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I recently came across an equation in a paper.

In short, suppose that $I(t)$ denotes a longevity index at time $t$. An informative indicator that is useful in the absence of any information about the market prices, would be the risk premium per annum $\delta$ derived by the following equation \begin{equation} \sum_{t=1}^{T}B(0, t)\Big\{\mathbb{E}^{\mathbb{Q}}[I(t)|\mathcal{F}_t] - \exp(\delta t)\mathbb{E}^{\mathbb{P}}[I(t)|\mathcal{F}_t]\Big\}= 0 \end{equation}

Where $B(0, t)$ is the zero-coupon bond price and $t=1, 2, \cdots, T$ is the period of payments. My question is why the above equation is used to extract the risk premium associated with the mortality risk? What does it mean exactly "risk premium of mortality risk" from an economics/finance perspective? It can be interpreted as the market price of risk?

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Longevity risk is the risk that pensioners (or policy holders) live longer than expected, which can lead to a considerable stress to (future) cash flows of a company.

The formula you present seems to imply that the risk-neutral expected value of the longevity index is (on average) higher than the physical ("true") expected future index level. Risk-neutral in this respect means that the expectation is estimated from (traded) instruments written on longevity, e.g. longevity insurance contracts.

The risk premium is a measure of the distance between the two expectations. If it is positive (the usual case), there is - on average - a premium added on top of the physical longevity expectation. The premium is driven by demands from companies / insurers who want to limit their exposure to longevity risk (see above), and the risk itself is not diversifiable beyond some point.

HTH?

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  • $\begingroup$ Thank you for your reply. If we look at the equation, it sounds like we are considering the expectation of payoff discounted by a risk-free rate under a risk-neutral measure, while under the physical measure we multiply the expectation of payoff by $\exp\{\delta t\}$. I mean for the computation of risk premium, do we always use such an equation? What if we consider a catastrophe risk instead of a mortality index risk? Then, is it possible to write a similar equation for the catastrophe loss index? $\endgroup$
    – user53249
    Mar 29, 2022 at 10:54
  • $\begingroup$ I think the formulation $\delta \times t$ is chosen so that the premium scales with time, IMO this is the modeller's choice. $\endgroup$ Mar 29, 2022 at 11:18

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