are the bull markets in which stock prices tend to increase, and there are the bear markets in which stock prices tend to fall.

3. Clusters of high & low volatility

Then, in terms of market stress,

4. Performance statistics in action

we have periods in which markets are calm with persistently low volatility,

5. Performance statistics in action

there are the more stressed regimes

6. Performance statistics in action

with big spikes in volatility.

7. Performance statistics in action

Note that those regimes are persistent.

8. Performance statistics in action

Once the volatility is higher than average,

9. Performance statistics in action

it tends to stay above average for some time.

10. Performance statistics in action

It follows that the current performance is better estimated,

11. Performance statistics in action

when we give more weight to the more recent observations than to the distant observations. The standard approach of doing this is by the use of

12. Rolling estimation samples

rolling estimation samples. Instead of estimating the performance measures on the full sample, we only take the K most recent observations. This implies that the performance estimate at time t is given by the performance statistic computed on the sample of returns at time t,

13. Rolling estimation samples

t-1 , t-2

14. Rolling estimation samples

up to t-K+1.

15. Rolling estimation samples

If we then move one observation further and estimate the performance for the next date t+1, then we will be using the observations from t+1

16. Rolling estimation samples

till t-K+2.

17. Rolling estimation samples

As such, we roll through time by adding the most recent observation

18. Rolling estimation samples

and discarding the most distant one. On each subsample,

19. Rolling performance calculation

any type of performance measure can be computed. As an example, you can see here the time series plot of annualized mean and volatility estimates obtained for the monthly S&P 500 returns using rolling samples of three years.

20. Choosing window length

Why did I choose three years and not one year when making this plot? This is a question of taste. We need to have a sufficiently large number of observations to reduce the effect of noise on the performance estimate. But, the longer the subperiod is, the more it smooths over the highs and lows in the data, and the less informative it becomes about the most recently observed performance. In the next exercises, you will get a feeling for this trade-off.

21. Let's practice!