Question

The force of interest δ(t) is a function of time and, at any time t (measured in years),

is given by the formula:

δ(t) =



0.07 0.005t , 0 t < 5

0.06 0.003t, 5 t < 10

0.03, t > 10

(a) Calculate the accumulated amount at time t = 15 of $100 invested at time

t = 0.

Answer 1

A=$100 x A(0,5) x A(5,10) x A(10,15)

$=100\times e^{\int_0^5(0.07-0.005t)dt} \times e^{\int_5^{10}(0.06-0.003t)dt} \times e^{0.03 \times 5}\\ =100 \times e^{0.07t-\frac{0.005t^2}{2}|_0^5} \times e^{0.06t-\frac{0.003t^2}{2}|_5^{10}} \times e^{0.15}\\ =100 \times e^{0.35-0.0625} \times e^{0.45-0.2925} \times e^{0.15}\\ =100 \times e^{0.2875} \times e^{0.1575} \times e^{0.15}\\ =100 \times 1.3331 \times 1.1706 \times 1.1618\\ =\$ 181.30$