Exercises in Life Insurance Mathematics

Exercises in Life Insurance Mathematics 

Edited by 

Bjarne Mess 

Jakob Christensen 

University of Copenhagen 

Laboratory of Actuarial Mathematics

Introduction 

This collection of exercises in life insurance mathematics replaces the collection of 

Steen Pedersen and all other exercises and problems in any text or article in the 

FM0L curriculum. 

The following abbreviations are being used for the contributors of exercises: 

AM Bowers et al. “Actuarial Mathematics”, 

Society of Actuaries, Itasca, Il 1986 

BM Bjarne Mess 

BS Bo Søndergaard 

FW Flemming Windfeld 

JC Jakob Christensen 

JH Jan Hoem “Elementær rentelære”, 

Universitetsforlaget, Oslo, 1971. 

MSC Michael Schou Christensen 

MS Mogens Steffensen 

RN Ragnar Norberg 

SH Svend Haastrup 

SK Stephen G. Kellison “The theory of Interest”, 

Richard D. Erwing, Inc., Homewood, Il 1970 

SP Steen Pedersen “Opgaver i livsforsikringsmatematik” 

SW Schwartz “Numerical Analysis”, Wiley, 1989 

Copenhagen, August 2, 1997 

Bjarne Mess 

Jakob Christensen

4 Exercises in Life Insurance Mathematics 

Exercises 

1. Interest 

FM0 S91, 1 FM0 S92, 1 FM0 S93, 1 

FM0 S93, 2 FM0 S94, 1 

Exercise 1.1 Show that an| < an| < än| when i > 0. 

Exercise 1.2 Show that an|/n decreases as n increases and i > 0. 

Exercise 1.3 Show that an|(i1)/an|(i2) decreases as n increases if i1 > i2. 

(JH(1), 1971) 

(JH(2), 1971) 

(JH(3), 1971) 

Exercise 1.4 A man needs approximately $2.500 and raises in that connection a 

loan in a bank. The principal, which is to be fully repaid after 6 months, is $2.600. 

From this amount the bank deducts the future interest $84,50 and other fees of $5,70, 

so that the bank pays out the man $2.509,80 cash. The interest rate of the bank is 

6.5%. 

(a) What is the effective interest rate p. a. for the bank? 

(b) What is the effective interest rate p. a. for the borrower? 

(JH(4 rev.), 1971) 

Exercise 1.5 One day a company receives an american loan offer: Principal of 

$5.000.000, rate of course 99% and nominal interest rate 6.5%. The loan is free of 

installments for 5 years and is after that to be amortized over 15 years: Interests 

and installments are due annually. Assume that the dollar-rate of exchange decreases 

exponentially from DKK 6,75 at the initial time to DKK 4,75 at the end of the loan. 

How can one determine the effective interest rate of the loan?

Interest 5 

(SP(19)) 

Exercise 1.6 By calculation of the interest rate for a fraction of a year, a bank will 

usually calculate with linear payment of interest instead of exponential payment of 

interest. If the interest rate is i p. a. and we have to calculate interest for a period 

of time α (0 < α < 1), the bank will calculate interest as αi per kr. 1. – in capital, 

instead of calculating the interest as 

(1 + i) α − 1 

per. kr. 1. – in capital. Is this for the benefit of the borrower? 

(JH(7), 1971) 

Exercise 1.7 A man is going to buy new furniture on an installment plan. In the 

hire-purchase agreement he finds the following account: 

What effective interest rate p. a. 

Cash payment for the furniture $6.306,00 

− In advance to the salesman $2.217, 00 

= Net balance $4.089,00 

+ Installment fee for 18 mths. $501, 00 

= Net balance for installment $4.590, 00 

The monthly installments are $255, 00 

(JH(10), 1971) 

Exercise 1.8 A man has been promised some money. He can choose from two 

alternatives for the payment. 

Under alternative (i) A5 = $4.495 and A9 = $5.548 are paid out after 5 and 9 years 

respectively. 

Under alternative (ii) B7 = $10.000 is paid out after 10 years. 

Denote the market interest rate by i. 

For which value (values) of i is (i) just as good as (ii), and when is (i) more profitable 

for the man? What if A9 = $5.562, i. e. $14 more? 

(JH(11), 1971)


Exercise 1.9 A says to B: “I would like to borrow $208 in one year from today. 

In return for your kindness I will pay $100 cash now, and $108,15 in two years from 

today by the end of the loan.” 

What is the effective interest rate p. a. for B if he accepts this? 

(JH(12), 1971) 

Exercise 1.10 Consider a usual annuity loan with principal H, interest rate i and 

n installments. Show that the installment which falls due in period t is 

Ft = 

i(1 + i)t−1 

(1 + i) n − 1 H, 

and find an expression for the remaining debt immediately after this period. 

(SP(6)) 

Exercise 1.11 Consider a linearly increasing annuity. At time t = 1, 2, . . . the 

amount t is being paid. The present value of this cash flow is denoted by Ian |. 

(a) Show that 

and interpret this equation intuitively. 

n−1 

Ian| = 

(b) Give an explicit expression for Ian|. 

t|an−t|, t=0 

(c) What does the symbol Ian| mean? Give expressions corresponding to the ones 

from (a) and (b). 

(SP(10)) 

Exercise 1.12 A person has a table of annual annutities with different interest rates 

and durations to his disposal. However, he needs some present values of half-year 

annuities. These annuities do all have the same interest rate and the corresponding 

whole-year annuities can be found in the table. 

What is the easiest way to find the desired annuities? 

(SP(11))

Interest 7 

Exercise 1.13 A debtor is going to pay an amount of 1 some time in the future. 

He does not know this point of time in advance; he only knows that it is a stochastic 

variable T with a known distribution. Now consider the expected present value 

denoted by 

A δ 

(T ) = E(e −δT ). 

(a) Show that the variance of the present value is given by 

Var(e −δT ) = A 2δ 

(T ) − (Aδ (T ) )2 . 

Now assume that creditor has to pay a continuous T -year annuity. The present value 

. Let the expected present value be denoted by 

is a δ 

T | 

(b) Show that 

and interpret the equation. 

a δ 

(T ) = E(aδ T | ). 

A δ 

(T ) = 1 − δa δ 

(T ) 

(c) Give an expression for the variance Var(aδ T | ) corresponding to the one from (a). 

(d) Show that 

A δ 

(T ) > v ET , 

and find a similar inequality for a δ (T ). (Hint: Use Jensen’s inequality.) 

(e) Find at least two situations where these considerations are relevant. 

(SP(16)) 

Exercise 1.14 Consider a loan, principal H, nominal interest rate i1, rate of course 

k and installment Ft in period t, where t = 1, . . . , N. Show that the effective interest 

rate ie satisfies 

where 

ie = 

S = 1 

H 

1 − S 

k − S i1 

N 

t=1 

Ftv t 

e . 

(SP(20)) 

Exercise 1.15 A loan with principal H and fixed interest rate i1 has to be amortized 

annually over a period of N years. The borrower can each year deduct half the interest


expenses on his tax declaration. Construct the installment plan in a way so that the 

amount of amortisation minus deductible (actual net payment) will be the same in 

all periods (assume tax is payable by the end of each year). Find the annual net 

installment. 

(RN “Opgaver til FM0 (rentelære)” 18.05.93) 

Exercise 1.16 Consider a general loan. Show that for t ≥ 1 we have 

At = (1 + i1)Rt−1 − Rt 

Rt = 

n−t 

j=1 

v t 

1Rt = H − 

At+jv j 

1 

t 

j=1 

Ajv j 

1 

(1.1) 

(1.2) 

with the use of standard notation. Formula (1.1) is called the prospective formula for 

the remaining debt. Formula (1.2) is called the retrospective formula for the remaining 

debt. 

(JH(20), 1971) 

Exercise 1.17 A loan is being repaid by 15 annual payments. The first five installments 

are $400 each, the next five $300 each, and the final five are $200 each. Find 

expressions for the remaining debt immediately after the second $300 installment – 

(a) prospectively, 

(b) retrospectively. 

(SK(1) p. 122, 1970) 

Exercise 1.18 A loan of $1.000 is being repaid with annual installments for 20 years 

at effective interest of 5% . Show that the amount of interest in the 11th installment 

is 

50 

. 

1 + v10 (SK(10) p. 123, 1970) 

Exercise 1.19 A borrower has mortgage which calls for level annual payments of 1 at 

the end of each year for 20 years. At the time of the seventh regular payment he also

Interest 9 

makes an additional payment equal to the amount of principal that according to the 

original amortisation schedule would have been repaid by the eighth regular payment. 

If payments of 1 continue to be made at the end of the eighth and succeding years 

until the mortgage is fully repaid, show that the amount saved in interest payments 

over the full term of the mortgage is 

1 − v 13 . 

(SK(16) p. 124, 1970) 

Exercise 1.20 A man has some money invested at an effective interest rate i. At 

the end of the first year he withdraws 162.5% of the interest earned, at the end of 

the second year he withdraws 325% of the interest earned, and so forth with the 

withdrawal factor increasing in arithmetic progression. At the end of 16 years the 

fund exhausted. Find i. 

(SK(40) p. 127, 1970) 

Exercise 1.21 A loan of a 25| is being repaid with continuous payments at the annual 

rate of 1 p. a. for 25 years. If the interest rate i is 0.05, find the total amount of interest 

paid during the 6th through the 10th years inclusive. 

(SK(42) p. 127, 1970) 

Exercise 1.22 After having made six payments of $100 each on a $1.000 loan at 

4% effective, the borrower decides to repay the balance of the loan over the next five 

years by equal annual principal payments in addition to the annual interest due on 

the unpaid balance. If the lender insists on a yield rate of 5% over this five-year 

period, find the total payment, principal plus interest, for the ninth year. 

(SK(45) p. 127, 1970) 

Exercise 1.23 A student has heard of a bank that offers a study loan of L = 10.000 

kr. The rate of interest is 3% p. a. and the student applicates for the loan on the 

following conditions: 

(i) The first m = 5 years he will only pay an interest of 300 kr. per year. 

(ii) After that period of time he will pay interests and installments of 900 kr. per year 

until the loan is fully amortized (the last installment may be reduced). 

(a) For how long N does he have to pay installments and how big is the last amount 

of amortisation?


(b) Which amount αt has to be paid at time t if the loan (with interest earned) is to 

be fully paid back at time t (t = 1, 2, . . . , N)? 

(Aktuarembetseksamen, Oslo 1960)

Aggregate Mortality 11 

2. Aggregate Mortality 

Exercise 2.1 Let T be a stochastic variable with distribution function F . Assume F 

is concentrated on the interval [a, b] and that F is continuous with continuous density 

f. Assume F (t) < 1 for t ∈ [a, b). Define 

We say that µ is the intensity of F . 

(a) Show that b 

a µ(t)dt = ∞. 

µ(t) = f(t) 

, t ∈ [a, b). 

1 − F (t) 

(b) Can we conclude that µ(t) → ∞ for t → b − ? 

Let T be the life length of a newly born. Let a = 0 and b = ω where ω is the maximum 

life length. 

(c) Show that µ is the force of mortality. 

(HRH(1)) 

Exercise 2.2 Use the decrement tables of G82M to find the following probabilities: 

(a) The probability that a 1 year old person dies after his 50th year, but before his 

60th year. 

(b) The probability that a 30 year old dies within the next 37 years. 

(c) The probability that two persons now 26 and 31 years old, and whose remaining 

life times are assumed to be stochastically independent, both are alive in 12 years. 

(SP(28)) 

Exercise 2.3 Explain why each of the following functions cannot serve in the role 

indicated by the symbol: 

F (x) = 1 − 22x 

12 

µx = (1 + x) −3 , x ≥ 0 

+ 11x2 

8 

7x3 

− , 0 ≤ x ≤ 3 

24


f(x) = x n−1 e −x/2 , x ≥ 0, n ≥ 1. 

(AM(3.4) p. 77, 1986) 

Exercise 2.4 Consider a population, where the distribution functions for a man’s 

and a woman’s total life lengths are xq M 0 and xq K 0 respectively. Assume that these 

probabilities are continuous so that the forces of mortality µ M x and µ K x are defined. 

Let s0 denote the probability that a newly born is a female. Assume moreover that s0 

and the forces of mortality are not being altered during the period of time considered 

in this exercise. 

(a) Find the distribution function xq0 for the total life time for a person of unknown 

sex and find tqx. Find the force of mortality µx for a person of unknown sex. 

(b) What is the probability sx that a person aged x is a woman? 

Using decrement series ℓM x and ℓKx for men and women respectively, work out a decrement 

serie ℓx for the total population: 

(c) How should one appropriately choose ℓ M 0 and ℓ K 0 ? 

(d) Express ax in terms of a M x and aK x . 

(SP(32)) 

Exercise 2.5 Consider a random survivorship group consisting of two subgroups: 

(1) The survivors of 1.600 births. 

(2) The survivors of 540 persons joining 10 years later at age 10. 

An excerpt from the appropriate mortality table for both subgroups follows: 

x ℓx 

0 40 

10 39 

70 26 

If γ1 and γ2 are the numbers of survivors under the age of 70 out of subgroups (1) 

and (2) respectively, estimate a number c such that P (γ1 + γ2 > c) = 0.05. Assume 

the lives are independent.


(AM(3.13) p. 78, 1986) 

Exercise 2.6 When considering aggregate mortality the probability that an x-year 

old person is going to die between x + s and x + s + t, is denoted by the symbol s|tqx. 

(a) Express this probability by the distribution function of the person’s remaining life 

time. 

(b) Is there a connection between s|tqx and tqx? 

(c) Show that 

and interpret this expression. 

When t = 1 we write s|qx = s|1qx. 

s|tqx = 

s+t 

(d) Show that for integer x and n we have 

s 

n|qx = dx+n 

upxµx+udu, 

(e) Show that s|tqx can be expressed similarly (use the function ℓ instead of d). 

