For the purposes of this blog, I’ve assumed that longevity risk is uncorrelated with investment risk, although whether that is a fair assumption is a very interesting question in itself.

The discussion is actually much broader than longevity risk – it applies to any uncorrelated risk. However, focusing on longevity risk makes the discussion a little more tangible.

### More bang for your buck…

Consider making the decision to invest in some growth assets in a pension scheme. The impact on the overall risk of the scheme depends on whether it is already exposed to longevity risk (i.e. longevity risk is unhedged). A simple example is shown in the table below:

*Source: LGIM calculations, October 2020. For illustrative purposes only.*

In the presence of longevity risk, the impact on overall risk from investing in growth assets is reduced (here halved from 4% to 2%), thanks to diversification, whereas the impact on expected return is unaffected^{[1]}. Growth assets’ “bang for your buck” therefore appears greater (here doubled) in the presence of longevity risk.

### …but reduced tolerance for further risk

However, it’s important not to jump to conclusions. By itself, such a calculation can be misleading because of another very important factor at play. This is that, as overall risk increases, investors generally become less tolerant of *further *increases in overall risk.** **

Under standard assumptions^{[2]}, unhappiness caused by risk goes with the square of the volatility, which is called variance. This means that the higher volatility is already, the more an incremental increase in volatility hurts. In our example, volatility increasing from 3% to 5% is a smaller change than moving from 0% to 4%, but we can check that the increase in variance is the same either way. Indeed, here the increases in variance are 5%^{2} – 3%^{2} and 4%^{2} – 0%^{2}, which both equal 0.16%.

The two effects (bang for buck but reduced tolerance for more risk) cancel each other out. The perhaps surprising conclusion is that longevity risk should not influence how much is held in growth assets. Statements such as “returns on growth assets act as a buffer against longevity risk” *sound* sensible, but are actually rather questionable when seen through this lens.** **

### Decumulation dilemmas

In the world of defined contribution (DC) and retail investing, longevity risk is much larger risk than in defined benefit (DB), so its potential influence is of much greater relevance and interest.

For a DC pensioner determined never to buy an annuity, the analysis above suggests their longevity risk shouldn’t impact their investment strategy^{[3]}.

For other DC pensioners, the question is when to buy an annuity. The key benefit of annuities is that they eliminate longevity risk, an unrewarded^{[4]} risk that increases with age. So why not remove this unrewarded risk as soon as retirement is reached? Due to capital constraints, purchasing an annuity means that the investor must also give up all rewarded sources of risk^{[5]}. This is because an annuity effectively combines a low-risk investment strategy with longevity insurance.

Nevertheless, we believe it still turns out to be quite hard to justify not purchasing an annuity, basically because the longevity risk faced by individuals is so big. Economists speak of the “annuity puzzle,” which is that far fewer people purchase an annuity than economic theory suggests ought to. I plan to explore this in future posts – watch this space!

*[1] For expected arithmetic return. When it comes to expected geometric return, matters are **a little more complicated** but the same idea applies.*

*[2] This is mean-variance optimisation, which corresponds to maximisation of quadratic utility.*

*[3] I have previously argued longevity risk promotes higher allocations to return-seeking assets on the grounds that if you die early the extra risk is unlikely to have been consequential, whereas if you live to 100+ the risk is very likely to pay off. There is merit to this argument but it implicitly relies on **time diversification**, which is an academically dubious concept, albeit something routinely adopted in practice perhaps due to **loss aversion**.*

*[4] Although in reality there will be some (generally relatively small) loadings for profit and expenses.*

*[5] This suggests an ideal product might combine rewarded risk taking with longevity insurance.*