Introducing max-GM (median@p), a new(?) performance statistic

Do you remember this post? https://qoppac.blogspot.com/2022/06/vol-targeting-cagr-race.html

Here I introduced a performance metric, the best annualised compounding return at the optimal leverage level for that strategy. This is equivalent to finding the highest geometric return once a strategy is run at it's Kelly optimal leverage.

I've since played with that idea a bit, for example in this more recent post amongst others I considered the implications of that if we have different tolerances for uncertainty and used bootstrapping of returns when optimising a stock and bond portfolio with leverage, whilst this post from last month does the same exercise in a bitcoin/equity portfolio without leverage.

In this post I return to the idea of focusing on this performance metric - the maximised geometric mean at optimal leverage, but now I'm going to take a more abstract view to try and get a feel for in general what sort of strategies are likely to be favoured when we use this performance metric. In particular I'd like to return to the theme of the original post, which is the effect that skew and other return moments have on the maximum achievable geometric mean. 

Obvious implications here are in comparing different types of strategies, such as trend following versus convergent strategies like relative value or option selling; or even 'classic' trend following versus other kinds.

Some high school maths

(Note I use 'high school' in honor of my US readers, and 'maths' for my British fans)

To kick off let's make the very heroric assumption of Gaussian returns, and assume we're working at the 'maximum Kelly' point which means we want to maximise the median of our final wealth distribution - same as aiming for maximum geometric return - and are indifferent to how much risk such a portfolio might generate.

Let's start with the easy case where we assume the risk free rate is zero; which also implies we pay no interest for borrowing (apologies I've just copied and pasted in great gobs of LaTeX output since it's easier than inserting each formula manually):

Now that is a very nice intuitive result!

Trivially then if we can use as much leverage as we like, and we are happy to run at full Kelly, and our returns are Gaussian, then we should choose the strategy with the highest Sharpe Ratio. What's more if we can double our Sharpe Ratio, we will quadruple our Geometric mean!

Truely the Sharpe Ratio is the one ratio to rule them all!

Now let's throw in the risk free rate:

We have the original term from before (although the SR now deducts the risk free rate), and we add on the risk free rate reflecting the fact that the SR deducts it, so we add it back on again to get our total return. Note that the higher SR = higher geometric mean at optimal leverage relationship is still true. Even with a positive risk free rate we still want to choose the strategy with the highest Sharpe Ratio!

Consider for example a classic CTA type strategy with 10% annualised return and 20% standard deviation, a miserly SR of 0.5 with no risk free rate; and contrast with a relative value fixed income strategy that earns 6.5% annualised return with 6% standard deviation, a SR of 1.0833

Now if the risk free rate is zero we would prefer the second strategy as it has a higher SR, and indeed should return a much higher geometric mean (since the SR is more than double, it should be over four times higher). Let's check. The optimal leverages are 2.5 times and 18.1 (!) times respectively. At those leverages the arithmetic means are 25% and 117% respectively, and the geometric means using the approximation are 12.5% for the CTA and 58.7% for the RV strategy.

But what if the risk free rate was 5%? Our Sharpe ratios are now equal: both are 0.25. The optimal leverages are also lower, 1.25 and 4.17. The arithmetic means come in at 12.5% and 27.1%, with geometric means of 9.4% and 24%. However we have to include the cost of interest; which is just 1.25% for the CTA strategy (borrowing just a quarter of it's capital at a cost of 5% remember) but a massive 15.8% for the RV. Factoring those in the net geometric means drop to 8.125% for both strategies - we should be indifferent between them, which makes sense as they have equal SR.

The horror of higher moments

Now there is a lot wrong with this analysis. We'll put aside the uncertainty around being able to measure exactly what the Sharpe Ratio of a strategy is likely to be (which I can deal with by drawing off more conservative points of the return distribution, as I have done in several previous posts), and the assumption that returns will be the same in the future. But that still leaves us with the big problem that returns are not Gaussian! In particular a CTA strategy is likely to have positive skew, whilst an RV variant is more likely to be a negatively skewed beast, both with fat tails in excess of what a Gaussian model would deliver. In truth in the stylised example above I'd much prefer to run the CTA strategy rather than a quadruple leveraged RV strategy with horrible left tail properties.

Big negative skewed strategies tend to have worse one day losses; or crap VAR if you prefer that particular measure. The downside of using high leverage is that we will be saddled with a large loss on a day when we have high leverage, which will significantly reduce our geometric mean.

There are known ways to deal with modifying the geometric mean calculation to deal with higher moments like skew and kurtosis. But my aim in this post is to keep things simple; and I'd also rather not use the actual estimate of kurtosis from historic data since it has large sampling error and may underestimate the true horror of a bad day that can happen in the future (the so called 'peso problem'); I also don't find the figures for kurtosis particuarly intuitive. 

(Note that I did consider briefly using maximum drawdown as my idiots tail effect here. However maximum drawdown is only relevant if we can't reduce our leverage into the drawdown. And perhaps counter intuitively, negative skewed strategies actually have lower and shorter drawdown)

Instead let's turn to the tail ratio, which I defined in my latest book AFTS. A lower tail ratio of 1.0 means that that the left tail is Gaussian in size, whilst a higher ratio implies a fatter left tail.

