Independent solution

How to solve this Nominal Discount Conversion question

Setup

Setup

Use quarterly periods for the account quoted at nominal discount. Its quarterly discount factor is 1 − d/4, while the other cash flows accumulate at 3% per period as shown.

10(1−d/4)−40(1.03)40+20(1.03)30=10010(1-d/4)^{-40}(1.03)^{40}+20(1.03)^{30}=100

Model

Model

Value every cash flow at the terminal date. Solving the equation of value first produces the forty-quarter accumulation factor 1.57738 for the discount-rate account.

(1−d/4)−40=1.57738(1-d/4)^{-40}=1.57738

Compute

Compute

Taking the negative fortieth root yields d = 4[1 − 1.57738 to the power −1/40] = 0.0453. This is a nominal discount rate, not an annual effective yield.

d=4(1−1.57738−1/40)=0.0453d=4(1-1.57738^{-1/40})=0.0453

Answer

Answer

The calculation gives 4.53% for nominal discount conversion, matching published choice C.

d(4)=4.53%(C)\boxed{d^{(4)}=4.53\%\quad\text{(C)}}