QABE_Notes04 - QABE Lecture 4 Geometric Progressions and...

QABE Lecture 4
Geometric Progressions and Annuities
School of Economics, UNSW
2011
Contents
1 Introduction 1
2 Geometric Progressions 2
3 Annuities 3
3.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Present Value, A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Spreadsheeting it . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 Future Value, S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Annuities Due 6
1 Introduction
Depending on your studies, you may never have heard of an ‘annuity’. However, chances
are, you have actually been thinking about, or seeing the work of, annuities in everyday
life for years. For example, when you are repaying your mobile phone in a number of
fixed installments you are paying an annuity . When you take out a loan for a car, or some
other purchase, and then begin repaying the loan, you are dealing with an annuity . If
you have an older friend who is receiving a regular pension amount from a superanuation
benefit such that there will be nothing left by their life’s end, they are getting money
through an annuity . Finally, if you have a savings account which you regularly transfer
money into (say, monthly) and don’t intend to touch for a number of years (as in the
film, The Bank ), you have an annuity . Enough said.
You see, annuities do live something of a secret life – we all use many financial services
which have their own specific names, but if we look a little deeper, we are just dealing
with an annuity . To understand these creatures better, we have to start with a bit of
maths to refresh ourselves on the nature of geometric progressions and series , which will
then enable us to investigate the many- and varied- forms of an annuity. If you can get
a handle on annuities, then doubt-less you’ll become a favoured expert amongst friends
and family as you crunch the numbers on their various financial options!
Agenda
1. Background: the geometric progression and series;
2. Annuities – present, future value;
1

ECON 1202/ECON 2291: QABE c School of Economics, UNSW
3. Annuities due.
2 Geometric Progressions
Consider the following number sequences:
2 ,-2,2,-2,2,-2,2,-2,2,-2,2,-2 (-1)
1.00 ,0.60,0.36,0.22,0.13,0.08,0.05 (0.6)
0.5 ,1.3,3.1,7.8,19.5,48.8,122.1,305.2 (2.5)
... in each, we have,
• An initial value , a ; and
• A ratio of terms , r ,
r =
x
i +1
x
i
Definition | Geometric Progression
A (finite) geometric progression (or sequence) is a list of numbers where
the first number a is chosen, and subsequent numbers are given by multiplying
the preceeding term by a constant factor r ,
a, ar, ar
2
, ar
3
, . . . , ar