After identifying new term structure patterns for reserve durations, we further
examine the effective convexities of reserves. The calculated results for policies maturing in different years are reported in Table 3 (Figure 6). They are new to the literature.
[Insert Table 3 and Figure 6 Here]
In Table 3 and Figure 6, we find three features of effective convexities. Firstly, many ECs are negative. Secondly, the signs of ECs may not be the same as those of EDs. It may not
18
be the same as those of reserves either. Thirdly, the term structure of ECs does not have the same patterns as those of EDs and seems to exhibit no general patterns.
We are not able to deduce general patterns for the term structure of ECs because reserves are affected by the initial interest rate in many ways. Reserves might decrease or increase with the initial interest rate in a convex or concave way and reserves themselves could be positive or negative. More specifically, there are eight permutations on the signs of reserves, EDs, and ECs: (+, +, +), (+, +, -), (+, -, +), (+, -, -), (-, +, +), (-, +, -), (-, -, +) and (-, -, -). Graphical representations of the above permutations are shown in Figure 7. Since reserves may be affected by the initial interest rate in so many ways, the term structures of ECs exhibit no general patterns. On the other hand, there are only four permutations on the signs of reserves and EDs. We are therefore able to sort out the patterns of EDs’ term structures with careful examination.
[Insert Figure 7 Here]
We could explain when and why the reserves of the policy would exhibit one of the eight conditions displayed in Figure 7 though. Table 4 summarizes our results about the correspondence of the eight conditions to the year(s) to maturity (YTM) of the policy under different long-run means of the interest rate and different interest sensitivities of surrenders.26 Let us look first at the case in which surrender rates have a higher sensitivity to interest
26 The ECs under more-sensitive and less-sensitive surrenders are listed in Table A4 and A5.
spreads than the benchmark case. More than half of the policies exhibit the typical condition of Figure 7(a). When μ is at 4%, the newly sold policy has a positive NPV but surrenders are not significant yet. It therefore exhibits the condition of Figure 7(h), a seemingly extending condition of Figure 7(a) into the negative domain of reserves. When μ ≥ 6%, significant
surrenders are triggered by high interest spreads and they turn the convexities of the policies that are at the middle of their lives (i.e. 9 ≤ YTM ≤ 17)27 into negative (Figure 7(b)). The
rationale behind this should be similar to that for the negative convexity of a callable bond.
Douglas (1990, pp. 275-280) demonstrated how the call provisions of a callable bond might result in negative convexities. The higher the value of the embedded call option is (e.g., when the call price is low, interest rates exhibit trends to lower levels, and interest rates have high volatilities), the more significant the effect of negative convexity from call provisions will be.
Since the surrender option of a life insurance policy is analogous to the call option of a callable bond, the surrender option may affect the ECs of reserves in similar ways.
[Insert Table 4 Here]
For new policies (YTM ≥ 18), the impact of surrenders overpowers the discounting
effect of interest rates and makes reserves an increasing function of the initial interest rate.
Reserves increase at a decreasing rate though. We thus observe the conditions of Figures 7(d) and 7(e) depending on whether reserves are positive or negative. Only when the policy is
27 The specific range of YTM changes with μ. For instance, a lower μ would allow a wider range of years to maturity to exhibit this condition. Please bear in mind this dependent relationship when reading similar statements in the following.
20
rather new (YTM ≥ 19) and when μ is at the saddle point of the arctangent function (5%
when surrenders are at the more sensitive case),28 the impact of surrenders would make reserves increase with the initial short term interest rate at an increasing speed (Figures 7(c) and 7(f)).29
The above reasoning also works for the case in which surrender rates exhibit the benchmark sensitivity to interest spreads.30 For instance, significant surrenders are triggered
by high interest spreads and turn the convexities of the policies that are at the middle of their lives (i.e. 7 ≤ YTM ≤ 15) into negative when μ ≥ 7% (Figure 7(b)). For newer policies (YTM
≥ 16), the impact of surrenders makes reserves increase with the initial interest rate at a diminishing rate (Figures 7(d) and 7(e)), unless the policy is rather new (YTM ≥ 18) and μ is
close enough to the saddle point of the arctangent function (6%) to make the rate of change increasing (Figures 7(c) and 7(f)).
