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程序代写案例-MATH 3281

时间：2021-04-12

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-

tions carefully. Fill in answers in designated spaces. Your work must

justify the answer you give. Answers without supporting work will

not be given credit.

5. Hand in your examination answer booklets.

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

insurer’s premium defining equation, or the insured’s premium defining equation) that

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

all contract premiums without interest. The return-of-premium feature applied both

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.

学霸联盟

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-

tions carefully. Fill in answers in designated spaces. Your work must

justify the answer you give. Answers without supporting work will

not be given credit.

5. Hand in your examination answer booklets.

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

insurer’s premium defining equation, or the insured’s premium defining equation) that

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

all contract premiums without interest. The return-of-premium feature applied both

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.

学霸联盟