The effective durations of reserves calculated under different levels of μ are
16 Although we do not have an explicit upper bound in the formula, the surrender rate is capped by the property of the arctangent function as Figure 3 will show later.
17 By the first-order and second-order differentiation of qt( )s at rta− , we get the saddle point of the rp arctangent surrender function atrt− =rp p4/p3, qt( )s = p1. At the saddle point, the marginal surrender rate is
2 3
p p .
18 This twenty-year period of 1969 - 1988 is chosen to be within the sampling period used in estimating the parameters of the interest rate model.
reported in Table 1.19 We see that some EDs are negative, and this is consistent with Tsai (2009). He argued that EDs are negative because the corresponding reserves are negative.
Policies with negative reserves are indeed assets to the insurance company, and designating these “assets” as “negative liabilities” on the balance sheet results in negative durations. The property that reserves decrease with the interest rate holds whether policies with negative reserves are treated as assets or negative liabilities.
[Insert Table 1 Here]
We have however found examples to counter the above argument of Tsai. Effective durations can be positive while the corresponding reserves are negative, and positive reserves may have negative EDs. The reserves corresponding to the EDs in Table 1 are presented in Table 2. We can see that the reserves maturing twenty years later are -$39,225, -$48,419 and -$52,458 when μ = 5%, 6% and 7%, respectively. This sold policy is an asset to the insurer mainly because the long-run mean of the interest rate is higher than the pricing rate. These negative reserves however have positive EDs of 3.21, 4.65 and 5.30. On the other hand, the reserves of the policy maturing eighteen years later are $6,246 and $229 when μ = 6% and
7%, but their EDs are -10.55 and -420.71 respectively. Similar cases can be found for the
19 The values of many EDs in Table 1 are rather small mainly due to the mean-reverting property of the CIR model. A shock to the initial interest rate fades away as (simulation) time goes by and leaves mid-run and long-run interest rate levels almost intact. Reserves hence do not change much and have small EDs.
Experimenting with alternative mean-reverting speeds, we confirmed that the values of EDs decreased with the speed. The interest rate sensitivity of surrenders also contributes to the small values of the EDs, as Babbel (1995) explained. When we remove the mean-reverting property of the interest rate model as well as the interest
sensitivity of surrenders and calculate the modified durations, the values become close to the years to maturity.
For instance, the modified durations for policies maturing 1 year and 5 years later are 0.96 and 4.86 respectively when the interest rate and surrender rate are set at 4%.
14
reserves maturing seventeen years later when μ = 7% and 8%.
[Insert Table 2 Here]
In Figure 1 (the results of Table 1), we find patterns that are rather different from Tsai (2009). In the cases of μ = 6%, 7% and 8%, the EDs increase from zero first but then decrease until they become negative as the policy’s maturity increases from one year to eighteen or nineteen years. The EDs jump to the positive domain for longer maturities and then start decreasing. We speculate that the general pattern of the term structure of the EDs in these cases can be depicted as the solid curves in Figure 2.
[Insert Figure 1 and Figure 2 Here]
We speculate that the three key conditions that have resulted in the above pattern are:
the long-run mean of the interest rate being higher than the pricing rate, the surrender rate being sensitive to the spread, and the policy having been issued a few years ago with small reserve values. When the first two conditions emerge, a positive interest rate shock will induce a significant number of policyholders to surrender their policies.20 Furthermore, the pre-determined cash values payable to these policyholders are larger than the reserves since the cash values are determined under the assumption that μ = 4% while the reserves are marked-to-market under a higher μ. These surrenders therefore will increase reserves. Since these policies have small reserve values (the third condition), the impact of these surrenders
20 Exceptions will occur when the spread is so large that the arctangent function turns into a relatively flat curve.
These exceptions do not show up in our tables or figures because the largest spread shown is 5%; that is still in the upward-sloping section of the arctangent function, as we will see from Figure 3 later.
may outweigh the discounting effect of a positive interest rate shock and cause reserves to increase.