The case in which surrender rates have a smaller sensitivity to interest spreads than the benchmark case further demonstrates the robustness of our reasoning. Most policies exhibit the typical condition of Figure 7(a). For the policies that are sold within three years
28 Please note that surrenders are rather sensitive to interest rate changes when μ is at the saddle point.
29 The condition of Figure 7(g), negative reserves decreasing with the initial interest rate at an increasing rate, never appears in our analyses. Reserves being negative implies that μ ≥ 4% and that YTM should be large (≥ 17), which in turn suggests the reserve value being small. On the other hand, reserves decreasing at an increasing rate implies active surrenders. When surrenders are active and reserve values are small, reserves should increase rather than decrease with the initial interest rate. We thus observe no cases like Figure 7(g).
30 We are aware of one exception: the brand new policy at μ = 2%. This exception probably results from the specifications of the interest rate model and the surrender rate function. When the long-run mean of the interest rate is merely 2% but the quarterly volatility of the interest rate is 8%, most surrender rates lie flat on the lower bound of the arctangent function while a few may jump to high levels. We conjecture that such irregular surrenders turn ECs into negative.
(YTM ≥ 17 years) with positive NPVs (i.e. μ ≥ 4%), the condition of Figure 7(h) may appear.
When the long-run mean of the interest rate is close to the saddle point (8%) and the policy is rather new (YTM ≥ 19), many surrenders are triggered by large interest spreads. These
surrenders make reserves increase with the initial interest rate at an increasing rate and thus exhibit the condition of Figure 7(f).
CONCLUSIONS
The reserves of life insurers are exposed to significant interest rate risk due to the long-term nature of life insurance. To quantify the interest rate risk, the popular pair of measures is duration and convexity. The extant insurance literature has argued strongly for the use of effective duration and effective convexity instead of the simpler Macaulay and modified measures due to the dependence of surrender rates on interest rates. Recently, Tsai (2009) identified a term structure of the effective durations of reserves using a VAR model of the interest rate and surrender rate.
In this paper we found that the term structure pattern of reserve durations identified in Tsai (2009) is not universal. His pattern is valid only when the long-run mean of the interest rate is not above the pricing rate and/or the surrender rate is not sensitive to the interest spread. The term structure pattern of reserve durations changes radically, as demonstrated in Figure 2, when the long-run mean of the interest rate is higher than the pricing rate and the surrender rate exhibits a certain degree of sensitivity to the spread. Tsai’s results need to be
22
revised even when the surrender rate is not sensitive to the spread, if the long-run mean of interest rates is significantly higher than the pricing rate.
Tsai (2009) did not detect the above newly identified patterns because his VAR model consisted of an interest rate process with little mean reversion and a surrender rate process featuring moderate interest rate sensitivity. Such an interest rate model obscures the distinction between an interest rate shock and the change in the long-run mean. The vector auto-regression structure also hinders Tsai (2009) from exploring alternative sensitivities of surrender rates to interest rates. His findings, therefore, do not apply universally.
Our findings signify the critical roles played by the long-run mean of the interest rate and the interest sensitivity of surrender rates in determining (the term structure pattern of) reserve durations. They have material implications to the life insurers selling savings-oriented products with low, fixed pricing rates in low interest rate eras. These life insurers will suffer severely when disintermediation takes place on the arrival of high interest rate regimes, if they do not correctly estimate the effective durations of their products and implement the asset-liability management accordingly. In the current economic situation, such damage may be caused by the awaiting recovery from ailing economies that is likely to be accompanied by interest rate rises.