(f) Prove the following identities: 

Exercise 2.7 Let e ◦ x:n| 

ℓx 

n|mqx = npx − n+mpx 

n|qx = npx · qx+n 

n+mpx = npx · mpx+n. 

. 

(SP(24)) 

denote the expected future lifetime of (x) between the ages 

of x and x + n. Show that 

e ◦ 

x:n| 

This is called the partial life expectancy. 

= 

= 

n 

ttpxµx+tdt + nnpx 

0 

n 

0 

tpxdt.


Exercise 2.8 The force of mortality µx is assumed to be 

µx = βc x . 

(AM(3.14) p. 78, 1986) 

Three persons are x, y and z years old respectively. What is the probability of dying 

in the order x, y, z? 

(Tentamen i försikringsmatematik, Stockholms Högskola 1954) 

Exercise 2.9 If F (x) = 1 − x/100, 0 ≤ x ≤ 100, find µx, F (x), f(x) and P (10 < 

X < 40). 

Exercise 2.10 If µx = 0.0001 for 20 ≤ x ≤ 25, evaluate 2|2q20. 

(AM(3.5) p. 77, 1986) 

(AM(3.7) p. 77, 1986) 

Exercise 2.11 Assume that the force of mortality µx is Gompertz-Makeham, i. e. 

µx = α+βc x . For at certain cause of death, the force of mortality is given by α1+β1c x . 

Show that the probability of dying from the above disease for an x-year old is 

β1 

β + α1β − αβ1 

β 

ex. 


Exercise 2.12 Show that constants a and b can be determined so that 

µx = a log(1 − qx) + b log(1 − qx+1). 

when µx can be put as a linear function for x < t < x + 2. 


Exercise 2.13 Assuming the force of mortality to be Gompertz-Makeham, i. e. 

µx = α + βc x , show that for each age x we have 

log c 

− 

c − 1 log(1 − qx) < µx < − log(1 − qx).


(Tentamen i försikringsmatematik, Stockholms Högskola, 1954) 

Exercise 2.14 Given that ℓx+t is strictly decreasing for t ∈ [0, 1] show that 

(a) if ℓx+t is concave down, then qx > µx, 

(b) if ℓx+t is concave up, then qx < µx. 

Exercise 2.15 Prove the following expressions: 

d 

dx ℓxµx < 0, when d 

dx µx < µ 2 

x 

d 

dx ℓxµx = 0, when d 

dx µx = µ 2 

x 

d 

dx ℓxµx > 0, when d 

dx µx > µ 2 

x. 

Exercise 2.16 Show the following identities: 

∂tpx 

∂t = −µx+t · tpx 

∂tpx 

∂x = (µx − µx+t) · tpx 

1 = 

ℓx = 

ω−x 

tpxµx+tdt 

0 

ω−x 

0 

ℓx+tµx+tdt. 

(AM, 1986) 

(AM(3.12) p. 77, 1986) 

(SP(26)) 

Exercise 2.17 If the force of mortality µx+t, 0 ≤ t ≤ 1, changes to µx+t − c where c 

is a positive constant, find the value of c for which the probability of (x) dying within 

a year will be halved. Express the answer in terms of qx. 

(AM(3.34) p. 80, 1986) 

Exercise 2.18 From a standard mortality table, a second table is prepared by 

doubling the force of mortality of the standard table. Is the rate of mortality, qx, at


any given age under the new table, more than double, exactly double or less than 

double the mortality rate, qx, of the standard table? 

(AM(3.35) p. 80, 1986) 

Exercise 2.19 If µx = Bc x , show that the function ℓxµx has a maximum at age x0, 

where µx0 = log c. (Hint: Exercise 2.15). 

Exercise 2.20 Assume 

for x > 0. 

(a) Find the survival function F (x). 

µx = Acx 

1 + Bc x 

(AM(3.36) p. 80, 1986) 

(b) Verify that the mode of the distribution of X, the age of death, is given by 

x0 = 

log(log c) − log A 

. 

log c 

(AM(3.37) p. 80, 1986) 

Exercise 2.21. (Interpolation in Life Annuity Tariffs) Consider a table with the 

present value a x:u−x| for an integer expiration age u with age at issue x = 0, 1, . . . , u 

and futhermore a table of the one year survival probabilities px for the same ages. Let 

t be a real number, 0 ≤ t < 1. We are trying to find a way to determine a x+t:u−x−t| 

from the data of the table; this method will, of course, depend on how the mortality 

varies with the age. 

Assume that the force of mortality is constant on one year age intervals, i. e. µx+t = µx 

for all t with 0 ≤ t < 1. 

(a) Find t−spx+s in terms of px for 0 ≤ s < t ≤ 1. 

(b) Find an expression for a x+t:1−t|. 

(c) Show that for every t there exists a λ so that 

a x+t:u−x−t| = λa x:u−x| + (1 − λ)a x+1:u−x−1|, 

and express λ in terms of the discount rate v, t and px. 

(d) How should one interpolate in a corresponding table for A x:u−x|.


(FM1 exam, summer 1983) 

Exercise 2.22 For insurances where the policies are issued on aggravated circumstances, 

one operates with excess mortality. Let µx be the force of mortality corresponding 

to the normal mortality. A person is said to have an excess mortality if his 

force of mortality is given by 

µ ′ 

x = (1 + k)µx. 

(a) Show that for all positive k, x and t we have 

tq ′ 

x < (1 + k)tqx. 

(b) Show that if there exists a constant ∆ so that µ ′ x = µx+∆ is valid for all x then 

a ′ 

x = ax+∆ 

for all x; in this case the insurance is issued with an increase of age ∆. 

(c) Show that the condition in (b) is fulfilled if the mortality satisfies Gompertz’s law, 

i. e. there exist constants β and γ so that µx = β exp(γx) for all x. 

(d) Show, oppositely, that if µx is strictly increasing in x and if there for any k ≥ 0 

exists a constant ∆k so that 

= (1 + k)µx, 

µx+∆k 

then the mortality satisfies Gompertz’s law. 

(SP(43))


Exercise 3.1 Prove the identities: 

3. Insurance of a Single Life 

FM0 S92, 3 

äx = 1 + vpxäx+1 

1 − nEx = äx:n| − ax:n| 

dax:n| 

dx 

= (µx + δ)ax:n| + nEx − 1. 

(SP(29)) 

Exercise 3.2 Let µx be a weakly increasing function of x and assume that µx → ∞ 

as x → ∞. 

(a) Show that ax → 0 as x → ∞. 

(b) Examine if Ax has a finite limit as x tends to infinity. 

(SP(31)) 

Exercise 3.3 Consider an insurance contract issued to an x-year old. At death 

within the first n years, the level continuous premium is paid back with interest and 

compound interest earned. Rewrite the integral expression for the present value after 

t years (t < h) of the future return of premium per unit of the premium in order to 

show that this value is 

(st| + ax+t:n−t|) − Dx+h 

sh| , 

Dx+t 

where s t| is defined by JH. Interpret the expression. 

Exercise 3.4 Show that 

(Eksamen i Forsikringsvidenskab og Statistik, KU, winter 1943-44) 

nEx = 1 − iax:n| − (1 + i)A 1 

x:n| , 

and interpret this formula (i is the interest rate). 

(Eksamen i Forsikringsvidenskab og Statistik (rev.), winter 1946-47)

Insurance of a Single Life 19 

Exercise 3.5 Assume there exist positive constants k and α, so that 

for all x ∈ [0, ω]. 

(a) Find an expression for µx. 

(b) Find an expression for e ◦ x 

Now assume that α = 1. 

ℓx = k(1 − x 

ω )α 

(see exercise 2.7). 

(c) Show that n|qx is independent of n. 

(d) Show that for n = ω − x we have 

ax = 

n − an| 

. 

nδ 

(SP(30)) 

Exercise 3.6 Ax:n| denotes the expected present value of a life insurance contract, 

where the amount of 1 is to be paid out by the end of the year in which the insured 

dies, not later than n years after the time of issue, or if he survives until the age of 

x + n. x is the age at entry. 

(a) Give an expression for Ax:n| and show that 

where d is the discount rate. 

A 1 x:n| 

Ax:n| = 1 − däx:n|, 

denotes the expected present value of a life insurance where the amount of 1 

is paid out by the end of the year, during which he dies if he dies before the age of 

x + n. x is the age at entry. 

(b) Give an expression for A 1 x:n| 

and show that 

A 1 

x:n| = 1 − nEx − däx:n|. 

(c) Try to interpret the formulas in (a) and (b). 

(SP(33)) 

Exercise 3.7 Assume that active persons have force of mortality µ a x as a function 

of age and force of disability νx as a function of age. Assume moreover that disabled


persons have force of mortality µ i x. There is no recovery. The force of interest is 

denoted by δ. 

The four quantities defined below are the single net premiums an insured with age 

at entry x has to pay for a level continuous annuity with sum 1 p. a. The insurance 

cancels n years after issue. 

a i x:n | 

a a x:n | 

a aa 

x:n | 

a ai 

x:n | 

The single net premium for an insured who is disabled at the time of issue. The 

annuity is payable from issue until the time of death of the insured. 

Single net premium for active persons. The annuity is payable from the time of 

issue until death of the insured. 

As above except that the premium is due for a contract that cancels by death 

or by disability of the insured. 

Single net premium for an active. The annuity is payable if the insured is being 

disabled within n years from time of issue. Expires if he dies. 

(a) Express aai x:n| in terms of µax , νx, µ i x and δ. 

(b) Assume that µ a x = µi x 

for all x. Express aai 

x:n| 

in terms of aaa 

x:n| and ai x:n| . 

(c) Assume that µ a x = µ i x + ε and νx = ν where ε and ν are independent of x and 

ε = ν. Express a ai 

x:n| 

by aaa 

x:n| and ai x:n| 

(and ε and ν). 

(Aktuarembetseksamen i Oslo (rev.), fall 1953) 

Exercise 3.8 Assume the force of mortality is a strictly increasing function of the 

age, when this is greater than or equal to a certain x0. 

Show that for x ≥ x0 and 0 < n ≤ ∞ the following inequalities hold: 

ax:n| < 1 − vn npx 

µx + δ , 

∂ax:n| 

∂x 

< 0, 

ax < 1 

δ . 

(SP(37))


Exercise 3.9 The quantity 

e ◦ 

x:n| = 

n 

tpxdt 

0 

is the expected period of insurance for a term insurance or an n-year temporary 

annuity, age of entry x and age of expiration x + n. 

Define 

ex:n| = 

ëx:n| = 

n 

tpx 

t=1 

n−1 

tpx. 

t=0 

Give a similar interpretation of these identities. Define 

a t| = 

ä t| = 

1 − vt 

i 

1 − vt 

d 

and show that for integer values of t the following inequalities are valid: 

Exercise 3.10 Show that 

ax:n| < ae x:n|| 

äx:n| < ä ëx:n|| . 

ax = 

∞ 

0 

tpxAx+tdt. 

(SP(39)) 

(SP(41)) 

Exercise 3.11 Assume that µx is a weakly increasing function of x and consider 

for given x two insurance contracts with initial age x: First consider a whole-life life 

insurance with sum insured 1 and secondly a whole-life continuous annuity with level 

payment intensity determined so that the expected present value of the two insurance 

contracts are equal. The one with the biggest variance of the present value is naturally 

the one with the biggest risk for the company. 

Show that there exists an x0 ≥ 0 so the annuity is more risky than the life insurance 

iff x > x0.


(SP(42)) 

Exercise 3.12. (Multiplicative Hazard Model) The mortality in a population varies 

from person to person; some has greater or lesser mortality than the average. This 

can be modelled as follows: 

There exists a underlying force of mortality µx and for each person a constant θ 

independent of age exists so that the force of mortality for a person aged x is θµx. 

The value of θ for a randomly chosen person is assumed to be a realisation of a 

stochastic variable Θ, and we assume moreover that EΘ = 1. 

Show that in this model the expected present value of a continuous temporary n-year 

annuity with payment intensity 1 is greater than or equal to 

ax:n| = 

n 

0 

e −δt tpxdt, 

and examine under which conditions the two present values are equal. 

(SP(44)) 

Exercise 3.13 Consider an n-year endowment insurance, sum insured S, age at 

entry x and premium paid continuously during the entire period with level intensity 

p. Expenses are disregarded. 

(a) What is the surplus of this contract for the company in terms of the remaining 

life time of the insured? 

(b) Find the mean and variance of the surplus. 

(c) Explain how p should be determined so that the probalility of getting a negative 

surplus is lesser than a certain ε. (Hint: Tchebychev’s inequality.) 

(d) Show (by applying the central limit theorem) how it is possible to obtain a probability 

of a negative surplus for the entire portfolio lesser than ε by using a smaller p 

than the one found in (c). 

(SP(50)) 

We have so far worked with continuous insurance benefits - annuities that are due 

continuously and life insurances that are due upon death. The pure endowment seems 

to be of another origin, because the time of possible single payment is determined in 

advance. In the next exercise we will consider more general kinds of non-continuous 

or discrete benefits. For at start consider an x-year old whose remaining life time T


is determined by the survival function 

F (t | x) = e − 

∞ 

0 µx+τ dτ , 

where µx+t is the force of mortality at the age of x + t, t > 0. As usual let v denote 

the annual discount rate. 

The results of the following exercise will show that continuous benefits can be concidered 

as limits for discrete benefits. We will also see that both continuous and discrete 

annuities and life insurances are closely related to pure endowment benefits. 

Exercise 3.14 The present value of a t-year pure endowment with sum 1 is 

C e 

t = v t 1{T >t}. 

(a) Find the expectation, tEx, of C e n and find Cov(C e s , Ce t ) for s = t. 