We illustrate the above reasoning using some examples. At μ = 6%, the reserves of the policies maturating twenty, nineteen, and eighteen years later are -$48,419, -$22,703 and
$6,246, respectively (Table 2). The corresponding surrender values are $0, $8,160.77 and
$39,789.39 (Table A1). A 0.25% interest rate shock will increase the surrender probability and cause reserves to increase by $641, $508 and $393 respectively, if we hold r ta
unchanged. On the other hand, a 0.25% interest rate shock will cause the present values of future cash flows to decline by $70, $174 and $228 when we assume that surrender
probability does not increase. The net changes in reserves are $571, $334 and $165 and result in effective durations of 4.72, 5.88 and -10.55, respectively (Table 1).21
Our findings and the above reasoning demonstrate the importance of the long-run interest rate level, the interest sensitivity of the surrender rate, and the policy’s time to maturity in determining the effective durations of reserves.22 Comparing the EDs across the
columns in Table 1 and/or examining the graphs in Figure 1, we see clearly the importance of μ in determining EDs. This implies that life insurers should pay much attention to their
estimates of the long-run interest rates when implementing asset-liability management. The
21 The duration figures are slightly different from those in Table 2 because we use only the positive interest rate shock to calculate EDs here but Table 2 employs Equation (4) that covers both positive and negative shocks.
22 The importance of time to maturity in determining Macaulay and modified durations is well known and self-evident. Tsai (2009) documented this importance in determining the effective durations of reserves. We confirm this importance again in this paper but do not elaborate it further for the sake of brevity.
16
importance of the long-run mean is obscure in Tsai (2009).23
The importance of the interest-sensitive surrender rate in determining the effective durations of reserves was not explored in Tsai (2009) either. We illustrate the importance by setting alternative sets of parameters of the arctangent function and examining the resultant term structures of EDs. The parameters (p1, p2, p3, p4) are set as (0.1, 0.05, 50, 0.5) to indicate the more sensitive behavior of surrenders and are chosen to be (0.05, 0.05, 50, 2) to represent the less sensitive surrender behavior.24 The arctangent functions associated with these two parameter sets along with the benchmark set are plotted in Figure 3, and the resultant term structures of EDs are shown in Figures 4 and 5.25
[Insert Figures 3, 4 and 5 Here]
The patterns of the term structures in Figure 4 are consistent with Figure 1 and conform to the general pattern depicted by the solid curves in Figure 2. Figure 5 contains patterns similar to those in Tsai (2009) but with a distinction: some recently issued policies have negative reserves but positive effective durations. This phenomenon emerges when μ is significantly larger than the pricing rate (e.g., μ = 8% or 9%). The large spread induces surrenders that have cash values much larger than the “fair” policy values and cause reserves to increase, even when the surrender rate is sensitive to the spread to a minor degree only. We
23 An obscure characteristic of the VAR model in Tsai (2009) is that changes in the initial interest rate cause similar changes in the mean of the simulated interest rates. Their effects on EDs therefore mingle together and are difficult to distinguish from each other.
24 The lower bound of the surrender rate is kept as 3% for these two types of surrender behaviors.
25 The corresponding values of these EDs are displayed in Table A2 and Table A3.
plot this modified pattern of Tsai (2009) as the dashed curves in Figure 2.
The above findings and reasoning are robust across different values of κ and σ . We experimented with κ = 0.1, κ = 0.18, and σ = 0.03. All patterns remained valid. We thus conclude that the pattern identified in Tsai (2009) represents the cases in which the long-run mean of the interest rate is not significantly above the pricing rate and the surrender rate is sensitive to the interest spread to a moderate degree only. If the long-run mean is significantly higher than the pricing rate, this pattern has to be modified (as the dashed curves in Figure 2) even when the surrender rate exhibits low sensitivity to the spread. Higher sensitivities of surrender rates will result in a pattern opposite to Tsai (2009) as the solid curves in Figure 2 display whenever the long-run mean of the interest rate is higher than the pricing rate. In short, the interest sensitivity of the surrender rate and the long-run interest rate level are critical in determining the term structure of effective durations.