Final Test MATH 3281 3.00 April 12, 2018
Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This exam is a ‘closed book’ exam, duration 180 minutes.
3. Only non-programmable calculators are permitted.
4. The text has 21 pages, and it contains 20 questions. Read the ques-
not be given credit.
Useful formulas:
1.) For t ∈ [0, 1), and non-negative integer k, we have that
P[T (u) ≥ k + t] UDD= (1− t)P[T (u) ≥ k] + tP[T (u) ≥ k + 1].
2.) For ∆f(k) := f(k + 1)− f(k), k ≥ 0, it holds that
n∑
k=m
f(k)∆g(k) =
[
f(k)g(k)|n+1m
]− n∑
k=m
g(k + 1)∆f(k).
GOOD LUCK!
Final Test MATH 3281 3.00 Page 2 of 21
Question 1 The random variable Z is the present value random variable for an n-year endowment
insurance. The insurance is issued to (x), and its n-year term part is payable upon
death. Let n = 20. If the force of interest is δ = 0.05 and the force of mortality is
µ(x + t) ≡ 0.01. Formulate the cumulative distribution function of Z and graph it.
Calculate Ax:n := E[Z] using the cumulative distribution function of Z.
Cont.
Final Test MATH 3281 3.00 Page 3 of 21
Question 2 Let Z1 and Z2 denote, respectively, the n-year term and the pure endowment insurance
coverages bearing \$1 face amounts and issued to (x). Assume that the n-year term
insurance is payable upon death. Show that
lim
n→0
Cov[Z1, Z2](n) = lim
n→∞
Cov[Z1, Z2](n) = 0.
Explain.
Cont.
Final Test MATH 3281 3.00 Page 4 of 21
Question 3 For t > 0, let b(t) = t, µ(x + t) ≡ µ and the force of interest be constant δ > 0, say.
Also, let T (x) denote the future life-time random variable of (x). Derive expressions
for (IA)x := E[b(T )v
T ] as well as the variance of the random variable b(T )vT ,
Cont.
Final Test MATH 3281 3.00 Page 5 of 21
Question 4 Recall that the Makeham law of mortality is given by the following special form of the
force of mortality
µ(x) = A+Bcx, x > 0, (1)
where A ≥ −B, B > 0, c > 1. Assume independence of the random future life-times of
(x) and (y). Also, assume that the corresponding forces of mortality follow Makeham
equation (1). Show that ax:y is equal to the actuarial present value of an annuity on a
single life (w) such that cw = cx + cy and the force of interest δ′ = δ + A. Use these
findings to demonstrate that
Ax:y = A

w + Aa

w,
where the primed functions are evaluated at the force of interest δ′.
Cont.
Final Test MATH 3281 3.00 Page 6 of 21
Question 5 Let I = 1 represent death by accidental means and I = 2 represent death by other
means. You are given that
• δ = 0.05;
• the force of decrement for death by accidental means is µ(1)(x + t) = 0.005 for
t ≥ 0;
• the force of decrement for death by other means is µ(2)(x+ t) = 0.02 for t ≥ 0.
A 20-year term insurance policy payable at the moment of death is issued to (x)
providing a benefit of \$2 if the death is by accidental means, and providing a benefit
of \$1 if the death is by other means. Find the expectation and the variance of the
present value of the benefits random variable.
Cont.
Final Test MATH 3281 3.00 Page 7 of 21
Question 6 Verify the formula:
δ(Ia)T (x) + T (x)v
T (x) = aT (x) ,
where T (x) represents the future life-time random variable of (x). Use it to prove that
δ(Ia)x + (IA)x = ax.
Here (Ia)x is the actuarial present value of a life annuity to (x) under which payments
are being made continuously at the rate of \$t per annum at time t > 0.
Cont.
Final Test MATH 3281 3.00 Page 8 of 21
Question 7 If g(X) is a non-negative function, and X is a non-negative random variable with
probability density function f(x), and if the expectation below is well-defined and
finite, justify the inequality
E[g(X)] =
∫ ∞
−∞
g(x)f(x)dx ≥ kP[g(X) ≥ k], k > 0.
Then use the above to show that
ax ≥ a◦
ex
P
[
aT (x) ≥ a◦ex
]
= a◦
ex
P
[
T ≥ ◦ex
]
,
where T (x) denotes as always the future life-time random variable of (x).
Cont.
Final Test MATH 3281 3.00 Page 9 of 21
Question 8 Consider the following portfolio of annuities-due currently being paid by a pension
fund
Age Number of annuitants
65 30
75 20
85 10
Each annuity has an annual payment of \$1 as long as the annuitant survives. As-
sume an earned interest rate of 0.06 and the Illustrative Life Table mortality. For
the present value of these obligations of the pension fund, calculate the expectation,
the variance, and the 95% percentile of the distribution. Assume that the lives have
mutually independent future life-times whenever required.
