# 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 $$$$\sum_{t=1}^{T}B(0, t)\Big\{\mathbb{E}^{\mathbb{Q}}[I(t)|\mathcal{F}_t] - \exp(\delta t)\mathbb{E}^{\mathbb{P}}[I(t)|\mathcal{F}_t]\Big\}= 0$$$$

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?

• 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? Commented Mar 29, 2022 at 10:54
• I think the formulation $\delta \times t$ is chosen so that the premium scales with time, IMO this is the modeller's choice. Commented Mar 29, 2022 at 11:18