Question

3. Given an amount function A(t)23t +4, t0, t in years, find the following: (a) The effective rate of interest in year 2. (b) The effective rate of discount in the 4th year (c) The corresponding accumulation function (d) I the amount of interest in the nth year.

Answer 1

ANSWER 3:

(a) Let i be the effective rate of interest.

$\dpi{100} A(1,2) = \frac{A(2)}{A(1)}= (1+i)$

$\dpi{100} (1+i) = \frac{2*2^{2}+3*2+4}{2*1^{2}+3*1+4}$

$\dpi{100} (1+i) = 2$

$\dpi{100} i = 100%$%

(b) Let d be the effective rate of discount in 4th year.

$\dpi{100} (1-d) = \frac{A(3)}{A(4)}$

$\dpi{100} (1-d) = \frac{2*3^{2}+3*3+4}{2*4^{2}+3*4+4}$

$\dpi{100} (1-d) = \frac{31}{48} =0.64583$

$\dpi{100} d = 35.417$%

(c) Let a(t) be the accumulation function.

A(t) = A(0) . a(t)

$\dpi{100} \frac{A(t)}{A(0)}= a(t)$

$\dpi{100} a(t) = \frac{2t^{2}+3t+4}{4}$

$\dpi{100} {\color{Red} a(t) = 0.5t^{2}+0.75t+1, t\geq 1, t \: in\: years}$

(d)

$\dpi{100} I_{n} = A(n) - A(n-1)$

$\dpi{100} I_{n} =[2(n)^{2}+3(n)+4]- [2(n-1)^{2}+3(n-1)+4]$

$\dpi{100} I_{n} = [2n^{2}+3n+4] - [2n^{2}-4n+2+3n-3+4]$

$\dpi{100} {\color{Red}I_{n} = 4n+1 }$