Suppose that the present discounted value of a stream of payments is $1,000. If the interest rate is 10%, what is the constant payment per year?
A) 100
B) 10
C) 1
D) 1,000

Question

Quizplus · Accepted Answer

To find the constant payment per year, we use the formula for present discounted value of an annuity:

PV = C * (1 - (1+r)^(-n))/r

where PV is the present value, C is the constant payment per year, r is the interest rate, and n is the number of years.

Substituting in PV = 1,000, r = 0.1, and solving for C, we get:

1,000 = C * (1 - (1+0.1)^(-n))/0.1
10,000 = C * (1 - (1+0.1)^(-n))

We don't know the value of n, but we do know that the present value of the stream of payments is $1,000, so the payments can't be too large. Let's try a payment of $100 per year and see if it works:

10,000 = 100 * (1 - (1+0.1)^(-n))
100 = (1 - (1+0.1)^(-n))
(1+0.1)^(-n) = 0.99
n = ln(0.99)/ln(1.1) ≈ 9.57

So a payment of $100 per year for 10 years (9 payments plus a final payment of $1.00) has a present discounted value of approximately $1,000. Therefore, the constant payment per year is 100.

Suppose That the Present Discounted Value of a Stream of Payments