With regard to effective convexities of reserves, we found that the convexities change significantly with the long-run mean of the interest rate and the policy maturity in complex
ways that further depend on the significance of surrenders. Insurers therefore should estimate reserve convexities vigilantly and dynamically for accurate and proper interest rate risk management. In addition, we unravel how reserves might change with interest rate shocks and work out all possible conditions. Our reasoning could serve as guidance to insurers in forming the strategies of asset-liability management.
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AFIR Conference, 375-395.
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Table 1: Effective Durations of Reserves
Long-Run Mean of the Interest Rate μ
Year(s) to Maturity 2% 3% 4% 5% 6% 7% 8% 9%
1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 3 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 4 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 5 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 6 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.03 7 0.06 0.06 0.06 0.05 0.04 0.04 0.04 0.04 8 0.09 0.08 0.08 0.07 0.06 0.05 0.05 0.05 9 0.12 0.12 0.10 0.09 0.08 0.07 0.07 0.07 10 0.17 0.16 0.14 0.12 0.10 0.08 0.08 0.08 11 0.23 0.22 0.19 0.15 0.12 0.10 0.10 0.10 12 0.32 0.31 0.26 0.20 0.15 0.12 0.12 0.13 13 0.44 0.43 0.36 0.26 0.18 0.14 0.13 0.15 14 0.62 0.60 0.50 0.34 0.21 0.14 0.13 0.16 15 0.87 0.85 0.70 0.44 0.21 0.07 0.06 0.11 16 1.25 1.23 1.00 0.55 0.09 -0.19 -0.24 -0.14 17 1.82 1.85 1.50 0.60 -0.54 -1.42 -1.70 -1.53 18 2.78 3.00 2.57 0.00 -10.55 -420.71 37.19 20.43 19 4.51 5.82 10.39 4.12 5.85 6.29 6.17 5.71 20 8.48 31.77 -1.17 3.21 4.65 5.30 5.50 5.43
28
Table 2: Reserves
Long-Run Mean of the Interest Rate μ
Year(s) to Maturity 2% 3% 4% 5% 6% 7% 8% 9%
1 980,343 970,743 961,246 951,851 942,588 933,427 924,376 915,434 2 920,948 903,010 885,571 868,700 852,516 836,866 821,689 806,901 3 862,527 837,393 813,363 790,619 769,391 749,338 730,263 711,931 4 805,170 773,883 744,460 717,213 692,471 669,653 648,369 628,193 5 748,792 712,297 678,529 647,940 620,930 596,628 574,409 553,623 6 693,508 652,697 615,530 582,598 554,330 529,535 507,325 486,823 7 639,209 594,893 555,164 520,741 492,042 467,528 446,024 426,430 8 585,861 538,795 497,267 462,087 433,614 409,963 389,667 371,412 9 533,521 484,410 441,773 406,468 378,754 356,402 337,666 321,036 10 482,159 431,662 388,520 353,627 327,107 306,389 289,453 274,617 11 431,721 380,447 337,364 303,359 278,377 259,520 244,521 231,562 12 383,366 331,859 289,288 256,479 233,191 216,246 203,173 192,053 13 335,779 284,552 242,902 211,573 190,115 175,115 163,944 154,614 14 289,012 238,555 198,207 168,599 149,072 136,028 126,720 119,126 15 243,090 193,867 155,169 127,489 109,966 98,861 91,360 85,432 16 198,995 151,432 114,661 89,020 73,478 64,223 58,419 54,059 17 156,260 110,735 76,125 52,599 38,985 31,454 27,208 24,285 18 115,004 71,836 39,532 18,091 6,246 229 -2,673 -4,340 19 77,691 37,101 7,129 -12,369 -22,703 -27,519 -29,394 -30,100 20 44,189 6,317 -21,411 -39,225 -48,419 -52,458 -53,748 -53,938
Table 3: Effective Convexities of Reserves
Long-Run Mean of the Interest Rate μ
Year(s) to Maturity 2% 3% 4% 5% 6% 7% 8% 9%