Cont.
Final Test MATH 3281 3.00 Page 10 of 21
Question 9 Assume that T (x) and T (y) are positively quadrant dependent future life-time random
variables, that is we have that
P[T (x) ≤ u, T (y) ≤ v] ≥ P[T (x) ≤ u]P[T (y) ≤ v]
for any u, v ≥ 0. Denote by ax:y the joint whole life annuity that is calculated under the
assumption of positively quadrant dependent future life-times of (x) and (y). Denote by
a⊥x:y the joint whole life annuity that is calculated under the assumption of independent
future life-times of (x) and (y). Check that
ax:y ≥ a⊥x:y.
Cont.
Final Test MATH 3281 3.00 Page 11 of 21
Question 10 If the force of mortality strictly increases with age, show that P (Ax) > µ(x), that is
the benefit premium is greater than the force of mortality evaluated at zero.
Cont.
Final Test MATH 3281 3.00 Page 12 of 21
Question 11 Define the notion of actuarial premium calculation principle. Show (using either the
every actuarial premium must be at least as large as the mathematical expectation of
the underlying risk random variable. State the condition for the feasibility of insurance
contracts.
Cont.
Final Test MATH 3281 3.00 Page 13 of 21
Question 12 A 20 payment life policy is designed to return, in the event of death, \$10, 000 plus
during the premium paying period and after. Premiums are annual, and death benefits
are paid at the end of the year of death. For a policy issued to (x), the annual contract
premium is to be 110% of the benefit premium plus \$25. Express in terms of the
actuarial present value symbols the annual contract premium.
Cont.
Final Test MATH 3281 3.00 Page 14 of 21
Question 13 Let T (x) ∼ Exp(µ), µ > 0. Derive the probability density function of the Loss-at-Issue
random variable L in terms of the probability density function of the random variable
T (x). Show that E[L] = 0 when the benefit premium is chosen to price the whole life
insurance on (x). Determine the premium pi such that P[L > 0] = 0.5.
Cont.
Final Test MATH 3281 3.00 Page 15 of 21
Question 14 A whole life insurance issued to (25) pays a unit benefit at the end of the year of death.
Premiums are paid annually to the age 65. The benefit premium for the first 10 years
is P25 followed by an increased level annual benefit premium for the rest 30 years. Use
the Illustrative Life Table to find the following
• The level annual benefit premium paid at ages 35 through 64.
• The tenth year benefit reserve.
• At the end of 10 years the policyholder has the option to continue with the benefit
premium P25 until age 65 in return for reducing the death benefit to \$y for the
death after age 35. Calculate y.
Cont.
Final Test MATH 3281 3.00 Page 16 of 21
Question 15 A fully discrete whole life insurance with a unit benefit issued to (x) has its first year’s
benefit premium equal to the actuarial present value of the first year’s benefit, and
the remaining benefit premiums are level and determined by the equivalence princi-
ple. Determine formulas for (a) the first year’s benefit premium, (b) the level benefit
premium after the first year, and (c) the benefit reserve at the first duration.
Cont.
Final Test MATH 3281 3.00 Page 17 of 21
Question 16 Consider the life insurance policy from Question 16. Calculate Cov[C0, Ch]. Repeat
your calculations for Cov[Cj, Ch], where 0 < j < h.
Cont.
Final Test MATH 3281 3.00 Page 18 of 21
Question 17 Let hLx(K(x);Px) represent the prospective loss random variable at time h = 0, 1, . . .
for a whole life insurance with unit benefit, premiums payable due if the insured is
alive, and the benefit payable upon death. Show that there exists random variable hCx
such that the identity
hLx =
∞∑
j=h
vj−hjCx
holds.
Cont.
Final Test MATH 3281 3.00 Page 19 of 21
Question 18 Under the assumption of the uniform distribution of death in each year of age, prove
or disprove
kv(Ax:n ) ≈ i
δ
kv(Ax:n ).
Cont.
Final Test MATH 3281 3.00 Page 20 of 21
Question 19 If 10v35 = 0.150 and 20v35 = 0.354, then what is the value of 10v45?
Cont.
Final Test MATH 3281 3.00 Page 21 of 21
Question 20 Bonus. LetX represent a random variable with finite second moment. It is well-known
that the variance of the random variable X is given, for µX := E[X], by
Var[X] = E[X2]− µ2X ,
and so the variance depends on the mean of X. This is somewhat misleading because
it is not clear why a good measure of variability should depend on the mean. Show
and prove an additional formulation of the variance that emphasizes that Var[X] in
fact does not depend on the mean of the random variable X.
The End. 