**2. Solvency Capital Requirement**

Let us denote the policy's net income in the interval [0, *t*] by *At*, which is defined as the cash flows up to time *t* that are generated under the real-world market. The mathematical definition is given by

$$A\_t = \int\_0^t e^{\mu(t-s)} \text{cashflow}(s, \Delta s) \text{ds}\_\prime \tag{1}$$

where *μ* is the expected return and "cashflow(*<sup>s</sup>*, <sup>Δ</sup>*s*)" denotes the generated cash flow over the interval [*<sup>s</sup>*,*<sup>s</sup>* + <sup>Δ</sup>*s*]. Similarly, we define the policy's liabilities by *Lt* and they are given by the discounted expected cash flows under the risk-neutral measure in the interval [*t*, *<sup>T</sup>*]:

$$L\_t = \mathbb{E}^{\mathbb{Q}\_t} \left[ \int\_t^T e^{-r(s-t)} \text{cashflow}(s, \Delta s) \text{ds} \, \middle| \, \mathcal{F}\_t \right], \tag{2}$$

with risk-free rate *r*. Note that a positive/negative cash flow corresponds to an income/liability for the insurer. We define

$$N\_t := A\_t - L\_t \tag{3}$$

which can be thought of as the policy's net value at time *t*. The Solvency Capital Requirement is now defined as the 99.5% Value-at-Risk (i.e., the 99.5% quantile) of the one-year loss distribution under the real-world measure, i.e.,

$$\text{SCR} = \text{VaR}\_{0.995} \left( N\_0 - \bar{N}\_1 \right) := \inf \left\{ \mathbf{x} \left| \mathbb{P} \left( N\_0 - \bar{N}\_1 < \mathbf{x} \right) > 0.995 \right\} . \tag{4}$$

Here, *N* ˜ 1 is defined as the discounted value of *N*1.
