An investment of $7,800 made March 31, 2001 pays $700 at the end of every six months for five years and pays $10,000 on March 31, 2006. What rate of interest was being earned every six months?
700/7800 = 0.09874 = 9.87%
You need to consult a "yield to maturity" formula. I get about 23% annual rate.
To find the rate of interest being earned every six months, we need to use the present value formula and solve for the interest rate.
First, let's break down the given information:
Investment (PV) = $7,800
Payment amount (PMT) = $700 (paid every six months for five years) and $10,000 (paid on March 31, 2006)
Now we can use the present value formula:
PV = PMT / (1 + r)^n
Where:
PV = Present Value
PMT = Payment amount
r = Interest rate per period
n = Number of periods
For the $700 payments every six months, we need to calculate the number of periods in 5 years:
Number of periods (n) = 5 years * 2 (since payments are made every six months)
Plugging in the values we know:
$7,800 = $700 / (1 + r)^(5 years * 2)
Now, let's solve for the interest rate (r):
Divide both sides of the equation by $7,800:
$7,800 / $7,800 = $700 / ($7,800 * (1 + r)^(5 years * 2))
1 = $700 / ($7,800 * (1 + r)^(5 * 2))
Simplify:
1 = $700 / ($7,800 * (1 + r)^10)
Cross multiply:
$7,800 * (1 + r)^10 = $700
Divide by $7,800:
(1 + r)^10 = $700 / $7,800
Take the 10th root of both sides to isolate (1 + r):
(1 + r) = ($700 / $7,800)^(1/10)
Now let's solve for (1 + r):
(1 + r) = 0.977
Subtract 1 from both sides to find r:
r = 0.977 - 1
r = -0.023 (approximately)
The interest rate being earned every six months is approximately -0.023 or -2.3%. Note that the negative sign indicates a decrease in value rather than an increase.