Exercises in Life Insurance Mathematics - kumogens/lifeexer.pdf · This collection of exercises in life insurance mathematics replaces the ... Exercise 1.15 A loan with principal

Exercises in Life Insurance Mathematics

Edited by

Bjarne MessJakob Christensen

University of CopenhagenLaboratory of Actuarial Mathematics

Introduction

This collection of exercises in life insurance mathematics replaces the collection ofSteen Pedersen and all other exercises and problems in any text or article in theFM0L 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 MessBS Bo SøndergaardFW Flemming WindfeldJC Jakob ChristensenJH Jan Hoem “Elementær rentelære”,

Universitetsforlaget, Oslo, 1971.MSC Michael Schou ChristensenMS Mogens SteffensenRN Ragnar NorbergSH Svend HaastrupSK Stephen G. Kellison “The theory of Interest”,

Richard D. Erwing, Inc., Homewood, Il 1970SP Steen Pedersen “Opgaver i livsforsikringsmatematik”SW Schwartz “Numerical Analysis”, Wiley, 1989

Copenhagen, August 2, 1997Bjarne Mess

Jakob Christensen

4 Exercises in Life Insurance Mathematics

Exercises

1. Interest

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

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

(JH(1), 1971)

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

(JH(2), 1971)

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

(JH(3), 1971)

Exercise 1.4 A man needs approximately $2.500 and raises in that connection aloan 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 is6.5%.

(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 ofinstallments for 5 years and is after that to be amortized over 15 years: Interestsand installments are due annually. Assume that the dollar-rate of exchange decreasesexponentially 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 willusually calculate with linear payment of interest instead of exponential payment ofinterest. If the interest rate is i p. a. and we have to calculate interest for a periodof 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 thehire-purchase agreement he finds the following account:

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

What effective interest rate p. a.

(JH(10), 1971)

Exercise 1.8 A man has been promised some money. He can choose from twoalternatives for the payment.

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

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 profitablefor 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 fromtoday 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 andn installments. Show that the installment which falls due in period t is

Ft =i(1 + i)t−1

(1 + i)n − 1H,

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

(a) Show that

Ian| =n−1∑

t=0

t|an−t| ,

and interpret this equation intuitively.

(b) Give an explicit expression for Ian | .

(c) What does the symbol Ian| mean? Give expressions corresponding to the onesfrom (a) and (b).

(SP(10))

Exercise 1.12 A person has a table of annual annutities with different interest ratesand durations to his disposal. However, he needs some present values of half-yearannuities. These annuities do all have the same interest rate and the correspondingwhole-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 stochas-tic variable T with a known distribution. Now consider the expected present valuedenoted by

Aδ(T ) = E(e−δT ).

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

Var(e−δT ) = A2δ(T ) − (Aδ

(T ))2.

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

T | . Let the expected present value be denoted by

aδ(T ) = E(aδ

T |).

(b) Show thatAδ

(T ) = 1 − δaδ(T )

and interpret the equation.

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

(d) Show thatAδ

(T ) > vET ,

and find a similar inequality for aδ(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 coursek and installment Ft in period t, where t = 1, . . . , N . Show that the effective interestrate ie satisfies

ie =1 − S

k − Si1

where

S =1

H

N∑

t=1

Ftvte.

(SP(20))

Exercise 1.15 A loan with principal H and fixed interest rate i1 has to be amortizedannually 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 theamount of amortisation minus deductible (actual net payment) will be the same inall periods (assume tax is payable by the end of each year). Find the annual netinstallment.

(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

At+jvj1 (1.1)

vt1Rt = H −

t∑

j=1

Ajvj1 (1.2)

with the use of standard notation. Formula (1.1) is called the prospective formula forthe remaining debt. Formula (1.2) is called the retrospective formula for the remainingdebt.

(JH(20), 1971)

Exercise 1.17 A loan is being repaid by 15 annual payments. The first five install-ments are $400 each, the next five $300 each, and the final five are $200 each. Findexpressions 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 yearsat effective interest of 5% . Show that the amount of interest in the 11th installmentis

50

1 + v10.

(SK(10) p. 123, 1970)

Exercise 1.19 A borrower has mortgage which calls for level annual payments of 1 atthe 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 theoriginal 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 yearsuntil the mortgage is fully repaid, show that the amount saved in interest paymentsover the full term of the mortgage is

1 − v13.

(SK(16) p. 124, 1970)

Exercise 1.20 A man has some money invested at an effective interest rate i. Atthe end of the first year he withdraws 162.5% of the interest earned, at the end ofthe second year he withdraws 325% of the interest earned, and so forth with thewithdrawal factor increasing in arithmetic progression. At the end of 16 years thefund exhausted. Find i.

(SK(40) p. 127, 1970)

Exercise 1.21 A loan of a25| is being repaid with continuous payments at the annualrate of 1 p. a. for 25 years. If the interest rate i is 0.05, find the total amount of interestpaid 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 at4% effective, the borrower decides to repay the balance of the loan over the next fiveyears by equal annual principal payments in addition to the annual interest due onthe unpaid balance. If the lender insists on a yield rate of 5% over this five-yearperiod, 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 = 10.000kr. The rate of interest is 3% p. a. and the student applicates for the loan on thefollowing 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 yearuntil 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 amountof amortisation?


(b) Which amount αt has to be paid at time t if the loan (with interest earned) is tobe 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 Fis concentrated on the interval [a, b] and that F is continuous with continuous densityf . Assume F (t) < 1 for t ∈ [a, b). Define

µ(t) =f(t)

1 − F (t), t ∈ [a, b).

We say that µ is the intensity of F .

(a) Show that∫ b

a µ(t)dt = ∞.

(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 maximumlife 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 his60th 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 remaininglife 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 roleindicated by the symbol:

µx = (1 + x)−3, x ≥ 0

F (x) = 1 − 22x

12+

11x2

8− 7x3

24, 0 ≤ x ≤ 3


f(x) = xn−1e−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’sand a woman’s total life lengths are xq

M0 and xq

K0 respectively. Assume that these

probabilities are continuous so that the forces of mortality µMx and µK

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 consideredin this exercise.

(a) Find the distribution function xq0 for the total life time for a person of unknownsex 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 `Mx and `K

x for men and women respectively, work out a decre-ment serie `x for the total population:

(c) How should one appropriately choose `M0 and `K

0 ?

(d) Express ax in terms of aMx 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 `x

0 4010 3970 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. Assumethe lives are independent.


(AM(3.13) p. 78, 1986)

Exercise 2.6 When considering aggregate mortality the probability that an x-yearold 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 lifetime.

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

(c) Show that

s|tqx =

∫ s+t

supxµx+udu,

and interpret this expression.

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

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

n|qx =dx+n

`x

.

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

(f) Prove the following identities:

n|mqx = npx − n+mpx

n|qx = npx · qx+n

n+mpx = npx · mpx+n.

(SP(24))

Exercise 2.7 Let e◦x:n| denote the expected future lifetime of (x) between the agesof x and x + n. Show that

e◦x:n| =

∫ n

0

ttpxµx+tdt + nnpx

=

∫ n

0tpxdt.

This is called the partial life expectancy.


(AM(3.14) p. 78, 1986)

Exercise 2.8 The force of mortality µx is assumed to be

µx = βcx.

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

(Tentamen i forsikringsmatematik, Stockholms Hogskola 1954)

Exercise 2.9 If F (x) = 1 − x/100, 0 ≤ x ≤ 100, find µx, F (x), f(x) and P (10 <X < 40).

(AM(3.5) p. 77, 1986)

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

(AM(3.7) p. 77, 1986)

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

βex.


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 + βcx, show that for each age x we have

− log c

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


(Tentamen i forsikringsmatematik, Stockholms Hogskola, 1954)

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

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

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

(AM, 1986)

Exercise 2.15 Prove the following expressions:

d

dx`xµx < 0, when

d

dxµx < µ2

x

d

dx`xµx = 0, when

d

dxµx = µ2

x

d

dx`xµx > 0, when

d

dxµx > µ2

x.

