*5.2. SCR Impact Study*

In this test, we assess the impact of assuming time-dependent risk-neutral parameters, according to three scenarios. Each scenario corresponds to a different initial market and consequently the initial expected liabilities, *L*0, will differ. We assume a no-arbitrage fee, i.e., a fee for which *L*0 is equal to zero. The general contract details of the variable annuity can be found in Table 3 and the initial values of the different scenarios accompanied with the fair premiums are presented in Table 4.


**Table 3.** General contract details of the GMAB rider.



We assume the fund value to follow a GBM with initial value *F*0 = 1000, so the fund value follows Equation (7). The VIX index is modelled simultaneously, following a mean-reverting path

$$\begin{cases} \text{ dvxis}\_{t} &= \kappa\_{\text{vix}} (\text{vix}\_{\text{Mean}} - \text{vix}\_{t}) + \gamma\_{\text{vix}} \text{vix}\_{t}^{\lambda\_{\text{vix}}} \left( \rho\_{\text{vix}} \text{d} \mathcal{W}\_{t}^{\mathbb{S}} + \sqrt{1 - \rho\_{\text{vix}}^{2}} \text{d} \mathcal{W}\_{t}^{\text{vix}} \right), \\ \text{ VIX} &= 100 \cdot \text{vix}. \end{cases} \tag{41}$$

The process parameters are estimated with the generalized Method of Moments (see Hansen (1982)),

$$\begin{cases} \text{ } \sigma = 0.21, \quad \mu = 0.05, \quad \text{ } \kappa\_{\text{vix}} = 4.964, \quad \text{ } \text{vix}\_{\text{Mean}} = 0.207, \\\ \gamma\_{\text{vix}} = 1.859, \quad \lambda\_{\text{vix}} = 1.271. \end{cases} \tag{42}$$

Moreover, we set *ρ*vix = −0.75, which is in line with observations in the risk-neutral market. The SDEs are discretized by the Milstein scheme.

### 5.2.1. Guaranteed Minimum Accumulation Benefit

We have simulated 100,000 real-world trajectories and evaluated the loss function as defined in Equation (4) under the two risk-neutral measures (with either constant or time-dependent parameters). This way, we are able to construct and compare the probability density functions under these measures. This process is repeated for the three different scenarios that are presented in Table 3. In Figure 4, we have graphically represented the impact on the probability density function of the loss distribution for the different scenarios. Moreover, the Solvency Capital Requirements associated with these distributions can be found in Table 5.

**Table 5.** Solvency Capital Requirement of the scenarios for the original and time-dependent risk-neutral measure.


**Figure 4.** Probability density functions of the one-year loss distribution for a variable annuity with the GMAB rider, under the original and time-dependent risk-neutral measure. Scenario 1 = average initial volatility; Scenario 2 = low initial volatility; and Scenario 3 = high initial volatility.

The impact on the probability density functions and the Solvency Capital Requirements is substantial and we wish to highlight a few noteworthy features.

The loss distribution under the original risk-neutral measure appears to be centred around 0, independent of the initial conditions. The one-year loss is defined as the difference between the policy value at the times *t* = 0 and *t* = 1. On average, the policy value will not change significantly if the risk-neutral parameters stay the same. Therefore, the loss distribution must be centred around 0, as long as the initial risk-neutral parameters do not change. The loss distribution under the time-dependent risk-neutral measure, on the other hand, heavily depends on the risk-neutral parameters at *t* = 0. Consider Scenario 2 for example. Initially, the volatility, *v*¯ and *γ* are relatively low, resulting in low initial expected liabilities. However, according to the mean-reverting VIX index, these parameters are more likely to increase over time and with them, the expected liabilities. Consequently, the one-year losses are much higher compared to those under the original risk-neutral measure, which still assumes the relatively low initial parameters at *t* = 1. In Figure 4, this effect is clearly visible where the loss function is shifted to the right. Conversely, the one-year losses under the time-dependent risk-neutral measure in Scenario 3 are much lower, as the expected liabilities are more likely to decrease. In conclusion, when the initial volatility is low (high), we can expect a higher (lower) SCR under the time-dependent risk-neutral measure. In the 2008 credit crisis, this resulted in a higher SCR before the crisis and a lower SCR during the crisis.

Besides the shifted mean, the loss distribution under the time-dependent risk-neutral measure also tends to have heavier tails, which is especially visible in Scenario 1. This is caused by the fact that *v*¯ and *γ* depend on the state of the market, which results in more extreme losses (or gains). If, for example, the market crashes, *v*¯ and *γ* are likely to increase. This will generate even higher expected liabilities, resulting in even higher losses. However, if the market flourishes, *v*¯ and *γ* tend to be much lower, leading to lower expected liabilities and lower losses (or higher gains). This feature is present in all scenarios of Figure 4, but is best visible in Scenario 1, where the probability of an extreme loss as well as the probability of an extreme gain is higher under the time-dependent risk-neutral measure. Consequently, the SCRs under the two risk-neutral measures are not necessarily equal, not even when the initial conditions are equal to the average market conditions (such as in Scenario 1).

To give a broad overview of the impact, we determine the SCR of a variable annuity with the GMAB rider for multiple points in time. The contract details presented in Table 3 remain unchanged, but the initial parameters depend on historical data. For computational purposes, the parameter *α* is assumed to be constant and equal to 0.01. In this test, we compare four different risk-neutral measures:


Thus far, we have applied the first two risk-neutral measures in our analysis. The latter two measures can only be applied on historical data (otherwise, the observed parameters at *t* = 1 are undefined) and are merely added for explanatory purposes. Ideally, the SCRs under the future and the future VIX risk-neutral measure are equal. The difference between these measures is caused by prediction errors of the regression model. Hence, the difference between the SCRs under these measures be is indication for the accuracy of the regression models.

In Figure 5, the results under the different risk-neutral measures are displayed. The difference between the original and time-dependent risk-neutral measure is also summarized in Table 6.

**Figure 5.** SCR over time under the different risk-neutral measures.

**Table 6.** Difference in SCR between original and time-dependent risk-neutral measure.

