Question

Suppose that you borrowed $400 from a friend and promised to repay the loan by making 3 annual payments of $100 at the end of each of the next 3 years, plus a final payment of $200 at the end of year 4. What is the interest rate implicit in this agreement?

Answer 1

The present value (PV) formula should be applied here, where the rate of interest is r.

PV = 400 = $100 / (1 + r)^1 + $100 / (1 + r)^2 + $100 / (1 + r)^3 + $200 / (1 + r)^4

Trial and error method:

At what r, the right hand side is equal to 400.

If there is 10% rate (r = 0.10),

$100 / (1 + r)^1 + $100 / (1 + r)^2 + $100 / (1 + r)^3 + $200 / (1 + r)^4

= 100/1.10 + 100/1.21 + 100/1.331 + 200/1.4641

= 90.91 + 82.64 + 75.13 + 136.60

= 385.28

This is smaller than 400; therefore, r should be lower than 10%.

NPV = 385.28 – 400 = - 14.72

Let us take 2% rate (r = 0.02),

$100 / (1 + r)^1 + $100 / (1 + r)^2 + $100 / (1 + r)^3 + $200 / (1 + r)^4

= 100/1.02 + 100/1.0404 + 100/1.061208 + 200/1.0824

= 98.04 + 96.12 + 94.23 + 184.77

= 473.16

NPV = 473.16 – 400 = 73.16

Therefore,

[(NPV at 2% rate) / (NPV at 2% rate – NPV at 10% rate)] × Difference in rate of interest

= [73.16 / (73.16 + 14.72)] × (10 – 2)

= (73.16 / 87.88) × 8

= 6.66

This is to be added to the 2% for getting answer; (2 + 6.66 =) 8.66%