(AM(3.12) p. 77, 1986)

Exercise 2.16 Show the following identities:

∂tpx

∂t= −µx+t · tpx

∂tpx

∂x= (µx − µx+t) · tpx

1 =

∫ ω−x

0tpxµx+tdt

`x =

∫ ω−x

0

`x+tµx+tdt.

(SP(26))

Exercise 2.17 If the force of mortality µx+t, 0 ≤ t ≤ 1, changes to µx+t − c where cis a positive constant, find the value of c for which the probability of (x) dying withina 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 bydoubling 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 thandouble the mortality rate, qx, of the standard table?

(AM(3.35) p. 80, 1986)

Exercise 2.19 If µx = Bcx, show that the function `xµx has a maximum at age x0,where µx0

= log c. (Hint: Exercise 2.15).

(AM(3.36) p. 80, 1986)

Exercise 2.20 Assume

µx =Acx

1 + Bcx

for x > 0.

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

(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 thepresent value ax:u−x| for an integer expiration age u with age at issue x = 0, 1, . . . , uand futhermore a table of the one year survival probabilities px for the same ages. Lett be a real number, 0 ≤ t < 1. We are trying to find a way to determine ax+t:u−x−t|

from the data of the table; this method will, of course, depend on how the mortalityvaries 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 ax+t:1−t| .

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

ax+t:u−x−t| = λax:u−x| + (1 − λ)ax+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 Ax:u−x| .


(FM1 exam, summer 1983)

Exercise 2.22 For insurances where the policies are issued on aggravated circum-stances, one operates with excess mortality. Let µx be the force of mortality corre-sponding to the normal mortality. A person is said to have an excess mortality if hisforce 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 ≥ 0exists a constant ∆k so that

µx+∆k= (1 + k)µx,

then the mortality satisfies Gompertz’s law.

(SP(43))


3. Insurance of a Single Life

FM0 S92, 3

Exercise 3.1 Prove the identities:

ax = 1 + vpxax+1

1 − nEx = ax: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 deathwithin the first n years, the level continuous premium is paid back with interest andcompound interest earned. Rewrite the integral expression for the present value aftert years (t < h) of the future return of premium per unit of the premium in order toshow that this value is

(st| + ax+t:n−t|) −Dx+h

Dx+t

sh| ,

where st| is defined by JH. Interpret the expression.

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

Exercise 3.4 Show that

nEx = 1 − iax:n| − (1 + i)A1x: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

`x = k(1 − x

ω)α

for all x ∈ [0, ω].

(a) Find an expression for µx.

(b) Find an expression for e◦x (see exercise 2.7).

Now assume that α = 1.

(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 insureddies, not later than n years after the time of issue, or if he survives until the age ofx + n. x is the age at entry.

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

Ax:n| = 1 − dax:n| ,

where d is the discount rate.

A1x: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 ofx + n. x is the age at entry.

(b) Give an expression for A1x:n| and show that

A1x:n| = 1 − nEx − dax:n| .

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

(SP(33))

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

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


persons have force of mortality µix. 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 ageat entry x has to pay for a level continuous annuity with sum 1 p. a. The insurancecancels n years after issue.

aix: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.

aax:n | Single net premium for active persons. The annuity is payable from the time of

issue until death of the insured.

aaax:n | As above except that the premium is due for a contract that cancels by death

or by disability of the insured.

aaix:n | 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 aaix:n| in terms of µa

x, νx, µix and δ.

(b) Assume that µax = µi

x for all x. Express aaix:n| in terms of aaa

x:n| and aix:n| .

(c) Assume that µax = µi

x + ε and νx = ν where ε and ν are independent of x andε 6= ν. Express aai

x:n| by aaax: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 theage, 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

0tpxdt

is the expected period of insurance for a term insurance or an n-year temporaryannuity, age of entry x and age of expiration x + n.

Define

ex:n| =n∑

t=1

tpx

ex:n| =n−1∑

t=0

tpx.

Give a similar interpretation of these identities. Define

at| =1 − vt

i

at| =1 − vt

d

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

ax:n| < aex:n||ax:n| < aex:n||.

(SP(39))

Exercise 3.10 Show that

ax =

∫ ∞

0tpxAx+tdt.

(SP(41))

Exercise 3.11 Assume that µx is a weakly increasing function of x and considerfor given x two insurance contracts with initial age x: First consider a whole-life lifeinsurance with sum insured 1 and secondly a whole-life continuous annuity with levelpayment intensity determined so that the expected present value of the two insurancecontracts are equal. The one with the biggest variance of the present value is naturallythe one with the biggest risk for the company.

Show that there exists an x0 ≥ 0 so the annuity is more risky than the life insuranceiff x > x0.


(SP(42))

Exercise 3.12. (Multiplicative Hazard Model) The mortality in a population variesfrom person to person; some has greater or lesser mortality than the average. Thiscan be modelled as follows:

There exists a underlying force of mortality µx and for each person a constant θindependent 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 astochastic variable Θ, and we assume moreover that EΘ = 1.

Show that in this model the expected present value of a continuous temporary n-yearannuity with payment intensity 1 is greater than or equal to

ax:n| =

∫ n

0

e−δttpxdt,

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 atentry x and premium paid continuously during the entire period with level intensityp. Expenses are disregarded.

(a) What is the surplus of this contract for the company in terms of the remaininglife 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 negativesurplus 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 prob-ability of a negative surplus for the entire portfolio lesser than ε by using a smaller pthan the one found in (c).

(SP(50))

We have so far worked with continuous insurance benefits - annuities that are duecontinuously and life insurances that are due upon death. The pure endowment seemsto be of another origin, because the time of possible single payment is determined inadvance. In the next exercise we will consider more general kinds of non-continuousor 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−∫

∞

0µx+τ dτ ,

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

The results of the following exercise will show that continuous benefits can be concide-red as limits for discrete benefits. We will also see that both continuous and discreteannuities 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

Cet = vt1{T>t}.

(a) Find the expectation, tEx, of Cen and find Cov(Ce

s , Cet ) for s 6= t.

A brute-forth generalisation of the pure endowment is produced by summing morebenefits like this. A simple variant is the n-year temporary deferred annuity payableannually with fixed amounts as long as the insured is alive. This is the sum of n pureendowments with deferment times 1, . . . , n. The present value at time t = 0 is

Ca(1)n =

n∑

t=1

Cet .

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

be

Ca(h)n =

hn∑

t=1

Cet/h.

(b) Find the expectation, a(h)x:n| , and the variance of the present value Ca(h)

n .

An n-year temporary life insurance with sum insured 1, payable at the end of theyear of death, has present value

Cti(1)n =

n∑

t=1

vt1{t−1<T≤t},

and the corresponding insurance payable at the end of the 1hth year, in which death

occurs, has present value

Cti(h)n =

hn∑

t=1

vtn 1{ t−1

h<T≤ t

h}.


(c) Express the present value C ti(h)n in terms of present values of annuities given by

Ca(h)n . Compare with similar expressions for continuous benefits.

(d) Find the expectation, A(h)x:n| , and the variance of the present value C ti(h)

n .

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

ax:n| =

∫ n

0

vtF (t | x)dt =

∫ n

0tExdt

A1

x:n| =

∫ n

0

vtF (t | x)µx+tdt

= 1 − δax:n| − nEx,

for the expectations of Can and Cti

n and also to find their variances. (Hint: By mono-tone convergence we have Cα(2m)

n ↗ Cαn 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 ofcontinuous benefits in the usual Markov model. Consider an annuity, payable withlevel intensity of 1 by staying in state j, and an insurance where a sum of 1 is paidupon every transition j → k.