1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.00 8 0.01 0.00 0.01 0.00 0.00 0.00 -0.00 -0.00 9 0.01 0.01 0.01 0.01 0.00 -0.00 -0.00 -0.00 10 0.01 0.02 0.02 0.01 0.00 -0.00 -0.00 -0.01 11 0.02 0.03 0.04 0.02 0.01 -0.01 -0.01 -0.01 12 0.04 0.05 0.06 0.04 0.01 -0.02 -0.03 -0.03 13 0.05 0.09 0.12 0.07 0.01 -0.05 -0.08 -0.08 14 0.08 0.17 0.23 0.14 0.00 -0.12 -0.18 -0.19 15 0.12 0.31 0.46 0.31 -0.01 -0.31 -0.48 -0.50 16 0.21 0.62 0.98 0.75 -0.01 -0.88 -1.38 -1.50 17 0.36 1.30 2.39 2.18 0.01 -2.98 -5.40 -6.36 18 0.58 3.18 7.57 11.61 4.75 -613.58 91.31 65.40 19 0.62 9.46 70.08 -33.66 -6.83 4.69 11.47 15.46 20 -2.50 82.09 -41.27 -22.84 -12.37 -4.31 1.93 6.74
30
Table 4: Categorization of the Sign Vectors of Reserves, Effective Durations, and Effective Convexities by Policy Years under the Three Sensitivities of Surrenders
Long-Run Mean of the Interest Rate μ
2% 3% 4% 5% 6% 7% 8% 9%
Condition Sign Vector Benchmark Case
(a) (+, +, +) 1-19 1-20 1-19 1-18 1-14 1-8 1-7 1-6
(b) (+, +, -) 20 15,16 9-15 8-15 7-15
(c) (+, -, +) 17, 18
(d) (+, -, -) 16-18 16,17 16,17
(e) (-, +, +) 19 18-20 18-20
(f) (-, +, -) 19-20 19-20 20
(g) (-, -, +)
(h) (-, -, -) 20
Condition Sign Vector More Sensitive Case
(a) (+, +, +) 1-20 1-20 1-19 1-18 1-10 1-8 1-8 1-8
(b) (+, +, -) 11-17 9-17 9-17 9-17
(c) (+, -, +) 19
(d) (+, -, -) 18 18 18 18
(e) (-, +, +) 19,20 19,20 19,20 19,20
(f) (-, +, -) 20
(g) (-, -, +)
(h) (-, -, -) 20
Condition Sign Vector Less Sensitive Case
(a) (+, +, +) 1-20 1-20 1-19 1-18 1-17 1-17 1-17 1-16 (b) (+, +, -)
(c) (+, -, +) (d) (+, -, -)
(e) (-, +, +)
(f) (-, +, -) 20 20 19,20
(g) (-, -, +)
(h) (-, -, -) 20 19,20 19,20 18.19 18,19 17,18 Note:
1. The signs in parentheses denote the signs of reserve, effective duration, and effective convexity of a policy in a given policy year.
2. The number (or the range of numbers) in the cells of the table denotes the policy year(s) in which the policy’s reserve, ED, and EC exhibit the corresponding sign vector. For instance, the “1-19” in the cell corresponding to condition (a) and μ = 2% in the “Benchmark Case” panel means that the sign vectors of the policies maturing between one and nineteen years are (+,+,+). The “20” in the cell below means that the sign vector of the policy maturing twenty years later is (+,+,-).
5 10 15 20
Figure 1: Term Structures of Effective Durations
Figure 2: Patterns of the Term Structure of Effective Durations
Years to Maturity Zero Reserve
Effective Durations
Modified Pattern of Tsai (2009)
New Pattern
Negative Reserves Positive Reserves
Pattern of Tsai (2009)
32
Figure 3: Arctangent Functions of the Surrender Rate to the Interest Spread
5 10 15 20
Figure 4: Term Structures of Effective Durations under More-Sensitive Surrenders
5 10 15 20
Figure 5: Term Structures of Effective Durations under Less-Sensitive Surrenders
5 10 15 20
Figure 6: Term Structures of Effective Convexities
34
(a) (+,+,+) (b) (+,+,− )
(c) (+,− ,+) (d) (+,− , − )
(e) (− ,+,+) (f) (− ,+, − )
(g) (− , − ,+) (h) (− , − , − )
Figure 7: Functions of Reserves to the Initial Interest Rate
Note:
1. The sign vector at the top of each graph indicates the signs of the function value, the first derivative, and the second derivative respectively. More specifically, the sign vector denotes the signs of reserve, effective duration, and effective convexity of the policy with a reserve function that can be represented by the graph.
