Suppose that the present discounted value of a stream of payments is $1,000. If the yearly payment is $50, what is the interest rate?
A) 50%
B) 20%
C) 5%
D) 0.5%

Question

Quizplus · Accepted Answer

The formula for present discounted value is:
Present Discounted Value (PDV) = Payment / (1 + r)^n
where r is the interest rate and n is the number of years.

Given PDV = $1,000 and Payment = $50 per year, we can use the formula to solve for r:
1,000 = 50 / (1 + r)^n

Taking the natural logarithm of both sides, we get:
ln(1,000) = ln(50) - n * ln(1 + r)

ln(1 + r) = (ln(50) - ln(1,000)) / (-n) = (ln(1/20)) / (-n)
(we used ln(a/b) = ln(a) - ln(b) and ln(a^n) = n ln(a))

Now we can solve for r:
1 + r = e^(ln(1/20) / (-n))
r = e^(ln(1/20) / (-n)) - 1

Putting n = 1 (because payments are yearly), we get:
r = e^(-ln(20)) - 1 = 0.05 or 5%

Therefore, the answer is C) 5%.

Suppose That the Present Discounted Value of a Stream of Payments