Some exercises in this and subsequent chapters are based on the Standard Select and Standard Ultimate Life Tables, which are given in Appendix D.
Although the full model underlying these tables was described inChapter 3, you are expected to use these life tables as if only the integer age information given in the tables is available. When you need to use a fractional age assumption, we specify which assumption to use in the exercise. We assume that an Excel version of the Standard Ultimate Life Table is available that can calculate annual functions at different rates of interest.
When we specify the use of the Standard Select Survival Model or Standard Ultimate Survival Model, then we assume knowledge of the full underlying distribution.
Shorter exercises
Exercise 4.1 You are given the following table of values for lx and Ax, assuming an effective interest rate of 6% per year.
x lx Ax
35 100 000.00 0.151375
36 99 737.15 0.158245
37 99 455.91 0.165386
38 99 154.72 0.172804
39 98 831.91 0.180505
40 98 485.68 0.188492
Calculate the following, assuming UDD between integer ages where necessary.
(a)5E35 (b)A1
35:5 (c)5|A35 (d)A¯35:5
Exercise 4.2 Using the Standard Ultimate Life Table, with interest at 5% per year effective, calculate the following, assuming UDD between integer ages where necessary.
(a)A1
30:20 (b)A¯40:20 (c)10|A25
Exercise 4.3 (a) Describe in words the insurance benefit with present value given by
Z =
T30vT30 ifT30≤25, 25vT30 ifT30>25.
(b) Write down an expression in terms of standard actuarial functions for E[Z].
4.9 Exercises 133 Exercise 4.4 Put the following functions in order, from smallest to largest, assumingi>0 andμx+t>0 for allt>0. Explain your answer from general reasoning (i.e., from general principles, not from calculating the values).
Ax A¯x A(4)
x:10 A¯x:10 A(4)1
x:10 A(12)
x:10
Exercise 4.5 An insurer issues a five-year term insurance to (65), with a benefit of $100 000 payable at the end of the year of death.
Calculate the probability that the present value of the benefit is greater than
$90 000, using the Standard Ultimate Life Table, with interest of 5% per year effective.
Exercise 4.6 Under an endowment insurance issued to a life agedx, letX denote the present value of a unit sum insured, payable at the moment of death or at the end of then-year term.
Under a term insurance issued to a life agedx, letYdenote the present value of a unit sum insured, payable at the moment of death within then-year term.
Given that
V[X]=0.0052, vn=0.3, npx=0.8, E[Y]=0.04, calculate V[Y].
Exercise 4.7 A whole life insurance with sum insured $50 000 is issued to (50). The benefit is payable immediately on death. Calculate the probability that the present value of the benefit is less than $20 000. Use the Standard Ultimate Survival Model with interest at 4% per year effective.
Exercise 4.8 Assuming a uniform distribution of deaths over each year of age, show that
A(xm)= i i(m)Ax. Exercise 4.9 CalculateA70given that
A50:20 =0.42247, A1
50:20 =0.14996, A50=0.31266.
Exercise 4.10 Using the Standard Ultimate Life Table, with interest at 5%
per year effective, calculate the standard deviation of the present value of the following benefits:
(a) $100 000 payable at the end of the year of death of (30), and
(b) $100 000 payable at the end of the year of death of (30), provided death occurs before age 50.
Exercise 4.11 You are given thatAx =0.25,Ax+20 =0.40,Ax:20 =0.55 and i=0.03. Calculate 10 000A¯x:20 using
(a) claims acceleration, and (b) UDD.
Exercise 4.12 Show that A¯x is a decreasing function of i, and explain this result by general reasoning.
Exercise 4.13 An insurer issues a whole life policy to a life aged 80, under which a benefit of $10 000 is payable immediately on death during the first year, and $100 000 is payable immediately on death during any subsequent year. Because the underwriting on the insurance is light, the insurer expects the mortality rate during the first year to be double the rate from the Standard Ultimate Life Table. After the first year, mortality is assumed to follow the Standard Ultimate Life Table with no adjustment.
Calculate the EPV of the benefit, using an interest rate of 5% per year effective. Assume UDD between integer ages.
Exercise 4.14 (I¯A)¯ x:n denotes the EPV of an increasing endowment insur- ance, where, ifTx<na benefit ofTxis payable immediately on death, and if Tx≥na benefit ofnis payable at timen.
Using the Standard Ultimate Survival Model, with 5% per year interest, calculate
d
dt(¯IA)¯ 40:t att=10.
Exercise 4.15 A whole life insurance policy issued to a life aged exactly 30 has an increasing sum insured. In thetth policy year,t =1, 2, 3,. . ., the sum insured is $100 000(1.03t−1), payable at the end of the year of death. Using the Standard Ultimate Survival Model with interest at 5% per year, calculate the EPV of this benefit.
Exercise 4.16 Show that
(IA)x:n1 =(n+1)Ax:n1 − n k=1
A1
x:k
and explain this result intuitively.
