**Disclaimer: **Views in this blog do not promote, and are not directly connected to any Legal & General Investment Management (LGIM) product or service. Views are from a range of LGIM investment professionals and do not necessarily reflect the views of LGIM. For investment professionals only.

Traditional models suggest that there is a very high chance equities outperform over the long term. But are they overconfident, and should long-term investors adjust their asset allocation accordingly? Let's consider this question from one particular technical angle.

After an uncommonly long period of market tranquillity, volatility was back with a bang at the start of 2018.

And yet despite the breathless coverage of the price swings, many investors would agree that equities are the least risky asset in the long run, at least in terms of their higher chance of outperformance. But how long is ‘long’, how large really are the risks and (what I commonly ask myself) are there any models that can help shed light on these questions?

One quantitative demonstration of the idea that ‘time conquers risk’ is to assume that equities generate a certain mean return above cash each year, or risk premium, and experience a certain degree of volatility. We then calculate the probability that they beat cash over different time horizons.

Most investors would agree that equities are likely to outperform in the long run, but how long is ‘long’?

Assuming an arithmetic (mean) risk premium of 4.5% per annum and volatility of 15% per annum, the probability that equities *underperform* cash over 40 years is around 8%. This is an unlikely, if far from negligible, possibility.

However, this calculation is potentially misleading. Take the equity risk premium (ERP) assumption of 4.5%, which is particularly important over the long term. The chart below summarises a variety of research papers’ best estimates of the ERP:

Clearly there is a wide range of estimates. Even these, however, do not reflect the degree of uncertainty held by each of the papers’ authors regarding their own estimates, or the risk that some authors are likely to see the world in a similar, but incorrect, way.

The table below shows the result of repeating the calculation conducted above, with a range of different ERP estimates, whose average is the 4.5% cited above:

Arithmetic ERP | 0.75% | 2.00% | 3.25% | 4.50% | 5.75% | 7.00% | 8.25% |

Chance developed market equities underperform cash over 40 years | 54.8% | 33.5% | 16.7% | 6.7% | 2.1% | 0.5% | 0.1% |

*Source: LGIM calculations*

If we, purely for illustrative purposes, assume that these ERP possibilities are equally likely the probability of equities underperforming cash over 40 years rises from 8% to around 17%.

By allowing for the uncertainty around our estimation of the ERP, the risk of underperforming cash has more than doubled. Even if we exclude the extreme ERP possibilities in red, there is a 13% chance equities underperform cash over 40 years – still significantly higher than 8%.

In general we would model uncertainty in return estimates and other parameters in a more sophisticated way than this but the above demonstrates the idea. The increase in risk reflects the fact that the sensitivities are asymmetric: overestimating the ERP generally leads to a smaller drop in the chance of equity underperformance than the increase in this probability by underestimating it to the same degree.

This uncertainty around expected returns is not specific to only the ERP. Investors allocating to different asset classes may rely on estimates for expected returns, volatilities or correlations. Some of these metrics, like asset-class volatilities, can be estimated relatively reliably over the long term, and relatively few people would argue that there is no ERP. But returns are notoriously difficult to estimate despite their importance to long-term outcomes.

Paying attention to the uncertainty surrounding the return estimates (and other parameters) used promotes a healthy degree of diversification even if one is investing over the course of decades.

*For a closer look at the impact of assumption uncertainty on asset allocation or funding plans we recommend reading our more in-depth article.*

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