This is my fourth post in my summer 2026 mini series on portfolio optimisation.
It will very much follow the format of (also with a sports alluding title) blog post number two, so it might be worth rereading that. A reminder if you can't be bothered, I used random data to compare some optimisation methods:
- monte carlo (random, parameteric)
- bootstrapping (random, non parametric)
- double shrinkage (shrinking SR towards average SR, and correlations to zero). This encompasses some other methods including:
- NMV naive mean variance (no shrinkage on anything)
- EW equal weights (both full shrinkage)
- MD maximum diversification (no shrinkage correlation, full shrinkage on SR)
- EPO (we just shrink the correlation matrix to some degree)
I found that MC/Bootstrap were the best, and didn't require any pesky estimation of the shrinkage meta-parameter. But they are SLOW. I worked out you'd need quite a few iterations to get the weights to converge, so each optimisation took quite a while. Should you wish to estimate that meta-parameter I found that for random data with a nice stable distribution that you didn't need much shrinkage. A little bit on the Sharpe Ratio was the most optimal; a little more wouldn't harm things much, but a lot was bad.
However as we know from post three, real data is not as nice as random data, and is much harder to forecast. It has a habit of doing annoying things, like changing it's distribution when you're not looking. So we're expecting that we will need, for example, more shrinkage to reflect this.
The real data we will be using will many different runs, each consisting of 9 randomly selected trading rules, chosen for a single randomly chosed instrument. Because we know from post one that fitting within instruments is the way to go. Although I currently have 40 trading rules in my actual portofolio, I am sticking with nine now for speed and intuition. Plus the results shouldn't be too different with more components - that is something I will be looking at later in the series. I'm sampling with replacement so it's feasible - but very unlikely- I'll get the same instrument/rule set more than once.
As per my previous posts I'm also going to compare the results for different lengths of data. In the random data post I could generate as much data as I want; that's tricky here when the absolute longest history I have for any instrument is just over 50 years and many are much less than that. So I'm going to use in sample lengths of 1 year, 5 years and 10 years; and out of sample lengths of 1 year and 5 years. If an instrument doesn't have sufficient data for a given pairing I won't use it; eg for 10 years/5 years I would need 15 years which will be tricky for many instruemnts whilst for 1 year/1 year I would just need 2 years obviously. If it has more data than required, then on a given random run I'll randomly select the required 2 to 15 year long period.
First some speed statistics. We already know that shrinkage will be darn quick, but as I'm using different data lengths from the prior post it's probably worth repeating the stats for montecarlo and bootstrap:
1 year in sample 5 years in sample 10 years in sample
BS 9.2 20.6 33.3
MC 5.1 6.6 8.0
Remember from the previous post that convergence is quicker with Monte Carlo than with Bootstrap, hence the substantially longer time taken to do BS which needs twice as many iterations; as well as the slight difference in implementation per iteration which explains the even worse performance of BS at longer iterations.
Results
One year in sample, One year out of sample
Let's begin with the median results. For the moment I'm going to present two data frames. The first is just Sharpe Ratios. Here is the one for an insample and out of sample period of just one year:
0.00 0.20 0.40 0.60 0.70 0.75 0.80 0.90 1.00
0 0.056 0.057 0.039 0.054 0.047 0.049 0.046 0.055 0.032
0.25 0.061 0.045 0.054 0.063 0.057 0.049 0.044 0.042 0.037
0.5 0.059 0.057 0.048 0.047 0.044 0.046 0.041 0.046 0.044
0.75 0.049 0.041 0.062 0.058 0.041 0.026 0.029 0.047 0.033
0.8 0.030 0.041 0.061 0.054 0.041 0.025 0.026 0.026 0.032
0.85 0.016 0.038 0.050 0.035 0.030 0.029 0.023 0.025 0.036
0.9 -0.002 0.022 0.043 0.030 0.041 0.038 0.029 0.024 0.052
0.95 0.015 0.022 0.045 0.043 0.049 0.043 0.049 0.034 0.056
1.0 0.014 0.003 0.038 0.060 0.056 0.058 0.043 0.032 0.004
MC -0.000 -0.000 -0.000 -0.000 -0.000 -0.000 -0.000 -0.000 -0.000
BS 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018
This will look very familiar if you looked at the previous post on random data, but there are a couple of extra rows. From the top then each column shows a different degree of correlation shrinkage. On the left 0.0 is no shrinkage where we used the estimated data. 1.0 is full shrinkage, where all correlations are set to zero. Apart from the diagonals. Obviously. Each row then is a different degree of SR shrinkage, from the top row where we use no shrinkage, down to the row labelled 1.0 where we fully shrink all SR to the average SR across assets.
