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¦ Black-Scholes-Merton hedging using 20 percent per year fixed volatility and 6 percent risk-free rate of interest.
¦ Monthly rebalancing of hedge.
¦ Transactions costs of 0.2 percent of change in stock holding at rebalancing dates.
Table 9.4 shows CTE and quantile risk measures for the two contracts. The results are shown separately for the actuarial and dynamic-hedging risk management strategies. The risk measures are illustrated in Figure 9.3.
The fixed GMDB carries relatively little risk, with more than a 95 percent estimated probability that the income is greater than the outgo. There is a slight tail risk from the fixed guarantee, with a 95 percent CTE of nearly 1 percent using actuarial risk management, but this is damped by using a hedging strategy, which virtually eliminates the risk.
Risk Measures for VA Death Benefits
TABLE 9.4 Risk measures for VA-type GMDB benefits, 30-year contract; percentage of initial fund value.
KisK Quantile CTE
Management ------------------------ ------------
Guarantee Strategy 90% 95% 90% 95%
Fixed Actuarial -0.350 -0.072 0.317 0.798
Fixed Hedging -0.294 -0.161 -0.119 -0.003
5% p.a. increasing Actuarial -0.086 1.452 1.857 3.044
5% p.a. increasing Hedging 0.071 0.579 0.706 1.102
The increasing GMDB is a more substantial risk, with a 95 percent CTE of around 3 percent of the initial single premium using actuarial risk management. Again, the hedging strategy significantly reduces the tail risk.
The comparisons provided in Figures 9.2 and 9.3 between actuarial and dynamic-hedging strategies give rise to the question: Which is better? The CTE curves show that, on average (i.e., at CTE0%), the actuarial approach is substantially more profitable than the dynamic-hedging approach. On the other hand, at the right tail the risk associated with the actuarial approach is greater than the dynamic-hedging approach, in some cases very substantially so. If solvency capital is to be determined using, for example, the 95 percent
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³ ³ ³ ² ² ¥ 0.0 0.2 0.4 0.6 0.8 1.0
ñë rt v
FIGURE 9.3 Risk measures for 30-year VA-GMDB benefits, comparing actuarial and dynamic-hedging risk management.
I I I I I Ã
0.0 0.2 0.4 0.6 0.8 1.0
FIGURE 9.3 (Continued)
0.0 0.2 0.4 0.6 0.8 1.0 Alpha
CTE, then the actuarial approach will require considerably more solvency capital to be maintained than the dynamic-hedging approach, and the cost of retaining this capital needs to be taken into consideration in determining whether to hedge or not. Indeed, it needs to be considered for all aspects of the management of equity-linked contracts, including decisions about commercial viability and pricing. Such decisions are the topic of the next chapter.
Emerging Cost Analysis
In this chapter, we show how to use the results of the analysis described in previous chapters to make strategic decisions about pricing and risk management for equity-linked contracts. The first decision is whether to sell the policy at all; if so, then at what price and with what benefits. If the contract has been sold, then the insurer must decide how much capital to hold in respect of the contract, and how that capital is to be managed. Market and competition issues are important in the decision process—for example, what are competitors charging for similar products? However, pure market considerations are not sufficient for actuarial pricing decisions. It is also essential to have some quantitative analysis available to ensure that business is sold with appropriate margins, to avoid following others on potentially ruinous paths.
Emerging cost analysis (also called profit testing) is a straightforward and intuitive approach to this analysis. It is very similar to the techniques of Chapters 6 and 8 in that it involves the projection of all the cash flows under the contract, according to the risk management strategy that the insurer proposes to adopt. The major difference between the projections in this chapter and those in earlier chapters is that here we take into account the capital requirements, so that the cash flows projected represent the loss or profit emerging each year after capital costs are taken into consideration. These cash flows are the returns to the shareholder funds and should be analyzed from the shareholders’ perspective.