A brute-forth generalisation of the pure endowment is produced by summing more 

benefits like this. A simple variant is the n-year temporary deferred annuity payable 

annually with fixed amounts as long as the insured is alive. This is the sum of n pure 

endowments with deferment times 1, . . . , n. The present value at time t = 0 is 

C a(1) 

n 

= 

If the annuity is payable h times a year with fixed amounts 1 , the present value will 

h 

be 

C a(h) 

n 

= 

n 

t=1 

hn 

t=1 

C e 

t . 

C e 

t/h. 

(b) Find the expectation, a (h) 

a(h) 

x:n| , and the variance of the present value Cn . 

An n-year temporary life insurance with sum insured 1, payable at the end of the 

year of death, has present value 

C ti(1) 

n 

= 

n 

t=1 

v t 1{t−1


(c) Express the present value C ti(h) 

n 

Ca(h) n 

in terms of present values of annuities given by 

. Compare with similar expressions for continuous benefits. 

(d) Find the expectation, A (h) 

ti(h) 

x:n| , and the variance of the present value Cn . 

(e) Use the results from (b) and (d) to prove the well known formulas 

ax:n| = 

A 1 

x:n| 

= 

n 

0 

n 

0 

v t F (t | x)dt = 

v t F (t | x)µx+tdt 

= 1 − δax:n| − nEx, 

n 

0 

tExdt 

for the expectations of C a ti 

n and Cn and also to find their variances. (Hint: By monotone 

convergence we have C α(2m ) 

n ↗ Cα n as m → ∞, α ∈ {a, ti}. Then use the 

monotone convergence for the expectation). 

(f) Use the technique in (e) to find formulas for the expectations and variances of 

continuous benefits in the usual Markov model. Consider an annuity, payable with 

level intensity of 1 by staying in state j, and an insurance where a sum of 1 is paid 

upon every transition j → k. 

(RN “Opgave E7” 29.01.90) 

Exercise 3.15 The functions that occur in insurance mathematics often depend on 

several variables, e. g. m|ax:n|, and are often hard to tabulate. In order to solve this 

problem, we introduce the so-called commutation functions. In connection with life 

insurances of one life we consider the following: 

Cx = vxdx Cx = x+1 

x vξℓξµξdξ Dx = vxℓx Dx = x+1 

x vξℓξdξ Mx = ω ξ=x vξdξ M x = ω 

x vξℓξµξdξ Nx = ω ξ=x vξℓξ N x = ω 

x vξℓξdξ Rx = ω ξ=x (ξ − x)vξdξ Rx = ω 

x (ξ − x)vξℓξµξdξ Sx = ω ξ=x (ξ − x)vξℓξ Sx = ω 

x (ξ − x)vξℓξdξ. (a) Show that 

m|ax:n| = N x+m − N x+m+n 

. 

(b) Find corresponding expressions for ax:n|, A 1 

x:n| , Ax, Ax:n|, nEx, ax:n| and äx:n|. 

(c) What can we possibly use Rx for? 

Dx


(SP(35))


4. The Net Premium Reserve and Thiele’s Differential Equation 

FM0 S91, 1 FM0 S92, 2 FM0 S93, 2 FM0 S94, 2 FM0 S95, 1 

Exercise 4.1 Consider an n-year pure endowment, sum insured S, premium payable 

continuously during the insurance period with level intensity π. Upon death two 

thirds of the premium reserve is being paid out. 

(a) Put up Thiele’s differential equation for Vt. What are the boundary conditions? 

(b) Find an expression for the premium reserve at time t, t ∈ [0, n) 

(c) Determine the premium intensity π by adopting the equivalence principle. 

(SH and MSC, 1995) 

Exercise 4.2 We have the choice of two different premium payment schemes. 

• For an insurance of a single life a level continuous premium is due with force 

p as long as the insured is alive at the most for n years from the issue of the 

contract. 

• Every year an annual premium of the size 

p (1) = p · a 1| 

is paid in advance. If the insured dies during the insurance period the return of 

premium is 

R = p (1) · a θ| 

a 1| 

= p · a θ| , 

where θ denotes the remaining part of the year at time of death. 

(a) Show that these two premium payment schemes are equivalent in the manner 

that regardless of when the insured is going to die, the present values of the premium 

payments under the two schemes are equal. 

(b) Find the expected present value of the return of premium at the time of issue. 

Let the prospective reserves at time t from the time of issue of the two premium 

schemes be denoted by V t and Vt.

The Net Premium Reserve and Thiele’s Differential Equation 27 

(c) Show that for t ≤ n 

where [t] is the integer part of t. 

Vt = V t + p · a [t]−t| 

(SP(53)) 

Exercise 4.3 Consider a linearly increasing n-year term insurance. If the insured 

dies at time t after the time of issue where t < n, the amount tS is paid out. If the 

insured is alive at time n, the amount nS is paid out at this time. The age of the 

insured at entry is x. The net premium determined by the equivalence principle, is 

due continuously with level intensity p. 

(a) Find an expression for p. 

(b) Find a prospective and a retrospective expression for the reserve Vt at any time 

t, 0 < t < n. 

(c) Show that the two expressions found in (b) are equal for all t, 0 ≤ t < n. 

(d) Derive Thieles differential equation. 

(e) Find the savings premium and the risk premium. 

(SP(54)) 

Exercise 4.4 An n-year insurance contract has been issued to a person (x). The 

premium is composed of a single premium π0 at the beginning of the contract and 

by a continuous intensity (πt)t∈(0,n) as long as (x) is alive, at the most for n years. 

The benefits are a pure endowment, sum insured Sn at time n, a term insurance, sum 

insured St at time t ∈ (0, n), and a continuous flow with intensity (st)t∈(0,n) as long 

as (x) is alive during the insurance period. 

(a) Put up Thiele’s differential equation. 

(b) Find a boundary condition without assuming the equivalence principle. 

(c) Find a prospective expression for the premium reserve by solving the differential 

equation. 

(d) Adopt the equivalence principle and find an alternative boundary condition. 

(e) Find, by applying the new boundary condition a retrospective expression for the 

premium reserve.


Now assume that the benefits moreover consist of a pure endowment, sum insured S 

at time m (0 < m < n). 

(f) Has this altered Thiele’s differential equation? 

(g) Which extra boundary condition are now to be added in order to solve the differential 

equation? 

Assume that π0 = 0, πt = π for t ∈ (0, m) and that π is determined by the equivalence 

principle. 

(h) Find the net premium. 

(i) Find the premium reserve at any time. 

(SH “Opgave til 14/10-94” (rev.)) 

Exercise 4.5. (Prospective Widow Pension in Discrete Time) Consider a policy 

with widow pension, insurance period n years, issued to a man aged x and his wife 

aged y. During the insurance period the premium Π falls due annually in advance 

as long as both are alive. If the man dies, the benefit is a widow pension of sum 1 

paid out on every following anniversary of the policy during the insurance period if 

the widow is still alive. All expenses are disregarded. 

(a) Put up an expression for the premium reserve for this policy at its tth anniversary. 

(b) Show how the premium reserve at any time t can be expressed in terms of the 

reserve at time t + 1 for t = 0, 1, . . . , n − 1 so that the premium reserves can be 

calculated recursively. 

(c) Define the savings premium and the risk premium and find an interpretable expression 

for the latter. 

(FM1 exam, summer 1977) 

Exercise 4.6 Consider an n-year term insurance, sum insured S, age at entry x. 

Continuous premium during the entire period with level intensity p determined by 

the equivalence principle. 

(a) Find Thieles differential equation for the premium reserve Vt. 

(b) Which initial conditions would be natural to use for t = 0 and t = n − respectively? 

(c) Solve the differential equation with each of the initial conditions and compare the 

solutions.


(SP(55)) 

Exercise 4.7 If the insured dies before time n (from the time of issue) the benefit is 

a continuous annuity with force 1, duration m from the time of death. If he is alive 

at time n, the benefit is a similar annuity from this time and if he is still alive at time 

n + m he receives a whole-life annuity with intensity 1. There is a single net premium 

at the time of issue and the equivalence principle is adopted. 

(a) Find the single net premium. 

(b) Find the prospective premium reserve at any time. 

(c) Derive Thieles differential equation. 

Let Rc denote the risk sum at time t. 

(d) Prove that 

⎧ 

⎪⎨ (1 − v 

Rt = 

⎪⎩ 

m )ax+t:n−t| − n+m−t|ax+t (t < n) 

−n+m−t|ax+t 

(n ≤ t < n + m) 

(n + m ≤ t). 

In particular we have 

−ax+t 

R0 = (1 − v m )ax:n| − n+m|ax, 

and since 

lim 

n→∞ R0 = (1 − v m ax) > 0, 

we have R0 > 0 if only n is big enough. Assume that R0 > 0. Because Rt is a strictly 

increasing continuous function on [0, n] and Rn < 0, there exists a unique τ ∈ (0, n) 

so that Rτ = 0. 

(e) Show that this τ is determined by 

N x+τ = N x+n + N x+n+m 

. 

1 − vm (SP(56)) 

Exercise 4.8 Consider a pension insurance contract, where the benefit is an n-year 

annuity of 1 deferred m years (expected present value m|nax). Premium is paid with 

level intensity c during the deferment period (expected present value cax:m|). 

(a) What is the equivalence premium c and the development of the reserve when 

x = 30, m = 30, n = 20 and the technical basis is G82M, i. e. i = 0.045 and µx = 

0.0005 + 10 −4.12+0.038x .


(b) Do similar calculations as in (a) for an extended contract where k times the 

premium reserve is being paid out by possible death during the deferment period, 

k = 0.5, k = 1. 

(RN “Opgave til FM0” 15.10.93) 

Exercise 4.9 Consider a single-life status (x) with force of mortality µx. Define the 

premium reserve by Vt = E(U[t,∞) | T > t) as usual. 

(a) Show that the premium reserve always is right continuous. 

(b) Discuss under which conditions it is left continuous. 

Consider the following benefits at time t 

• stdt1{T >t}, annuities 

• St1{T ∈dt}, life insurances 

• Bt1{T >t}, pure endowment, 

and show that if the premium is being paid with level intensity π and there isno lump 

sum at time t then Vt is left continuous. 

(c) If there is a lump sum at time t, what does Vt − Vt− look like? 

NB: Assume that µx+t, st, St are continuous, and assume that there only exist a finite 

number of t’s where Bt = 0. 

(SH “Opgave til 13/10-95”) 

Exercise 4.10 Consider an insurance of a single life, age at entry x. At time t 

(t = 0, 1, 2, . . .) the premium Pt is being paid if the insured is still alive, and if he dies 

during [t − 1, t) then St is the benefit. The equivalence principle is adopted for the 

insurance and all expenses are disregarded. Let the premium reserve at time t be Vt 

and let the stochastic variable Gt be given by 

⎧ 

⎪⎨ 0 (the insured is dead at time t 

Gt = 

⎪⎩ 

− ) 

Vt + Pt − vSt+1 (the insured dies during the interval [t, t + 1)) 

Vt + Pt − vVt+1 (the insured is alive at time t + 1). 

Let the stochastic variable Y be the present value at time 0 of the company’s surplus 

of the insurance.


(a) Interpret Gt and show that 

(b) Show that 

Y = 

VarY = 

∞ 

v t Gt. 

t=0 

∞ 

Var(v t Gt). 

t=0 

G1, G2, . . . are not necessarily stochastically independent, but note that (b) is valid 

regardless of whether G0, G1, . . . are stochastically independent or not. 

(c) Show that Hattendorf’s Formula 

is valid. 

VarY = 

∞ 

tpxv 

t=0 

2t+2 px+tqx+t(St+1 − Vt+1) 2 

(SP(58)) 

Exercise 4.11 In this exercise we are to study an endowment insurance with return 

of premium paid at death before expiration. Consider a person of age x who wishes 

to buy an insurance with age of expiration x + n, and where the premium is paid 

continuously with level intensity p as long as he is alive during the insurance period. 

The lump sum S = 1 is paid if he is alive at age x + n and if he dies before that 

the premium paid so far will be returned with interest (basic interest i) earned. We 

disregard expenses. 

(a) Show that the variable payment at death is given by 

Bt = pa t|(1 + i) t . 

(b) Determine the continuous premium intensity p. 

(c) Find the net premium reserve Vt at time t, t ∈ [0, n). 

(d) Put up Thiele’s differential equation and determine the risk sum. 

(e) Comment on the results and evaluate whether or not you will recommend the 

insurance company to issue this kind of insurance. 

(FM1 exam, summer 1985)


Exercise 4.12 An n-year deferred whole-life annuity on the longest lasting life has 

been issued to two persons (x) and (y). Annual amount of 1 and level continuous 

premium on the longest lasting life with intensity π during the deferment period. The 

forces of mortality are denoted by µx and νy 

Put up a retrospective expression for the net reserve at time t assuming that both 

are alive. 

(Eksamen i Forsikringsvidenskab og Statistik (rev.), winter 1944-45) 

Exercise 4.13 A family annuity is an insurance contract of one life that assures 

payment of a continuous annual annuity from the possible death of the insured during 

the insurance period until the expiration of the contract after n years. Force of 

mortality µx, interest rate i, age at entry x. 

(a) Put up the formulas for the net premium reserves, prospectively and retrospectively, 

with level continuous premium payment π. Show that if the insured does not 

die during the insurance period the reserve will at least once become negative. 

(b) Discuss how the total premium reserve of this contract for a portfolio of identical 

contract issued at the same time will develop during the insurance period. Show that 

this reserve never becomes negative. 

(The students of 1946 had 10 hours to complete this exercise!) 

(Aktuarembetseksamen (rev.), Oslo fall 1946) 

Exercise 4.14 During construction of a technical basis with mortality of death and 

mortality of survival it is a problem that the premium for an insurance can depend 

on whether the insurance stands alone or it is combined with other insurances. 

This exercise describes the attempts made under construction of G82 in order to solve 

this problem of additivity. 

Consider an insurance 

issued against a single premium. 

