Consider the accumulation function a(t) = (1 i)t for integer t ? 0. Interest accruing according to this function is called compound interest. We call i the rate of compound interest.

1
Sections 1.5, 1.6
Consider the accumulation function a(t) (1
i)t for integer t ? 0. Interest accruing
according to this function is called compound
interest. We call i the rate of compound
interest.
Observe that this constant rate of compound
interest implies a constant rate of effective
interest, and the two are equal
a(n) a(n 1) (1 i)n (1 i)n1 in
1 i 1 i .
a(n 1) (1 i)n1
Suppose we want to define a differentiable
function a(t) so that for non-integer t, we
preserve the following property a(t s)
1 a(t) 1 a(s) a(s) 1
amount of interest earned over t s periods, for
one unit
amount of interest earned over t periods, for
one unit, immediately re-invested for s periods
amount of interest earned over s periods, for one
unit
2
In other words, we want a(t s) a(t) a(s) .
Observe that this property is true for the
compound interest accumulation function a(t) (1
i)t but not for the simple interest
accumulation function a(t) 1 it . That is,
(1 i)t s (1 i)t (1 i)s , and 1
i(t s) ? (1 it)(1 is) .
3
Are compound interest accumulation functions the
only ones which preserve the property? For a(t)
to be differentiable, we must have
a(t s) a(t) a(t) a(s) a(t) a ? (t)
lim lim s?0 s
s?0 s
a(t) (a(s) 1) a(s) a(0) lim
a(t) lim a(t) a ? (0) s?0
s s?0 s
a ? (t) a ? (0) ? a(t)
d lna(t) a ? (0) ? dt
a ? (t) a(t) a ? (0) ?
t
t
t
t
d lna(r) dr a ? (0) dr ? dr
lna(r) r a ? (0) ?
0
0
0
0
t a ? (0)
lna(t) ln(1) t a ? (0) ?
a(t) e ?
4
t a ? (0)
lna(t) ln(1) t a ? (0) ?
a(t) e ?
a ? (0)
a(1) 1 i e ?
a ? (0) ln(1 i) ?
a(t) et ln(1i) ?
We have a(t) (1 i)t for all t ?
0. Consequently, compound interest accumulation
functions are the only ones which preserve the
property.
5
Observe that (1) (2)
With simple interest, the absolute amount of
growth is constant, that is, a(t s) a(t) does
not depend on t.
With compound interest, the relative rate of
growth is constant, that is, a(t s) a(t) /
a(t) does not depend on t.
What is the amount A(0) which must be invested to
obtain a balance of 1 at the end of one period?
1 A(0) . 1 i
Since we want 1 A(1) A(0) a(1) A(0) (1
i), then
1 v is called the discount factor.
1 i
What is the amount A(0) which must be invested to
obtain a balance of 1 at the end of t periods?
Since we want 1 A(t) A(0) a(t), then A(0)
a(t)1.
a(t)1 is called the discount function.
6
1 for t ? 0. 1 it
With a simple interest accumulation function,
a(t)1
With a compound interest accumulation
function, a(t)1
1 vt for t ? 0. (1 i)t
a(t) is said to be the accumulated value of 1 at
the end of t periods, and a(t)1 is said to be
the present value (or discounted value) of 1 to
be paid at the end of t periods.
7
Find the present (discounted) value of 3000 to
be paid at the end of 5 years (i.e., the amount
which must be invested in order to accumulate
3000 at the end of 5 years) (a) with a rate of
simple interest of 7 per annum. (b) with a
rate of compound interest of 7 per annum. (c)
with the accumulation function a(t)
3000 2222.22 1 (0.07)(5)
3000a(5)1
3000 2138.96 (1 0.07)5
3000a(5)1 3000v5
t2 1 . 25
3000 1500 1 (52/25)
3000a(5)1