### The basics

There are many ways to measure risk. My favourite is the expected daily standard deviation of your portfolio returns, and I usually look at the annualised version of this - multiply by 16. Until recently I was targeting 50% a year (or about 4% a day). I hasten to add that I only have a fraction of my net worth in my futures trading system - this would be an inappropriately high level if it was my entire asset base. I've now cut it to 25% a year, which is closer to the 10 - 20% that most systematic trend following firms target.

It's a well known result that to be consistent with the continuous Kelly criterion you should target the same standard deviation as you expect your Sharpe Ratio (SR) to be. I'll talk about what the right Sharpe Ratio might be in a moment.

So if you expect a SR of 0.5, you should run at 50% annualised risk. However that is a little bit rich for me, and for any sensible person. Consider the following graph:

Recovering the Kelly criterion from simulated data. Source: Author's research. |

Focusing on the blue line for the moment (apologies to the colour blind, it's the middle one once we get to the right) you can see it peaks at around 0.50, or 50%, which is Kelly optimal for a portfolio (trading system, or long only portfolio with one or more assets in it) with a true Sharpe Ratio of 0.5 as we have here. However suppose you don't know what your true Sharpe is, which is the normal state of affairs.

Suppose you think that your SR is 1.0, in which case you would be betting at a risk target of 100% annualised risk. As the picture shows if the true SR is really 0.5 you would on average lose money in the long run, and in many cases you'd lose a lot. A far safer bet is to run at 'Half-Kelly'. Expecting a SR of 1.0 you'd run at 50% risk. If you thought you'd get a SR of 0.50 as in the graph then 25% annualised risk is fine. This isn't optimal, your average annual return will be about a third less than 'Full Kelly', but it's better than risking too much and ending up on the right hand side of the peak.

That is the result for a normal asset with symmetric returns. The other two lines show you the results of different kinds of assets. The green line (bottom line on the right) is for a negative skew asset - like a trading system that sells volatility either directly or through running relative value type trades. The red line (top line on the right) is for a positive skew asset like trend following. As you can see the negative skew asset becomes toxic much quicker than the other two.

Overestimating the Sharpe ratio of a negative skew asset and Kelly betting accordingly is a one way ticket to bankruptcy. This is made worse by the fact that these strategies normally have quite low natural volatility, so to get up to the likes of 50 or 100% annualised risk they will need enormous leverage.

In contrast the positive skew asset is relatively benign at larger risk percentages. It's still better to run at the optimal Kelly, and safer to run at half Kelly, but running too much risk isn't quite as damaging.

### What is a reasonable Sharpe Ratio to expect?

All this is well and good but what sort of Sharpe should we expect? Most people would at this point just get some estimates of past returns and volatility, or if you run a trading system you fire up some back test software. Two reasons why you should take what comes of this with a pinch of salt.

Firstly asset returns in the future are unlikely to be as high as they were in the past. Take stocks. Even with the financial crisis over the last 40 years they have done pretty well. A good chunk of that comes from effects that won't be repeated (falling inflation) or could well reverse (rising proportion of GDP as corporate profits, rerating of earnings:price ratios). This also affects trading systems, since if assets have generally been going up then trend following for example will work better.

The second problem is most back tests are overfitted. Unless you've genuinely put in the first set of trading rules you thought of, not looked at the performance, not thrown anything away; and done a pure backward looking optimisation. Even if you do all of these things chances are you're still using trading rules that somebody else has come up, using past data or experience.

You can either apply a very sophisticated method, adjusting past asset class performance to take out secular effects and using statistical techniques to estimate the effect of overfitting, or just use a reasonable rule of thumb which is to cut the expected back test performance in half.

In long only world for a single average stock a SR of 0.2 is likely. For a diversified portfolio of equities you could get up to 0.3. Diversifying across asset classes might get you up to a SR of 0.5. Adding a trading system on top of these numbers could half again; with a mixture of styles you could probably double this.