2. The dashed line represents a tangent line of the reserve function at an initial interest rate.
Reserves
0
Initial Interest Rate
Reserves
0
Initial Interest Rate
Reserves
0
Initial Interest Rate
Reserves
0
Initial Interest Rate Initial Interest Rate
Reserves
0
Initial Interest Rate
Reserves
Initial Interest Rate
Reserves
0
Initial Interest Rate
Reserves
APPENDIX
Table A1: Actuarial Assumptions about the Twenty-Year Endowment Product
Insured’s Age
Mortality Rate of age a
At the Beginning of Policy Year
32 0.0010481 3 39789.39 20.60% 1,359 0.001
33 0.0011075 4 73766.54 14.00% 1,359 0.001
34 0.0011826 5 110191.76 13.00% 1,359 0.001
35 0.0012712 6 149173.28 12.00% 1,359 0.001
36 0.0013711 7 190830.52 10.00% 1,359 0.001
37 0.0014807 8 235294.26 10.00% 1,359 0.001
38 0.0015989 9 282706.96 10.00% 1,359 0.001
39 0.0017291 10 333222.85 10.00% 1,359 0.001
40 0.0018749 11 387003.56 7.00% 1,359 0.001
1. The death benefit and survival benefit is $1,000,000. The policy is issued to a 30-year-old male, and the annual premium expected to be paid at the beginning of each surviving year is $45,300 under the pricing rate of 4%.
2. Notation 20* is used to denote the end of policy year 20.
3. If the policy is surrendered at the beginning of the first policy year, it has no surrender value. Neither mortality nor expenses apply when the policy matures. We denote all these values as N/A.
4. Variable cost is assumed to be 0.1% flat.
36
Table A2: Effective Durations of Reserves under More-Sensitive Surrenders
Long-Run Means of the Interest Rate μ
Year(s) to Maturity 2% 3% 4% 5% 6% 7% 8% 9%
1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 3 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 4 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 5 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 6 0.04 0.04 0.04 0.03 0.03 0.03 0.03 0.03 7 0.06 0.06 0.05 0.05 0.04 0.04 0.04 0.04 8 0.09 0.08 0.07 0.06 0.06 0.05 0.05 0.05 9 0.12 0.11 0.09 0.08 0.07 0.07 0.07 0.07 10 0.17 0.15 0.13 0.11 0.10 0.09 0.09 0.09 11 0.23 0.20 0.17 0.14 0.12 0.12 0.12 0.12 12 0.32 0.28 0.23 0.19 0.16 0.15 0.15 0.16 13 0.45 0.39 0.31 0.25 0.21 0.19 0.20 0.21 14 0.63 0.55 0.43 0.33 0.27 0.25 0.25 0.27 15 0.90 0.77 0.59 0.43 0.33 0.30 0.31 0.34 16 1.30 1.11 0.83 0.55 0.38 0.33 0.35 0.40 17 1.93 1.66 1.18 0.67 0.34 0.21 0.24 0.35 18 3.00 2.64 1.79 0.62 -0.36 -0.92 -1.04 -0.86 19 5.09 4.97 3.52 -3.87 42.71 14.04 9.44 6.82 20 10.85 26.71 -0.95 3.94 4.94 5.12 4.95 4.63
Table A3: Effective Durations of Reserves under Less-Sensitive Surrenders
Long-Run Means of the Interest Rate μ
Year(s) to Maturity 2% 3% 4% 5% 6% 7% 8% 9%