The bottom two rows are the results for Monte Carlo and Bootstrapping. There is no shrinkage here, so for consistency I've just copied the single value for each across all columns.
Some elements of interest in the main part of the table, the top left corner (0.0, 0.0) is naive mean variance with no shrinkage, the top right (0.0, 1.0) is full correlation shrinkage, the bottom left (1.0, 0.0) is full SR shrinkage, and the bottom right (1.0, 1.0) is full shrinkage on both which leads to equal weights. The EPO empirical optimal is (0, 0.75).
The optimum value here has some shrinkage: 0.25 on SR and 0.60 on correlations.
Compare and contrast that with the results for random data. The optimal shrinkage was barely nothing: 0.25 SR, 0 correlations or thereabouts. It isn't surprising we need more shrinkage in general. Remember from the previous post in this series on random data:
Essentially random data sets a lower bound on robustness calibration. For example, suppose we determine that the correct shrinkage for the vector of expected SR on a Bayesian portfolio optimisation using random data is 0.1 (which means we average using 90% of the estimated SR, and 10% of the prior SR). Then it's likely the correct shrinkage on real data will be higher than 0.1.
However the amount of optimal SR versus correlation shrinkage might seem surprising. Quoting now from post three in this series, on forecasting statistical parameters with real data:
In simple terms, we are a little bit worse than forecasting Sharpe Ratios in real data one year ahead than we would be with random data, but a LOT worse with correlations. Partly this is because we are pretty terrible at forecasting SR one year ahead anyway even with a stable underlying distribution; we don't do much worse with real data. However it does seem that correlations are far more unstable in reality than in randomly generated data.... If we recall from the prior post that the optimal shrinkage is zero on correlations with random data; we can now see why with actual data we'd probably want to opt for some correlation shrinkage; purely because the sampling error is much larger in practice. That is the empirical finding of the EPO paper. It does feel a bit weird since up to now my gut feeling has been that we have to shrink means a lot because they are much harder to forecast and because they have an outsized effect on portfolio weights compared to differences in correlation. Whilst the latter is still true it seems the former is not.
There are two different effects here remember: predicability of each estimate compared to random data (where correlation is worse), and more about their outright predictability (where SR is worse), and the different effects each has on MV optimisation (small differences in SR affect the outcome more).
Another surprise might be the relatively poor performance of MC and BS. Remember that the only difference between them is the assumption of joint Gaussian returns in one case and not in the other. In the random data round each method was the best performing. Both however are making an implicit assumption that there is a stable distribution (parameteric in one case, not in the other), and that any variance in outcome over the out of sample period will be the same as would be expected from the sampling distribution of each parameter. Which is exactly what happens with random data. But we know from post three that the parameter estimates we're making have a wider distribution with real data; and this is especially true for correlations. Hence, the MC/BS methods are too optimistic about predictability and their weights are suboptimal compared to those produced by high shrinkage optimisations.
Note: I have ideas to fix that, which may or may not in a subsequent blog post. Briefly they involve playing with the MC parameter inputs to reflect the higher RMSE of real versus random data.