For a very well diversified system like mine (45 futures markets over all major asset classes, 8 types of signal over three different styles) then backtested SR of 2.0 translate to an expectation of 1.0.

Unless you're in high frequency world, and benefiting from low latency technology or have market maker advantages, then I don't believe a SR above this is realistic.

So far I haven't justified why I cut my annual risk percentage by half, since if I was expecting a Sharpe of 1.0 then my 50% target was probably okay. So now we need to think about how wealth influences risk taking.

### Should wealth determine the amount of risk you take, and the kinds of investments you have?

Economic theory generally assumes

*. This would imply that wealth doesn't affect your desire for risk. A bricklayer who somehow managed to come a billionaire would maintain the same level of risk as a percentage of their portfolio. Financial theory also assumes that everyone should have the same portfolio of investments, with the highest possible Sharpe Ratio, and then leverage as required to get the risk they want.*

**constant relative risk aversion***I am not picking on bricklayers for any reason, except for the alliterative opportunities they offer here.*

In practise this doesn't seem to happen. For example under prospect theory the bricklayer would probably become more risk averse as they get richer, for fear of losing their new found gains. Secondly most people also aren't comfortable using leverage, except when buying residential property.

Imagine you're a 64 year old bricklayer, who will be retiring next week. You only have a state pension and no other investments, except £10,000 in cash. Economically you own an annuity (the pension) worth perhaps £180,000 plus the cash which is 5.3% of your net worth.

Is the best use of £10,000 to invest it in a Sharpe ratio 1.0 opportunity which will return 10%, or to buy lottery tickets? The latter is more likely and also makes more sense. £1,000 isn't going to make any difference at all (adding 0.53% to wealth, and if invested risk free about the same to income). But in the 2 million to one or so chance of a lottery jackpot and winning £10 million the bricklayer could be much better off.

Point one: people who don't / can't use leverage and need / want high returns will pay for risky investments - lottery tickets, growth story stocks, 100-1 horses - even if they have a negative expectation.

If the bricklayer could infinitely leverage up his £10,000 the lottery ticket would make no sense as it would be dominated by a leveraged form of the SR 1.0 investment. This could net him £10 million (a leverage factor so large I can't be bothered to work it out) with a positive expectation. But that would be well beyond half or even full Kelly. Betting at half Kelly - five times leverage - would still only expect to earn £5,000 again - not enough to make a huge difference (2.5% of wealth). It's more likely the builder will bet beyond half or even full Kelly, even if they don't go all the way to lottery like levels.

Point two: people who have a low level of financial wealth, which is dominated by other income, will often use too much leverage or go for riskier investments.

Now suppose you are a billionaire, with a billion quid, and 5.3% or £53 million spare. You could certainly afford to throw it away on lottery tickets, or buy a football team, both of which have negative expectation. However it's much more likely that you will put it into the SR 1.0 investment - that after all is how you became rich, not by making stupid financial decisions but by making good ones.

*Or maybe you inherited the money, in which case good decision to be born to the right parents. Go you!*

I also think it's much more likely that you will be very cautious, investing at most half-Kelly, and probably not even leveraging at all. You don't need the extra income, so preserving your wealth is more important than taking additional risk to get it. This is why rich people like investments with consistent returns. Prospect theory tells us that fear of losing new found wealth makes people more risk averse than if they are trying to recover gains.

This also opens things up for the billionaires. They can invest in high SR, but low return, investments that other people would spurn.

This effect applies to all levels of wealth. Now hopefully you understand why after a good run on the futures markets I wanted to lower the risk of my portfolio, by scaling back on my leveraged derivative exposure and putting the money into relatively low risk bonds.

Point three: As people get more wealth they become risk averse, able to invest in low risk but high SR investments, and they use less leverage.

### Let's get a bit more sophisticated...

Apart from risk preferences can we say anything else about preferences for different wealth levels. I was inspired to write this post by the following which also generated some discussion with my ex colleague Matt. The paper argues that wealthier investors are more likely to be 'value' investors, whereas others are 'momentum' investors.

