Actuarial Mathematics II

Hand-Out 1: Effective & Nominal Interest Rates; Annuities & Perpetuities; Repaying a Debt

Frank Coolen (CM206 - [e-mail]), January 2008

The first topics in this course are very basic concepts underlying actuarial and financial mathematics, which will

be used later in this course in combination with probabilities on individuals’ future lifetimes, to study life insurance
and life annuities. We stay close to Chapter 1 of the course book by Gerber (‘Life Insurance Mathematics’), which
however contains some more material than we present. Notation used in the lectures will also remain very close to
Gerber’s notation. The hand-outs include exercises, some of which will be presented in the problems classes (in weeks
12, 14, 16 and 18), and some will be set as homework (weekly): questions will be set and collected at the Thursday
lecture (15.15), that is a strict deadline (due to practicalities of the marking arrangements no late homeworks will
be accepted). The remainder of the exercises are useful as well, and you can also work on (some of) these during
the tutorials (weeks 13, 15, 17 and 19). Short hints and/or solutions to all exercises (including detailed solutions to
homework exercises) will be made available once we have gone through the whole question sheet.

Course web page: http://maths.dur.ac.uk/stats/courses/AMII/am.html
Hand-outs, and some more information about this course (e.g. homework and suggestions with regard to exercises),
will become available on the course web page, which you can access via the Maths Departments 2H Teaching page,
by clicking on my name. (There is more material on this webpage, for students doing a 3H project on follow-up
material.)

Exercises
1-1. Suppose you want to invest capital C > 0 in a savings account. You have two choices: (1) C in one account
with annual effective interest rate (AER) i > 0; (2) C/2 in an account with AER i +  and C/2 in an account with
AER i − , for  ∈ (0, i). You want to leave the money in the account(s) for n ≥ 1 years. Which of these is the best
choice?
Hint: it may help to show first that, for α ≥ 0 and for n ≥ 1: (1 + α)

n

+ (1 − α)

n

≥ 2.

1-2. You invest £5,000 in a savings account at the start of 2005. For parts (a) and (b), assume that the AER is
fixed at 4%.
(a) What is the balance in the account at the end of 2007?
(b) How many years will it take for the initial amount to have doubled?
(c) What should the fixed AER have been to double the initial amount in y years?
(d) Calculate the balance in the account at the end of 2007, with 4% nominal interest rate, and with interest
compounded m times per year, for m = 1, 12, 52, 365. Also calculate this balance in this case but with continuous
compounding.
(e) With continuous compounding, what should the force of interest be to double the initial amount in y years?

1-3. George decides to set up a savings account, with his initial savings of £1,000. At the end of each of the next 4
years, he will add a further £500 to this account. Calculate the total value at the end of year 5 if the bank offers
(a) An annual effective interest rate of 3% for all years; repeat this for AER 5%, and for AER 6%.
(b) AER 3% for years 1 and 2, and AER 5% thereafter.
(c) AER 3% in year 1, with 0.5% added to the AER each following year for the total balance in the account.

1-4. Thelma wishes to put £2,000 in a savings account at the start of year 1, and to add £c to it at the end of years
1 to 8. She wishes the account to reach £15,000 at the end of year 8. Assuming that the AER i is constant over this
period, determine c as a function of i, and give c explicitly for i = 0.04 and for i = 0.08.

1-5. Louise considers putting £200 in a savings account at the start of each month, for a whole year. Calculate the
total value of this account, at the end of the year, if the annual nominal interest rate is 4% with a conversion period
of 1 month.

1-6. Calculate the present values of the following annuities and perpetuities, for each case with AER 2%, 4% and
6%.
(a) An immediate annuity with 10 annual payments of £5,000.
(b) An immediate annuity with 10 annual payments of £10,000.
(c) An annuity-due with 10 annual payments of £10,000.
(d) A perpetuity-due with annual payments of £5,000.
(e) A perpetuity-due with annual payments of £10,000.
(f ) An immediate perpetuity with annual payments of £10,000.
Give some comments on the differences between these present values.

1-7. Consider an annuity or perpetuity with present value £100,000. Calculate the annual payments for each of the
following cases, for each case with AER 2%, 4% and 6%.
(a) An immediate annuity with 10 annual payments.
(b) An immediate annuity with 20 annual payments.
(c) An annuity-due with 10 annual payments.
(d) A perpetuity-due.
(e) An immediate perpetuity.
Give some comments on the differences between these payments.

1-8. Calculate the final values of the annuities in Exercise 1-6.

1-9. At the start of year 1, Stan buys an immediate annuity with present value £50,000, giving 10 equal payments
at the end of years 1 to 10, with AER 3%. At the end of year 5, immediately after the 5th payment, Stan wishes to
stop this annuity, to invest the 5th payment and the remaining amount of the annuity in a new similar immediate
annuity, for a further 5 years, but with AER 4%. Stan is allowed to stop the first annuity at a penalty cost of £1,000.
(a) Calculate all annual payments Stan receives according to this combination of annuities.
(b) Suppose the penalty cost had been £C, with C ≥ 0. Calculate all values for C for which this switch of annuities
is to Stan’s benefit.

1-10. Consider a perpetuity which consists of yearly increasing payments, such that the payment at the end of year
j ≥ 1 is equal to (1 + k)

j

, for 0 < k < 0.04. The AER is equal to 4%. At the start of year 1, the present value of this

perpetuity is 51.
(a) Determine k.
(b) Explain why, for such a perpetuity with increasing payments of this form, k must be less than the AER.

1-11. Dagobert wishes to be remembered forever by his old university, and decides to offer a perpetuity which allows
the Dagobert-studentship to be offered. The grant payments for a full year are required at the start of each year,
and for year 1 the total grant needed is £15,000 (fees and maintenance costs). Thereafter, the grant payment is
scheduled to increase by 3% each year, to cover inflation. Dagobert is sure, however, that the clever people at his
old university will be able to invest the money and get an 8% return on the invested total per year.
(a) Calculate the amount of money Dagobert must now transfer to the university to fund this indefinitely continuing
studentship.
(b) Let V

j

be the value of the investment, financing the perpetuity, right at the start of year j ≥ 1, just before the

grant for year j is paid to the student. Let g

j

be the grant paid to the student in year j. Derive recursive formulae

for g

j

and V

j

, and determine g

j

and V

j

for j = 1, . . . , 10.

1-12. Suppose you have a debt of £10,000 at the start of year 1. You wish to repay this in 10 payments, one at the
end of each of years 1 to 10.
(a) Suppose you opt for 10 equal payments. Determine these payments if the AER is 5%, and also if the AER is
10%. For each of these cases, provide a detailed breakdown of all the annual payments, in interest payments and
reductions of principal.
(b) Suppose you pay £1,000 at the end of each of years 1 to 9. What is the required final payment if the AER is
5%, and what if the AER is 10%?
(c) Suppose that the AER is 10%, and you follow the scheme of 10 equal payments (part (a)). But, at the start of
year 6, you are offered the opportunity, at a penalty cost of £C ≥ 0 (which is added to the outstanding debt at that
moment in time), to switch to AER of 8%. Determine all values of C for which this switch is to your benefit.