(RN “Opgave E7” 29.01.90)

Exercise 3.15 The functions that occur in insurance mathematics often depend onseveral variables, e. g. m|ax:n| , and are often hard to tabulate. In order to solve thisproblem, we introduce the so-called commutation functions. In connection with lifeinsurances of one life we consider the following:

Cx = vxdx Cx =∫ x+1

x vξ`ξµξdξ

Dx = vx`x Dx =∫ x+1

x vξ`ξdξMx =

∑ωξ=x vξdξ Mx =

∫ ω

x vξ`ξµξdξNx =

∑ωξ=x vξ`ξ Nx =

∫ ω

x vξ`ξdξRx =

∑ωξ=x(ξ − x)vξdξ Rx =

∫ ω

x (ξ − x)vξ`ξµξdξ

Sx =∑ω

ξ=x(ξ − x)vξ`ξ Sx =∫ ω

x (ξ − x)vξ`ξdξ.

(a) Show that

m|ax:n| =Nx+m − Nx+m+n

Dx

.

(b) Find corresponding expressions for ax:n| , A1

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

(c) What can we possibly use Rx for?


(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 payablecontinuously during the insurance period with level intensity π. Upon death twothirds 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 forcep as long as the insured is alive at the most for n years from the issue of thecontract.

• Every year an annual premium of the size

p(1) = p · a1|

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

R = p(1) ·aθ|

a1|

= p · aθ| ,

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 mannerthat regardless of when the insured is going to die, the present values of the premiumpayments 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 premiumschemes be denoted by V t and Vt.

The Net Premium Reserve and Thiele’s Differential Equation 27

(c) Show that for t ≤ n

Vt = V t + p · a[t]−t|

where [t] is the integer part of t.

(SP(53))

Exercise 4.3 Consider a linearly increasing n-year term insurance. If the insureddies at time t after the time of issue where t < n, the amount tS is paid out. If theinsured is alive at time n, the amount nS is paid out at this time. The age of theinsured at entry is x. The net premium determined by the equivalence principle, isdue 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 timet, 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). Thepremium is composed of a single premium π0 at the beginning of the contract andby 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, suminsured St at time t ∈ (0, n), and a continuous flow with intensity (st)t∈(0,n) as longas (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 differentialequation.

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

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


Now assume that the benefits moreover consist of a pure endowment, sum insured Sat 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 differ-ential equation?

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

(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 policywith widow pension, insurance period n years, issued to a man aged x and his wifeaged y. During the insurance period the premium Π falls due annually in advanceas long as both are alive. If the man dies, the benefit is a widow pension of sum 1paid out on every following anniversary of the policy during the insurance period ifthe 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 thereserve at time t + 1 for t = 0, 1, . . . , n − 1 so that the premium reserves can becalculated recursively.

(c) Define the savings premium and the risk premium and find an interpretable ex-pression for the latter.


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 bythe 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 thesolutions.


(SP(55))

Exercise 4.7 If the insured dies before time n (from the time of issue) the benefit isa continuous annuity with force 1, duration m from the time of death. If he is aliveat time n, the benefit is a similar annuity from this time and if he is still alive at timen+m he receives a whole-life annuity with intensity 1. There is a single net premiumat 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

Rt =

(1 − vm)ax+t:n−t| − n+m−t|ax+t (t < n)−n+m−t|ax+t (n ≤ t < n + m)−ax+t (n + m ≤ t).

In particular we haveR0 = (1 − vm)ax:n| − n+m|ax,

and sincelim

n→∞R0 = (1 − vmax) > 0,

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

Nx+τ = Nx+n +Nx+n+m

1 − vm.

(SP(56))

Exercise 4.8 Consider a pension insurance contract, where the benefit is an n-yearannuity of 1 deferred m years (expected present value m|nax). Premium is paid withlevel intensity c during the deferment period (expected present value cax:m|).

(a) What is the equivalence premium c and the development of the reserve whenx = 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 thepremium 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 thepremium 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 lumpsum 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 finitenumber of t’s where Bt 6= 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 diesduring [t − 1, t) then St is the benefit. The equivalence principle is adopted for theinsurance and all expenses are disregarded. Let the premium reserve at time t be Vt

and let the stochastic variable Gt be given by

Gt =

0 (the insured is dead at time t−)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 surplusof the insurance.


(a) Interpret Gt and show that

Y =∞∑

t=0

vtGt.

(b) Show that

VarY =∞∑

t=0

Var(vtGt).

G1, G2, . . . are not necessarily stochastically independent, but note that (b) is validregardless of whether G0, G1, . . . are stochastically independent or not.

(c) Show that Hattendorf’s Formula

VarY =∞∑

t=0

tpxv2t+2px+tqx+t(St+1 − Vt+1)

2

is valid.

(SP(58))

Exercise 4.11 In this exercise we are to study an endowment insurance with returnof premium paid at death before expiration. Consider a person of age x who wishesto buy an insurance with age of expiration x + n, and where the premium is paidcontinuously 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 thatthe premium paid so far will be returned with interest (basic interest i) earned. Wedisregard expenses.

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

Bt = pat|(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 theinsurance company to issue this kind of insurance.



Exercise 4.12 An n-year deferred whole-life annuity on the longest lasting life hasbeen issued to two persons (x) and (y). Annual amount of 1 and level continuouspremium on the longest lasting life with intensity π during the deferment period. Theforces of mortality are denoted by µx and νy

Put up a retrospective expression for the net reserve at time t assuming that bothare alive.

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

Exercise 4.13 A family annuity is an insurance contract of one life that assurespayment of a continuous annual annuity from the possible death of the insured duringthe insurance period until the expiration of the contract after n years. Force ofmortality µx, interest rate i, age at entry x.

(a) Put up the formulas for the net premium reserves, prospectively and retrospec-tively, with level continuous premium payment π. Show that if the insured does notdie 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 identicalcontract issued at the same time will develop during the insurance period. Show thatthis 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 andmortality of survival it is a problem that the premium for an insurance can dependon 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 solvethis problem of additivity.

Consider an insurance

Ax:n| + s · n|ax

issued against a single premium.

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 forcesof mortality µ and µ which satisfy

µx+t < µx+t < µx+t,

and replace (4.1) by

∂Vx(t)

∂t=

{

(δ + µx+t)Vx(t) − µx+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 safeside”.

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

Consider the special caseµx = (µx + g2)(1 + g1),

and letµ∗

x = µx + g2

andδ∗ = 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 specialcase 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 gt and cg in terms of V n

t and cn and α′ in the case whereα′′ = β′′ = γ′′ = γ′′′ = 0.


Exercise 5.3 Treat the case of a level annuity payable upon death in (m,n) againstlevel premiums in (0,m).


Exercise 5.4 Consider an n-year deferred whole-life annuity, age at entry x, payablewith level continuous intensity S. Level gross continuous premium intensity p payableduring the deferment period. The premium is determined by the equivalence princi-ple. For now assume that the expenses are initial expenses αS, loading for collectionfees due continuously with level intensity βp and administration costs also due con-tinuously with level intensity γS.

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

Because of inflation, loading for collection fees and administration expenses are paidwith intensities βf(t)p and γf(t)S at time t.

(b) Put up exspressions for p and V gt .

(b) Find p and V gt 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

B− 1t

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 attime 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 isindependent of the age at entry.

A simple capital insurance only makes sense if it is not paid by a single payment (whendealing with insurance). Assume that the premium is paid continuously during theentire insurance period with level intensity p, but only if the insured is alive. Thepremium 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 collec-tion fees βp and administration expenses γS paid continuously.

(c) Determine p.

(d) Put up an expression for the prospective gross premium reserve, both when theinsured is alive as well as when he is dead.

(SP(61))

The following exercise examines what happens to the insurance technical quantitieswhen we bring surrender into consideration.

Exercise 5.7 Consider an n-year endowment insurance, age of entry x, benefits areS1 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 ageof entry). Moreover assume that surrender can take place at any time during thepremium payment period, and that the present value of the conventionally calculatedgross premium reserve, liquidated by surrender, is positive. By surrender at time tthe company pays out G(t).

(a) Now disregard all expenses and assume that G(t) is lesser than or equal to theconventionally calculated (net) premium reserve at time t. Instead of using a con-ventional technique, the company could itself bring surrender into consideration and


into its own technical basis. Show that the equivalence principle then would lead to apremium P ′ ≤ P . Discuss conditions for P ′ = P and give a lower limit for how smallP ′ can get when G(·) varies. Here and in the following it might be useful to studyThiele’s differential equation.

