1. 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) =a(t) is (usually) 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 a(t) =versus a(t) = 1 if 0  t < 1 26 / 25 if 1  t < 2 29 / 25 if 2  t < 3 36 / 25 if 3  t < 4 41 / 25 if 4  t < 5 2 if 5  t < 6 1increasing. Sections 1.1, 1.2, 1.3, 1.4 t 2 1 + — 25

2. If amount k units (k dollars) is invested at time t = 0, the amount function is 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, The effective rate of interest during the nth period from the date of investment is I n = A(n) – A(n – 1) = is called the amount of interest earned during the nth period. t 2 If a(t) = 1 + —, then i 1 =,i 2 =, and i 3 =. 25 1 — 25 3 — 26 5 — 29 A(t) = A(0) a(t) = k a(t) for t  0. i = a(1) – a(0) = a(1) – 1. a(n) – a(n – 1)A(n) – A(n – 1) i n =——————=—————— a(n – 1) A(n – 1)

3. Observe that i = i 1, and for any accumulation function, it must be true that a(1) = Note that there are an infinite number of accumulation functions for which 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) i n =—————— = a(n – 1) which is a decreasing function of integer n. 1 If a(t) = ———— for t < 20, then i 1 =, i 2 =, and i 3 =. 1 – 0.05t 1 — 25 1 — 19 20 — 171 1 + i. a(0) = 1 and a(1) = 1 + i. 1 + in – [1 + i(n – 1)] ————————— = 1 + i(n – 1) i ————— 1 + i(n – 1)

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) = t 2 1 + —. 25 That is, i(t + s)= it+ is, and i(t + s) 2 ——— . 25 is 2 — 25 it 2 — 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  0s s  0 s a(s) – 1a(s) – a(0) = lim——— = lim———— = a  (0) s  0 s s  0 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.