I'm going to struggle to rewrite the relatively simple 0.5SR^2 formulae to include a skew and left tail term, which in case will require me to make some distributional assumptions. Instead I'm going to use some bootstrapping to generate some distributions, measure the tail properties, find the optimal leverage, and then work out the geometric return at the optimal leverage point. We can then plot maximal geometric means against empirical tail ratios to get a feel for what sort of effect these have.

Setup

To generate distributions with different tail properties I will use a mixture of two Gaussian distributions; including one tail distribution with a different mean and standard deviation* which we draw from with some probability<0.5. It will then be straightforward to adjust the first two moments of the sample distribution of returns to equalise Sharpe Ratio so we are comparing like with like.

* you will recognise this as the 'normal/bliss' regime approach used in the paper I discussed in my prior post around optimal crypto allocation, although of course it will only be bliss if the tail is nicer which won't be the case half the time.

As a starting point then my main return distribution will have daily standard deviation 1% and mean 0.04% which gives me an annualised SR of 0.64, and will be 2500 observations (about 10 years) in length - running with different numbers won't affect things very much. For each sample I will draw the probability of a tail distribution from a uniform distribution between 1% and 20%, and the tail distribution daily mean from uniform [-5%, 5%], and for the tail standard deviation I will use 3% (three times the normal). 

All this is just to give me a series of different return distributions with varying skew and tail properties. I can then ex-post adjust the first two moments so I'm hitting them dead on, so the mean, standard deviation and SR are identical for all my sample runs. The target standard deviation is 16% a year, and the target SR is 0.64, all of which means that if the returns are Gaussian we'd get a maximum leverage of 4.0 times.

As always with these things it's probably easier to look at code, which is here (just vanilla python only requirements are pandas/numpy).

Results

Let's start with looking at optimal leverage. Firstly, how does this vary with skew?

Ignore the 'smearing' effect this is just because each of the dots in a given x-plane will have the same skew and first two moments, but slightly different distribution otherwise. As we'd expect given the setup of the problem the optimal leverage with zero skew is 4.0

For a pretty nasty skew of -3 the leverage should be about 10% lower - 3.6; whilst for seriously positive skew of +3 you could go up to 4.5. These aren't big changes! Especially when you consider that few real world trading strategies have absolute skew values over 1 with the possible exception of some naked option buying/selling madness. The most extreme skew value I could find in my latest book was just over 2.0, and that was for single instruments trading carry in just one asset class (metals).

What about the lower tail? I've truncated the plot on the x-axis for reasons I will explain in a second.

Again for the base case with a tail ratio of 1.0 (Gaussian) the optimal leverage is 4.0; for very thin lower tails again it goes up to 4.4, but for very fat lower tails it doesn't really get below about 3.6. Once again, tail ratios of over 4.0 are pretty rare in real strategies (though not individual instruments), although my sampling does sometimes generate very high tail ratios the optimal leverages never go below 3.6 even for a ratio of nearly 100.

Upper tail, again truncated:

Again a tail ratio of 1.0 corresponds to optimal leverage of roughly 4.0; and once again even for very fat upper tail ratios (of the sort that 'classical' trend followers like to boast about, the optimal leverage really isn't very much higher.

Now let's turn to Geometric mean, first as affected by skew:

As we'd expect we can generate more geometric mean with more skew... but not a lot! Even a 3 unit improvement in skew barely moves the return needle from 22.5% to 24.5%. To repeat myself it's rare to find trading strategies above or below 1.0, never mind 3.0. For example the much vaunted Mulvaney capital has a return skew on monthly returns of about 0.45. The graph above shows that will only add about 0.25% to expected geometric return at optimal leverage versus a Gaussian normal return distribution (this particular fund has extremely large standard deviation as they are one of the few funds in the world that actually runs at optimal leverage levels).

For completeness, here are the tail ratios. First the lower tail ratio:

Fat left tails do indeed reduce maximum optinmal geometric means, but not a lot.

Now for the upper tail:

Summary

I do like neat formulae, and the results for Gaussian normal distributions do have the property of being nice. Of course they are based on unreal expectations, and anyway who runs full Kelly on a real strategy expecting the returns to be normal and the SR to be equal to the backtested SR is an idiot. A very optimistic idiot, no doubt with a sunny disposition, but an idiot nonetheless.

For the results later in the post it does seem surprising that even if you have something with a very ugly distribution that you'd not really adjust your optimal leverage much, and hence see barely on impact on geometric mean. But remember here that I'm fixing the first two moments of the distribution; which means I'm effectively assuming that I can measure these with certainty and also that the future will be exactly like the past. These are not realistic expectations! 

And that is why in the past rather than choose the leverage that maximised geometric mean, I've chosen to maximise some more conservative point on the distribution of terminal wealth (where geometric mean would be the median of that distribution). Doing that would cause some damage to the negative skewed, fat left tail distributions resulting in lower optimal leverage and thus lower geometric means.

