Question

Jeff puts $100 into a fund that pays an effective annual rate of discount of 20% for the first two years and the force of interest δ_t = (2t)/(t^2+8), 2 ≤ t ≤ 4 for the next two years. At the end of four years, the amount in Jeff’s account is the same as what it would have been if he put $100 into an account paying interest at the nominal rate of i per annum compounded quarterly for four years. Calculate i.

Answer 1

Jeff starts with a sum P₁ = $100 . The effective annual rate of discount is 20% for the first two years. This means Sum of money earned after Year 2 is P₂= $100e^20%*2 = $149.1825. The force of interest t is given as 2t/(t² +8) for t=2 to t=4.

P₃ = $149.1825e^(t) here substituting t=3 we get, $149.1825e^(6/17) = $212.3236

Similarly, by substituting t=4 , we get P₄= $212.3236e^(8/24)= $296.3214

so the equivalent rate i per annum compounded quarterly for four years will be such that

$296.3214=$100e⁴ⁱ

Solving for i we get i = 6.79% quarterly rate or 27.16% annual rate (Answer)