Question

The present value today of K payable at the end of 4 years from now

is 81.87 if the compound model is assumed and the interest is credited at a

force of interest of δ.

The present value today of K payable at the end of 6 years from now is 83.53

if the interest is credited at a force of interest of δt = t for t ≥ 0. 100

Calculate δ and the effective annual discount rate which is equivalent to δ.

Answer 1

Present value of K payable at the end of 4 years from now is 81.87

81.87e^4δ = K ............equation(1)

Present value of K payable at the end of 6 years from now is 83.53 and δt = t for t ≥ 0

$\dpi{100} \mathbf {83.53 \, exp (\int_{0}^{6} t\, dt) = K}$

83.53 $mathbf{e^{[rac{t^{2}}{2}]_{0}^{6}}}$ = K

83.53 exp[18-0] = K

Substituting value of K from eq(1)

83.53 e^[18-0] = 81.87e^4δ

$\dpi{100} \mathbf {e^{4\delta} = \frac{83.53\, e^{18}}{81.87}}$

$\dpi{100} \mathbf {4\delta = ln( \frac{83.53\, e^{18}}{81.87})}$

δ = 4.505

Annual Discount Rate:

Let d = Annual discount rate

d = 1-e^-δ

d = 0.9889