I have thought of a way of doing this analysis with distributional points, but it's quite computationally intensive and this post is already on the long side, so let's stop here. 

Skew preferences for crypto degens

@investingidiocy looking for my favorite statistics wonk to critique this. https://t.co/xWQMEl5LUU

— bengraham (@bengraham) January 5, 2024

An old friend asking for help... how can I resist? Here is the perplexing paper:

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4042239

And here is the (not that senstional) abstract:

Bitcoin (BTC) returns exhibit pronounced positive skewness with a third central moment of approximately 150% per year. They are well characterized by a mixture of Normals distribution with one “normal” regime and a small probability of a “bliss” regime where the price appreciation is more than 100 times at the annual horizon. The large right-tail skew induces investors with preferences for positive skewness to add significant BTC holdings to equity-bond portfolios. Even when BTC is forecast to lose half of its value in the normal regime, investors with power utility optimally add 3% allocations to BTC when the probability of the bliss regime is around 1%. Cumulative Prospect Theory investors are even more sensitive to positive skewness and hold BTC allocations of around 3% when the probability of the bliss regime is 0.0006 and the mean of BTC in the normal regime corresponds to a loss of 90%.

3% in BTC doesn't sound too crazy to me, but what has been really setting the internet on fire is this out of context quote from later in the paper:

Starting with a 60-40 equity-bond portfolio, which is produced with a risk aversion of 𝛾 = 1.50, the optimal BTC allocation is a large 84.9%! The remainder of the portfolio, 15.1% is split 60-40 between equities and bonds. Although BTC has an extremely large volatility of 1.322 (see Exhibit 1), the pronounced positive skewness leads to large allocations and dominates in the utility function (see equation (9)). The certainty equivalent compensation required to not invest in BTC is close to 200%. [my emphasis]

Here's my English translation of this:

- Bitcoin has pronounced positive skew 

- Some people really like positive skew (people  with 'power utility' and 'cumulative prospect theory' preferences)

- This justifies a higher allocation to Bitcoin than they would otherwise have, since it has lots of positive skew (both on an outright basis, and as part of a 60:40 portfolio).

- There is a 'Bliss' regime when Bitcoin does really well ('goes to the moon') but which isn't very likely

- Even if there is a tiny probaility of this happening, and if things are generally terrible in the non bliss regime, then people who like positive skew should have more Bitcoin. Some of them should have a lot!

Now, I could just as easily write this:

- Lottery tickets have (very!) pronounced positive skew 

- Some people really like positive skew 

- This justifies a higher allocation to lottery tickets than they would otherwise have

- There is a 'Bliss' regime when lottery tickets do very well ('winning the jackpot') but which isn't very likely

- Even if there is a tiny probaility of this happening, and if things are generally terrible in the non bliss regime, then people who like positive skew should have more lottery tickets. Some of them should have a lot!

I see nothing here that I can argue with (sorry Ben)! And it certainly doesn't require an academic to make the argument that people who like lottery ticket type payoffs, and think that there is a chance that Bitcoin will go up a lot, should buy more Bitcoin. But I think there is a blogpost to be written about the interaction of skew prefences and allocations; and hopefully one that is perhaps easier to interpret. Two key questions for me are:

- to what extent does the expectation of return distributions affect allocations?

- just how far from 'skew neutral' does ones prefence have to be before we allocate significant amounts to Bitcoin

Luckily, I already have an intuitive framework for analysing these problems, which I used in a fairly complete way in my previous post - bootstrapping the return distribution. 

Setup

The goal then, is to understand the asset allocation that comes out of (a) a set of return distributions and (b) a preference for skew.

For the return distributions we have two broad approaches we can use. Firstly, we can use actual data. Secondly, we can use made up return distributions fitted to the actual data. This is what the paper does, mostly "We use monthly frequency data at the annual horizon from July 2010 to December 2021 for BTC and from January 1973 to December 2021 for stocks and bonds. The univariate moments for each asset are computed using the longest available sample, and the correlation estimates are computed with the common sample across the assets."

The paper also uses a third approach, which is to see what happens if they mess with the return distributions once fitted by changing the probability of 'Bliss'.

I'm going to use the first approach, which is to use real return data at least initially. Other slight differences, I will use returns from July 2010 to November 2023 for all three assets, I will use excess rather than total returns (which given the low interest rates in the period makes almost no difference) with futures prices for S&P 500 (equity proxy) and US 10 year bonds (bond proxy), with Bitcoin total return deflated by US 3 month treasury yields, and I'm going to use daily rather than monthly data to improve my sample size.

The next consideration is the utility preference of the investor. I am going to assume that the investor wants to maximise the Nth percentile point of the distribution of geometric returns. This is the approach I have used before which requires no assumptions about utility function and allows an intuitive measure of risk preference to be used by modifying N. 

As I have noted at length, someone with N=50 is a Kelly optimiser. That is the absolute maximum you should bet, irrespective of your appetite for skew or risk. Thus the Kelly bettor must have the maximum possible appetite for skew. Someone with N<50 would be very nervous about the downside and much more worried about small losses than the potential for large gains; and hence they would have less of a preference for positive skewed assets.

