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?
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%
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