*Guaranteed Minimum Accumulation Benefit*

This section is dedicated to deriving the fund dynamics and the SCR of a frequently used guarantee in the insurance industry, namely the Guaranteed Minimum Accumulation Benefit (GMAB) variable annuity rider.

We assume that the fund only contains stocks, but the derivation is similar when different assets are combined. We denote the stock and fund value by *St* and *Ft*, respectively, and define the initial premium by *G*. We assume the payou<sup>t</sup> at maturity *T* is at least equal to the initial premium, in other words,

$$\text{Payout}\_T = \max\left(F\_{T'}, G\right). \tag{5}$$

The dynamics of the fund are very similar to the stock dynamics, except for a fee *α* which is deducted from the fund as a paymen<sup>t</sup> to the insurer. This fee can be thought of as a dividend yield. Following Milevsky and Salisbury (2001), we obtain

$$\mathbf{d}F\_t = \mathbf{d}S\_t \frac{F\_t}{S\_t} e^{-\mathbf{a}t}, \qquad F\_0 = G. \tag{6}$$

The specific dynamics depend on the assumptions regarding stock price *St*. Here, we assume the Black–Scholes model (geometric Brownian motion, GBM) under the *observed real-world measure* P, which yields:

$$\mathbf{d}\mathbf{F}\_t^{\mathbb{P}} = (\mu - \mathfrak{a})F\_t^{\mathbb{P}}\mathbf{d}t + \sigma F\_t^{\mathbb{P}}\mathbf{d}\mathcal{W}\_t^{\mathbb{P}}, \quad F\_0 = \mathbf{G}. \tag{7}$$

Moreover, for the valuation, a risk-neutral Heston model is implemented, leading to

$$\begin{cases} \mathbf{d}F\_{l} = (r - \kappa)F\_{l}\mathbf{d}t + \sqrt{\upsilon\_{l}}F\_{l}\mathbf{d}\mathcal{W}\_{l}^{F}, \quad F\_{0} = \mathbf{G},\\ \mathbf{d}v\_{t} = \kappa(\overline{v} - v\_{l})\mathbf{d}t + \gamma\sqrt{\upsilon\_{l}}\mathbf{d}\mathcal{W}\_{l}^{v},\\ \langle \mathbf{d}\mathcal{W}\_{l}^{S}, \mathbf{d}\mathcal{W}\_{l}^{v} \rangle = \rho\mathbf{d}t. \end{cases} \tag{8}$$

The income is generated by the accumulated fees, hence

$$A\_t = \int\_0^t \mathfrak{a} F\_s^{\mathbb{P}} e^{\mu(t-s)} \, \mathrm{d}s. \tag{9}$$

The liabilities, on the other hand, depend on the final value of the fund, as follows,


Moreover, the insurer continues to claim future fees, hence, according to Equation (2), we can write the liabilities as

$$L\_t = \mathbb{E}^{\mathbb{Q}\_t} \left[ e^{-r(T-t)} \max\left( G - F\_T, 0 \right) - \int\_t^T e^{-r(s-t)} a F\_s \text{ds} \bigg| \mathcal{F}\_t \right]$$

$$= \text{Put}(F\_t, G) + F\_t \left( e^{-a(T-t)} - 1 \right), \tag{10}$$

where Put(*Ft*, *G*) denotes the value of a European put option on the fund at time *t* with strike price *G* and dividend yield *α*. We can substitute these definitions into Equation (4) to obtain

$$\begin{aligned} \text{SCR} &=& \text{VaR}\_{0.995} \left( N\_0 - e^{-r} N\_1 \right) \\ &=& \text{VaR}\_{0.995} \left( A\_0 - L\_0 - e^{-r} (A\_1 - L\_1) \right) \\ &=& e^{-r} \text{VaR}\_{0.995} \left( \text{Put} (F\_1, G) - \text{g} (F\_1, 1) \right) - \text{Put} (F\_0, G) + \text{g} (F\_0, 0), \end{aligned} \tag{11}$$

with

$$\log(F\_t, t) = \int\_0^t a F\_s e^{\mu(t-s)} \, \text{ds} + F\_t \left(1 - e^{-a(T-t)}\right),\tag{12}$$

which can be thought of as the sum of the realized and expected fees. In this case, the SCR depends on the real-world distribution of *F*1, which determines *g*(*<sup>F</sup>*1, 1) and also influences the risk-neutral valuation of Put(*<sup>F</sup>*1, *<sup>G</sup>*).

Variable annuity riders require a risk-neutral valuation at (future) time *t* = 1, as the liabilities are, by definition, conditional expectations under the risk-neutral measure. In the case of a GMAB rider, there is an analytic expression available, but often there is no such expression for *Lt*. Hence, the evaluation of the conditional expectation typically requires an approximation. In that case, often the Least-Squares Monte Carlo algorithm is used to approximate the conditional expectations at *t* = 1. The experiments presented in this paper were also performed for the Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuity rider. As the results and conclusions were very similar as for the GMAB, in this paper, we restrict ourselves to the GMAB variable annuity.

The distributions of *g*(*<sup>F</sup>*1, 1) and Put(*<sup>F</sup>*1, *G*) in Equation (12) can be obtained by means of a *Monte Carlo simulation*,


Some more detail about the Monte Carlo simulation is provided in Appendix A.

### **3. Dynamic Stochastic Volatility Model**

When valuing options, one typically wishes to calibrate a risk-neutral model according to the market's expectations, which are quantified by the implied volatility surface. This implied volatility surface can be used to extract European option prices for a wide range of maturities and strikes. The market expectation (and so the volatility surface), is however unknown at *t* = 1 and therefore practitioners typically use the implied volatility surface at *t* = 0 to calibrate the risk-neutral model parameters. In this *standard* Q *in* P *approach*, these parameters are assumed to be constant over time, i.e., the risk-neutral measure is independent of the real-world measure. Note that this approach is also used to compute risk measures from the Basel accords, such as credit value adjustment (CVA), capital valuation adjustment (KVA) and potential future exposure (PFE) (see Kenyon et al. (2015); Jain et al. (2016); Ruiz (2014)). However, regarding the computation of CVA, this quantity is typically hedged, and so using only the market expectation at *t* = 0 is sufficient.

In the P Q approach, we relax the assumption of independence. The calibrated risk-neutral model parameters are related to the simulated real-world scenarios.
