1. Sections 1.1, 1.2, 1.3, 1.4
If one unit (one dollar) is invested at time t = 0, the accumulation function a(t) gives the accumulated value at time t  0.
a(0) = 1
a(t) is (usually)
increasing.
a(t) is often continuous, although in some applications, it may have discontinuities (reflecting points in time when interest payments are actually made versus the instantaneous “value” of the investment). To illustrate, consider
1 if 0  t < 1
26 / if 1  t < 2
29 / if 2  t < 3 t2
1 + — 25
34 / if 3  t < 4
a(t) = versus a(t) =
41 / if 4  t < 5
2 if 5  t < 6 •

2. If amount k units (k dollars) is invested at time t = 0, the amount function is
A(t) = A(0) • a(t) = k • a(t) for t  0.
The effective rate of interest i is the amount of money that one unit (one dollar) invested at the beginning of a (the first) period will earn during the period, with interest being paid at the end of the (first) period. That is,
i = a(1) – a(0) = a(1) – 1 .
The effective rate of interest during the nth period from the date of investment is
a(n) – a(n – 1) A(n) – A(n – 1)
in = —————— = ——————
a(n – 1) A(n – 1)
In = A(n) – A(n – 1) = is called the amount of interest earned during the nth period. t2
If a(t) = 1 + — , then i1 = , i2 = , and i3 = 25 1 — 25 3 — 26 5 — 29

3. 1
If a(t) = ———— for t < 20, then i1 = , i2 = , and i3 =
1 – 0.05t 1 — 19 1 — 18 1 — 17
Observe that i = i1, and for any accumulation function, it must be true that a(1) =
Note that there are an infinite number of accumulation functions for which
1 + i.
a(0) = 1 and a(1) = 1 + i.
Consider the accumulation function a(t) = 1 + it for integer t  0. Interest accruing according to this function is called simple interest. We call i the rate of simple interest.
Observe that this constant rate of simple interest does not imply a constant rate of effective interest:
a(n) – a(n – 1)
in = —————— =
a(n – 1)
1 + in – [1 + i(n – 1)]
————————— =
1 + i(n – 1) i
—————
1 + i(n – 1)
which is a decreasing function of integer n.

4. 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) – 1)
amount of interest earned over t + s periods, for one unit
amount of interest earned over t periods plus amount of interest earned over s periods, for one unit
In other words, we want a(t + s) = a(t) + a(s) – 1 .
Observe that this property is true for the simple interest accumulation function a(t) = 1 + it but not for the accumulation function a(t) = t2
1 + — . 25
That is,
i(t + s) = it is ,
and
(t + s)2
———  25 t2 — 25 s2 — 25

5. Are simple 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) – 1 – a(t)
a  (t) = lim —————— = lim ————————
s0 s s0 s
a(s) – 1 a(s) – a(0)
= lim ——— = lim ———— = a  (0)
s s s s
a  (t) = a  (0) 
a(t) = 1 + t • a  (0) 
a(1) = 1 + i = 1 + a  (0) 
i = a  (0)
We now have a(t) = 1 + it for all t  0.
Consequently, simple interest accumulation functions are the only ones which preserve the property.

6. Next, we shall 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.