Exercise 4.17 You are given that the time to first failure, denoted T, of an industrial robot, has a probability density function for the first 10 years of operations of
fT(t)=
0.1 for 0≤t<2, 0.4t−2 for 2≤t<10.
4.9 Exercises 135 Consider a supplemental warranty on this robot which pays $100 000 at timeT, if 2≤T <10 , with no benefits payable otherwise.
Calculate the 90th percentile of the present value of the benefit under the warranty, assuming a force of interest ofδ=0.05.
(Copyright 2017. The Society of Actuaries, Schaumburg, Illinois. Repro- duced with permission.)
Longer exercises
Exercise 4.18 You are given the following excerpt from a select life table.
[x] l[x] l[x]+1 l[x]+2 l[x]+3 lx+4 x+4 [40] 100 000 99 899 99 724 99 520 99 288 44 [41] 99 802 99 689 99 502 99 283 99 033 45 [42] 99 597 99 471 99 268 99 030 98 752 46 [43] 99 365 99 225 99 007 98 747 98 435 47 [44] 99 120 98 964 98 726 98 429 98 067 48
Assuming an interest rate of 6% per year, calculate (a) A[40]+1: 4 ,
(b) the standard deviation of the present value of a four-year term insurance, deferred one year, issued to a newly selected life aged 40, with sum insured
$100 000, payable at the end of the year of death, and
(c) the probability that the present value of the benefit described in part (b) is less than or equal to $85 000.
Exercise 4.19 A select life aged 50 purchases a whole life insurance policy, with sum insured $1 000 000. The benefit is valued using the Standard Select Life Table, with interest of 5% per year. Assume UDD between integer ages where necessary.
(a) Assume the claim is payable four months after the death of the policy- holder. Calculate the mean and standard deviation of the present value of the benefit.
(b) Now assume that there are two types of claims. A claim is either ‘straight- forward’ or ‘complex’. The straightforward claims are settled two months after the death of the policyholder. The complex claims are settled one year after the death of the policyholder. The probability that an individual claim is straightforward is 80% and the probability that it is complex is 20%. Calculate the mean and standard deviation of the present value of the benefit.
Exercise 4.20 (a) An insurer issues a 20-year term insurance with sum insured $100, payable immediately on death, to a life currently aged 50.
Calculate the 95th percentile of the present value of the benefit, assuming interest of 5% per year, and that mortality follows the Standard Ultimate Life Table, with UDD between integer ages.
(b) The insurer issues 10 000 identical policies to independent lives. Using a normal approximation, estimate the 95th percentile of the present value of the aggregate payment over all the policies.
(c) Explain why the normal approximation is reasonable for part (b) but would not be reasonable for part (a)
Exercise 4.21 (a) Describein wordsthe insurance benefits with the present values given below.
(i) Z1=
20vTx ifTx≤15, 10vTx ifTx>15.
(ii) Z2=
⎧⎪
⎨
⎪⎩
0 ifTx≤5,
10vTx if 5<Tx≤15, 10v15 ifTx>15.
(b) Write down in integral form the formula for the expected value for (i)Z1
and (ii)Z2.
(c) Derive expressions in terms of standard actuarial functions for the expected values ofZ1andZ2.
(d) Derive expressions in terms of standard actuarial functions for the variance of bothZ1andZ2.
(e) Derive an expression in terms of standard actuarial functions for the covariance ofZ1andZ2.
(f) Assume now thatx=40. Calculate the standard deviation of bothZ1and Z2, using the Standard Ultimate Life Table, with UDD between integer ages, and interest at 5% per year.
Exercise 4.22 Consider a five-year deferred, 20-year endowment insurance policy issued to (40). The policy pays no benefit on death before age 45.
A death benefit of $100 000 is payable immediately on death between ages 45 and 65. On survival to age 65 a benefit of $50 000 is payable.
Assume mortality follows the Standard Ultimate Life Table, with UDD between integer ages, and with interest at 5% per year.
(a) Write down an expression for the present value of the benefits in terms ofT40.
4.9 Exercises 137 (b) Sketch a graph of the present value of the benefit as a function of the time
of death. Clearly label the axes and show all key values.
(c) Calculate the EPV of the benefit.
(d) Calculate the probability that the present value of the benefit is more than
$16 000.
(e) Calculate the 60% quantile of the present value of the benefit.
Exercise 4.23 Assume Gompertz’ law,μx=Bcx, withB=2.5×10−5, and c=1.1, with interest at 4% per year effective.
(a) Evaluate
d
dtA¯50:t1 att=10.
(b) Hence, or otherwise, evaluate d
dtt| ¯A50 att=10.
Exercise 4.24 Show that if νy = −logpy fory = x,x+1,x+2,. . ., then under the assumption of a constant force of mortality between integer ages,
A¯x= ∞
t=0
vttpx νx+t(1−vpx+t) δ+νx+t
.
Exercise 4.25 For three insurance policies on the same life, you are given:
(i) Z1is the present value of a 20-year term insurance with sum insured $200, payable at the end of the year of death, with E[Z1] =6.6 and V[Z1] = 748.