1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 3 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 4 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 5 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 6 0.05 0.05 0.04 0.04 0.04 0.04 0.04 0.04 7 0.06 0.07 0.06 0.06 0.06 0.06 0.06 0.06 8 0.09 0.09 0.08 0.08 0.08 0.08 0.08 0.08 9 0.13 0.13 0.12 0.12 0.11 0.11 0.11 0.11 10 0.18 0.18 0.16 0.16 0.16 0.16 0.15 0.15 11 0.24 0.25 0.23 0.22 0.22 0.22 0.21 0.21 12 0.33 0.34 0.31 0.31 0.30 0.30 0.29 0.29 13 0.46 0.48 0.44 0.44 0.43 0.43 0.42 0.42 14 0.64 0.67 0.62 0.63 0.62 0.63 0.61 0.62 15 0.90 0.95 0.89 0.92 0.92 0.97 0.95 1.00 16 1.27 1.39 1.34 1.43 1.49 1.67 1.72 2.01 17 1.83 2.11 2.15 2.54 2.99 4.60 6.54 -88.83 18 2.72 3.49 4.21 7.84 115.08 -4.55 -2.39 -1.09 19 4.21 6.94 23.40 -6.76 -2.30 -0.82 -0.36 0.04 20 7.10 37.93 -5.65 -1.47 -0.35 0.28 0.53 0.76
38
Table A4: Effective Convexities of Reserves under More-Sensitive Surrenders
Long-Run Means of the Interest Rate μ
Year(s) to Maturity 2% 3% 4% 5% 6% 7% 8% 9%
1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 8 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 9 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 10 0.02 0.02 0.02 0.01 0.00 0.00 0.00 0.00 11 0.03 0.03 0.03 0.01 0.00 -0.01 -0.01 0.00 12 0.05 0.05 0.04 0.02 0.00 -0.01 -0.01 -0.01 13 0.08 0.09 0.08 0.03 -0.01 -0.03 -0.03 -0.03 14 0.14 0.17 0.14 0.05 -0.03 -0.08 -0.08 -0.07 15 0.25 0.33 0.27 0.08 -0.10 -0.19 -0.21 -0.19 16 0.47 0.65 0.54 0.14 -0.28 -0.52 -0.59 -0.54 17 0.94 1.37 1.17 0.23 -0.87 -1.62 -1.91 -1.86 18 2.01 3.22 3.01 0.46 -3.53 -7.24 -9.73 -10.97 19 4.66 9.32 12.96 4.61 116.18 54.70 45.13 37.63 20 13.20 82.57 -29.37 -3.56 7.00 13.11 16.84 18.86
Table A5: Effective Convexities of Reserves under Less-Sensitive Surrenders
Long-Run Means of the Interest Rate μ
Year(s) to Maturity 2% 3% 4% 5% 6% 7% 8% 9%
1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 8 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 9 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 10 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 11 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 12 0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.02 13 0.06 0.06 0.06 0.05 0.04 0.04 0.04 0.04 14 0.10 0.11 0.11 0.09 0.09 0.09 0.08 0.08 15 0.19 0.20 0.20 0.18 0.18 0.19 0.18 0.19 16 0.35 0.40 0.39 0.39 0.41 0.46 0.47 0.56 17 0.69 0.82 0.86 0.98 1.17 1.87 2.69 -38.41 18 1.38 1.88 2.36 4.43 68.26 -2.97 -1.67 -0.90 19 2.96 5.28 19.18 -6.03 -2.38 -1.18 -0.79 -0.47 20 6.98 42.14 -7.43 -2.62 -1.36 -0.67 -0.38 -0.12
The time line below describes the relations among policy year k, insured’s age 30+k-1, and evaluation time t. It also illustrates where the net cash flows are.
30 31 32 30+ k -2 30+ k -1 30+ k 30+ k -1+t-1 30+ k -1+t 30+ k +t Time -1 0 1 t-1 t t+1
Figure A1: Illustrative Time Line
Insured' s Age
Policy Year 1 2 k-1 k k-1+t k+t