I personally think this is a much more intuitive way to proceed than randomly choosing utility functions and risk aversion parameters, and choosing from a menu of theoretical distributions. The downside is that isn't possible to decompose skew and risk preferences, since both have been replaced with a different measure - the 'appetite for uncertainty'.

An important point is that maximising CAGR will naturally lead to a higher allocation to crypto than you would get from the more classical method of maximising mean subject to some standard deviation constraint or risk aversion penalty. 

The method I will use then is:

- sample the returns data repeatedly to create multiple new sets of data.The new set of data would be the same length as the original, and we'd be sampling with replacement (or we'd just get the new data in a different order). 

- from this new set of data and a given set of possible portfolio allocations, estimate the geometric return

- for a given set of allocations, take the Nth percentile of the distribution of geometric means

- plot the Nth percentile for each allocation to work out roughly where the optimal might be

I say 'roughly', because as readers of previous posts on this subject are aware, we never know exactly where the optimal is when bootstrapping, which is a much better reflection of reality than the precise analytical calculations done by the original authors. Still, we can get a feel for how the optimal changes as we vary N (skew preference).

Note: As a fan of Red Dwarf, the use of the term 'Bliss' in this context is very confusing!

The data

Since we're pretending to be proper academics, here are the summary statistics of the real data:

Annualised mean:
equity     0.256
bonds      0.000
bitcoin    1.280

Annualised standard deviation:
equity     0.160
bonds      0.064
bitcoin    0.816

Correlation:
          equity  bonds  bitcoin
equity    1.000 -0.245    0.069
bonds    -0.245  1.000   -0.014
bitcoin   0.069 -0.014    1.000

Sharpe ratio:
equity     0.783
bonds      0.166
bitcoin    1.612

Skew:
equity    -0.431
bonds      0.294
bitcoin    0.970

Note that if anything the statistics here are more favourable to Bitcoin than in the original paper. Importantly, we are assuming that as in the past Bitcoin will more than double every year on average (the figure in bold), and that it will have a Sharpe Ratio well north of 1.0. Given these raw statistics, it isn't then very suprising regardless of skew preferences that we would potentially dump a large part of our portfolio into Bitcoin. And indeed, if I run these numbers through my optimisation the optimal position is 100% in Bitcoin for a Kelly maximiser.

To add another line to my 'dumb' bullet point translation of the paper earlier:

- if you think Bitcoin will go up a lot like it did in the past, you should only own Bitcoin

To make things more realistic and interesting, I'm doing to massage the data to reflect what I think is a fairly conservative forward looking position: All assets will have the same Sharpe Ratio (which I will set arbitrarily at an annualised 0.5). I achieve this by shifting the mean returns up or down respectively, which means all the other return characteristics remain the same - only Sharpe Ratio and means are affected. Note that this also means that bonds will look better relative to equities.

This still implies that Bitcoin will, on average, go up by 40% a year, which means it will double every two years. Personally I still think this is extremely optimistic, but I'm going to put my own views to one side for this exercise.

Note: even if you are not a Bitcoin skeptic, it seems unlikely that Bitcoin will behave in the same way going forward as it did when it was worth less than $1,000 and had the market cap of a penny stock rather than a decent sized country; both the mean, skewness, and the standard deviation have reduced in the last few years since Bitcoin has become a bigger market.

Results

Right, let's see some pictures. 

The following heatmap shows what happens to the median of the distribution of bootstrapped geometric returns (Kelly maximiser, with maximum appetite for skew) as we allocate to equities (y-axis) and Bitcoin (x-axis). The allocation to bonds will be whatever is leftover. The white area is where we can't allocate, since we are putting more than 100% into the portfolio, and my working assumption is here is that leverage isn't allowed (if it was, we'd have much more bonds, much less equities and Bitcoin, and use leverage to maximse CAGR).

The optimal allocation to Bitcoin is somewhere around 50% with equities taking most of the rest. So even the most gung-ho optimistic skew loving nutjob shouldn't put more than half their wealth in BTC. For context, a 50% equity and BTC portfolio would have a standard deviation of around 42%, nearly 3 times the risk of equities. To be Kelly optimal, that implies the Sharpe Ratio would need to be at least 0.42. This is a much higher risk target than pretty much every hedge fund uses.

Now let's see what happens if we reduce our N to the 25% percentile point. Importantly: this is roughly the N that produces a 60:40 portfolio considering a portfolio with only equities and bonds. So we can think of this as the 'base case' for risk and skew preference. Again with CAGR below 4% washed out to produce a more granular z axis:

You can see the optimal allocation to Bitcoin is lower here, around 30%, with perhaps 50% in equities and the rest in bonds.

What about N=10%?

Again, we are looking at a bit less again in Bitcoin; with something around 20% with perhaps 70% in equities and the rest in bonds. This would give you something with a standard deviation not much higher than equities, at least in theory.