Now let's run a paired t-test comparision of that optimum median value against all other values. Here are the p=values from doing those tests:
0.00 0.20 0.40 0.60 0.70 0.75 0.80 0.90 1.00
0 0.91 0.64 0.39 0.63 0.86 0.83 0.90 0.56 0.22
0.25 0.84 0.83 0.99 NaN 0.27 0.24 0.64 0.75 0.37
0.5 0.62 0.37 0.34 0.20 0.07 0.19 0.17 0.54 0.79
0.75 0.76 0.81 0.70 0.70 0.79 0.93 0.85 0.99 0.93
0.8 0.81 0.95 0.81 0.84 0.95 1.00 0.99 0.67 0.97
0.85 0.99 0.97 0.72 0.67 0.76 0.69 0.84 0.60 0.99
0.9 0.92 0.68 0.84 0.85 0.66 0.64 0.74 0.71 0.91
0.95 0.73 0.64 0.91 0.73 0.72 0.60 0.56 0.66 0.94
1.0 0.71 0.68 0.87 0.46 0.41 0.42 0.34 0.42 0.47
MC 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
BS 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06
We can see that the optimum value itself is NaN since the p-value is undefined. We can also see that statistically, there isn't much difference in how shrinkage is used. MC and BS are definitely worse
Now, how do things change if we are more pessimistic? As before I'm going to look at the 5% distributional point of outcomes from my multiple random results. If I do this, the optimal shrinkage is 0.8 on correlations, but a massive 1.0 on SR. At the 25% point it's 0.75 on SR but 1.0 on correlations. We want more shrinkage for sure!
Now let's think about a nice graphical way of showing these values. I'll start with a heatmap of the median SR:
Now I'm going to do something familar to the students on my course. I'm going to replace every value that is statistically insigificant from the optimal median with the optimal media value. Here I will use a 90% critical value:
Here the result looks like a really shit piece of modern art. Since almost all shrinkage values are not significantly different from the optimal; except weirdly correlation shrinkage 0.7 and SR shrinkage 0.5 (which is adjacent to the optimum); it's just a sea of blue. But we can see the MC/BS methods are inferior.
One year in sample, Five years out of sample
It isn't obvious but I used the same procedure for this plot which shows SR, with all values that can't be distinguished from the optimum in the same colour as that optimum. But every single value other than the optimum, which is full shrinkage or equal weights, is inferior to that optimum.
Five years in sample, One years out of sample
A very interesting picture here. There's clearly a shrinkage area that doesn't work. Note that the results overall are quite poor.
Five years in sample, Five years out of sample
A little clearer here. Modest shrinkage would work well, but then so would random data. Just don't shrink the SR too much.
Ten years in sample, One years out of sample
Importantly here the critical value is 80%, not 90%. With 90% the whole plot goes one colour. Pretty much any amount of shrinkage works. Again the SR results are very poor.
Ten years in sample, Five years out of sample
Summary of results
Well that was messy. I'd conclude that shrinkage of SR 0.5 and correlation 0.75 (the EPO value) is in the optimum region in almost all time periods. That's a reversal of what my original intuition suggested and I've used before, with more shrinkage on the SR. I've explained at length why my intuition was wrong. The random methods (MC/BS) are also inferior in many cases, as well as being slow.
The exception is one year / five years where you need full shrinkage (equal weights). Using 0.5/0.75 isn't so bad however. Although it's significantly worse, the actual loss in SR is small. Still it does seem logical to use more shrinkage with more data; and we can see from the one year/five year plot that we're better off shrinking SR more. So here is my heuristic rule of thumb:
Five or more years of data: SR shrinkage 0.5, correlation 0.75
Four to five years of data: SR shrinkage 0.6, correlation 0.75
Three to four years of data: SR shrinkage 0.7, correlation 0.80
Two to three years of data: SR shrinkage 0.8, correlation 0.85
One to two years of data: SR shrinkage 0.9, correlation 0.90
One or less than one year of data: SR shrinkage 1.0, correlation 1.0 (equal weights)
These results are very domain specific. In particular, I'm mostly dealing with holding periods in the weeks and months. A faster trading system would be able to compress the periods above. But the main lesson is that it's very hard to state categoricially what the exact amount of shrinkage should be. The surface is mostly too noisy. So don't sweat it. Use a vaguely okay value and you'll do vaguely ok.