Now assume that some administration costs are not neglectible. The expenses consistof the amount α in initial expenses, of γ in administration costs per time unit andof a fraction β of the actual annual gross premium P in loading for collection fees, Pcalculated conventionally. Upon surrender 100θ% of the gross premium reserve is paidout if the reserve is positive, 0 ≤ θ ≤ 1. By surrender where the gross premium reserveis positive, a fixed percentage of the reserve is deducted to cover the loss experiencedby surrender where the gross premium reserve is negative. For now disregard expensesthat fall upon surrender. This gross premium reserve is calculated without respect tosurrender.

(b) Show that you will get a lesser gross premium reserve if you bring surrenderinto consideration. Assume that θ is chosen so that the equivalence principle can beapplied anyway.

(c) Now assume that in the above situation a constant expense ξ is associated withthe actual payment of G(t). If G(t) calculated in (b) is smaller than ξ, nothing is paidout by surrender. When the value upon surrender mentioned exceeds ξ the differenceis paid out. Show that the actual gross premium reserve still will be lesser than theconventional when surrender is brought into consideration. Can θ still be determinedso 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 calcu-lated as being proportional to the gross premium reserve instead of being proportionalto the sum insured. Now consider an n-year endowment insurance, level continuousgross 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 grosspremium 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− andshow that the solutions can be expressed by expected present values for annuitieswith 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 arecalculated 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 hedies during the insurance period, the company returns the up till now paid premiumswithout interest earned. The level gross premium intensity p is payable during theentire insurance period. The expenses are initial expenses αS, loading for collectionfees due with continuous level intensity βp and continuous administration costs duewith 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’sinsurance: If the insured dies during [x, y) the gross premium is paid back withearned interest according to the technical basis. If he dies during [y, u) the sum S isimmediately paid out and if he is alive at age u, then S is paid out. The level grosspremium p, the administration costs γS and loading for collection fees βp fall duecontinuously 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 insuranceperiod.

(c) What is the gross (equivalence) premium intensity, and what is the risk sum atany time with this premium.

(SP(68))

Exercise 5.11 Consider an n-year endowment insurance, sum insured S, age atissue x, premium payable until time m, initial expenses αS, administration costs andloading for collection fees due during the entire insurance period continuously withlevel intensities γS and βpg respsctively, where pg is the level gross premium intensity.Assume that m ≤ n and γ < δ.

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


be cast as

pg =Ax:n| + γax:n|

(1 − β)ax:m|

S.

(b) Show that β > β and γ > γ.

The numerator of the expression is the so-called passive with added sum, because itis produced from the net passive Ax:n| increased by the present value of γ during the

entire insurance period. This passive is denoted by AS

x:n| .

Let Vt be the net premium reserve at time t (calculated from the time of issue) andlet V 1

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

(c) Give expressions for Vt and V 1t .

The company ought to set aside the reserve V 1t but normally the reserve

V 2t = SA

S

x+t:n−1| − (1 − β)pgax+t:m−t|

is set aside.

(d) Compare Vt, V1

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

t insteadof V 1

t .

If the insured wishes to surrender his contract at time t, the company pays him thesurrender value of the contract Gt, which is the reserve V 1

t less the part of the initialexpenses that have not yet been amortized.

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

Gt = S(Ax+t:n−t| + γax+t:n−t|) − (1 − β)pgax+t:m−t| .

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

x:n| .

(f) Find an expression for the difference between the net premium reserve and thesurrender 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 surrenderthe contract, it is called a premium free policy. The size of this policy is determined byletting the surrender value of the new policy equal the surrender value of the originalpolicy at the time of change.

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

pg = εpg,

where ε is called the continuity factor.

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

SAg

x+t:n−t| − ζpgax+t:m−t|

and

SAS

x+t:n−t| − ηpgax+t:m−t| ,

respectively.

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

He furthermore considers a supplementary pension, also with level intensity s which isdue 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 timen from the time of issue, n > m + r.

As a second alternative he considers a survival annuity, also with payment intensitys. Here he considers two options:

(iv) The annuity initiates if the man dies before time r from the time of issue and isdue as long as his wife lives.

(v) The annuity initiates if the man dies before time r from the time of issue and isdue 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 paidin advance as long as the man and his wife are alive, at the most until time r fromthe time of issue. If one of the two dies during the insurance period, the amount(aθ|/a1|)P is being returned (in Danish: Ristorno), where θ is the remaining part ofthe 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 insuranceperiod for the last insurance (v) applying the recently described premium paymentprinciples.

When calculating the gross premiums, the company uses the following expense rates:Initial expenses αS, loading for collection fees β times the gross premium, adminis-tration costs due continuously with intensityand γ times the gross premium reserveat 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 describedbefore (b).

(SP(76) rev.)

Select Mortality 41

6. Select Mortality

Exercise 6.1

(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 ` under the assumption that theperiod 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 diesbetween 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 aninsurance 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 mortalityfor 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 disabledaccording to the figure below where the course of events is modelled by a Markovprocess {Xt}t≥0, and t is the age of the insured.

1. Active

σ(t)//

µ(t)KK

KKKK

%%KKKK

KK

2. Disabledρ(t)

oo

ν(t)ss

ssss

yysssss

s

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) = limh↘0

pij(t, t + h)

h, i 6= j.

We assume that the intensities are continuous functions. Define

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

andµ1(t) = µ(t) + σ(t), µ2(t) = ρ(t) + ν(t).

We take it thatµ(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 afterentry (as an active), one gets a filtration (of the above Markov model) which isdetermined by the force of mortality for a random insured. Let µ(τ, x) denote thisintensity 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)p12

p11 + p12

(τ, x)

= µ(x) + {ν(x) − µ(x)} p12

p11 + p12

(τ, x)

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

We obviously want τ → µ(τ, x) to be decreasing for fixed x which will be shown bydifferentiation in the following.

(b) Explain that τ → µ(τ, x) is decreasing iff the fraction τ → (p12/p11)(τ, x) isdecreasing and give an interpretation of this.

(c) Show thatd

dτ

(

p12

p11

)

(τ, x) =σ(τ)(p12p21 − p11p22)(τ, x)

p211(τ, x)

andd

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

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

Let pij(s, t) denote the transition probabilities corresponding to the model withoutrecovery, i. e. ρ(t) = 0, ∀t ≥ 0.

Select Mortality 43

(d) Find expressions for p11(τ, x) and p12(τ, x) as a function of the intensities andshow that

p12

p11

(τ, x) <p12

p11

(τ, x).

Give an interpretation of this and explain how µ is affected by changing to the modelwithout recovery.

It is a common opinion that the selection the insured goes through at entry disappearsafter a period of time, called the period of selection. We will try to explain thisphenomenon mathematically. Lad τ0 be the age of entry for an insured.

(e) Show that

x → exp

(∫ x

τ0

µ1(s)ds

)

p11(τ0, x), x ≥ τ0

is an increasing function. Assume that µ2(t) ≥ µ1(t), ∀t ≥ 0 and that there exists aw > τ0 so that

∫ w

τ0

(µ2(t) − µ1(t))dt = ∞.

Show that

limx↗w

d

dτ

(

p12

p11

)

(τ0, x) = 0

monotonically withd

dτ

(

p12

p11

)

(τ0, x) = 0, ∀x ≥ w,

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 fromthe mortality of the general population because the insured lives are a selected partof the population. We will in this exercise study one relationship that is assumed tocontribute a great deal to the effect of selection, i. e. the fact that people with ill-nesses 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 thepopulation can be divided according to the figure below.

0. Not insured, not ill

κx

((QQQQQQQQQQQQQQQ

σx

��

ρx // 1. Insured, not ill

σx

��

κx

vvnnnnnnnnnnnnnnn

4. Dead

3. Not insured, ill

λx

66mmmmmmmmmmmmmmm

2. Insured, ill

λx

hhPPPPPPPPPPPPPPP


Now assume that every person enters state “0” at birth and that the transitionsbetween the states afterwards go on as a time continuous Markov chain with transitionintensities only dependent on the age x as indicated in the figure. Excess mortalityfor the ill persons means that

λx ≥ κx, x > 0, (6.1)

with “>” for some values of x.

