You are the valuation actuary for Glorious Life Insurance Company. You are implementing a new principle-based reserving methodology for your company. You have created a set of 10 economic scenarios that represent an appropriate range of possible outcomes for your company’s future, so that you can treat them as a complete probability space with equal probability assigned to each scenario. Your company has just received a $1000 single premium for a deferred annuity policy. Your analysis has shown that the present value at time 0 of the accumulated deficiency (i.e., the excess of future interest credited to customer’s account over the amount earned on investments, after all expenses) at the end of each year is as given in the table below. The regulation requires the reserve to be calculated as the current account balance plus the 70-th percentile Conditional Tail Expectation (CTE) of greatest present value of accumulated deficiency. Calculate that reserve

Scenario Present value of accumulated deficiency at time t =

0 1 2 3 4 5 6 7 8 9 10 1 0 5 6 11 6 4 7 3 2 –1 –5 2 0 8 6 5 3 8 10 2 –5 1 7 3 0 –2 –4 –5 –6 –8 –9 –7 –10 –4 –3 4 0 –6 –4 1 2 7 3 4 5 0 –3 5 0 7 8 5 2 6 1 –3 –5 2 –6 6 0 15 20 11 16 18 9 19 23 21 8 7 0 2 4 3 1 –3 –5 0 1 –2 –1 8 0 3 4 2 6 1 –3 –2 0 1 5 9 0 8 10 7 14 9 5 6 2 4 1 10 0 –4 –6 –5 –6 –7 –3 –5 –6 –8 –9 A. Less than 1000

B. At least 1000 but less than 1005 C. At least 1005 but less than 1010 D. At least 1010 but less than 1015 E. At least 1015

Solution.

The greatest present values (GPV) of accumulated deficiencies for scenarios 1 through 10 are:

11, 10, 0, 7, 8, 23, 4, 6, 14, 0.

CTE70 calculation requires us to average the largest 30 percent of Scenario GPVs of initial account balance plus GPV of accumulated deficiency. That means we need to add the starting asset amount 1000 to each GPV for 3 scenarios with greatest GPV and calculate their average:

1000+11+ 23+ 14 3 = 1011+ 1023+ 1014

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3 = 1016. Answer E.**Exercise. **

We have an example of scenarios runs for 3 years, with 10 scenarios each year shown below. The starting value of assets, equal to the single premium paid, is 1000. Calculate the CTE70 reserve.

Scenario

Value of accumulated deficiency at each year

Annual effective
*interest i (%) for *
each year
*V0 * *V1 * *V2 * *V3 * *i1 * *i2 * *i3 *
1 0 50 60 110 1 3 2
2 0 80 60 50 0 4 2
3 0 –20 –40 –50 1 2 3
4 0 –60 –40 10 4 4 4
5 0 70 72 50 2 3 6
6 0 150 200 110 5 5 2
7 0 20 40 30 0 2 6
8 0 40 41 20 4 3 4
9 0 80 100 70 5 6 1
10 0 –40 –60 –50 1 2 1
A. Less than 1000

B. At least 1000, less than 1050 C. At least 1050, less than 1100 D. At least 1100, less than 1150 E. At least 1150

Solution.

GPV (greatest present value) for scenarios 1 through 10: 103.7, 80, 0, 0, 68.6, 181.4, 39.2, 38.5, 89.8, 0.

CTE70 requires us to average the largest 30 percent of current account value plus Scenario GPVs of accumulated deficiency. That means we need to add the starting

Scenario PV at time 0 of accumulated deficiency at time t = 0 1 2 3 1 0 49.5 57.7 103.7 2 0 80 57.7 47.1 3 0 –19.8 –38.8 –47.1 4 0 –57.7 –37 8.9 5 0 68.6 68.5 44.9 6 0 142.9 181.4 97.8 7 0 20 39.2 27.7 8 0 38.5 38.3 18 9 0 76.2 89.8 62.3 10 0 –39.6 –58.2 –48.1

1103.7 + 1089.8 + 1181.4

3 = 1125.