Summary- ignore everything I have said

The original paper has been toted around the internet to say that you should have 85% of your portfolio in Bitcoin ('this is optimal'). But:

- this is a single figure taken out of context from a much more nuanced paper; note again that the abstract does not include such an extreme figure

- it assumes that historic Bitcoin performance is matched going forward, including performance from 2010 back when BTC cost less than $1 and the total 'market cap' was less than $200,000. 

- it assumes particular risk aversion, preferences for skew and utility functions; such that you would hold quite a lot of Bitcoin even if you thought it's performance would generally be bad except in rare 'Bliss' regimes. Basically it says 'if you like lottery tickets, you are going to love Bitcoin!'.

In this post I take a different approach which hopefully is more intuitive for the non economist, and gives a bit more insight into the interplay between return skew and skew preference, which is also useful beyond the narrow problem of allocating to crypto currency. But what you couldn't or shouldn't do is take anything I or anyone else has written, and claim it 'proves' that the 'optimal' allocation to Bitcoin is x%. All it can do is say based on these assumptions and assuming this set of preferences what your allocation should be. That can quite easily come out to 85%, or 100%. It can also quite easily come out to less than 1%, or even zero.  

What's my own personal allocation to Bitcoin, I hear you ask? On a long only basis it is zero, and nothing I have written here will change that. Partly this is because of my long standing and well known aversion to this 'asset class', both in principle* and in practice**.

* to summarize it's a ponzi that wastes energy with the ownership structure of a pyramid scheme, and which will never be useful for anything except the current use cases: 1% illegal money transfer, 99% gambling

** it's a real pain and very expensive to buy Bitcoin 'properly' i.e. owning your own coins and putting them into cold wallet storage 

But it's also because unlike in this example, there are more than three assets in the world! Concretely, I trade well over 100 futures; of which just a couple are crypto coins. Accordingly it also makes no sense to me to put more than a few % of my trading account into crypto - an account where I can go long and short and hence my personal biases are irrelevant.

My allocation to Bitcoin and Ether in my futures trading strategy is a touch under 5%. And those are risk weights; the equivalent cash weight would be lower: as I write this my position in Bitcoin is long 3 micro futures with a notional value of perhaps £12K or around 3% of my trading capital. Of course it could just as easily be zero, or a short position...

Portfolio optimisation, uncertainty, bootstrapping, and some pretty plots. Ho, ho, ho.

Optional Christmas themed introduction

Twas the night before Christmas, and all through the house.... OK I can't be bothered. It was quiet, ok? Not a creature was stirring... literally nothing was moving basically. And then a fat guy in a red suit squeezed through the chimney, which is basically breaking and entering, and found a small child waiting for him (I know it sounds dodgy, but let's assume that Santa has been DBS checked*, you would hope so given that he spends the rest of December in close proximity to kids in shopping centres)

* Non british people reading this blog, I could explain this joke to you, but if you care that much you'd probably care enough to google it.

"Ho ho" said the fat dude "Have you been a good boy / girl?"

"Indeed I have" said the child, somewhat precociously if you ask me.

"And what do you want for Christmas? A new bike? A doll? I haven't got any Barbies left, but I do have a Robert Oppenheimer action figure; look if you pull this string in his stomach he says 'Now I am become Death destroyer of worlds', and I'll even throw in a Richard Feynman lego mini-figure complete with his own bongo drums if you want."

"Not for me, thank you. But it has been quite a long time since Rob Carver posted something on his blog. I was hoping you could persuade him to write a new post."

"Er... I've got a copy of his latest book if that helps" said Santa, rummaging around in his sack "Quite a few copies actually. Clearly the publisher was slightly optimistic with the first print run."

"Already got it for my birthday when it came out in April" said the child, rolling their eyes.

"Right OK. Well I will see what I can do. Any particular topic you want him to write about in this blog post?"

"Maybe something about portfolio optimisation and uncertainty? Perhaps some more of that bootstrapping stuff he was big on a while ago. And the Kelly criterion, that would be nice too."

"You don't ask for much, do you" sighed Santa ironically as he wrote down the list of demands.

"There need to be really pretty plots as well." added the child. 

"Pretty... plots. Got it. Right I'll be off then. Er.... I don't suppose your parents told you to leave out some nice whisky and a mince pie?"

"No they didn't. But you can have this carrot for Rudolf and a protein shake for yourself. Frankly you're overweight and you shouldn't be drunk if you're piloting a flying sled."

He spoke not a word, but went straight to his work,And filled all the stockings, then turned with a jerk. And laying his finger aside of his nose, And giving a nod, up the chimney he rose! He sprang to his sleigh, to his team gave a whistle, And away they all flew like the down of a thistle. But I heard him exclaim, ‘ere he drove out of sight,

"Not another flipping protein shake..."

https://pixlr.com/image-generator/ prompt: "Father Christmas as a quant trader"

Brief note on whether it is worth reading this

I've talked about these topics before, but there are some new insights, and I feel it's useful to combine the question of portfolio weights and optimal leverage into a single post / methodology. Basically there is some familar stuff here but now in a coherent story, plus some new stuff.

And there are some very nice plots.