(ii) Z2is the present value of a 20-year deferred whole life insurance with sum insured $1000, payable at the end of the year of death, with E[Z2]=107.5 and V[Z2]=5078.
(iii) Z3is the present value of a whole life insurance with sum insured $2000, payable at the end of the year of death.
Calculate the mean and standard deviation ofZ3. Exercise 4.26 (a) Show that
Ax:n =
n−2
k=0
vk+1k|qx+vnn−1px.
(b) Compare this formula with formula (4.18) and comment on the differ- ences.
Exercise 4.27 A life insurance policy issued to a life aged 50 pays $2000 at the end of the quarter year of death before age 65 and $1000 at the end of the quarter year of death after age 65. Use the Standard Ultimate Life Table, with UDD between integer ages, and assuming interest at 5% per year, in the following.
(a) Calculate the EPV of the benefit.
(b) Calculate the standard deviation of the present value of the benefit.
(c) The insurer charges a single premium of $500. Assuming that the insurer invests all funds at exactly 5% per year effective, what is the probability that the policy benefit has greater value than the accumulation of the single premium?
Exercise 4.28 LetZ1 denote the present value of an n-year term insurance benefit, issued to (x). Let Z2 denote the present value of a whole of life insurance benefit, issued to the same life.
Express the covariance ofZ1andZ2in actuarial functions, simplified as far as possible.
Exercise 4.29 Show that
(IA(m))x=A(xm)+vpxA(xm+)1+v22pxA(xm+)2+ · · · and explain this result intuitively.
Exercise 4.30 (a) Derive the following recursion formula for an n-year increasing term insurance:
(IA)x:n1 =vqx+vpx
(IA)x1+1:n−1 +A 1
x+1:n−1
. (b) Give an intuitive explanation of the formula in part (a).
(c) You are given that(IA)50 = 4.99675,A1
50:1 =0.00558,A51 =0.24905 andi=0.06. Calculate(IA)51.
Exercise 4.31 Assuming a uniform distribution of deaths over each year of age, find an expression for(¯IA)¯ xin terms ofAxand(IA)x.
Exercise 4.32 A two-year term insurance is issued to (70), and the sum insured is payable immediately on death. The amount payable on death at time tis $100 000(1.05t), for 0<t≤2.
(a) Calculate the EPV of the benefit, assuming mortality follows the Standard Ultimate Life Table with interest at 5% per year.
(b) Calculate the EPV of the benefit, assumingμ70+t=0.012 for 0≤t≤2, with interest at 4% per year.
4.9 Exercises 139 Excel-based exercises
Exercise 4.33 Suppose that Makeham’s law applies with A = 0.0001,B = 0.00035 andc =1.075. Assume also that the effective rate of interest is 6%
per year.
(a) Use Excel and backward recursion in parts (i) and (ii).
(i) Construct a table of values ofAxfor integer ages, starting atx=50.
(ii) Construct a table of values ofA(x4) for x = 50, 50.25, 50.5,. . .. (Do not use UDD for this.)
(iii) Hence, write down the values ofA50,A100,A(504)andA(1004) .
(b) Use your values forA50andA100to estimateA(4)50andA(4)100using the UDD assumption.
(c) Compare your estimated values for theA(4)functions (from part (b)) with your accurate values (from part (a)). Comment on the differences.
Exercise 4.34 The force of mortality for a survival model is given by μx=A+BCxDx2,
where
A=3.5×10−4, B=5.5×10−4, C=1.00085, D=1.0005.
Use the repeated Simpson’s rule to calculate (a) tp60fort=0, 1/40, 2/40,. . ., 2, and (b) A¯1
60:2 using an effective rate of interest of 5% per year.
Answers to selected exercises
4.1 (a) 0.735942 (b) 0.012656 (c) 0.138719 (d) 0.748974 4.2 (a) 0.00645 (b) 0.38163 (c) 0.05907
4.5 0.012494 4.6 0.01 4.7 0.887277 4.9 0.59704
4.10 (a) 7 186 (b) 6 226 4.11 (a) 5 507.44 (b) 5 507.46 4.13 56 270
4.14 0.3120 4.15 33 569.47 4.17 81 873
4.18 (a) 0.79267 (b) $7 519.71 (c) 0.99825
4.19 (a) EPV=190 693, SD=123 938 (b) EPV=190 718, SD=124 009
4.20 (a) 44.492 (b) 43 657
4.21 (f) SD[Z1]=1.6414, SD[Z2]=0.3085 4.22 (c) 15 972 (d) 0.0448 (e) 14 765 4.23 (a) 0.004896 (b)−0.004896 4.25 E[Z3]=281 SD[Z3]=258.33 4.27 (a) $218.88 (b) $239.88 (c) 0.04054 4.30 (c) 5.07307
4.32 (a) 2196.19 (b) 2394.18
4.33 (a)(iii) 0.33587, 0.87508, 0.34330, 0.89647 (b) 0.34333, 0.89453
4.34 (a) Selected values are 1/4p60 = 0.999031, p60 = 0.996049 and
2p60 =0.991885 (b) 0.007725