# What is risk (II)?

It is a truism of investing that achieving higher returns requires taking more risk. On average those who take risk are rewarded for doing so – but ‘on average’ provides no guarantees and there remains the possibility that the outcome will be worse than expected. Determining whether the average expected reward is sufficient to compensate for the risks taken requires an understanding of precisely what is meant by risk.

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Following on from last months’ Compass where I explained why risk is not the same as volatility, in this edition I describe our Behavioural Risk measure in more detail.

To recap: standard deviation is a poor measure of risk. To see why, consider Figure 1. This shows how investors would evaluate the risk of possible future returns if they really did interpret risk as standard deviation. As shown, there would be no contribution to perceived risk associated with the expected outcome (here 0% to keep things simple).

Any possible outcomes less than this, however, add to the perceived risk of the investment. The worse the outcome, the more it adds to the perceived risk – for example, the possibility of getting -1% instead of 0% increases the risk by a small amount; -5% by somewhat more; and the thought of -10% causes the investor considerable stress.

**Figure 1: Implied psychology of risk as volatility**

However, note that the curve is symmetric around 0%. This means that the possibility of getting plus 5% instead of 0% actually adds to the risk of the investment – in fact it adds the same amount to perceived risk as getting -5%. And as returns get better and better they add more and more to the measured risk of the investment! Clearly, assuming that good outcomes increase the risk to investors is unrealistic … and yet this assumption still underpins the majority of optimisation techniques used in the industry. This definition of risk means that any technique used to minimise risk will penalise potential good outcomes as much as it will potential negative outcomes. Investors’ risk budgets will be used up protecting against potential outcomes that they just do not see as risky.

## Reflecting risks that matter

Our behavioural risk measure is, by contrast, a much more psychologically accurate and intuitive response to risk. It is based on psychologically plausible assumptions from research into the psychology of risk and financial decision making. Figure 2 shows the intuition behind this measure without going into the mathematical formulation (for which see our Asset Allocation White Paper). As before, potential outcomes that are worse than expected, add to risk, and at an increasing rate. However, outcomes that are better than expected actually detract from the perceived risk of the investment. Investments with potential upside thus increase the risk budget so real risks can be taken elsewhere in the portfolio.

**Figure 2: Actual psychology of risk**

Another difference between the psychological approach to risk and the traditional volatility measure is more subtle but just as important: for most of the downside the two curves are fairly close to each other. However, the psychological measure gets steeper at a faster rate as outcomes get worse and worse. This means that extra emphasis is placed on those possible outcomes that people most fear, the potential for catastrophic losses in the left tail of the returns distribution. Using this measure means that portfolios are optimised to reflect the risks that are most important to investors, but also to take on the upside variation that they would rather embrace than avoid.

## Accounting for fat tails

This last aspect of the behavioural risk measure is extremely important, since it enables us to overcome a further significant failing of standard deviation as a risk measure: it cannot fully measure the risks of distributions of different shapes. In particular, the distributions of the possible future outcomes of many investments, tend to have features that cannot be captured by standard deviation. Firstly they are typically not symmetric: instead they are usually somewhat negatively skewed, meaning that large negative outcomes happen slightly more frequently than large positive outcomes. And secondly they exhibit fat tails: extreme outcomes, whether negative or positive happen more often than is predicted by using standard deviation as the risk measure.1

**Figure 3: Standard deviation can’t detect fat tails**

Because the behavioural risk measure is deliberately asymmetric and downward focussed, and because it places greater attention on the far left tail, it accounts for both negative skewness and fat tails. That is, a distribution of possible future outcomes that has these common features will be measured as being more risky on our behavioural measure, whereas standard deviation will not pick up these features at all.

To illustrate this briefly, examine the two simple distributions in Figure 3. The one on the left is designed so that all the variability is picked up by standard deviation: it is perfectly symmetric around 0% and does not have fat tails.2 It has just five possible future outcomes: the most likely is the average (expected) outcome of 0%, which has a 46% probability of occurring; the outcomes of -5% and 5% are both equally likely, each with a 22% chance of occurring; and least likely (each at 5% chance) are the two extreme outcomes of -10% and 10%. This distribution has a standard deviation of 4.6%.

The behavioural risk of this distribution is also 4.6%, since there are no fat tails or skewness to compensate for. If the distribution of outcomes follows the theoretical ideal of normal then behavioural risk is identical to standard deviation. However, is reality this is seldom the case.

Let us examine a simple variation on this distribution, one that introduces fat tails.3 This distribution has the same mean, or expected outcome, of 0%. However, the good and bad outcomes are shifted. Instead of getting either -5% or 5%, these two are now closer to the average: there is the same 22% chance for each, but now on -1.3% and 1.3%. And the extreme outcomes, though occurring with the same probability, are now much more extreme: there is a 5% chance of getting either 14.3% or -14.3%. This distribution has very fat tails, and because investors naturally focus on the extreme negative outcome when evaluating risk, we should very naturally expect that this second distribution is riskier than the first.

However, these two distributions have the same standard deviation! Any traditional portfolio optimisation based on standard deviation as a risk measure would treat them exactly the same, as though they have the same risk for the investor.

Our behavioural risk measure, by contrast recognises that the danger of -14.3%, rather than -10% matters more to investors than the other changes in the distribution, and therefore appropriately indicates an increase in risk. Indeed doing the calculation for a low risk tolerance investor shows that the risk increases to 4.6% from 4.8%. This risk increase means that the optimisation process will penalise the fat tail investment more, and reduce the optimal holding, making more space in the portfolio for investments with less tail risk.

To build portfolios that give investors the best possible returns for the risk they need to take, it makes no sense to strive to suppress outcomes that aren’t actually risky (good/positive outcomes), or to ignore outcomes that really matter to investors (the extreme negative outcomes). Our use of behavioural risk enables us to ensure that our asset allocations fully reflect the risks that count, and therefore deliver greater efficiency in our solutions.

^{1} We have only to think of the absurd claims that September 2008 was a ‘one in three thousand year event’ to see the failure of standard risk measures to account for the likelihood of extreme movements, and extreme downward movements in particular.

^{2} It follows what is known as the normal distribution, though as noted it’s actually rather uncommon in reality!

^{3} For simplicity we’ll ignore skewness here, but behavioural risk accounts for differences in skewness just as easily as differences in extreme outcomes.