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)

p12(x − t, x) =

∫ x

x−t

e−∫

z

x−t(σu+κu)du

σze−∫

x

zλududz, 0 < t < x; (6.3)

p00(0, x) = e−∫

x

0(σu+κu+ρu)du, (6.4)

p01(0, x) = e−∫

x

0(σu+κu)du

(1 − e−∫

x

0ρudu

), (6.5)

p02(0, x) =

∫ x

0

e−∫

z

0(σu+κu)du

(1 − e−∫

z

0ρudu

)σze−∫

x

zλudu

dz, (6.6)

p03(0, x) =

∫ x

0

e−∫

z

0(σu+κu+ρu)duσze

−∫

x

zλududz, 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” receivedthe insurance contract before they were struck by illness). The insurance companydoes not observe in which of the two states the insured is. All the company knows isthe time of entry and age. Let µ[x−t]+t denote the force of mortality for an insured ofage 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 isa 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 whohas no knowledge of actuarial science?

(d) Discuss the formula for µ[x−t]+t to find theoretical explanations as to why insurancecompanies operate with a period of selection s so that the mortality is considered tobe aggregate for t > s.

Let µx denote the force of mortality for a randomly chosen person of age x in thepopulation (that is, we do not observe in which of the states “0” – “3” the person is).

Select Mortality 45

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

µx ≥ µ[x−t]+t, 0 < t < x

is satisfied. The result clarifies the preliminary remarks of this exercise. Try to give anexplanation that is comprehensible for a person without any knowledge of actuarialmathematics.

(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” and1, . . . , k denoting death from k different reasons; denote the partial probabilities ofdeath by

tq(j)x = 1 − exp

(

−∫ t

0

µ0jx+τdτ

)

,

and define

tp(j)x = 1 − tq

(j)x .

(a) Prove that

tp00x =

k∏

j=1

tp(j)x .

(b) Prove that

k∑

j=1

tp0jx =

k∑

j=1

tq(j)x −

∑

1≤i<j≤k

tq(i)x tq

(j)x

+∑

1≤h<i<j≤k

tq(h)x tq

(i)x tq

(j)x − · · · + (−1)k+1

tq(1)x · · · tq

(k)x .

Now assume that for given j there exist constants c and t0 > 0 so that

tp(j)x µ0j

x+t = c,

for all t ∈ [0, t0].

(c) Prove that for all t ∈ [0, t0]

tq(j)x =

t

t0· t0q

(j)x

tp0jx =

1

t· tq

(j)x

∫ t

0

πh6=jτpx(h)dτ.

(SP(83))

Exercise 7.2. (Life Insurance with Exemption from Payment of Premium by Dis-ability) In this exercise we will consider a policy that can attend one of N states

Markov Chains in Life Insurance 47

enumerated 1, . . . , N . The development of the policy is being described by a Markovprocess with transition intensities µjk

t for transition from state j to state k at time tfrom issue.

In the contract it is stated that the amount B jkt is payable upon transition from state

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

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

is paid out. Assume that all amounts Bjkt 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 jt . How should negative values

of Bjt be interpreted?

Let V jt 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 untiltime t, given that the policy is in state j at time t.

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

−Bjt =

d

dtV j

t − δV jt +

∑

k 6=j

µjkt (Bjk

t + V kt − V j

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

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

A special case of the this general Markov model is the disability model. This modelhas three states, 1, 2 and 3, corresponding to active, disabled and dead. The insured’sage at entry is x and the intensities are denoted by

µ12t = σx+t (from active to disabled),

µ21y = ρx+t (recovery),

µ13y = µx+t (dead as active),

µ23t = νx+t (dead as disabled).

All other transition intensities are 0.

Apply this model for an investigation of an endowment insurance with exemptionfrom payment of premiums by disability. Assume the insured is active at entry. Theinsurance period is n, so the insurance cancels when the insured has reached the ageof x + n or if he dies before that.

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

B1t = −π,B2

t = 0, B13t = B23

t = S.

(d) Come up with formulas for π, V 1t and V 2

t for 0 ≤ t < n in terms of the transitionprobabilities and transition intensities in the model and the discounting rate v, bydirect prospective reasoning.

If the premium and the benefits depend on the reserves, the premium and the premiumreserve cannot be determined directly as in (d); instead the differential equationsystem from (b) must be solved with appropriate boundary conditions. Now it isassumed 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 ispaid out as a supplement to the sum insured; B j3

t = S+V jt for j = 1, 2 and 0 < t < n.

(e) Show that the differential equation system from (b) gives the grounds for a dif-ferential equation of first order in V 1

t − V 2t . What is the initial condition? Solve the

differential equation.

(f) What are π, V 1t and V 2

t for 0 ≤ t < n.

(g) Show that if νx+t ≥ µx+t for all t < n then V 2t > V 1

t for all t < n. Interpret thisresult.

Assume that we have the following expenses: Initial expenses due at time 0 withan amount of αS, loading for collection fees βp, where p is the level gross premiumintensity and administration costs due with an intensity at time t equal to γV j

t if thepolicy is in state j.

(h) Explain, without performing any detailed calculations, what changes would followfrom 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 duecontinuously with level intensity b upon disability before the age of x+n. Premium ispayable at rate π as long as the person is active during the contract period. Assumethat the state of the policy is S(t) at time t after the time of issue where {S(t)}t≥0 isa time continuous Markov model with state space and transitions as follows


0. Active σ(x+t) //

µ(x+t)KK

KKKK

%%KKKK

KK

1. Disabled

ν(x+t)tt

tttt

yyttttt

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

s

vt−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

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

and the conditional variance

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 andthe functions (7.2) and (7.3).

(d) Now assume that the equivalence principle is being adopted, i. e. V0(0) = 0, whereπ is the net premium intensity and Vj(s) in (7.2) above is the net premium reserve instate 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 inexercise 7.3. There are two states of disability i1 and i2 representing two degrees ofdisability. Assume that ν2(x+t) ≥ ν1(x+t). Let {S}t≥0 be the corresponding Markovchain and define the stochastic process {S(t)}t≥0 by S(t) = S(t) for S(t) ∈ {a, d} andS(t) = i for S(t) ∈ {i1, i2}. If one does not know the degree of disability, {S(t)} isthe observable process.


i1.

ν1(x+t)

��

a.

µ(x+t)LL

LLLL

L

&&LLLLLL

σ1(x+t)rrrrrrr

88rrrrrr

σ2(x+t) // i2.

ν2(x+t)rr

rrrr

xxrrrrrr

d.

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

µjk(t, {S(s)}s<t) = limdt↘0

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

dt

for j 6= k and any specification of {S(s)}s<t.

(a) Prove thatµai(t, {S(s)}s<t) = σ1(x + t) + σ2(x + t)

andµad(t, {S(s)}s<t) = µ(x + t)

(both independent of {S(s)}s<t) and

µid(t;S(s) = a for 0 ≤ s < τ, S(s) = i for τ ≤ s < t)

=

∑2j=1 σj(x + τ) exp(− ∫ t−τ

0 νj(x + τ + u)du)νj(x + t)∑2

j=1 σj(x + τ) exp(−∫ t−τ

0 νj(x + τ + u)du)(7.4)

(a function of x+τ and t−τ). Give a sufficient condition for {S(t)}t≥0 being Markov.

(b) Interpret the expression in (7.4). Give sufficient conditions for finding an ap-proximating function to (7.4) depensing only on x + t for sufficiently great t − τ , sayt − τ ≥ s0 (the time of selection).

(FM1 exam opgave 2, summer 1989)

Exercise 7.5. (Markov Chains in Life Insurance) Give a detailed proof of theKolmogorov backward differential equations (2.15) and the corresponding integralequations (2.21).

