What annual interest rate is earned by a 14​-week ​T-bill with a maturity value of ​$2,000 that sells for ​$1,972.81​?

Question

What annual interest rate is earned by a 14​-week ​T-bill with a maturity value of ​$2,000 that sells for ​$1,972.81​?

The annual interest rate is ___%

​(Type an integer or decimal rounded to three decimal places as​ needed.)

Answer 1

Apply the Future-Value equation: F = P(1 + i)ⁿ

where F = Future Value, P = Present Value, i = Interest Rate, n = number of compounding periods

F = 2000

P = 1972.81

n = 14 weeks / 52 (weeks / year) = 0.26923 year

Equation Rearrangement 1:

P(1 + i)ⁿ = F

Equation Rearrangement 2:

(1 + i)ⁿ = F / P

Equation Values Assignment:

(1 + i)^0.26923 = 2000 / 1972.81

Apply logarithms:

Log[(1 + i)^0.26923] = Log[1.013782] = 0.005945

0.26923 * Log[(1 + i)] = 0.005945 Power property of logarithms: Log[aⁿ] = n * Log[a]

Log[(1 + i)] = 0.005945 / 0.26923 = 0.0220804

1 + i = 10^0.0220804

Reduce to the solution:

i = 10^0.0220804 - 1 = 1.05216 - 1

i = 0.05216

i = 0.05216 * 100% = 5.216% equivalent annual rate of return (ANS)

Check the solution:

$1972.81 * (1 + 0.05216)^{14/52 year} = $2000.00 (OKAY)

I hope this helps you,

Gary