1. 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) n–1 i n =—————— =————————— =1 + i – 1 = i. a(n – 1) (1 + i) n–1 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 Sections 1.5, 1.6

2. 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, In other words, we want a(t + s) = a(t) a(s). (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  0s s  0s 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(t) a  (0)  a  (t) —— = a  (0)  a(t) d — ln[a(t)] = a  (0)  dt 0 t d — ln[a(r)] dr = a  (0) dr  dr 0 t ln[a(r)] = r a  (0)  0 t 0 t ln[a(t)] – ln(1) = t a  (0)  a(t) = e  t a  (0)

4. a(1) = 1 + i = e  a  (0) a  (0) = ln(1 + i)  a(t) = e t ln(1+i)  We have a(t) = (1 + i) t for all t  0. Consequently, compound interest accumulation functions are the only ones which preserve the property. ln[a(t)] – ln(1) = t a  (0)  a(t) = e  t a  (0)

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? Since we want 1 = A(1) = A(0) a(1) = A(0) (1 + i), then 1 A(0) =——. 1 + i 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. With a simple interest accumulation function, [a(t)] –1 = 1 ——for t  0. 1 + it With a compound interest accumulation function, [a(t)] –1 = 1 —— = v t 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) = t 2 1 + —. 25 3000[a(5)] –1 = 3000 —————= $2222.22 1 + (0.07)(5) 3000[a(5)] –1 = 3000v 5 = 3000 ———— = $2138.96 (1 + 0.07) 5 3000[a(5)] –1 = 3000 ———— = $1500 1 + (5 2 /25)