Somewhat messy python code is available here (with some data here or use your own), and it has no dependency on my open source trading system pysystemtrade so everyone can enjoy it.

Bootstrapping

I am a big fan of bootstrapping. Some definitional stuff before I explain why. Let's consider a couple of different ways to estimate something given some data. Firstly we can use a closed form. If for example we want the average monthly arithmetic return for a portfolio, we can use the very simple formula of adding up the returns and dividing by the number of periods. We get a single number. Although the arithmetic mean doesn't need any assumptions, closed form formula often require some assumptions to be correct - like a Gaussian distribution. And the use of a single point estimate ignores the fact that any statistical estimate is uncertain. 

Secondly, we can bootstrap. To do this we sample the data repeatedly to create multiple new sets of data. Assuming we are interested in replicating the original data series, the new set of data would be the same length as the original, and we'd be sampling with replacement (or we'd just get the new data in a different order). So for example, with ten years of daily data (about 2500 observations), we'd choose some random day and get the returns data from that. Then we'd keep doing that, not being bothered about choosing the same day (sampling with replacement), until we had done this 2500 times. 

Then from this new set of data we estimate our mean, or do whatever it is we need to do. We then repeat this process, many times. Now instead of a single point estimate of the mean, we have a distribution of possible means, each drawn from a slightly different data series. This requires no assumptions to be made, and automatically tells us what the uncertainty of the parameter estimate is. We can also get a feel for how sensitive our estimate is to different variations on the same history. As we will see, this will also lead us to produce estimates that are more robust to the future being not exactly like the past.

Note: daily sampling destroys any autocorrelation properties in the data, so it wouldn't be appropriate for example for creating new price series when testing momentum strategies. To do this, we'd have to sample larger chunks of time period to retain the autocorrelation properties. For example we might restrict ourselves to sampling entire years of data. For the purposes of this post we don't mind about autocorrelation, so we can sample daily data.

Bootstrapping is particularly potent in the field of financial data because we only have one set of data: history. We can't run experiments to get more data. Bootstrapping allows us to create 'alternative histories' that have the same basic character as our actual history, but aren't quite the same. Apart from generating completely random data (which itself will still require some assumptions - see the following note), there isn't really much else we can do.

Bootstrapping helps us with the quant finance dilemma: we want the future to be like the past so that we can use models calibrated on the past in the future, but the future will never be exactly like the past. 

Note: that bootstrapping isn't quite the same as monte carlo. With that we estimate some parameters for the data, making an assumption about it's distribution. Then we randomly sample from that distribution. I'm not a fan of this. We have all the problems of making assumptions about distribution, and of uncertainty about the parameter estimates we use for that distribution. 

Portfolio optimisation

With all that in mind, let's turn to the problem of portfolio opimisation. We can think as this as making two decisions:

• Allocating weights to each asset, where the weights sum to one
• Deciding on the total leverage for the portfolio

Under certain assumptions we can seperate out these two decisions, and indeed this is the insight of the standard mean variance framework and the 'security market line'. The assumption is that enough leverage is available that we can get to the risk target for the investor. If the investor has a very low risk tolerance, we might not even need leverage, as the optimal portfolio will consist of cash + securities.

So basically we choose the combination of asset weights that maximises our Sharpe Ratio, and then we apply leverage to hit the optimal risk target (since SR is invariant to leverage, that will remain optimal). 

To begin with I will assume we can proceed in this two phase approach; but later in the post I will relax this and look at the effect of jointly allocating weights and leverage.

I'm going to use data for S&P 500 and 10 year Bond futures from 1982 onwards, but which I've tweaked slightly to produce more realistic forward looking estimates for means and standard deviations (in fact I've used figures from this report- their figures are actually for global equities and bonds, but this is all just an illustration). 

My assumptions are:

• Zero correlation (about what it has been in practice since 1982)
• 2.5% risk free rate (which as in standard finance I assume I can borrow at)
• 3.5% bond returns @ 5% vol
• 5.75% equity returns @ 17% vol

This is quite a nice technique, since it basically allows us to use forward looking estimates for the first two moments (and first co-moment - correlation) of the distribution, whilst using actual data for the higher moments (skew, kurtosis and so on) and co-moments (co-skew, co-kurtosis etc). In a sense it's sort of a blend of a parameterised monte-carlo and a non parameterised bootstrap.

Optimal leverage and Kelly

I'm going to start with the question of optimal leverage. This may seem backwards, but optimal leverage is the simpler of the two questions. Just for illustrative purposes, I'm going to assume that the allocation in this section is fixed at the classic 60% (equity), 40% (bonds). This gives us vol of around 10.4% a year, a mean of 4.85%, and a Sharpe Ratio of 0.226

The closed form solution for optimal leverage which I've written about at some length, is the Kelly Criterion. Kelly will maximise E(log(final wealth)) or median(final wealth), or importantly here it will maximise the geometric mean of your returns.