By cutting my exposure to momentum (which I did before reading about the Lettau et al paper) I have definitely followed this track, at least to a degree.

The authors postulate that investors with different wealth levels are hedging different risk exposures that they already have.

"Thus
shareholders in
the bottom 90% of the wealth distribution may seek to hedge risks
associated with an increase
in the capital share by chasing returns and sticking to stocks whose
prices have appreciated
most recently. On the other hand, those in the top 10%, such as
corporate executives
whose fortunes are highly correlated with recent stock market gains,
may have compensation
structures that are already momentum-like. These shareholders may
seek to hedge
their compensation structures by undertaking contrarian investment
strategies that go long
in stocks whose prices are low or recently depreciated."

There may be other reasons. It's possible we can wrap this up with what we already know from above. Pure value strategies are relative value, exactly the kind of high SR, naturally low risk strategy that rich people like. Momentum strategies tend to be have higher natural risk, due to low futures margin and the positive skew that means you can safely run higher risk targets.

Another explanation relates to liquidity. Billionaires are more likely to be owners of 'patient capital', money that can be tied up for years or decades in family trusts. Value strategies - buying stuff that's cheap - particularly illiquid stuff like private equity or land - do better if they don't have to suddenly liquidate after losses due to redemption's by impatient investors. Again momentum strategies tend to be in more liquid futures which for the common or garden retired investor who relies on regular returns for income is a good thing.

### Concluding thought

Although the story in the paper is an interesting one, and might have some truth to it, ultimately having a good mix of investment styles is undoubtedly better than favouring one or another, and will give you a higher Sharpe Ratio overall. So although getting a little bit richer might be a good excuse for reducing your risk appetite and leverage, it doesn't justify trusting all your money to one investing style.

Great article, Rob. Thanks for the long-winded reply to my (relatively speaking) off-the-cuff remark!

ReplyDeleteThe Kelly criterion study is pretty fascinating here. So suppose you have a slightly negatively-skewed investment strategy (say, a fund of hedge funds) which annualises a pretty solid SR of 1.5, but runs about 5% annualised vol; Kelly says go all-in or more? For very cash-rich investors, maybe the SR is the main thing; but for retail investors like myself, I can't get cheap enough leverage to turn that into a palatable strategy (forgive me for wanting an expected return > 7.5%).

In the end, I really appreciate the cheap/available leverage on things like options and futures to provide the necessary mix of SR and bet size to improve the portfolio.

Yes Kelly says you should run that baby at 30 times leverage (optimal vol 150%). This is clearly nuts since a 3.3% drop would wipe you out. At 150% vol you'd expect to lose six times this - 18% - about once a month!

ReplyDeleteWhat would you be comfortable leveraging to, even if someone would let you? Double leverage? That's still a 15% return. Not bad.

The existence of economically high SR, low vol, opportunities is probably one of the best 'free lunches' finance. If you have low risk appetite / low return appetite you're always going to do better than other people, at least on a SR basis.

Thx for the article Rob, it's very insightful. May I ask how did you produce the data to generate the graph? I'd like to try it out myself. Cheers

ReplyDeleteThis comment has been removed by the author.

DeletePseudo code:

DeleteSharpe ratio=.5

N=10

X=1000

Loop over skews [-2, 0, 1]: {

Loop over risk targets [0.05,0.1,0.15, .... 1.60] {

Loop X times {

Generate N years of random daily returns from distribution with skew, risk target, sharpe ratio

Cumulate returns and log of final wealth

}

Take average log wealth of all X log final wealth values

}

Plot risk targets on x axis, average log wealth on y axis; for skew

This comment has been removed by the author.

ReplyDeleteI would also really like to do the above but need a little more guidance! I read Kurt's article you mentioned elsewhere about the Norm.Inv function in excel to get the normal distributions. Can you enlighten me on how you would get the distributions with different skew, risk targets and sharpe ratios .. many thanks

ReplyDeleteChris

Sorry since I don't use excel I can't help you.