Ax:n| + s · n|ax 

Thiele’s differential equation for the net premium reserve is 

∂Vx(t) 

∂t = 

 

δVx(t) − ˜µx+t(1 − Vx(t)) (0 < t < n) 

δVx(t) − s + ˜µx+tVx(t) (n < t) 

(4.1)


where ˜µx+t is the actual expected force of mortality. Introduce the first order forces 

of mortality µ and ˆµ which satisfy 

and replace (4.1) by 

∂Vx(t) 

∂t = 

 

µx+t < ˜µx+t < ˆµx+t, 

(δ + µx+t)Vx(t) − ˆµx+t (0 < t < n) 

(δ + µx+t)Vx(t) − s (n < t). 

(4.2) 

Thus we get a smaller increase of the reserve and we get a technical basis “on the safe 

side”. 

(a) Determine the single premium on the first order technical basis by solving (4.2). 

Consider the special case 

and let 

and 

ˆµx = (µx + g2)(1 + g1), 

µ ∗ 

x = µx + g2 

δ ∗ = δ − g2. 

(b) What will the single premium be in this case? Comment on the result. 

Now consider an educational endowment ax|n|. 

(c) Put up the differential equations (4.1) and (4.2) and solve (4.2) for the special 

case above. What is the problem in this case? 

Finally consider a survival annuity ax|y. 

(d) Answer the same question as in (c). 

(SP(99))


5. Expenses 

Exercise 5.1 Work out the details in RN in the case where α ′′ = β ′′ = γ ′′ = 0. 

(RN(1) “Expenses, gross premiums and reserves” 12.10.90 rev. 13.03.93) 

Exercise 5.2 Express V g 

t and c g in terms of V n 

t and c n and α ′ in the case where 

α ′′ = β ′′ = γ ′′ = γ ′′′ = 0. 


Exercise 5.3 Treat the case of a level annuity payable upon death in (m, n) against 

level premiums in (0, m). 


Exercise 5.4 Consider an n-year deferred whole-life annuity, age at entry x, payable 

with level continuous intensity S. Level gross continuous premium intensity p payable 

during the deferment period. The premium is determined by the equivalence principle. 

For now assume that the expenses are initial expenses αS, loading for collection 

fees due continuously with level intensity βp and administration costs also due continuously 

with level intensity γS. 

(a) Find p and the prospective gross premium reserve V g 

t . 

Because of inflation, loading for collection fees and administration expenses are paid 

with intensities βf(t)p and γf(t)S at time t. 

(b) Put up exspressions for p and V g 

t . 

(b) Find p and V g 

t when f(t) = 1 + kt and where f(t) = exp(ct). 

(SP(62)) 

Exercise 5.5 Consider a whole-life life insurance, sum insured 1, age at entry x, 

single net premium B. 

(a) Show that the expected effective interest rate for the insured is 

∞ 

0 

1 − 

B t 

tpxµx+tdt − 1.

Expenses 35 

Assume that the company has some initial expenses α, but no other expenses. 

(b) What is the expected effective interest rate? 

(SP(60)) 

Exercise 5.6 A simple capital insurance, sum insured S, duration n, pays out S at 

time n from the time of issue no matter if the insured is alive or not. 

(a) Put up an expression for the net payment for this insurance and explain why it is 

independent of the age at entry. 

A simple capital insurance only makes sense if it is not paid by a single payment (when 

dealing with insurance). Assume that the premium is paid continuously during the 

entire insurance period with level intensity p, but only if the insured is alive. The 

premium is determined by the equivalence principle. 

(b) What will the net premium be? 

In the gross premium p, initial expenses αS are included as well as loading for collection 

fees βp and administration expenses γS paid continuously. 

(c) Determine p. 

(d) Put up an expression for the prospective gross premium reserve, both when the 

insured is alive as well as when he is dead. 

(SP(61)) 

The following exercise examines what happens to the insurance technical quantities 

when we bring surrender into consideration. 

Exercise 5.7 Consider an n-year endowment insurance, age of entry x, benefits are 

S1 if one dies during the insurance period and S2 if one obtains the age of x + n. 

Life conditioned equivalence premium is paid continuously until time n (from the age 

of entry). Moreover assume that surrender can take place at any time during the 

premium payment period, and that the present value of the conventionally calculated 

gross premium reserve, liquidated by surrender, is positive. By surrender at time t 

the company pays out G(t). 

(a) Now disregard all expenses and assume that G(t) is lesser than or equal to the 

conventionally calculated (net) premium reserve at time t. Instead of using a conventional 

technique, the company could itself bring surrender into consideration and


into its own technical basis. Show that the equivalence principle then would lead to a 

premium P ′ ≤ P . Discuss conditions for P ′ = P and give a lower limit for how small 

P ′ can get when G(·) varies. Here and in the following it might be useful to study 

Thiele’s differential equation. 

Now assume that some administration costs are not neglectible. The expenses consist 

of the amount α in initial expenses, of γ in administration costs per time unit and 

of a fraction β of the actual annual gross premium P in loading for collection fees, P 

calculated conventionally. Upon surrender 100θ% of the gross premium reserve is paid 

out if the reserve is positive, 0 ≤ θ ≤ 1. By surrender where the gross premium reserve 

is positive, a fixed percentage of the reserve is deducted to cover the loss experienced 

by surrender where the gross premium reserve is negative. For now disregard expenses 

that fall upon surrender. This gross premium reserve is calculated without respect to 

surrender. 

(b) Show that you will get a lesser gross premium reserve if you bring surrender 

into consideration. Assume that θ is chosen so that the equivalence principle can be 

applied anyway. 

(c) Now assume that in the above situation a constant expense ξ is associated with 

the actual payment of G(t). If G(t) calculated in (b) is smaller than ξ, nothing is paid 

out by surrender. When the value upon surrender mentioned exceeds ξ the difference 

is paid out. Show that the actual gross premium reserve still will be lesser than the 

conventional when surrender is brought into consideration. Can θ still be determined 

so that the equivalence premium still can be applied? 

(FM1 exam (rev.), summer 1979) 

Exercise 5.8 It has been proposed that the administration expenses should be calculated 

as being proportional to the gross premium reserve instead of being proportional 

to the sum insured. Now consider an n-year endowment insurance, level continuous 

gross premium intensity p, sum insured S, age at entry x. Acquisition expenses αS. 

Loading for collection fees βp and γVt at time t, respectively. Vt denotes the gross 

premium reserve. Administration costs fall due continuously with level intensity γVt 

at time t. 

(a) Put up Thiele’s differential equation for Vt. 

(b) Solve the differential equation with initial conditions for t = 0 and t = n − and 

show that the solutions can be expressed by expected present values for annuities 

with another interest rate than the interest rate of the technical basis. 

(c) Determine the equivalence premium.

Expenses 37 

In G82 the interest rate is i = 5% p. a. but gross premiums and gross reserves are 

calculated with an interest rate of 4.5% p. a. 

(d) What is the corresponding value of γ? 

(SP(64)) 

Exercise 5.9. (Equipment Insurance) By an equipment insurance, sum insured S, 

insurance period n, the sum S is paid out at time n if the insured is still alive; if he 

dies during the insurance period, the company returns the up till now paid premiums 

without interest earned. The level gross premium intensity p is payable during the 

entire insurance period. The expenses are initial expenses αS, loading for collection 

fees due with continuous level intensity βp and continuous administration costs due 

with level intensity γS. 

Put up an expression for p applying the equivalence principle. 

(SP(66)) 

Exercise 5.10. (Child’s Insurance) A person aged x has been issued a child’s 

insurance: If the insured dies during [x, y) the gross premium is paid back with 

earned interest according to the technical basis. If he dies during [y, u) the sum S is 

immediately paid out and if he is alive at age u, then S is paid out. The level gross 

premium p, the administration costs γS and loading for collection fees βp fall due 

continuously during the insurance period. Acquisition expenses are αS. 

(a) Put up Thiele’s differential equation for this insurance. 

(b) Find the prospective gross premium reserve at any time during the insurance 

period. 

(c) What is the gross (equivalence) premium intensity, and what is the risk sum at 

any time with this premium. 

(SP(68)) 

Exercise 5.11 Consider an n-year endowment insurance, sum insured S, age at 

issue x, premium payable until time m, initial expenses αS, administration costs and 

loading for collection fees due during the entire insurance period continuously with 

level intensities γS and βp g respsctively, where p g is the level gross premium intensity. 

Assume that m ≤ n and γ < δ. 

(a) Give an expression for p g applying the equivalence principle and prove that it can


be cast as 

(b) Show that ˜ β > β and ˜γ > γ. 

p g = Ax:n| + ˜γax:n| 

(1 − ˜ S. 

β)ax:m| 

The numerator of the expression is the so-called passive with added sum, because it 

is produced from the net passive Ax:n| increased by the present value of ˜γ during the 

entire insurance period. This passive is denoted by A S 

x:n| . 

Let Vt be the net premium reserve at time t (calculated from the time of issue) and 

let V 1 

t be Vt increased by the reserve of the future administration costs. 

(c) Give expressions for Vt and V 1 

t . 

The company ought to set aside the reserve V 1 

t 

is set aside. 

but normally the reserve 

V 2 

t = SA S 

x+t:n−1| − (1 − ˜ β)p g a x+t:m−t| 

(d) Compare Vt, V 1 

t and V 2 

t and try to explain why one prefers to set aside V 2 

t instead 

of V 1 

t . 

If the insured wishes to surrender his contract at time t, the company pays him the 

surrender value of the contract Gt, which is the reserve V 1 

t less the part of the initial 

expenses that have not yet been amortized. 

(e) Show that the surrender value can be cast as 

Gt = S(A x+t:n−t| + γa x+t:n−t|) − (1 − β)p g a x+t:m−t|. 

The coefficient for S is the surrender value passive and is denoted by A g 

x:n| . 

(f) Find an expression for the difference between the net premium reserve and the 

surrender value and prove that for m = n it is α(S − Vt). 

If the insured wishes to cancel the payment of premiums without entirely to surrender 

the contract, it is called a premium free policy. The size of this policy is determined by 

letting the surrender value of the new policy equal the surrender value of the original 

policy at the time of change. 

(g) Give an expression for the sum of the premium free policy and show that its 

reserve at the time of change is lesser than the reserve of the original policy.

Expenses 39 

Assume that the annual gross premium is given by 

where ε is called the continuity factor. 

¨p g = εp g , 

(h) Explain why the surrender value and the reserve can be cast as 

and 

respectively. 

SA g 

x+t:n−t| − ζ ¨p g a x+t:m−t| 

SA S 

x+t:n−t| − η¨p g a x+t:m−t|, 

(SP(70 rev.)) 

Exercise 5.12 A married man considers a life insurance on the following conditions: 

(i) If he dies before time r from time of issue of the contract, the company has to pay 

a continuous pension with level intensity s in a period of time m. 

He furthermore considers a supplementary pension, also with level intensity s which is 

due to initiate right after the expiration of the pension (i). He considers two options: 

(ii) The supplementary pension is due as long as his wife lives. 

(iii) The supplementary pension is due as long as his wife lives, at the most until time 

n from the time of issue, n > m + r. 

As a second alternative he considers a survival annuity, also with payment intensity 

s. Here he considers two options: 

(iv) The annuity initiates if the man dies before time r from the time of issue and is 

due as long as his wife lives. 

(v) The annuity initiates if the man dies before time r from the time of issue and is 

due as long as his wife lives, at the most until time n from the time of issue, n > r. 

(a) Put up an expression for the single net premium for these five contracts. 

Assume that the above contracts are issued against an annual premium payment paid 

in advance as long as the man and his wife are alive, at the most until time r from 

the time of issue. If one of the two dies during the insurance period, the amount 

(a θ| /a 1|)P is being returned (in Danish: Ristorno), where θ is the remaining part of 

the last premium payment period and P is the term premium.


(b) How would you calculate the annual net premiums? 

(c) Put up an expression for the net premium reserve at any time during the insurance 

period for the last insurance (v) applying the recently described premium payment 

principles. 

When calculating the gross premiums, the company uses the following expense rates: 

Initial expenses αS, loading for collection fees β times the gross premium, administration 

costs due continuously with intensityand γ times the gross premium reserve 

at any time, and finally payment costs of ε times the amount paid out. 

(d) Find the continuous gross premium intensity, applying the equivalence principle. 

(e) Find the gross premium when the premium payment takes place as described 

before (b). 

(SP(76) rev.)

Select Mortality 41 

Exercise 6.1 

6. Select Mortality 

(a) What could be the meaning of the symbol s|tq[x]+u? 

(b) Put up an expression for 2|6q[30]+2 in terms of ℓ under the assumption that the 

period of selection is 5 years. 

(c) Express the following three quantities with one symbol: 

• The probability that a person now 50 years old who got insured 3 years ago dies 

between the ages of 58 and 59, presuming the period of selection is 5 years, 

• the probability that a new born dies between 67 and 72 years of age, 

• the number of deaths between the ages of 29 and 30 in the third year of an 

insurance portfolio, presuming the period af selection now is 3 years. 

(SP(25)) 

Exercise 6.2 In this exercise we will try to explain the presence of select mortality 

for a portfolio of insured and study its properties. 

For two functions f and g we shall use the obvious notation 

fg(t) = f(t)g(t), (f + g)(t) = f(t) + g(t). 

The portfolio is assumed to be divided between the two states active and disabled 

according to the figure below where the course of events is modelled by a Markov 

process {Xt}t≥0, and t is the age of the insured. 