Under the assumption of i.i.d. Gaussian returns optimal Kelly leverage is achieved by setting your risk target as an annual standard deviation equal to your Sharpe Ratio. With a SR of 0.226 we want to get risk of 22.6% a year, which implies running at leverage of 22.6 / 10.4 = 2.173

That of course is a closed form solution, and it assumes that:

• Return parameters are Guassian i.i.d. (which financial data famously is not!)
• The return parameters are fixed
• That we have no sampling uncertainty of the return parameters
• We are fine running at fully Kelly, which is a notoriously aggressive amount of leverage

Basically that single figure - 2.173 - tells us nothing about how sensitive we would be to the future being similar to, but not exactly like, the past. For that we need - yes - bootstrapping. 

Bootstrapping optimal leverage 

Here is the bootstrap of my underlying 60/40 portfolio with leverage of 1.

Each point on this histogram represents a single bootstrapped set of data, the same length as the original. The x-axis shows the geometric mean, which is what we are trying to maximise. You can see that the mean of this distribution is about 4.1%. Remember the arithmetic mean of the original data was 4.85%, and if we use an approximation for geometric mean that assumes Gaussian returns then we'd get 4.31%. The difference between 4.1% and 4.31% is because this isn't Guassian. In fact, mainly thanks to the contribution of equities, it's left tailed and also has fat tails. Left fat tails result in lower Geometric returns - and hence also a lower optimal leverage, but we'll get to that in a second.

Notice also that there is a fair bit of distributional range here of the geometric mean. 10% of the returns are below 2%, and 1% are below 0.4%.

Now of course I can do this for any leverage level, here it is for leverage 2:

The mean here is higher, as we'd probably expect since we know the optimal leverage would be just over 2.0 if this was Gaussian. It comes in at 4.8%; versus the 7.2% we'd expect if this was the arithmetic mean, or the 5.04% that we would have for Gaussian returns.

Now we can do something fun. Repeating this exercise for many different levels of leverage, we can take each of the histograms that are producing and pull various distributional points off them. We can take the median of each distribution (50% percentile, which in fact is usually very close to the mean), but also more optimistic points such as the 75% and 90% percentile which would apply if you were a very optimistic person (like SBF, as I discussed in a post about a year ago), and perhaps more usefully the 25% and 10% points. We can then plot these:

How can we use this? Well, first of all we need to decide what our tolerance for uncertainty is. What point on the distribution are you optimising for? Are you the sort of person who worries about the bad thing that will happen 1 in 10 times, or would you instead be happy to go with the outcome that happens half the time (the median)?

This is not the same as your risk tolerance! In fact, I'm assuming that your tolerance for risk is sufficient to take on the optimal amount of leverage implied by this figure. Of course it's likely that someone with a low tolerance for risk in the form of high expected standard deviation would also have a low tolerance for uncertainty. And as we shall see, the lower your tolerance for uncertainty, the lower the standard deviation will be on your portfolio.

(One of the reasons I like this framing of tolerance is that most people cannot articulate what they would consider to be an appropriate standard deviation, but most people can probably articulate what their tolerance for uncertainty is, once you have explained to them what it means)

Next you should focus on the relevant coloured line, and mentally remove the odd bumps that are due to the random nature of bootstrapping (we could smooth them out by using really large bootstrap runs - note they will be worse with higher leverage since we get more dispersion of outcomes based on one or two bad days eithier being absent or repeated in the sample), and then find the optimium leverage.

For the median this is indeed at roughly the 2.1 level that theory predicts (in fact we'd expect it to be a little lower because of the negative skew), but this is not true of all the lines. For inveterate gamblers at 90% it looks like the optimum is over 3, whilst for those who are more averse to bad outcomes at 10% and 25% it's less than 2; in fact at 10% it looks like the optimium could easily be 1 - no leverage. These translate to standard deviations targets of somewhere around 10% for the person with a 10% risk tolerance . 

Technical note: I can of course use corrections to the closed form Kelly criterion for non Gaussian returns, but this doesn't solve the problem of parameter estimation uncertainty - if anything it makes it worse.

The final step, and this is something you cannot do with a closed form solution, is to see how sensitive the shape of the line is to different levels of leverage, thus encouraging us to go for a more robust solution that is less likely to be problematic if the future isn't exactly like the past. Take a slightly conservative 25% quantile person on the red line in the figure. Their optimium could plausibly be at around 1.75 leverage if we had a smoother plot, but you can see that there is almost no loss in geometric mean from using less leverage than this. On the other hand there is a steep fall off in geometric mean once we get above 1.75 (this assymetry is a property of the geometric mean and leverage). This implies that the most robust and conservative solution would be to choose an optimal leverage which is a bit below 1.75. You don't get this kind of intuition with closed form solutions.

Optimal allocation - mean variance

Let's now take a step backwards to the first phase of solving this problem - coming up with the optimal set of weights summing to one. Because we assume we can use any amount of leverage, we want to optimise the Sharpe Ratio. This can be done in the vanilla mean-variance framework. The closed form solution for the data set we have, which assumes Gaussian returns and linear correlation, is a 22% weight in equities and 78% in bonds. That might seem imbalanced, but remember the different levels of risk. Accounting for this, the resulting risk weights are pretty much bang on 50% in each asset. 

