Actuarial Mathematics - Lecture 1

Frank Coolen

Durham University

14 January 2013

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 1 / 10

Outline of the course

Compound interest; annuities; repaying a debt

Lifetime models

Life insurance; life annuities

Calculating premiums

Family income; expenses; reserves; possible further topics

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 2 / 10

Course details

Books: Life Insurance Mathematics by Hans Gerber (3rd edition,1997), Fundamentals of Actuarial Mathematics by DavidPromislow (2nd edition 2010, 1st edition also fine)

Support classes: tutorials (weeks 12, 14, 16, 18) and problemsclasses (weeks 13, 15, 17, 19)

Homework: set once every two weeks, hand-in strict deadline oneweek later

Webpage: http://maths.dur.ac.uk/stats/courses/AMII/am.html

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 3 / 10

1. Compound interest

1 Effective interest rates

2 Nominal interest rates

3 Perpetuities and annuities

The closely related topic of ‘Repayment of a Debt’ is Homework 1.

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 4 / 10

1.1 Effective interest rates

To specify an interest rate, one needs to define the basic time unitsuch as ‘year’ (e.g. annual interest rate of 6%) and the conversionperiod at the end of which interest is paid (credited, compounded).

If the conversion period is equal to the basic time unit, then the interestrate is called effective, e.g. 6% annual interest rate credited at the endof each year. Such rates are called Annual Effective Rates: AER anddenoted by i (e.g. i = 0.06).

We assume throughout that i > 0.

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 5 / 10

Initial (time 0) amount F0 with AER i leads to Fn = (1 + i)nF0 after nyears

If you invest a further rk ≥ 0 at the end of year k = 1, . . . ,n, then

Fk = (1 + i)Fk−1 + rk , k = 1, . . . ,n

which leads to

Fn = (1 + i)nF0 +n∑

k=1

(1 + i)n−k rk

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 6 / 10

(1 + i) is called the accumulation factor and (1 + i)k is called theaccumulation value after k years

The discount factor v = 11+i is used to determine the present value

(value at time 0) of future amounts. If AER i remains unchanged forthe whole period concerned, then the present value of an amount C atthe end of time unit k is vkC

Clearly

vnFn = F0 +n∑

k=1

vk rk

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 7 / 10

1.2 Nominal interest rates

If the basic time unit and the conversion period differ, the interest rateis called nominal.

For example, nominal annual interest rate 0.06 with a conversionperiod of three months (m = 4) means interest of 0.06

4 = 0.015 (so1.5%) is credited at the end of each quarter, so initial capital F0becomes (1 + 0.015)4F0 = 1.06136F0 after one year. So nominalinterest rate of 6% with a quarterly conversion period is equivalent toAER of 6.136%.

(Note: the word ‘nominal’ is often deleted if it is clear what the maintime unit is - usually year. For the same reason ‘annual’ is oftendeleted.)

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 8 / 10

Let i(m) be the equivalent nomimal interest rate to AER i with m(equal-length) conversion periods per year (so i(4) = 0.06 correspondsto i = 0.06136 in example above).

(1 +i(m)

m)m = 1 + i

i(m) = m[(1 + i)1/m − 1]

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 9 / 10

For i = 0.06

m i(m)

1 0.062 0.059133 0.058844 0.058706 0.05855

12 0.0584152 0.05830

365 0.05829∞ 0.05827

Exercise: prove that i(m) is decreasing.

Challenge: Derive a formula for limm→∞

i(m).

Frank Coolen (Durham University) Actuarial Mathematics - Lecture 1 14 January 2013 10 / 10