σ(t) 

 

1. Active 

 

2. Disabled 

 

ρ(t) 

 

µ(t) 

ν(t) 

 

 

 

3. Dead 

The transition probabilities of the model are denoted by 

pij(s, t) = P (Xt = j | Xs = i), s ≤ t, i, j = 1, 2, 3


and the intensities µij(t) are assumed to exist and are given by 

µij(t) = lim 

h↘0 

pij(t, t + h) 

, i = j. 

h 

We assume that the intensities are continuous functions. Define 

and 

We take it that 

µ(t) = µ13(t), ν(t) = µ23(t), σ(t) = µ12(t), ρ(t) = µ21(t) 

µ1(t) = µ(t) + σ(t), µ2(t) = ρ(t) + ν(t). 

µ(t) < ν(t), ∀t ≥ 0, 

i. e. the mortality for a disabled is always greater than for an active person. 

When we cannot observe whether an insured is active or disabled at any time after 

entry (as an active), one gets a filtration (of the above Markov model) which is 

determined by the force of mortality for a random insured. Let ˜µ(τ, x) denote this 

intensity for an insured of age x with age of entry τ, τ ≤ x. 

(a) Explain that ˜µ is given by 

˜µ(τ, x) = µ(x) 

p11 

p11 + p12 

(τ, x) + ν(x) 

= µ(x) + {ν(x) − µ(x)} 

p12 

p11 + p12 

p12 

p11 + p12 

and thus give the grounds for the presence of select mortality. 

(τ, x) 

(τ, x) 

We obviously want τ → ˜µ(τ, x) to be decreasing for fixed x which will be shown by 

differentiation in the following. 

(b) Explain that τ → ˜µ(τ, x) is decreasing iff the fraction τ → (p12/p11)(τ, x) is 

decreasing and give an interpretation of this. 

(c) Show that 

d 

dτ 

p12 

p11 

 

(τ, x) = σ(τ)(p12p21 − p11p22)(τ, x) 

p 2 11(τ, x) 

and 

d 

dτ (p12p21 − p11p22)(τ, x) = {µ1(τ) + µ2(τ)}(p12p21 − p11p22)(τ, x), 

and explain why τ → ˜µ(τ, x) is decreasing. 

Let ˆpij(s, t) denote the transition probabilities corresponding to the model without 

recovery, i. e. ρ(t) = 0, ∀t ≥ 0.


(d) Find expressions for ˆp11(τ, x) and ˆp12(τ, x) as a function of the intensities and 

show that 

p12 

(τ, x) < ˆp12 

(τ, x). 

p11 

Give an interpretation of this and explain how ˜µ is affected by changing to the model 

without recovery. 

It is a common opinion that the selection the insured goes through at entry disappears 

after a period of time, called the period of selection. We will try to explain this 

phenomenon mathematically. Lad τ0 be the age of entry for an insured. 

(e) Show that 

x 

x → exp 

τ0 

ˆp11 

 

µ1(s)ds p11(τ0, x), x ≥ τ0 

is an increasing function. Assume that µ2(t) ≥ µ1(t), ∀t ≥ 0 and that there exists a 

w > τ0 so that w 

(µ2(t) − µ1(t))dt = ∞. 

Show that 

τ0 

d 

lim 

x↗w dτ 

p12 

p11 

 

(τ0, x) = 0 

monotonically with 

 

d p12 

(τ0, x) = 0, ∀x ≥ w, 

dτ p11 

and explain why this verifies the presence of a period of selection of w. 

(FM1 exam, 1989-ordning, opgave 1, summer 1995) 

Exercise 6.3 Mortality in a portfolio of insured lives will usually be different from 

the mortality of the general population because the insured lives are a selected part 

of the population. We will in this exercise study one relationship that is assumed to 

contribute a great deal to the effect of selection, i. e. the fact that people with illnesses 

that cause severe excess mortality are not allowed to underwrite life insurances 

(under the usual terms). In the following such persons will be called “ill”. Thus the 

population can be divided according to the figure below. 

 

 

 

 

 

 

 

0. Not insured, not ill 

 

κx 

 

 

ρx 

1. Insured, not ill 

κx 

 

σx 

4. Dead 

 

 

 

λx 

λx 

 

 

 

σx 

3. Not insured, ill 

2. Insured, ill


Now assume that every person enters state “0” at birth and that the transitions 

between the states afterwards go on as a time continuous Markov chain with transition 

intensities only dependent on the age x as indicated in the figure. Excess mortality 

for the ill persons means that 

with “>” for some values of x. 

λx ≥ κx, x > 0, (6.1) 

With the usual notation for the transition probabilities the following are satisfied 

p11(x − t, x) = e − 

x 

x−t (σu+κu)du , (6.2) 

x z 

p12(x − t, x) = e 

x−t 

− 

x−t (σu+κu)du σze − 

x 

z λudu dz, 0 < t < x; (6.3) 

p00(0, x) = e − 

x 

0 (σu+κu+ρu)du p01(0, x) = 

, 

e 

(6.4) 

− 

x 

0 (σu+κu)du (1 − e − 

x 

0 ρudu p02(0, x) = 

), 

x 

e 

(6.5) 

− 

z 

0 (σu+κu)du (1 − e − 

z 

0 ρudu )σze − 

x 

z λudu dz, (6.6) 

p03(0, x) = 

0 

x 

0 

e − 

z 

0 (σu+κu+ρu)du σze − 

x 

z λudu dz, 0 < x. (6.7) 

(a) Prove the formulas (6.2) and (6.3) by putting up and solving differential equations. 

(b) Assume that (6.4) is given. Give direct, informal grounds for the expressions (6.5) 

– (6.7). 

The insured lives are either in state “1” or in state “2” (those in state “2” received 

the insurance contract before they were struck by illness). The insurance company 

does not observe in which of the two states the insured is. All the company knows is 

the time of entry and age. Let µ[x−t]+t denote the force of mortality for an insured of 

age x who received the insurance t years ago. 

(c) Derive an expression for µ[x−t]+t. Show that under the condition (6.1), µ[x−t]+t is 

a non-decreasing function of t for constant x (it might be desirable to express µ[x−t]+t 

as a weighted average of κx and λx). How will you explain this result to a person who 

has no knowledge of actuarial science? 

(d) Discuss the formula for µ[x−t]+t to find theoretical explanations as to why insurance 

companies operate with a period of selection s so that the mortality is considered to 

be aggregate for t > s. 

Let µ x denote the force of mortality for a randomly chosen person of age x in the 

population (that is, we do not observe in which of the states “0” – “3” the person is).


(e) Find an expression for µ x. Show that under the condition (6.1) the inequality 

µ x ≥ µ[x−t]+t, 0 < t < x 

is satisfied. The result clarifies the preliminary remarks of this exercise. Try to give an 

explanation that is comprehensible for a person without any knowledge of actuarial 

mathematics. 

(FM1 exam (1), winter 1985/86)


7. Markov Chains in Life Insurance 

FM0 S94, 1 

Exercise 7.1 Consider a model for competing risks with k + 1 states, 0: “alive” and 

1, . . . , k denoting death from k different reasons; denote the partial probabilities of 

death by 

and define 

(a) Prove that 

(b) Prove that 

k 

j=1 

tp 0j 

x 

= 

+ 

k 

j=1 

tq (j) 

x 

t 

= 1 − exp − µ 

0 

0j 

 

x+τdτ , 

tp (j) 

x = 1 − tq (j) 

x . 

tp 00 

x = 

tq (j) 

x − 

 

1≤h

Markov Chains in Life Insurance 47 

enumerated 1, . . . , N. The development of the policy is being described by a Markov 

process with transition intensities µ jk 

t for transition from state j to state k at time t 

from issue. 

In the contract it is stated that the amount B jk 

t is payable upon transition from state 

j to state k at time t. As long as the policy stays in state j, a continuous payment 

with intensity B j 

t is due, i. e. in the time interval [t, t + ∆t) the amount B j 

t ∆t + o(∆t) 

is paid out. Assume that all amounts B jk 

t are non-negative and that the force of 

interest δ is independent of time. For now we disregard administration expenses. 

(a) There are no assumptions regarding the sign of B j 

t . How should negative values 

of B j 

t be interpreted? 

Let V j 

t denote the premium reserve in state j at time t. It is defined as the expected 

present value of the out payments in the time interval [t, ∞) discounted back until 

time t, given that the policy is in state j at time t. 

(b) Show that the premium reserve satisfies the differential equation system 

−B j 

t = d j 

Vt − δV 

dt j 

t + 

k=j 

µ jk 

t (B jk 

t + V k 

t − V j 

t ), j = 1, . . . , N. 

(c) Interpret this differential equation system intuitively in terms of savings premium 

and risk premium. 

A special case of the this general Markov model is the disability model. This model 

has three states, 1, 2 and 3, corresponding to active, disabled and dead. The insured’s 

age at entry is x and the intensities are denoted by 

All other transition intensities are 0. 

µ 12 

t = σx+t (from active to disabled), 

µ 21 

y = ρx+t (recovery), 

µ 13 

y = µx+t (dead as active), 

µ 23 

t = νx+t (dead as disabled). 

Apply this model for an investigation of an endowment insurance with exemption 

from payment of premiums by disability. Assume the insured is active at entry. The 

insurance period is n, so the insurance cancels when the insured has reached the age 

of x + n or if he dies before that. 

In question (d) it is assumed that the equivalence premium is paid continuously with 

level intensity π as long as the insured is active and that a constant sum insured S


is payable upon the expiration of the policy; hence, with the notation from above we 

have 

B 1 

t 

= −π, B2 

t 

= 0, B13 

t 

= B23 

t 

= S. 

(d) Come up with formulas for π, V 1 

t and V 2 

t for 0 ≤ t < n in terms of the transition 

probabilities and transition intensities in the model and the discounting rate v, by 

direct prospective reasoning. 

If the premium and the benefits depend on the reserves, the premium and the premium 

reserve cannot be determined directly as in (d); instead the differential equation 

system from (b) must be solved with appropriate boundary conditions. Now it is 

assumed that the premium and the payment of benefit at age x + n are due as above, 

but upon death of the insured before time x + n, the premium reserve of the policy is 

paid out as a supplement to the sum insured; B j3 

t = S +V j 

t for j = 1, 2 and 0 < t < n. 

(e) Show that the differential equation system from (b) gives the grounds for a differential 

equation of first order in V 1 

t − V 2 

t . What is the initial condition? Solve the 

differential equation. 

(f) What are π, V 1 

t and V 2 

t for 0 ≤ t < n. 

(g) Show that if νx+t ≥ µx+t for all t < n then V 2 

t 

result. 

> V 1 

t for all t < n. Interpret this 

Assume that we have the following expenses: Initial expenses due at time 0 with 

an amount of αS, loading for collection fees βp, where p is the level gross premium 

intensity and administration costs due with an intensity at time t equal to γV j 

t if the 

policy is in state j. 

(h) Explain, without performing any detailed calculations, what changes would follow 

from these assumptions in the theory discussed in (e) – (g). 

(FM1 exam , summer 1984) 

Exercise 7.3 An active person aged x considers a disability annuity, which falls due 

continuously with level intensity b upon disability before the age of x+n. Premium is 

payable at rate π as long as the person is active during the contract period. Assume 

that the state of the policy is S(t) at time t after the time of issue where {S(t)}t≥0 is 

a time continuous Markov model with state space and transitions as follows


0. Active σ(x+t) 

 

1. Disabled 

 

 

µ(x+t) 

ν(x+t) 

 

 

 

 

2. Dead 

(a) Give, without any proof, expressions for the transition probabilities 

pjk(s, t) = P (S(t) = k | S(s) = j), 0 ≤ s ≤ t, j, k ∈ {0, 1, 2}. (7.1) 

The present value at time s of benefits less premiums in [s, n] can be cast as 

C(s) = 

n 

v 

s 

t−s (b1{S(t)=1} − π1{S(t)=0})dt, 

where 1A is the indicator function for the event A. 

(b) Find, for 0 ≤ s ≤ n and j = 0, 1, 2, the conditional expection 

and the conditional variance 

Vj(s) = E(C(s) | S(s) = j) (7.2) 

Zj = Var(C(s) | S(s) = j) (7.3) 

as expressions of integrals of functions of the transition probabilities from (7.1). 

(c) Find EC(s) and VarC(s) in terms of the transition probabilities (7.1) above and 

the functions (7.2) and (7.3). 

(d) Now assume that the equivalence principle is being adopted, i. e. V0(0) = 0, where 

π is the net premium intensity and Vj(s) in (7.2) above is the net premium reserve in 

state j at time s. Does the net reserve ever become negative? 

(FM1 exam opgave 1, summer 1989) 

Exercise 7.4 The figure below illustrates an expansion of the model discussed in 

exercise 7.3. There are two states of disability i1 and i2 representing two degrees of 

disability. Assume that ν2(x+t) ≥ ν1(x+t). Let { ˜ S}t≥0 be the corresponding Markov 

chain and define the stochastic process {S(t)}t≥0 by S(t) = ˜ S(t) for ˜ S(t) ∈ {a, d} and 

S(t) = i for ˜ S(t) ∈ {i1, i2}. If one does not know the degree of disability, {S(t)} is 

the observable process.


a. 

 

 

 

 

 

 

σ1(x+t) 

µ(x+t) 

 

 

 

 

 

 

i1. 

ν1(x+t) 

 

d. 

σ2(x+t) 

 

The transition intensities for the S(t)-process are 

 

i2. 

ν2(x+t) 

 

P (S(t + dt) = k | S(t) = j, {S(s)}s


(a) Find the second order differential equation for p00(s, ·) as outlined in the text, 

making appropriate assumptions about differentiability of the intensities. 

From now on consider the special case with constant intensities: 

(b) Find explicit solutions for p00(s, t) and p01(s, t). Note that the solution depends 

on s and t only through t − s, hence put s = 0. Discuss how the probabilities depend 

on t and the intensities. 

(c) Calculate for t = 0, 10, 20, . . . , 100 and draw graphs of the probabilities for some 

different choices of the intensities, including as key references 

1. σ = 0.01, ρ = 0.01, µ = 0.005, ν = 0.005, 

2. σ = 0.01, ρ = 0.01, µ = 0.005, ν = 0.01, 

3. σ = 0.01, ρ = 0.01, µ = 0.005, ν = 0.1. 