As well as the problems we had with Kelly, we know that mean variance has a tendency to produce extreme and not robust outcomes, especially when correlations are high. If for example the correlation between bonds and equities was 0.65 rather than zero, then the optimal allocation would be 100% in bonds and nothing in equities.

(I actually use an optimiser rather than a single equation to calculate the result here, but in principal I could use an equation which would be trivial for two assets - see for example my ex colleague Tom's paper here - and not that hard for multiple assets eg see here).

So let's do the following; boostrap a set of return series with different allocations to equities (bond allocation just 100% - equity allocation), then measure the Sharpe Ratio of each allocation/bootstrapped return series, and then measure the distribution of those Sharpe Ratios for different distributional points.

Again, each of these coloured lines represents a different point on the distribution of Sharpe Ratios. The y-axis is the Sharpe Ratio, and the x-axis is the allocation to equities; zero in equities on the far left, and 100% on the far right. 
Same procedures as before: first work out your tolerance for uncertainty and hence which line you should be on. Secondly, find the allocation point which maximises Sharpe Ratio. Thirdly, examine the consequences of having a lower or higher allocation - basically how robust is your solution.
For example, for the median tolerance (green line) the best allocation comes in somewhere around 18%. That's a little less than the closed form solution; again this is because we haven't got normally distributed assets here. And there is a reasonably symettric shape to the gradient around this point, although that isn't true for lower risk tolerances.
You may be surprised to see that the maximum allocation is fairly invarient to uncertainty tolerance; if anything there seems to be a slightly lower allocation to equities the more optimistic one becomes (although we'd have to run a much more granular backtest plot to confirm this). Of course this wouldn't be the case if we were measuring arithmetic or even geometric return. But on the assumption of a seperable portfolio weighting problem, the most appropriate statistic is the Sharpe Ratio. 
This is good news for Old Skool CAPM enthusiasts! It really doesn't matter what your tolerance for uncertainty is, you should put about 18% of your cash weight - about 43% of your risk weight in equities; at least with the assumption that future returns have the forward looking expectations for means, standard deviations, and correlations I've specified above; and the historic higher moments and co-moments that we've seen for the last 40 years.

 

Joint allocation

Let's abandon the assumption that we can seperate our the problem, and instead jointly optimise the allocation and leverage. Once again the appropriate statistic will be the geometric return. We can't plot these on a single line graph, since we're optimising over two parameters (allocation to equities, and overall leverage), but what we can do is draw heatmaps; one for each point on the return distribution.
Here is the median:

The x-axis is the leverage; lowest on the left, highest on the right. The y-axis is the allocation to equities; 0% on the top, 100% on the bottom. And the heat colour on the z-axis shows the geometric return. Dark blue is very good. Dark red is very bad. The red circle shows the highest dark blue optimum point. It's 30% in equities with 4.5 times leverage: 5.8% geometric return.
But the next question we should be asking is about robustness. An awful lot of this plot is dark blue, so let's start by removing everything below 3% so we can see the optimal region more clearly:

You can now see that there is still quite a big area with a geometric return over 5%. It's also clear from the fact there is variation of colour within adjacent points that the bootstrapped samples are still producing enough randomness to make it unclear exactly where the optimium is; and this also means if we were to do some statistical testing we'd be unable to distinguish between the points that are whiteish or dark blue. 
In any case when we are unsure of the exact set of parameters to use, we should use a blend of them. There is a nice visual way of doing this. First of all, select the region you think the optimal parameters come from. In this case it would be the banana shaped region, with the bottom left tip of the banana somewhere around 2.5x leverage, 50% allocation to equities; and the top right tip around 6.5x leverage, 15% allocation. And then you want to choose a point which is safely within this shape, but further from steep 'drops' to much lower geometric returns which means in this case you'd be drawn to the top edge of the banana. This is analogous to avoiding the steep drop when you apply too much leverage in the 'optimal leverage' problem. 
I would argue that something around the 20% point in equities, leverage 3.0 is probably pretty good. This is pretty close to a 50% risk weight in equities, and the resulting expected standard deviation of 15.75% is a little under equities. In practice if you're going to use leverage you really should adjust your position size according to current risk, or you'd get badly burned if (when) bond vol or equity vol rises.
Let's look at another point on the distribution, just to get some intuition. Here is the 25% percentile point, again with lower returns taken out to better intuition:

The optimal here stands out quite clearly, and in fact it's the point I just chose as the one I'd use with the median! But clearly you can see that the centre of gravity of the 'banana' has moved up and left towards lower leverage and lower equity allocations, as you would expect. Following the process above we'd probably use something like a 20% equity allocation again, but probably with a lower leverage limit - perhaps 2.

Conclusion

Of course the point here isn't to advocate a specific blend of bonds and equities; the results here depend to some extent on the forward looking assumptions that I've made. But I do hope it has given you some insight into how bootstrapping can give us much more robust outcomes plus some great intuition about how uncertainty tolerance can be used as a replacement for the more abstract risk tolerance. 
Now go back to bed before your parents wake up!