(RN “Problems in Life Insurance” 14.10.93, problem 7)

Exercise 7.6. (Markov Chains in Life Insurance) consider the model for disabilities,recoveries, and deaths in Paragraph H in RN “Markov Chains in Life Insurance”04.03.94.


(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 dependson s and t only through t− s, hence put s = 0. Discuss how the probabilities dependon t and the intensities.

(c) Calculate for t = 0, 10, 20, . . . , 100 and draw graphs of the probabilities for somedifferent 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 probabilitesfor the case σ = 0.1, ρ = 0.1, µ = 0.05, ν = 1. (The ratios between the intensities arethe essential feature.)

(e) The probability p02(s, t) gives the mortality law for a person who is known to beactive at age s. Discuss how it depends on the intensities, with special attention tothe case where µ = ν.

(f) Consider the special case with no mortality (µ = ν = 0), whereby the number ofstates essentially becomes 2. Find p01(0, t), and discuss how the expression dependson t and the intensities. Find the limit as t → ∞ and discuss the expression.


Exercise 7.7. (Markov Chains in Life Insurance) Formulate suitable continuoustime Markov chain models for solution of the following problems, where N is assumedto 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 theproblem.



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 thesample 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 pensioninsurance policy specifying that, conditional on survival, premiums are payable withlevel intensity c from time 0 to time m and pensions are payable continuously withlevel 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 thatinterest is earned at a constant rate δ so that v(t) = vt, with v = e−δ the annualdiscount rate. Throughout b is taken as fixed and c is to be determined by theequivalence principle.

(a) Put up prospective and retrospective expressions for the reserves in both statesat any time t ∈ [0, n) after issue of the policy. As an exercise, find the reserves alsoby solving the appropriate differential equations. Determine c.

(b) Find the conditional variances of the individual reserves (the present values offuture 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 tothose in (b).

Consider another modified policy ’R’, by which the retrospective reserve in state 0 isto 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’, correspond-ing 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 whichthe retrospective reserve in state 0 is to be repaid upon the death of (x) during thepremium period [0,m], whilst the prospective reserve in state 0 is to be repaid upondeath 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 specifyingthat an amount b (the sum insured) is provided immediately upon death before timen and premiums are payable with level intensity c as long as the person is aliveand active up to time n (premium waiver by disability). Assuming that recoveryis impossible, the relevant Markov model can be sketced below. Assume that thediscount function is v(t) = vt = e−δt. The premium c is to be determined by theequivalence principle.

0. Activeσ(x)

//

µ(x)JJ

JJJJ

%%JJJJ

JJ

1. Disabled

ν(x)tt

tttt

yyttttt

t

2. Dead

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

(b) Put up expressions for the prospective and retrosprective reserves in all statesat any time t ∈ [0, n). Find the reserves also by solving the appropriate differentialequations. Determine the premium intensity c.

Suppose that instead of full premium waiver, the premium during disability is madedependent on the past savings on the contract. More specifically, assume that premi-ums during disability fall due with intensity c − c′V −

1 (t) at time t if the policy thenis in state 1.

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



Exercise 7.11 Consider the usual disability model without recovery:

0. Activeσ(x)

//

µ(x)JJ

JJJJ

%%JJJJ

JJ

1. Disabled

ν(x)tt

tttt

yyttttt

t

2. Dead

An insurance is issued to a person (x). Premium is payed continuously with levelintensity 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 ageof x + n the sum D is paid out. If (x) is active at the time m ∈ (0, n) the sum S ispaid 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 amountS is paid immediately upon death before time of expiration n. If the insured is aliveat time n he is also provided the sum insured S. Premiums are payable with levelintensity p as long as the insured is active up to time n. That is, the insured has theright to exemption from payment of premium if he is disabled. Interest is earned ata constant rate δ. The Markov model used is illustrated below.

0. Active

σ(x)//

µ(x)JJ

JJJJ

%%JJJJ

JJ

1. Disabledρ(x)

oo

ν(x)tt

tttt

yyttttt

t

2. Dead

(a) Put up the following:

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 p00(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 p00(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 differentialequation.

(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 theresult 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 allresults.

(e) Find expressions for the safety loadings for the two states and consider the differ-ence. Which sign does it have with a reasonable choice of parameters?

(f) Expand the model with the state “surrender”. What would you consider a rea-sonable payment in connection with surrender from the state of active and disabledrespectively. How should the first order intensity of surrender be put in relation tothe 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 inten-sities µ, ν, σ and ρ. Let α = µ + σ, κ = ν + ρ and assume that α 6= κ.

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

(c) Explain how one relying, on information about death, disability and recovery insome 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 ofmortality µx = 0.0005+105.88+0.038x−10 (G82M), interest rate i = 5% p. a., acquisitioncosts 3% of the sum insured, loading for collecting fees 5% of the premium andadministration expenses due continuously with intensity γV g

t where V gt is the gross

premium reserve and

γ = log1, 05

1, 045.

The technical basis of second order has interest rate i = 5.5% p. a. and force ofmortality µx = µx − (δ − δ), where δ and δ are the forces of interest corresponding toi and i 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 continuouslywith level intensity during the entire period of insurance.

(a) Calculate the gross premium intensity and the gross premium reserve after 20 and40 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 insuranceperiod.

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

Now assume that the bonus scheme consists of two discrete payments: One after 20years 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 20years 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 ac-cording 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 passesaway 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 remaininglife 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) isalive 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 annu-ity in terms of one-life and both-life annuity expressions. Give a direct interpretationof this expression.

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

(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 forcesof 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 ofsecond order? How would you determine the elements of the technical basis of secondorder 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 dueeither. If (x) dies before (y), a continuous amount B is added to the annuity paymentof 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 thepresent 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 asthe whole-life annuity is due.

(i) What is the difference between the two bonus schemes? Which Bonus schemewould 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 deathbefore 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 ofinterest is δ, the force of mortality µx, expenses αS by issue and continuous expenseswith intensity γS + βB per year during the insurance period.

(a) Determine the premium B by applying the equivalence principle. Determine thenet and gross premium reserves at any time during the insurance period. Find aconnection 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 areassumed to be safety loadings κ and λ included in the force of interest and in theforce of mortality respectively. Hence, a realistic force of interest of second order isδ = 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 timeduring the insurance period.

The return of premium scheme is as follows: (i) An amount (κ − λ)Vt is paid outduring (t, t + dt) during the insurance period, where Vt is the premium reserve forcovering the outstanding claims and return of premiums (i. e. the insured earns aninterest (κ− λ)dt of his part of the total fund); (ii) He receives an additional amountK 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 ofentry x. The company adopts the following technical basis for calculation of premiumsand premium reserves: Force of interest δ, force of mortality µx and expenses βπ paidcontinuously where π is the level net premium, paid continuously during the entireperiod. In the technical basis of first order we assume the force of surrender to bezero.

Bonus Schemes 59

(a) Put up Thiele’s differential equation and an expression for the equivalence pre-mium intensity.

The actual development with respect to the technical basis of second order for thecompany is as follows: The interest rate has dropped to δ ′ and the administrationexpenses have dropped to β ′. Now assume that the force of surrender exists and isdenoted by νx+t t years after the time of issue where νx is independent of the timethe 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 premi-ums 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 everywhole year or by death or surrender.

(c) Put up Thiele’s differential equation for the premium reserve on the technicalbasis 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 boundaryconditions 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 insur-ance is described by a continuous Markov Chain on a state space J = {0, . . . , J},and the contract specifies that a0

g(t)dt is payable if the policy stays in state g in thetime interval (t, t+dt) and a0

gh(t) is payable upon transition from state g to state h attime t. (More general annuity payments can easily be dealt with, but the expressionbecomes more messy.) Assume for the time being that the discount function v isdeterministic.

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

(∫ ∞

0

v(τ)A(dτ)

)2

= E

(∫ ∞

0

v2(τ)(A(dτ))2 + 2

∫ ∞

0

v(τ)A(dτ)

∫

ϑ>τ

v(ϑ)A(dϑ)

)

. (9.1)

(a) Prove that (9.1) can be recast as

EV 2 =

∫ ∞

0

v2(τ)∑

g

p0g(0, τ){2a0gV

+g (τ)

+∑

h6=g

µgh(τ)a0gh(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 fromthe mean of the square. Formula (9.2) appears to offer an escape from the doubleintegration that has to be performed in (9.1). It requires that the prospective reservein different states be computed (they are essentially the inner integral, of course) andstored 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 simplelife annuity, for which the results are well-known.