(You can program the formulas and compute directly or you can employ the program 

’retres’.) 

(d) Explain how the result 3. in item (c) above can be used to find the probabilites 

for the case σ = 0.1, ρ = 0.1, µ = 0.05, ν = 1. (The ratios between the intensities are 

the essential feature.) 

(e) The probability p02(s, t) gives the mortality law for a person who is known to be 

active at age s. Discuss how it depends on the intensities, with special attention to 

the case where µ = ν. 

(f) Consider the special case with no mortality (µ = ν = 0), whereby the number of 

states essentially becomes 2. Find p01(0, t), and discuss how the expression depends 

on t and the intensities. Find the limit as t → ∞ and discuss the expression. 

(RN “Problems in Life Insurance” 14.10.93, problem 8) 

Exercise 7.7. (Markov Chains in Life Insurance) Formulate suitable continuous 

time Markov chain models for solution of the following problems, where N is assumed 

to be a Poisson variate with parameter 1: 

(a) Find the probability that N ∈ {0, 2, 4, . . .}. 

(b) Find the probability that N ∈ {0, 3, 6, . . .}. Think of other variations of the 

problem.



Exercise 7.8. (Markov Chains in Life Insurance) Prove that the four definitions 

(2.1), (2.2), (2.3), and (2.5) of the Markov property are equivalent (assuming that the 

sample paths of the process are as stated in Paragraph 1A). 


Exercise 7.9. (Reserves) At time 0 a person (x) aged x buys a standard pension 

insurance policy specifying that, conditional on survival, premiums are payable with 

level intensity c from time 0 to time m and pensions are payable continuously with 

level intensity b from time m to time n, m < n. There are two states, 0: “alive” 

and 1: “dead”. Let µx+t be the force of mortality at age x + t, and denote by 

tpx = exp(− t 

0 µx+sds) the probability that (x) survives to age x + t. Assume that 

interest is earned at a constant rate δ so that v(t) = v t , with v = e −δ the annual 

discount rate. Throughout b is taken as fixed and c is to be determined by the 

equivalence principle. 

(a) Put up prospective and retrospective expressions for the reserves in both states 

at any time t ∈ [0, n) after issue of the policy. As an exercise, find the reserves also 

by solving the appropriate differential equations. Determine c. 

(b) Find the conditional variances of the individual reserves (the present values of 

future and past payments) at time t, given the state of the policy at time t. 

Henceforth the standard policy is referred to as policy ’S’. Consider a modified policy 

’P’, by which the prospective reserve in state 0 is to be repaid upon death of (x) 

during the period [0, n). 

(c) Find the statewise reserves for ’P’. (Differential equations must now be used.) 

Determine c. 

(d) Find the conditional variances of the individual reserves for ’P’, corresponding to 

those in (b). 

Consider another modified policy ’R’, by which the retrospective reserve in state 0 is 

to be paid upon the death of (x) during [0, n). 

(e) Find the statewise reserves for ’R’ and determine c. 

(f) Find the conditional variances of the individual reserves by policy ’R’, corresponding 

to those in (b). 

(g) Compare the results obtained for ’S’, ’P’, and ’R’.


Finally, consider a policy ’M’ with a mixed rule for repayment of the reserve, by which 

the retrospective reserve in state 0 is to be repaid upon the death of (x) during the 

premium period [0, m], whilst the prospective reserve in state 0 is to be repaid upon 

death during the pension period [m, n]. 

(h) Find the statewise reserves for the policy ’M’, and determine c. 


Exercise 7.10. (Reserves) At time 0 a person buys a life insurance policy specifying 

that an amount b (the sum insured) is provided immediately upon death before time 

n and premiums are payable with level intensity c as long as the person is alive 

and active up to time n (premium waiver by disability). Assuming that recovery 

is impossible, the relevant Markov model can be sketced below. Assume that the 

discount function is v(t) = v t = e −δt . The premium c is to be determined by the 

equivalence principle. 

σ(x) 

0. Active 

 

µ(x) 

 

 

 

2. Dead 

 

1. Disabled 

ν(x) 

(a) Put up integral expressions for the transition probabilities. 

(b) Put up expressions for the prospective and retrosprective reserves in all states 

at any time t ∈ [0, n). Find the reserves also by solving the appropriate differential 

equations. Determine the premium intensity c. 

Suppose that instead of full premium waiver, the premium during disability is made 

dependent on the past savings on the contract. More specifically, assume that premiums 

during disability fall due with intensity c − c ′ V − 

1 (t) at time t if the policy then 

is in state 1. 

(c) Which relation must c and c ′ satisfy? 

(RN “Problems in Life Insurance” 14.10.93, problem 16)


Exercise 7.11 Consider the usual disability model without recovery: 

σ(x) 

0. Active 

 

µ(x) 

 

 

 

2. Dead 

 

1. Disabled 

ν(x) 

An insurance is issued to a person (x). Premium is payed continuously with level 

intensity π as long as (x) is alive, at the most for n years. As long as (x) is disabled, 

he receives a payment with level intensity c until his death. If (x) dies before the age 

of x + n the sum D is paid out. If (x) is active at the time m ∈ (0, n) the sum S is 

paid out. 

(a) Put up the statewise reserves. 

(b) Put up Thiele’s differential equation and determine the risk sums. 

(BS “Opgave til FM0” 1995) 

Exercise 7.12 A person considers a life insurance policy specifying that an amount 

S is paid immediately upon death before time of expiration n. If the insured is alive 

at time n he is also provided the sum insured S. Premiums are payable with level 

intensity p as long as the insured is active up to time n. That is, the insured has the 

right to exemption from payment of premium if he is disabled. Interest is earned at 

a constant rate δ. The Markov model used is illustrated below. 

(a) Put up the following: 

σ(x) 

 

0. Active 

 

1. Disabled 

 

ρ(x) 

 

µ(x) 

ν(x) 

 

 

 

 

2. Dead 

1. Thiele’s differential equation for the statewise reserves V0(t) and V1(t). 

2. The statewise reserve V0(t) as an integral function of p 00(t, u) and V1(t). 

3. The statewise reserve V0(t) as an integral function of p00(t, u) and p01(t, u). 

4. The statewise reserve V1(t) as an integral function of p 00 (t, u) and V0(t). 

5. The statewise reserve V1(t) as an integral function of p11(t, u) and p10(t, u).


(b) Differentiate both expressions for V0(t) in order to verify Thiele’s differential 

equation. 

(c) How will you in practice find the equivalence premium for this kind of insurance? 

(d) Let µt = νt. Put up a differential equation for V1(t) − V0(t). Solve it and use the 

result to find out how σt and ρt ought to be chosen in relation to σ ′ t and ρ′ t so that 

the effect will be the desired increase of the premium. Give an interpretation of all 

results. 

(e) Find expressions for the safety loadings for the two states and consider the difference. 

Which sign does it have with a reasonable choice of parameters? 

(f) Expand the model with the state “surrender”. What would you consider a reasonable 

payment in connection with surrender from the state of active and disabled 

respectively. How should the first order intensity of surrender be put in relation to 

the intensity on the second order basis with your choice of surrender value? 

(MS “Opgave til FM1” 19.09.95) 

Exercise 7.13 Consider the usual disability model with four level transition intensities 

µ, ν, σ and ρ. Let α = µ + σ, κ = ν + ρ and assume that α = κ. 

(a) Show that the probability for an active person being active in t years, after having 

been disabled once and only once is 

 

σρ 

κ − α 

te −αt + e−αt − e −κt 

α − κ 

(b) Find the probability for an active person being disabled for the second time in t 

years. 

(c) Explain how one relying, on information about death, disability and recovery in 

some population, can estimate the probabilities in (a) and (b) in two different ways. 

 

. 

(HRH “Opgaver til FM1”)


8. Bonus Schemes 

FM0 S95, 1 

Exercise 8.1 An insurance company uses the following technical basis: Force of 

mortality µx = 0.0005+10 5.88+0.038x−10 (G82M), interest rate i = 5% p. a., acquisition 

costs 3% of the sum insured, loading for collecting fees 5% of the premium and 

administration expenses due continuously with intensity γV g 

t where V g 

t is the gross 

premium reserve and 

γ = log 

1, 05 

1, 045 . 

The technical basis of second order has interest rate ĩ = 5.5% p. a. and force of 

mortality ˜µx = µx − ( ˜ δ − δ), where δ and ˜ δ are the forces of interest corresponding to 

i and ĩ respectively. Same expenses as above. 

A 30-year old person considers an endowment insurance, insurance period n years. 

The premium, calculated according to the equivalence principle, is due continuously 

with level intensity during the entire period of insurance. 

(a) Calculate the gross premium intensity and the gross premium reserve after 20 and 

40 years, respectively. 

(b) Put up a differential equation for the safety loading St 

Assume for the time being that the bonus fund is entirely paid out after 40 years. 

(c) Confirm that this gives the greatest possible bonus fund during the entire insurance 

period. 

(d) Calculate this bonus fund after 20 and 40 years. 

Now assume that the bonus scheme consists of two discrete payments: One after 20 

years and one again after 40 years. 

(e) Calculate these two payments. 

Finally assume that the bonus scheme is as previously, but that the payment after 20 

years is used as a deposit for a life conditioned capital insurance, duration n years, 

calculated on the technical basis of second order without expense contributions. 

(f) Show that the amount, payable at the 40th year, is the same as the payment according 

to the first bonus scheme and explain the difference between the two schemes.

Bonus Schemes 57 

(g) Critisize the considered bonus schemes especially related to the insured that passes 

away during the insurance period and try to suggest a more reasonable bonus scheme. 

(SP(74)) 

Exercise 8.2 Consider two lives (x) and (y) ages x and y respectively with remaining 

life times Tx and Ty. Assume Tx and Ty to be stochastically independent. 

For premium calculation the company has established a technical basis of first order; 

force of interest δ, force of mortality µ (x) 

x+t for (x) at age x + t and force of mortality 

µ (y) 

y+t for (y) at age y + t. (x) and (y) consider a widow insurance (whole-life annuity). 

The possible whole-life annuity is due continuously with intensity 1 as long as (y) is 

alive and (x) is dead. 

(a) What is the present value of the possible whole-life annuity? 

(b) Derive an expression for the expected present value of the possible whole-life annuity 

in terms of one-life and both-life annuity expressions. Give a direct interpretation 

of this expression. 

For the possible annuity a level continuous premium is due as long as they both are 

alive. 

(c) What is the equivalence premium intensity? 

(d) Give an expression for the premium reserve by prospective reasoning. 

(e) Derive Thiele’s differential equation. 

Introducing the technical basis of second order, the interest rate is ˜ δ and the forces 

of mortality are ˜µ (x) 

x+t and ˜µ (y) 

y+t. 

(f) Which principles in general should be the basis for choosing the technical basis of 

second order? How would you determine the elements of the technical basis of second 

order compared to the ones of the first order for the possible whole-life annuity? 

(g) Derive an expression for the safety loading. 

We now introduce the following bonus scheme: As long as (x) and (y) both are alive, 

no return of safety margin is due to payment. If (y) dies before (x) no return is due 

either. If (x) dies before (y), a continuous amount B is added to the annuity payment 

of one. The amount B is paid out as a continuous bonus. 

(h) Put up an expression for determination of B.


Instead of the bonus scheme above, we would like a return of an amount equal to the 

present value of the added amount B at the time of (x)’s death if (x) dies before (y). 

The payment of bonus consists of a continuous payment of an amount B as long as 

the whole-life annuity is due. 

(i) What is the difference between the two bonus schemes? Which Bonus scheme 

would be preferable from a safety margin view? 

(FM100, Oslo 1987 05.12.87 (ex. 1)) 

Exercise 8.3 By an endowment insurance an amount of S is paid either upon death 

before year n after the time of issue or at the latest n years after the time of issue. 

The premium is due with level intensity B during the insurance period, the force of 

interest is δ, the force of mortality µx, expenses αS by issue and continuous expenses 

with intensity γS + βB per year during the insurance period. 

(a) Determine the premium B by applying the equivalence principle. Determine the 

net and gross premium reserves at any time during the insurance period. Find a 

connection between the two reserves and explain the reason for this difference. 

The expenses α, β, γ do not contain any safety loadings. On the contrary there are 

assumed to be safety loadings κ and λ included in the force of interest and in the 

force of mortality respectively. Hence, a realistic force of interest of second order is 

˜δ = δ + κ and a realistic force of mortality is ˜µx = µx − λ. 

(b) Put up the expression for the safety loading, the policy contributes at any time 

during the insurance period. 

The return of premium scheme is as follows: (i) An amount (κ − λ)Vt is paid out 

during (t, t + dt) during the insurance period, where Vt is the premium reserve for 

covering the outstanding claims and return of premiums (i. e. the insured earns an 

interest (κ − λ)dt of his part of the total fund); (ii) He receives an additional amount 

K to the sum insured upon death during the insurance period. 

(c) Determine K so the total safety loadings are being allotted to the insured. 

(FM1, winter 1985-86 (ex. 2)) 

Exercise 8.4 Consider an endowment insurance, sum insured S, duration n, age of 

entry x. The company adopts the following technical basis for calculation of premiums 

and premium reserves: Force of interest δ, force of mortality µx and expenses βπ paid 

continuously where π is the level net premium, paid continuously during the entire 

period. In the technical basis of first order we assume the force of surrender to be 

zero.

Bonus Schemes 59 

(a) Put up Thiele’s differential equation and an expression for the equivalence premium 

intensity. 

The actual development with respect to the technical basis of second order for the 

company is as follows: The interest rate has dropped to δ ′ and the administration 

expenses have dropped to β ′ . Now assume that the force of surrender exists and is 

denoted by νx+t t years after the time of issue where νx is independent of the time 

the person has been insured. 

