Interpreting Longevity Risk Premiums Through Physical and Risk-Neutral Expectations
Summary
The document interprets a formula that extracts an annual longevity risk premium by comparing discounted risk-neutral and physical expectations of a longevity index across payment dates. The risk-neutral expectation is described as inferred from traded longevity-linked instruments, while the physical expectation represents the expected index under the real-world probability measure. The premium summarizes the difference between these two expectation profiles in the equation.
The answer connects this difference to insurers' and companies' demand to reduce exposure to people living longer than expected, which can strain future cash flows. It suggests that the premium may be positive when the risk-neutral expectation exceeds the physical expectation and links this to longevity risk being difficult to diversify fully. This is an interpretation of the equation, not a complete derivation of the pricing model or a general claim that the premium must be positive. It also does not establish that the parameter is identical to a market price of risk in every framework.
Key ideas
- Longevity risk arises when people live longer than expected, increasing future obligations.
- The formula compares risk-neutral and physical expectations of the longevity index, with payments discounted by bond prices.
- The implied premium measures the difference between those expectation profiles under the equation's setup.
- Demand from institutions seeking to hedge longevity exposure can contribute to the premium.
- The interpretation is framework-dependent and does not prove the premium is always positive or identical to a market price of risk.
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Full text
# Risk premium of insurance risk
# Risk premium of insurance risk
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?
## Answer by Kermittfrog (score 1)
https://quant.stackexchange.com/a/70334
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.