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!