(b) Discuss if this assumption of independence i reasonable. 

Until now the company has used the safety loadings to bring down the future premiums 

every year, or bonus has been paid out in connection with death or surrender, 

where the total reserve is allotted. A new bonus scheme is allotting bonus cash every 

whole year or by death or surrender. 

(c) Put up Thiele’s differential equation for the premium reserve on the technical 

basis of second order. What are the boundary conditions? Intuitively, why does νx 

vanish? 

(d) Put up the differential equations for the bonus funds. What are the boundary 

conditions for the two bonus schemes? 

(e) Try to figure out the variance of the bonus funds of the 2 bonus schemes. 

(FW (rev.), 1994)


9. Moments of Present Values 

Exercise 9.1 Consider the standard setup, where the development of the life insurance 

is described by a continuous Markov Chain on a state space J = {0, . . . , J}, 

and the contract specifies that a 0 g(t)dt is payable if the policy stays in state g in the 

time interval (t, t+dt) and a 0 gh(t) is payable upon transition from state g to state h at 

time t. (More general annuity payments can easily be dealt with, but the expression 

becomes more messy.) Assume for the time being that the discount function v is 

deterministic. 

To calculate the variance of 

V = 

∞ 

0 

v(τ)A(dτ), 

the present value at time 0 of future benefits less premiums, one needs to find 

EV 2 = E 

= E 

∞ 

v(τ)A(dτ) 

2 

0 

∞ 

v 

0 

2 (τ)(A(dτ)) 2 ∞ 

+ 2 

0 

(a) Prove that (9.1) can be recast as 

EV 2 = 

∞ 

0 

+ 

h=g 

v 2 (τ) 

g 

 

 

v(τ)A(dτ) v(ϑ)A(dϑ) . (9.1) 

ϑ>τ 

p0g(0, τ){2a 0 

gV + 

g (τ) 

µgh(τ)a 0 

gh (τ)(a0 

+ 

gh (τ) + 2V h (τ))}dτ, (9.2) 

where V + 

g denotes the prospective reserve in state g at time t. 

The variance is obtained by subtracting the square of the mean present value from 

the mean of the square. Formula (9.2) appears to offer an escape from the double 

integration that has to be performed in (9.1). It requires that the prospective reserve 

in different states be computed (they are essentially the inner integral, of course) and 

stored in memory beforehand. However, we have standard programs for that. 

(b) Use (9.2) to calculate the variance for a simple term insurance and for a simple 

life annuity, for which the results are well-known. 

Another simple formula for the variance, which shall not be proved here, is 

∞ 

VarV = v 2 (τ) 

p0g(0, τ)µgh(τ)(a 0 

+ 

+ 

gh (τ) + V h (τ) − V g (τ))2dτ (9.3) 

0 

g=h 

(c) Prove that (9.2) essentially remains valid by stochastic discount function v(t) = 

exp(−∆(t)) if the process {∆(t)}t≥0 has independent increments.

Moments of Present Values 61 

(RN “Problems in Life Insurance Mathematics” 14.10.93 (Problem 1)) 

Exercise 9.2 Consider a temporary life insurance issued to a person at age x. The 

policy specifies that the sum insured S is payable immedieately upon (possible) death 

of the insured before time n and that premium is due with fixed amount c at times 

0, 1, . . . , n − 1 as long as the insured is alive. Assume that administration costs incur 

continuously with constant intensity γS throughout the duration of the policy and 

that the force of interest δ is fixed. 

(a) Determine the premium c as a function of S and γ by the equivalence principle. 

Find an expression for the prospective reserve of the policy at time t ∈ [0, n). 

(b) Find the variances and the covariances of the present values at time 0 of the 

insurance payment, the premiums and the administration costs. Find the variance of 

the present value at time 0 of the total cash flow of payments generated by the policy. 

To be continued as exercise 11.3, page 71. 


Exercise 9.3 Consider an n-year term insurance with equivalence premium payable 

continuously with level intensity π throughout the duration of the policy. 

(a) Put up the prospective reserve and the variance of the present value at issue at 

time 0 of benefits less premiums. 

(b) Suppose that in case of surrender the insured immedieately gets the current net 

value of the policy defined as the prospective reserve at the time of surrender. Assume 

that surrender takes place with intensity γ(t) at time t < n. (Thus we consider an 

extended model with three states “insured”, “withdrawn” and “dead”.) Find the 

prospective reserve in state “insured” and the variance of the present value at issue 

of benefits less premiums. Compare with the results in (a). 


It is possible to find differential equations which determines any n-order moment 

V (n) 

k (t) recursively in any state k for a generalized Markov-model. In the next exercise 

we consider the 2nd order moment for an endowment insurance. 

Exercise 9.4 Assume that a person at age x buys an n-year endownment insurance 

with level premium intensity π payable as long as the insured is alive. Upon death 

the amount S is paid out immediately. The force of mortality is given by µx at age x. 

Let as usual Ut denote the present value at time t of benefits less premiums in the


future. 

(a) Find an expression for V (2) (t) = E(U 2 t 

| T > t). 

(b) Derive the expression from (a) with respect to t to find that you will get a differential 

equation which determines V (2) (t) recursively from V (t). 

(BM and JC, 1995)

Inference in the Markov Model 63 

10. Inference in the Markov Model 

Exercise 10.1 The time continuous Markov Model shown below 

0 

 

 

 

 

 

 

µ1(x) 

µ2(x) 

µh(x) 

 

 

 

 

 

 

 

is called the model for competing risks with h reasons for resignation. The function 

µ = h 

i=1 µi is the total intensity of resignation. 

Define tpx = p00(x, x + t) and tq (k) 

x = p0k(x, x + t). 

(a) Prove that 

and 

tpx = p00(x, x + t) = e − 

t 

o µ(x+s)ds 

tq (k) 

x+t 

x = p0k(x, x + t) = 

x 

p00(x, s)µk(s)ds, for k = 1, 2, . . . , h. 

Assume that L independent persons of the same age are being observed during one 

year. Let the state “0” in the model above represent the state “alive” and let the 

h reasons of resignation be reasons of deaths. It is possible to assume the forces of 

mortality, µ1, . . . , µh, to be constant and the same for all L persons. The number of 

deaths from the h reasons are denoted D1, . . . , Dh and the sum of life time is T . 

(b) Determine the maximum likelihood estimators ˆµ1, . . . , ˆµh and the asymptotic distributions 

of the estimators. 

1 

2 

. 

h 

(HRH “Opgaver til FM3” (ex. 2)) 

Exercise 10.2 Consider the time continuous Markov model as follows


σ(x) 

 

0. Working 

 

 

1. Not working 

 

ρ(x) 

 

µ(x) 

µ(x) 

 

 

 

 

2. Dead 

(a) Find explicit expressions for the transition probabilities. 

(b) Assume that all members in a portfolio of N independent lives are being observed 

during the period of time [0, 1] and assume that all transition intensities are constant 

during the time of observation. How would you estimate σ, ρ and µ? Determine the 

asymptotic distributions of the estimators. 

(HRH “Opgaver til FM3” (ex. 12)) 

Exercise 10.3 An insurance company is to perform a mortality study based on 

complete records for n life insurance policies with unlimited term period. Policy 

number i was issued zi years ago to a person who was then aged xi. The actuary sets 

out to maximize the likelihood 

n 

µ(xi + Ti, θ) Di 

xi+Ti 

exp µ(s, θ)ds , 

i=1 

where the notation is obvious. 

One employee in the department objects that the method represents a neglect of 

information; it is known that the insured have survived, not only the period they 

were insured, but also the period from birth until entry into the scheme. Thus, he 

claims, the appropriate likelihood is rather 

n 

µ(xi + Ti, θ) Di 

xi+Ti 

exp µ(s, θ)ds . 

i=1 

Settle this apparent paradox. (A suitible framework for discussing the problem is an 

enriched model with three states, “uninsured”, “insured” and “dead”.) 

xi 

0 


Exercise 10.4 Reference is to Paragraph 2C in the paper RN “Inference in the 

Markov Model”, 09.02.93, the Gompertz-Makeham mortality study.


(a) Modify the formulas to the situation where person number i entered the study zi 

years ago at age xi. 

(b) Find explicit expressions for the entries of the asymptotic covariance matrix of 

the MLE. 


Exercise 10.5. (Transformation of Data and Analytical Smoothening) Consider the 

mortality model where n independent persons are being observed during the interval 

of age [x, x + z), where x and z are integers. Exact times of death are observed and 

we disregard censoring during the period except by age z. The force of mortality µ(t) 

at the age of t is assumed to be piecewise constant over one-year intervals of age so 

that 

µ(t) = µk, for t ∈ [k, k + 1), k = x, . . . , x + z + 1. 

We introduce µ = (µx, . . . , µx+z−1) t . Let further more Dk and Tk denote the number 

of deaths occured and the observed time of risk in [k, k + 1) respectively and let 

ˆµ = ( ˆµx, . . . , ˆµx+z−1) t , where 

(a) Assume that µk can be cast as 

ˆµk = Dk 

, k = x, . . . , x + z − 1. 

Tk 

µk = gk(θ), k = x, . . . , x + z − 1, 

where gk(θ) = g(ξk; θ) and ξk ∈ [k, k + 1). Here g(t; θ) denotes a function of the age 

t and of an unknown parameter θ = (θ1, . . . , θp) t , p < z. Assume moreover that the 

exists and has full rank p for g(θ) = (gx(θ), . . . , gx+z−1(θ)) t . Let 

Jacobian Dg = Dg(θ) 

dθ 

R z 

+ = {η = (ηx, . . . , ηx+z−1) ∈ R z | ηk > 0, k = x, . . . , x + z − 1}, 

and let L : R z + → R z be a differentiable mapping, so that the z × z Jacobian DL = 

DL(η) 

dη 

and 

has full rank z. Define at last ˆα = (ˆαx, . . . , ˆαx+z−1) t and ˜g(θ) by 

ˆα = L(ˆµ) 

˜g(θ) = L ◦ g(θ) = L(g(θ)). 

The parameter θ can be determined in two ways by analytical smoothening. You can 

either by modified χ 2 -minimizing determine the value of θ that brings g(θ) “closest” 

to ˆµ, or you can by modified χ 2 -minimizing determine the value of θ that brings ˜g(θ) 

“closest” to ˆα. Denote these two estimators by ˆ θ and ¨ θ respectively and show that ˆ θ 

and ¨ θ has the same asymptotic variance. Is ˆ θ = ¨ θ?


(b) Now assume that 

µk = βc ξk , k = x, . . . , x + z − 1. 

Show how it is possible to use the theory in (a) to construct an estimator for (β, c) 

that is easy to calculate and give the asymptotic variance of the estimators. 

(FM3 exam, winter 1985-86) 

Exercise 10.6 Verify (1.13) – (1.16) in the paper RN “Inference in the Markov 

Model”, 09.02.93. 

(RN “Inference in the Markov Model” 09.02.93 (Problem 1)) 

Exercise 10.7 In the situation of paragraph 1E in the paper RN “Inference in the 

Markov Model”, 09.02.93, consider the problem of estimating µ from the Di alone, 

the interpretation being that it is only observed whether survival to z takes place or 

not. Show that the likelihood based on Di, i = 1, . . . , n, is 

q N (1 − q) n−N , 

with q = 1 − e −µz , the probability of death before z. (Trivial: It is the binomial 

situation.) 

Note that N is now sufficient, and that the class of distributions is a regular exponential 

class. The MLE of q is 

q ∗ = N 

n 

with the first two moments 

Eq ∗ = q, Varq ∗ = 

q(1 − q) 

. 

n 

The MLE of µ = − log(1 − q)/z is µ ∗ = − log(1 − q∗ )/z. Apply (6.6) in the Appendix 

of the paper to show that 

µ ∗ 

q 

∼as N µ, 

nz2 

. 

(1 − q) 

The asymptotic efficiency of ˆµ relative to µ ∗ is 

asVarµ ∗ 

asVarˆµ = 

µz 

e 2 − e 

µz 

µz 2 

− 2 

 

sinh(µz/2) 

= 

µz/2 

(sinh is the hyperbolic sine function defined by sinh(x) = (e x −e −x )/2). This function 

measures the loss of information suffered by observing only death/survival by age z 

2


as compared to inference based on complete obervation throughout the time interval 

(0, z). It is ≥ 1 and increases from 1 to ∞ as µz increses from 0 to ∞. Thus, for small 

µz, the number of deaths is all that matters, whereas for large µz, the life lengths are 

all that matters. Reflect over these findings. 


Exercise 10.8 Use the general theory of Section 2 of the paper RN “Inference in 

the Markov Model”, 09.02.93, to prove the special results in Section 1. 


Exercise 10.9 Work out the details leading to (2.9) – (2.11) in the paper RN 

“Inference in the Markov Model”, 09.02.93. 

(RN “Inference in the Markov Model” 09.02.93 (Problem 5))


11. Numerical Methods 

In this chapter we will use some numerical methods on the theory of life inurance that 

we have already seen. One does not always have a program that can calculate the 

premiums, the development of the reserves etc. and it can be necessary to develop 

your own programs. 

Inspired by SW let us at first deal with numerical integration. We wish to calculate 

the definite integral 

I = 

b 

a 

f(x)dx. (11.1) 

A commonly used numerical method for evaluating (11.1) is the summed Simpson’s 

rule. In all it’s simplicity is states that 

b 

I = f(x)dx h 

 

 

N−1 

f(a) + 4f(x1) + f(b) + 2 (f(x2k) + 2f(x2k+1)) 

3 

a 

with h = (b − a)/2N, xj = a + jh, j = 1, 2, . . . , 2N − 1. The accuracy is good for small 

values of h and the implementation of this method is straightforward. 