Another simple formula for the variance, which shall not be proved here, is

VarV =

∫ ∞

0

v2(τ)∑

g 6=h

p0g(0, τ)µgh(τ)(a0gh(τ) + V +

h (τ) − V +g (τ))2dτ (9.3)

(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. Thepolicy specifies that the sum insured S is payable immedieately upon (possible) deathof the insured before time n and that premium is due with fixed amount c at times0, 1, . . . , n− 1 as long as the insured is alive. Assume that administration costs incurcontinuously with constant intensity γS throughout the duration of the policy andthat 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 theinsurance payment, the premiums and the administration costs. Find the variance ofthe 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 payablecontinuously 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 attime 0 of benefits less premiums.

(b) Suppose that in case of surrender the insured immedieately gets the current netvalue of the policy defined as the prospective reserve at the time of surrender. Assumethat surrender takes place with intensity γ(t) at time t < n. (Thus we consider anextended model with three states “insured”, “withdrawn” and “dead”.) Find theprospective reserve in state “insured” and the variance of the present value at issueof benefits less premiums. Compare with the results in (a).


It is possible to find differential equations which determines any n-order momentV

(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 insurancewith level premium intensity π payable as long as the insured is alive. Upon deaththe 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 2t | T > t).

(b) Derive the expression from (a) with respect to t to find that you will get a differ-ential 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

1

0

µ1(x)kkkkk

55kkkkk

µ2(x) //

µh(x)E

EE

EE

EE

EE

E

""EE

EE

EE

EE

EE

2

...

h

is called the model for competing risks with h reasons for resignation. The functionµ =

∑hi=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

tpx = p00(x, x + t) = e−∫

t

oµ(x+s)ds

and

tq(k)x = p0k(x, x + t) =

∫ 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 oneyear. Let the state “0” in the model above represent the state “alive” and let theh reasons of resignation be reasons of deaths. It is possible to assume the forces ofmortality, µ1, . . . , µh, to be constant and the same for all L persons. The number ofdeaths from the h reasons are denoted D1, . . . , Dh and the sum of life time is T .

(b) Determine the maximum likelihood estimators µ1, . . . , µh and the asymptotic dis-tributions of the estimators.

(HRH “Opgaver til FM3” (ex. 2))

Exercise 10.2 Consider the time continuous Markov model as follows


0. Working

σ(x)//

µ(x)KK

KKKK

%%KKKK

KK

1. Not workingρ(x)

oo

µ(x)rr

rrrr

yyrrrrrr

2. Dead

(a) Find explicit expressions for the transition probabilities.

(b) Assume that all members in a portfolio of N independent lives are being observedduring the period of time [0, 1] and assume that all transition intensities are constantduring the time of observation. How would you estimate σ, ρ and µ? Determine theasymptotic distributions of the estimators.

(HRH “Opgaver til FM3” (ex. 12))

Exercise 10.3 An insurance company is to perform a mortality study based oncomplete records for n life insurance policies with unlimited term period. Policynumber i was issued zi years ago to a person who was then aged xi. The actuary setsout to maximize the likelihood

n∏

i=1

µ(xi + Ti, θ)Di exp

(

∫ xi+Ti

xi

µ(s, θ)ds

)

,

where the notation is obvious.

One employee in the department objects that the method represents a neglect ofinformation; it is known that the insured have survived, not only the period theywere insured, but also the period from birth until entry into the scheme. Thus, heclaims, the appropriate likelihood is rather

n∏

i=1

µ(xi + Ti, θ)Di exp

(

∫ xi+Ti

0

µ(s, θ)ds

)

.

Settle this apparent paradox. (A suitible framework for discussing the problem is anenriched model with three states, “uninsured”, “insured” and “dead”.)


Exercise 10.4 Reference is to Paragraph 2C in the paper RN “Inference in theMarkov 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 ofthe MLE.


Exercise 10.5. (Transformation of Data and Analytical Smoothening) Consider themortality model where n independent persons are being observed during the intervalof age [x, x + z), where x and z are integers. Exact times of death are observed andwe 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 sothat

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 letµ = (µx, . . . , µx+z−1)

t, where

µk =Dk

Tk

, k = x, . . . , x + z − 1.

(a) Assume that µk can be cast as

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 aget and of an unknown parameter θ = (θ1, . . . , θp)

t, p < z. Assume moreover that theJacobian Dg = Dg(θ)

dθexists and has full rank p for g(θ) = (gx(θ), . . . , gx+z−1(θ))t. Let

Rz+ = {η = (ηx, . . . , ηx+z−1) ∈ Rz | ηk > 0, k = x, . . . , x + z − 1},

and let L : Rz+ → Rz be a differentiable mapping, so that the z × z Jacobian DL =

DL(η)

dηhas full rank z. Define at last α = (αx, . . . , αx+z−1)

t and g(θ) by

α = L(µ)

andg(θ) = L ◦ g(θ) = L(g(θ)).

The parameter θ can be determined in two ways by analytical smoothening. You caneither by modified χ2-minimizing determine the value of θ that brings g(θ) “closest”to µ, or you can by modified χ2-minimizing determine the value of θ that brings g(θ)“closest” to α. Denote these two estimators by θ and θ respectively and show that θand θ has the same asymptotic variance. Is θ = θ?


(b) Now assume thatµk = βcξk , 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 MarkovModel”, 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 theMarkov 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 ornot. Show that the likelihood based on Di, i = 1, . . . , n, is

qN(1 − q)n−N ,

with q = 1 − e−µz, the probability of death before z. (Trivial: It is the binomialsituation.)

Note that N is now sufficient, and that the class of distributions is a regular expo-nential 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 Appendixof the paper to show that

µ∗ ∼as N

(

µ,q

nz2(1 − q)

)

.

The asymptotic efficiency of µ relative to µ∗ is

asVarµ∗

asVarµ=

(

eµz

2 − e−µz

2

µz

)2

=

(

sinh(µz/2)

µz/2

)2

(sinh is the hyperbolic sine function defined by sinh(x) = (ex−e−x)/2). This functionmeasures the loss of information suffered by observing only death/survival by age z


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 smallµz, the number of deaths is all that matters, whereas for large µz, the life lengths areall that matters. Reflect over these findings.


Exercise 10.8 Use the general theory of Section 2 of the paper RN “Inference inthe 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.



11. Numerical Methods

In this chapter we will use some numerical methods on the theory of life inurance thatwe have already seen. One does not always have a program that can calculate thepremiums, the development of the reserves etc. and it can be necessary to developyour own programs.

Inspired by SW let us at first deal with numerical integration. We wish to calculatethe definite integral

I =

∫ b

a

f(x)dx. (11.1)

A commonly used numerical method for evaluating (11.1) is the summed Simpson’srule. In all it’s simplicity is states that

I =

∫ b

a

f(x)dx ' h

3

[

f(a) + 4f(x1) + f(b) + 2N−1∑

k=1

(f(x2k) + 2f(x2k+1))

]

with h = (b−a)/2N,xj = a+ jh, j = 1, 2, . . . , 2N −1. The accuracy is good for smallvalues 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, ageof expiration 60. The technical basis of the company is G82M, i. e. the interest rateis i = 4.5% and the force of mortality is

µx = 0.0005 + 105.88+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 pre-mium 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) toevaluate the premium reserve at times t = 10, 20, 30.

Some expenses are, however, not neglectible and the insured has to pay some ex-penses 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 acontinuous 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 nowthe gross premium) modified?

