Realistic example

Recall that the critical levels of risk aversion are strongly influenced by the existence of background risk. Thus, it is important to assess if a realistic scenario implies that the loss rate of the additional rider is decreasing in the level of risk aversion or not. Based on realistic input data, we illustrate the utility loss for different types of investors. We consider three types of GMAB-investors which are defined by the proportion of retirement income stemming from the GMAB. These are called high, medium and lowGMAB–investors. We still assume that the background value X_TB is lognormally distributed, cf. Equation (4.25). Based on the scenario w (w ∈ {high, medium, low}), we define the initial amount P(w)invested in the GMAB. Along the lines of the above reasonings, the optimization is based on constant mix strategies. Here, the terminal wealth TW(w)can be represented as follows

TW_T(w)= X_TB+ GMAB(w)_T = X_TB+ P(w)max{V_T(1), egT} = P(w) X B T P(w)+ max{V (1) T , e gT_}  .

V_T(1) denotes the payoff of a constant mix strategy with an initial investment of 1. For a CRRA investor, it is enough to consider24

˜

TW_T(w)= Z_TB,w+ max{V_T(1), egT} where Z_TB,W = X

B T

P(w). A realistic parametrization for ZT is motivated by

ln Z_TB∼ N((µ_ZB− 1 2σ 2 ZB)T, σ_Z2B)T ) with µw ZB_T = 1 T ln E " fretI_T a_T s(w) ∑T_i=0−1e−rti_It i #! = 1 T ln 1 s(w)+ ln E " fretI_T a_T ∑T_i=0−1e−rti_It i #! .

fret denotes the retirement factor, It denotes the income at time t of the accumulation

phase with initial value I0. We benchmark on the average income of the year 2010 in

Germany, i.e. I0= 32003 . Using the mean of the growth rate of the average income

since 1952, the initial income is projected to time ti. The standard deviation σZB is ap-

proximated by the standard deviation of the historical growth rate, i.e. it is set equal to 0.035. s(w) denotes the savings factor of the investor with w (w ∈ {high, medium, low). a_T =R∞

0 e−ruup65du is the annuity factor of an individual aged 65 at the retirement

time T . As usual, tpx denotes the probability of a living aged x to survive the next

t years. In particular, we vary the saving factor s(w) to model our three types of in- vestors.25

Income and Pension Parameters

µw ZB T − T ln 1 s(w) -0.0403971 T 20 σ_ZB 0.035 fret 0.43 aT 18.36 I0 32.003 w s(w) low 7.2% medium 8.5% high 11.3%

Table 4.2 Income and Pension Parameters.

The relevant income and pension parameters are summarized in Table 6.1. The pa- rameters are set according to German data on income, pension, saving rates as well as demographic data. The age of retirement is set to 65, i.e. the investor starts con- tributing in the GMAB with 45, i.e. we set T = 20. According to the German mortality tables from the Statistisches Bundesamt, the retirement factor (which is adjusted for longevity risk) is given by fret = 0.43, and the annuity factor is aT = 18.36.

Table 6.3 shows the corresponding utility loss for the three types of investors and varying levels of risk aversion γ. Independent of the risk aversion the investor who relies most heavily on the GMAB at retirement, incurs the highest loss in utility. For non binding borrowing constraints it still holds: the higher the risk aversion the higher

25 _{Notice that it is the proportion between investment in the GMAB and other sources of retirement}

income that drives the results. Therefore, the impact on the utility loss remains the same fixing the amount invested in the GMAB and varying the level of other income sources.

4.5 Conclusion 79

the loss in utility. For binding borrowing constraints the loss rate decreases, i.e. for rea- sonable risk averse investors the utility loss is lower due to the borrowing constraints. This implies that moderate risk averse investors which are the typical clientele of these products have to pay a price in terms of the utility loss for the additional flexibility. However, compared to lower risk averse investors or very risk averse investors the loss is smaller.

Loss rates for low, medium and high GMAB-investor investors investor γ = 1.5 γ = 2.0 γ = 2.5 γ = 3.0 high 37.1 36.7 36.0 35.3 medium 32.1 31.9 31.4 30.9 low 29.2 29.2 28.8 28.3 investor γ = 3.5 γ = 4.0 γ = 4.5 γ = 5.0 high 34.5.7 33.7 32.9 32.3 medium 30.2 29.6 29.0 28.4 low 27.8 27.3 26.8 26.2 investor γ = 5.5 γ = 6.0 γ = 15 γ = 20 high 31.5 30.8 31.6 33.0 medium 27.9 27.3 26.7 28.1 low 25.7 25.2 23.9 25.4

Table 4.3 The Table gives the loss rates of the low, medium and high GMAB-investor for varying risk aversion levels γ in basis points. The parameters are set as stated in Table 4.1 and 6.1.

4.5 Conclusion

This chapter has analyzed the investor’s incentive to deviate from an optimal diver- sified investment strategy due to the insurance companies price setting. We consider products where the payoff is linked to the performance of an investment strategy and includes a minimum interest rate guarantee. In the case that the investor also receives the rider to decide on the investment strategy, the provider takes into account for the

highest possible guarantee value. This is given by the strategy which maximizes the insurance put, i.e. the most risky one. The most risky strategy implies that the whole portfolio is invested in one asset only, i.e. it is a purely non-diversifying strategy. A risk averse investor who maximizes her expected utility faces two opposing effects. On one hand, she looses risk capital by paying an unfair price for the guarantee. On the other hand, deviating from the diversified strategy means taking more risk than optimal. In consequence, the investments become more aggressively which might be considered as an unwanted incentive effect. In addition, the investor herself faces a utility loss.

Surprisingly, it turns out that the loss rate which is implied by the contract rider is not monotonously increasing in the level of risk aversion. This is due to the borrowing constraints which are imposed on a private investor. While the loss rate is increasing in the case that the risk aversion is sufficiently high such that the borrowing constraints are not relevant for the expected utility maximization, this is not true as long as the bor- rowing constraints are binding. The critical level of risk aversion where the borrowing constraints are not binding any more also depends on the background risk of the in- vestor. By means of realistic data, we quantify both, the increase in the risk structure and the utility losses which are caused by the additional contract rider.