Exercise 11.1 A man aged 25 years considers a pure endowment of 1.000.000, age 

of expiration 60. The technical basis of the company is G82M, i. e. the interest rate 

is i = 4.5% and the force of mortality is 

k=1 

µx = 0.0005 + 10 5.88+0.038x−10 . 

The premium is a net continuous premium with level intensity π. Disregard expenses. 

(a) Put up Thiele’s differential equation and the expression for the equivalence premium 

π. 

(b) Apply the summed Simpson’s rule in order to calculate π numerically. 

(c) Solve Thiele’s differential equation and apply the same algorithm as in (b) to 

evaluate the premium reserve at times t = 10, 20, 30. 

Some expenses are, however, not neglectible and the insured has to pay some expenses 

during the insurance period in order to cover the administration expenses β, 

some fraction of the net premium intensity π. Administration costs are due with a 

continuous intensity γVt. Assume that γ is lesser than the force of interest δ. 

(d) How is Thiele’s differential equation and the equivalence premium (which is now 

the gross premium) modified? 

(e) Use your program in (c) to calculate the gross premium and the gross premium 

reserve at times t = 10, 20, 30.

Numerical Methods 69 

(BM og JC, 1995) 

The following is inspired by SW. Now consider the function f. When evaluating the 

net premium reserve in the above we solved the differential equation theoretically and 

then applied Simpson for evaluation. This is not always possible. Another way is to 

solve Thiele’s differential equation numerically; methods for this are plentyful and the 

Runge-Kutta method of fourth degree is highly recommended. Let d y = f(x, y(x)), 

dx 

xk = x0 + kh, k integer and define 

then 

k1 = hf(xk, yk), 

k2 = hf(xk + 0.5h, yk + 0.5k1), 

k3 = hf(xk + 0.5h, yk + 0.5k2), 

k4 = hf(xk + h, yk + k3) 

y(xk+1) y(xk) + (k1 + k2 + 2k3 + k4) 

. 

6 

The initial condition is y0 = x0. It turns out that a surprisingly big h can be chosen 

when the slope is not too big. A sixth order Runge-Kutta can also be applied, but 

the difference from the fourth order R-K is really not that great. It is possible to use 

the difference between the fourth and the sixth order Runge-Kutta in order to find 

appropriate and varying h’s. The sixth order Runge-Kutta looks like this: 

k1 = hf(xk, yk), 

k2 = hf(xk + 0.5h, yk + 0.5k1), 

k3 = hf(xk + 0.5h, yk + 0.5k2), 

k4 = hf(xk + h, yk + k3), 

k5 = hf(xk + 2 

3 h, yk + 7 

27 k1 + 10 

27 k2 + 1 

27 k4), 

k6 = hf(x + 1 

5 h, yk + 28 

625 k1 − 1 

5 k2 + 546 

625 k3 + 54 

625 k4 − 378 

625 k5), 

and hence 

y(xk+1) y(xk) + 1 

24 k1 + 5 

48 k4 + 27 

56 k5 + 125 

336 k6. 

In both the fourth and the sixth order R-K, we have initial condition y0 = x0. 

It is easy to generalisize the R-K to a system of differential equations. For such a 

system of differential equations 

⎛ 

y1(x) 

⎜ y2(x) 

y(x) = ⎜ 

⎝ . 

⎞ 

⎟ 

⎠ 

⎛ 

⎜ 

f(x, y) = ⎜ 

⎝ 

f1(x, y1(x), . . . , yn(x)) 

f2(x, y1(x), . . . , yn(x)) 

. 

⎞ 

⎟ 

⎠ 

yn(x) 

fn(x, y1(x), . . . , yn(x)) 

y 0 = 

⎛ 

y 

⎜ 

⎝ 

(1) 

0 

y (2) 

0 

⎞ 

⎟ 

⎠ 

. 

y (n) 

0


we have, similar to the fourth order R-K, 

k 1 = hf(xk, y k ), 

k 2 = hf(xk + 0.5h, y k + 0.5k1), 

k 3 = hf(xk + 0.5h, y k + 0.5k2), 

k 4 = hf(xk + h, y k + k3), 

y(xk+1) y(xk) + (k1 + k2 + 2k3 + k4) , 

6 

where d 

dx y(x) = f(x, y(x)) and y(x0) = y 0 . Remember this method when evaluating 

a system of simultaneous differential equations. It is obvious that this method for 

solving differential equation systems simultaneously has a great applicability when 

studying the simultaneous development of the statewise reserves in a general Markov 

model for a policy. The functions are allowed to depend on each other just as the 

statewise reserves depend on each other, compare with the expression for the statewise 

reserves. 

Exercise 11.2 Consider an endowment insurance, duration 20 years, age at entry 

x = 30, interest rate i = 4.5%, force of mortality 

µx = 0.0005 + 10 5.88+0.038x−10 . 

The sum insured is 500.000. The only premium is a single net premium Π upon issue 

of the contract. The equivalence principle is adopted. 

(a) Use the fourth order Runge-Kutta to find this single net premium Π. Use the 

same program to study the development of the net premium reserve. 

Instead the man does not want to compose any initial capital, but a level continuous 

premium with intensity π during the insurance period. 

(b) Use Runge-Kutta to calculate this premium. 

(c) Our man is having a hard time deciding, but he chooses to compose an initial 

capital of 5.000 and then a level continuous premium with intensity π during the 

insurance period. What will π be, again applying Runge-Kutta? 

(BM and JC, 1995) 

Exercise 11.3. (Continued from exercise 9.2) Compute quantities in items (a) and 

(b) numerically in the case with the G82M mortality, δ = log(1.045), x = 30, n = 10 

and b = 1. (A numerical integration must be performed to find the second order moments. 

Recall formulas (9.1)-(9.2) from exercise 9.1, page 61. Formula (9.1) requires


integration in two dimensions. Formulas (9.2) and (9.3) require integration in one 

dimension when a table of reserves has been generated.) 


Exercise 11.4. (Moments of Present Values) Consider the Markov model for disabilities, 

recoveries and deaths with intensities as in G82M technical basis; no recoveries, 

non-differential mortality intensity µ(x) = 0.0005 + 10 −4.12+0.038x at age x and disability 

intensity σ(x) = 0.0004 + 10 −5.46+0.06x at age x. As annual interest rate use 

4.5%. 

(a) Compute the expected value and standard deviation of the present value at time 0 

of benefits less premiums for a disability pension insurance issued to an active person 

at age x = 30, with insurance period n = 20 years and specifying that pensions are 

payable continuously with intensity 1 during disability and premiums determined by 

the equivalence principle. Perform the calculations also for x = 30, n = 40 and for 

x = 50, n = 20. 

(b) Now consider a portfolio of I insurance contracts and let V denote the present 

value of future benefits less premiums for the entire insurance portfolio. Suppose 

V + 2 √ V is to be provided as a reserve. Assume all I contracts are identical pension 

insurance policies as described above, with x = 30 and n = 20 and that the individual 

life histories are stochastically independent. Study e. g. the “fluctuation loading per 

policy”, 2 √ V /I, as function of I. 

(c) Perform calculations parallel to those in (a) for a modified contract where the 

benefit, instead of pensions during disability, consists of a lump sum payment of 

n 

v u−t e − 

u 

t µ(x+s)ds du 

t 

upon onset of disability at time t < n. (That is, the sum paid is the value of the 

pension described in (a), capitalised upon onset of disability.) Take x = 30 and 

n = 20. Compare with the corresponding result in (a) and comment. 


Exercise 11.5 Consider a 30-year term insurance, issued on G82M, sum insured 

DKK 100.000, age at entry 40. The equivalence principle is adopted. Assume at first 

that we have a single net premium. 

(a) What is the single net premium? 

(b) Put up an expression for the premium reserve Vt at time t and calculate it for 

t = 0, 5, 10, 15, 20, 25 and 30.


(c) Now assume that the premium is paid continuously during the entire insurance 

period with level intensity π. What is π? 

(d) Put up an expression for the premium reserve and evaluate it using the same 

values of t as in (b). 

(SP(57)) 

Exercise 11.6 A man aged 40 years has been issued a 20-year term insurance with 

sum insured 200.000 and level continuous premium intensity during the insurance 

period. 

Find the premium intensity and the net premium reserve 10 years after the time of 

issue of the contract using the technical basis G82, 4.5% net, assuming the equivalence 

princple is adopted. 

(SP(52)) 

Exercise 11.7 Consider the disability model outline below with recovery and excemption 

from payment of premium by disability. The insurance contract is a 40-year 

term insurance, sum insured S = 800.000, age of entry x = 25, level premium intensity 

π as long as the person is active at the most for 40 years. The interest rate is 

4.5%. 

Assume that the intensities are 

σ(x) 

 

0. Active 

 

1. Disabled 

 

ρ(x) 

 

µ(x) 

ν(x) 

 

 

 

 

2. Dead 

µx = 0.0005 + 10 5.88+0.038x−10 , 

σx = 0.0004 + 10 4.54+0.06x−10 , 

ρx = 0.15, 

νx = 10 · µx, 

where µx and σx correspond to the technical basis G82M (including GA82M). 

(a) Put up differential equations for the statwise reserves and for the transition probabilities. 

What are the boundary conditions?


(b) Find the equivalence premium intensity π applying Runge-Kutta to finde the 

transition probabilities and some numerical integration method for evaluating the 

integrals. 

(c) Study the development of the reserves simultaneously, again applying Runge- 

Kutta. What are the statewise reserves at the times t = 10, 20, 30? 

Exercise 11.8 

(a) Construct graphs for the reserves for the following four life insurances 

1. Pure endowment, sum 1 against single net premium. 

(BM and JC, 1995) 

2. Pure endowment, sum 1 against level continuous premium during the entire 

period. 

3. Term insurance, sum 1 upon death before time n against level continuous premium. 

4. Endowment insurance, sum 1 upon death or at time x + n, if the insured is still 

alive, against level continuous premiums. 

All insurances are issued on G82, i. e. µx = 0.0005 + 10 −4.12+0.038x and i = 0.045. 

(b) Find the premiums above. 

(c) What is the expected insurance period for product 3? 

(d) Consider product 1. Assume that 50% of the reserve is paid out upon death 

before the age of x + n. Construct graphs of the reserve and calculate the single net 

premium. 

(MSC “Tillæg til opgave E3” FM0 12.10.94) 

Exercise 11.9 We consider a force of mortality which is Gompertz-Makeham, i. e. 

µx = α + βc x . 

As usual the survival function is denoted by F . Consider an x-year old person with 

remaining life time T . The survival function for the person is 

F (t | x) = 

F (x + t) 

. 

F (x) 

The person can sign different kinds of insurance contracts:


• A pure endowment, where an amount of 1 is payable at time x + n if the person 

then is alive. Expected present value for this benefit is 

nEx = v n F (n | x), 

where v = 1/(1 + i) is the one-year discount rate at interest rate i. 

• An n-year temporary annuity, payable continuously with force 1 per year until 

death, at the most for n years. Expected present value for this contract is 

ax:n| = 

n 

0 

v t F (t | x)dt = 

n 

0 

tExdt. 

• An n-year term insurance, where an amount of 1 is payable upon death before 

age x + n. The expected present value of this contract is 

A 1 

x:n| 

= 

n 

0 

v t F (t | x)µx+tdt 

= 1 − δax:n| − nEx, 

where δ = log(1 + i) is the force of interest. 

• An endowment insurance which is the sum of a pure endowment and a term 

insurance. The expected present value of this benefit is 

Ax:n| = 1 − δax:n|. 

(a) Work out tables for µx, F (x) and the density f(x) = F (x)µx for x = 0, 1, . . . , 100. 

Use the values in the Danish technical basis G82, i. e. α = 0.0005, β = 10 −4.12 , 

c = 10 0.038 and interest rate i = 0.045. 

(b) Calculate the expected present values for the contracts above for x = 30, i = 0.045. 

(c) Redo the calculations in (b) using other values of x and n. In particular, let n 

vary for fixed x = 30 and let x vary for fixed n = 30. 

(d) Construct tables that show how the expected present values above depend on 

i, α, β and c. 

(RN FM1 89/90 opg. E3 05.10.89)

Numerical Methods 75


Insurance Terms 

Aggravated circumstance Skærpede vilk˚ar 

Allot Tilbageføre 

Amount allotted Tilbageføringsbeløb 

Annuity Annuitet, rente 

Bonus scheme Bonusplan 

Capital insurance Kapitalforsikring 

Child’s insurance Børneforsikring 

Collection costs Inkassoomkostninger 

Deduct At fratrække (skat) 

Discount rate Diskonteringsfaktor 

Down payment Kontant udbetaling (p˚a l˚an) 

Duration Varighed 

Endowment insurance Livsforsikring med udbetaling 

evt. sammensat livsforsikring 

Entry Indtrædelse 

Exemption from payment of premium Præmiefritagelse 

Excess mortality Overdødelighed 

Force of interest Renteintensitet 

Force of mortality Dødelighedsintensitet 

Hire-purchase agreement Købskontrakt 

Installment Afdrag 

Insurance period Forsikringstid 

Interest rate Rentefod 

Issue Udstedelse 

Life annuity Livrente 

Loading for collection costs Inkassotillæg 

Portfolio Portefølje, bestand 

Premium free policy Fripolice 

Principal Hovedstol 

Pure endowment Ren oplevelsesforsikring 

Rate of course Kurs 

Return of premium Ristorno 

Second order technical basis Teknisk grundlag af anden orden 

Single net premium Nettoengangspræmie 

Sum insured Forsikringssum 

Surrender Tilbagekøb, genkøb 

Surrender value Tilbagekøbsværdi, genkøbsværdi 

Technical basis Teknisk rundlag, beregningsgrundlag 

Term insurance Ophørende livsforsikring 

Time of expiry Udløbsdato 

Underwriting Processen at tegne en forsikring

Exercises in Life Insurance Mathematics

Create successful ePaper yourself

Delete template?

Save as template?