(e) Use your program in (c) to calculate the gross premium and the gross premiumreserve 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 thenet premium reserve in the above we solved the differential equation theoretically andthen applied Simpson for evaluation. This is not always possible. Another way is tosolve Thiele’s differential equation numerically; methods for this are plentyful and theRunge-Kutta method of fourth degree is highly recommended. Let d

dxy = f(x, y(x)),

xk = x0 + kh, k integer and define

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)

then

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 chosenwhen the slope is not too big. A sixth order Runge-Kutta can also be applied, butthe difference from the fourth order R-K is really not that great. It is possible to usethe difference between the fourth and the sixth order Runge-Kutta in order to findappropriate 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

3h, yk +

7

27k1 +

10

27k2 +

1

27k4),

k6 = hf(x +1

5h, yk +

28

625k1 −

1

5k2 +

546

625k3 +

54

625k4 −

378

625k5),

and hence

y(xk+1) ' y(xk) +1

24k1 +

5

48k4 +

27

56k5 +

125

336k6.

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 asystem of differential equations

y(x) =

y1(x)y2(x)

...yn(x)

f(x, y) =

f1(x, y1(x), . . . , yn(x))f2(x, y1(x), . . . , yn(x))

...fn(x, y1(x), . . . , yn(x))

y0

=

y(1)0

y(2)0

...

y(n)0


we have, similar to the fourth order R-K,

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,

where ddx

y(x) = f(x, y(x)) and y(x0) = y0. Remember this method when evaluating

a system of simultaneous differential equations. It is obvious that this method forsolving differential equation systems simultaneously has a great applicability whenstudying the simultaneous development of the statewise reserves in a general Markovmodel for a policy. The functions are allowed to depend on each other just as thestatewise reserves depend on each other, compare with the expression for the statewisereserves.

Exercise 11.2 Consider an endowment insurance, duration 20 years, age at entryx = 30, interest rate i = 4.5%, force of mortality

µx = 0.0005 + 105.88+0.038x−10.

The sum insured is 500.000. The only premium is a single net premium Π upon issueof the contract. The equivalence principle is adopted.

(a) Use the fourth order Runge-Kutta to find this single net premium Π. Use thesame program to study the development of the net premium reserve.

Instead the man does not want to compose any initial capital, but a level continuouspremium 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 initialcapital of 5.000 and then a level continuous premium with intensity π during theinsurance 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 = 10and b = 1. (A numerical integration must be performed to find the second order mo-ments. 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 onedimension when a table of reserves has been generated.)


Exercise 11.4. (Moments of Present Values) Consider the Markov model for disabil-ities, 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 dis-ability intensity σ(x) = 0.0004 + 10−5.46+0.06x at age x. As annual interest rate use4.5%.

(a) Compute the expected value and standard deviation of the present value at time 0of benefits less premiums for a disability pension insurance issued to an active personat age x = 30, with insurance period n = 20 years and specifying that pensions arepayable continuously with intensity 1 during disability and premiums determined bythe equivalence principle. Perform the calculations also for x = 30, n = 40 and forx = 50, n = 20.

(b) Now consider a portfolio of I insurance contracts and let V denote the presentvalue of future benefits less premiums for the entire insurance portfolio. SupposeV + 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 individuallife histories are stochastically independent. Study e. g. the “fluctuation loading perpolicy”, 2

√V /I, as function of I.

(c) Perform calculations parallel to those in (a) for a modified contract where thebenefit, instead of pensions during disability, consists of a lump sum payment of

∫ n

t

vu−te−∫

u

tµ(x+s)dsdu

upon onset of disability at time t < n. (That is, the sum paid is the value of thepension described in (a), capitalised upon onset of disability.) Take x = 30 andn = 20. Compare with the corresponding result in (a) and comment.


Exercise 11.5 Consider a 30-year term insurance, issued on G82M, sum insuredDKK 100.000, age at entry 40. The equivalence principle is adopted. Assume at firstthat 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 fort = 0, 5, 10, 15, 20, 25 and 30.


(c) Now assume that the premium is paid continuously during the entire insuranceperiod with level intensity π. What is π?

(d) Put up an expression for the premium reserve and evaluate it using the samevalues of t as in (b).

(SP(57))

Exercise 11.6 A man aged 40 years has been issued a 20-year term insurance withsum insured 200.000 and level continuous premium intensity during the insuranceperiod.

Find the premium intensity and the net premium reserve 10 years after the time ofissue of the contract using the technical basis G82, 4.5% net, assuming the equivalenceprincple is adopted.

(SP(52))

Exercise 11.7 Consider the disability model outline below with recovery and ex-cemption from payment of premium by disability. The insurance contract is a 40-yearterm insurance, sum insured S = 800.000, age of entry x = 25, level premium inten-sity π as long as the person is active at the most for 40 years. The interest rate is4.5%.

0. Active

σ(x)//

µ(x)JJ

JJJJ

%%JJJJ

JJ

1. Disabledρ(x)

oo

ν(x)tt

tttt

yyttttt

t

2. Dead

Assume that the intensities are

µx = 0.0005 + 105.88+0.038x−10,

σx = 0.0004 + 104.54+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 prob-abilities. What are the boundary conditions?


(b) Find the equivalence premium intensity π applying Runge-Kutta to finde thetransition probabilities and some numerical integration method for evaluating theintegrals.

(c) Study the development of the reserves simultaneously, again applying Runge-Kutta. What are the statewise reserves at the times t = 10, 20, 30?

(BM and JC, 1995)

Exercise 11.8

(a) Construct graphs for the reserves for the following four life insurances

1. Pure endowment, sum 1 against single net premium.

2. Pure endowment, sum 1 against level continuous premium during the entireperiod.

3. Term insurance, sum 1 upon death before time n against level continuous pre-mium.

4. Endowment insurance, sum 1 upon death or at time x+n, if the insured is stillalive, 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 deathbefore the age of x + n. Construct graphs of the reserve and calculate the single netpremium.

(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 + βcx.

As usual the survival function is denoted by F . Consider an x-year old person withremaining 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 personthen is alive. Expected present value for this benefit is

nEx = vnF (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 untildeath, at the most for n years. Expected present value for this contract is

ax:n| =

∫ n

0

vtF (t | x)dt =

∫ n

0tExdt.

• An n-year term insurance, where an amount of 1 is payable upon death beforeage x + n. The expected present value of this contract is

A1

x:n| =

∫ n

0

vtF (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 terminsurance. 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 = 100.038 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 nvary for fixed x = 30 and let x vary for fixed n = 30.

(d) Construct tables that show how the expected present values above depend oni, α, β and c.

(RN FM1 89/90 opg. E3 05.10.89)



Insurance Terms

Aggravated circumstance Skærpede vilkarAllot TilbageføreAmount allotted TilbageføringsbeløbAnnuity Annuitet, renteBonus scheme BonusplanCapital insurance KapitalforsikringChild’s insurance BørneforsikringCollection costs InkassoomkostningerDeduct At fratrække (skat)Discount rate DiskonteringsfaktorDown payment Kontant udbetaling (pa lan)Duration VarighedEndowment insurance Livsforsikring med udbetaling

evt. sammensat livsforsikringEntry IndtrædelseExemption from payment of premium PræmiefritagelseExcess mortality OverdødelighedForce of interest RenteintensitetForce of mortality DødelighedsintensitetHire-purchase agreement KøbskontraktInstallment AfdragInsurance period ForsikringstidInterest rate RentefodIssue UdstedelseLife annuity LivrenteLoading for collection costs InkassotillægPortfolio Portefølje, bestandPremium free policy FripolicePrincipal HovedstolPure endowment Ren oplevelsesforsikringRate of course KursReturn of premium RistornoSecond order technical basis Teknisk grundlag af anden ordenSingle net premium NettoengangspræmieSum insured ForsikringssumSurrender Tilbagekøb, genkøbSurrender value Tilbagekøbsværdi, genkøbsværdiTechnical basis Teknisk rundlag, beregningsgrundlagTerm insurance Ophørende livsforsikringTime of expiry UdløbsdatoUnderwriting Processen at tegne en forsikring

Documents

Exercises in Life Insurance Mathematics - kumogens/lifeexer.pdf · This collection of exercises in life insurance mathematics replaces the ... Exercise 1.15 A loan with principal