DeleteHi Rob,

ReplyDeleteYour book is very insightful, each time I read it it become more and more intuitive. Just one thing I was work out how you calculated table 20 in chapter 9 that provides the expected loss of a 200% volatility target with 100,000 trading capital and a sharpe ratio of 0.5

this is my understanding from you book, most likely I have totally the wrong end of the stick!

ReplyDeleteWith a shape ratio of 0.5, my average return is half the standard deviation, so with an annual volatility target of 200K (16x Daily target), my expected average return is 100K. Assuming gaussian normal, then 68% of the time my returns will be in the range of -100K and 300K i.e average (100K) +/- standard deviation (200K). So for losses, (1-0.68)/2, 16% of the time will be greater than -100K per annum. For daily, the deviation is 200K/16 =12500, giving an average return of 6250. So for 16% of days I will have a loss that is greater than -6250, which is around once every 41 days a year, or around once every 6 days the loss will be greater than -6250 with a volatility target of 200K and a sharpe ratio of 0.5. For 2.5% of days the loss will be (6250 - (2x 12500)) greater than -18,750 or around once every 2 months.

Your understanding is correct.

DeleteHi Rob,

ReplyDeleteI am bit confused by this article and also in the book cap 9 page 146 "a realistic SR of 0.75.With full kelly betting that would be a 75% volatility target"

as far as I understand it Kelly for continuous time is SR/vol..not SR only...

lets say a portfolio has a SR a 2, that doesn't tell us what the volatility is ..it can be with returns of 2 and vol of 1 or in another case returns of 1 and vol of ret of 0.5..

in the first case kelly tells you to leverage 200% and in the 2nd case 400 %

I guess I don't understand you correctly you seem to be saying SR is equal do Kelly?

Kelly is equal to SR to find optimal vol target (but to scale positions you divide by vol again perhaps your confusion comes from this...?).

DeleteSo if you have a portfolio with SR of 0.75 then it doesn't matter what the 'natural' vol is, the optimal full kelly would 75% annualised vol. If it has a 'natural' vol of 150% then you should run it an absolute maximum of half its 'natural' leverage [and of course ideally much less].

Hi Rob,

ReplyDeletemany thanks for your quick reply.

I guess the volatility target is working like a VaR with 84% Confidence? 16% of the time the vol will be larger (got this reading Douggie's post above) and the vol will get even bigger if the dist of returns has kurtosis and a negative skew. Maybe it would be safer to do monte carlos on the backtest trades and see the lowest quantiles

You're not exactly right. 16% of the time the RETURNS will show a loss of greater than one vol (assuming Gaussian).

DeleteBut 50% of the time the [expected] vol will be below the vol target, and 50% it will be above the vol target.

See the difference?

Not sure why you need to look at individual trades since normally you'd have several on and should look at the entire portofolio (doing individual trades means you will throw away corelation data that might make your VAR estimate too aggressive or too lackluster depending on the nature of your system).

But sure if you want to monte carlo the returns of the whole system to get a better idea of it's properties thats fine.

many thanks Rob! you raised good points, I didn't think about loosing the correlation info, maybe what I can do is to look at the daily volatility of the portfolio and then do monte carlos on that and then see the VaR at 95%...I think I would feel safer with that. Also correlation is a bit tricky, it has the nasty habit of changing all the time and when you really need it , in times of crisis, it is then when it helps you the least,everything gets highly correlated, as know, of course

ReplyDeleteHi Rob,

ReplyDeleteyou state in the article and the book too, that most of the gains in the stock market in the past came from falling inflation.

Can you please elaborate on the mechanics of the correlation between falling inflation and higher stock markets.

As inflation falls interest rates fall, leading to higher equilibrium PE ratios. Although future nominal earnings growth is lower this doesn't seem to affect valuations.

Delete