**Exercise **

You are the valuation actuary for the Hard Knocks Life Insurance Company offering
life annuities and life insurance to guinea pigs. The valuation interest rate is 2%, and
*mortality of guinea pigs follows De Moivre’s Law with l _{x}*

*= 4 − x, 0 ≤ x ≤ 4. You*can assume that times after age 4 are ignored. Your company issues fixed deferred annuities to guinea pigs. The interest rate is guaranteed at 3% for the first two years, and 1% thereafter. The surrender charge is 10% the first year, 5% the second year, and 0% thereafter. Calculate the reserve for a $1000 single premium deferred annuity at issue sold to a newborn. There are no front-end loads. Assume that you use

*Commissioners Annuity Reserve Valuation Method and you are not doing this in the*state of New York. Regulations require that surplus must be 3% of assets at the policy issue date. Find the additional surplus investment requirement for this policy under this standard reserving policy. Then consider a new principles-based reserving requirement that the valuation rate must be replaced by three possible yield curves of forward rates:

*f*0,1= 1.80%, *f*1,2 = 1.85%, *f*2,3= 1.95%, *f*3,4 = 2.15%;
*f*0,1= 1.80%, *f*1,2 = 1.95%, *f*2,3= 2.15%, *f*3,4 = 1.75%;
*f*_{0,1}= 1.80%, *f*_{1,2} = 1.75%, *f*_{2,3}= 1.65%, *f*_{3,4} = 1.55%.

Calculate the CARVM reserve defined as the maximum under all three scenarios and the new capital requirement. Find the difference between the two capital requirements.

A. Less than 5

B. At least 5 but less than 7 C. At least 7 but less than 9 D. At least 9 but less than 11 E. 11 or more

Solution.

Because you are not doing business in the state of New York, you only need to do the calculations at the end of each policy year. Under constant valuation rate, the values are:

Time Fund Value Cash Surrender Value PV of Cash Value 1 1030.00 927.00 908.82

2 1060.90 1007.86 968.72

3 1071.51 1071.51 1009.71

4 1082.23 1082.23 999.81

The CARVM reserve is given by the present value of the guaranteed cash value at the end of the second policy year, when the surrender charge wears off, and it equals 1009.71. Without a surplus contribution, the company only has 1000 worth of assets,

then we must have
1000*+ x − 1009.71*
1000*+ x* = 0.03,
and
*x*= 39.71
0.97 = 40.94.

Under the first possible yield curve, we have

*f*0,1= 1.80%, *f*1,2 = 1.85%, *f*2,3= 1.95%, *f*3,4 = 2.15%,
and

Time Fund Value Cash Surrender Value PV of Cash Value 1 1030.00 927.00 910.61

2 1060.90 1007.86 972.06

3 1071.51 1071.51 1013.68

4 1082.23 1082.23 1002.27 Under the second possible yield curve

*f*0,1= 1.80%, *f*1,2 = 1.95%, *f*2,3= 2.15%, *f*3,4 = 1.75%,
and

Time Fund Value Cash Surrender Value PV of Cash Value 1 1030.00 927.00 910.61

2 1060.90 1007.86 971.10

3 1071.51 1071.51 1010.70

4 1082.23 1082.23 1003.26 Under the third possible yield curve

*f*0,1= 1.80%, *f*1,2 = 1.75%, *f*2,3= 1.65%, *f*3,4 = 1.55%,
and

Time Fund Value Cash Surrender Value PV of Cash Value 1 1030.00 927.00 910.61

2 1060.90 1007.86 973.01

3 1071.51 1071.51 1017.67

4 1082.23 1082.23 1012.16

Thus the new CARVM reserve is 1017.67. To find the capital requirement, we set
1000*+ x −1017.67*

1000*+ x* = 0.03,
resulting in

*x*=1017.67− 970

0.97 ≈ 49.14.

The difference in capital requirements is 49.14 – 40.94 = 8.20. Answer C.