Expected returns: boosting our binoculars

In a blog earlier this year, Martin Dietz and I outlined our framework for setting long-term strategic expected returns.

In the meantime, we’ve been making some modest refinements. We’re always looking to ensure we have the best possible approach. The focus of this blog is an enhancement to how we appropriately compensate for the shape of returns.

Bringing moments into focus

As described in our previous blog, a possible starting point for excess returns on asset classes is the Capital Asset Pricing Model (CAPM), an equilibrium model. This model indicates the relative undiversifiable (or systematic) risk of different asset classes. It assumes expected excess returns are proportional to these risks. 

CAPM makes various simplifying assumptions. A key assumption is that investors:

• only care about the covariance of returns, relating to undiversifiable volatility, when it comes to risk and/or that 
• returns are normally distributed.

In reality, returns are not normally distributed, and investors generally should care about aspects of risk beyond volatility. These other aspects are called higher ‘moments’ that relate to aspects of the shape of returns. The question is, how should our strategic return estimates reflect them?

The next moment up from variance (the square of volatility) is called skewness, which is a measure of asymmetry. Positive skewness of a return distribution indicates that an investor may expect frequent small losses and a few large gains from the investment. In contrast, negative skewness indicates frequent small gains punctuated by a few large losses, akin to picking up pennies in front of a steamroller. Under some very reasonable assumptions you can show that investors prefer positive skew (all else equal), and so ought to be compensated by a higher expected return if there is more negative skew.  

The moment after skew is called kurtosis and relates to fat tails. Fatter tails are a bad thing, all else equal, and so investors ought to be compensated by a higher expected return if there is more positive kurtosis. In fact, there’s an alternating pattern in terms of preferences:

All the moments greater than or equal to two correspond to different aspects of risk. The non-diversifiable aspects of these risks are called covariance, coskewness, cokurtosis, cohypertailedness and so on. In a rational world, the expected return should compensate for more positive values of the even co-moments and more negative values of the odd co-moments. CAPM only allows for covariance, yet really we should allow for all of them. But how?

Adjusting for precision

As a starting point, we note that CAPM assumes investors have ‘quadratic utility’ – only caring about the first two moments. To allow for all the higher moments, we need a more sophisticated utility function.

The utility function we opted to replace it is called ‘power utility’ (otherwise known as isoelastic utility) and has several nice properties. Its defining feature is that it has something called constant relative risk aversion (CRRA). This means appetite for risk doesn’t depend on the level of projected wealth. This is desirable in sense that we don’t generally want strategic risk premia to change if interest rates happen to change, or there is a run of good or bad investment performance. CRRA makes sense from some other perspectives too[1]. 

Zooming in on impact

There are other choices we need to make before we can compute expected returns. One of these is the time horizon[2] over which risk is measured. We opted to use one year, for various reasons[3]. 

I’m keen to stress that this enhanced process for capturing systematic risk remains just the starting point for setting expected returns. As discussed in our previous blog, there are good reasons to deviate from such an approach, such as allowances for illiquidity, or leverage constraints impacting investor behaviour. Investors may not behave entirely rationally either.

However, the chart below shows some examples[4] of the impact from moving from quadratic to power utility before any overrides are applied:

As you can see, the changes tend to be modest, especially if they are considered in the context of general assumption uncertainty. 

At the same time, asset allocation optimisations can be notoriously sensitive to return estimates, which can justify a more nuanced approach. As any binocular user knows, a gentle twist of the focus wheel can unveil hidden detail.

 
[1] For example, it avoids the ‘split period paradox’ described here.
[2] Higher moments can decrease in importance over longer horizons.
[3] We are not using prospect theory but Benartzi and Thaler’s 1995 paper suggests the size of the ERP is consistent with investors evaluating their portfolios annually. 
[4] You may notice the absence of IG corporate credit. This is because it is an example of an asset class where we override expectations based on estimated spreads and expected losses.