a (0) =

1. Sections 1.1, 1.2, 1.3, 1.4 If one unit (one dollar) is invested at time t = 0, the accumulation functiona(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 / 25 if 1 t < 2 29 / 25 if 2 t < 3 t2 1 + — 25 34 / 25 if 3 t < 4 a(t) = versus a(t) = 41 / 25 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 interesti 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 ss0 s a(s) – 1 a(s) – a(0) = lim ——— = lim ———— = a (0) s0 ss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.