Adam wants to invest $40,000 in a pension plan. One investment offers 7% compounded quarterly. Another offers 6.5% compounded continuously. Which investment will earn more interest in 10 years?
1. P = Po(1+r)^n
Po = $40,000.
r = 0.07/4 = 0.0175. = Quarterly % rate.
n = 4Comp./yr. * 10yrs. = 40 Compounding periods.
P = ?
2. P = Po * e^rt.
Po = $40,000.
r * t = 0.065 * 10 = 0.65.
P = ?
To determine which investment will earn more interest in 10 years, we can compare the future value of the investment in each case.
For the investment offering 7% compounded quarterly, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment,
P = the principal amount (initial investment),
r = annual interest rate (in decimal form),
n = number of times the interest is compounded per year, and
t = number of years.
Plugging in the values:
P = $40,000,
r = 0.07 (since 7% is equal to 0.07),
n = 4 (compounded quarterly),
t = 10,
A = 40,000(1 + 0.07/4)^(4*10)
Now, let's calculate the future value (A) for the first investment:
A = 40,000(1 + 0.0175)^40
A = 40,000(1.0175)^40
A ≈ 40,000 * 1.99204
A ≈ $79,681.60
For the investment offering 6.5% compounded continuously, we use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the future value of the investment,
P = the principal amount (initial investment),
r = annual interest rate (in decimal form),
t = number of years, and
e = Euler's number (approximately equal to 2.71828).
Plugging in the values:
P = $40,000,
r = 0.065 (since 6.5% is equal to 0.065),
t = 10,
A = 40,000 * e^(0.065*10)
Now, let's calculate the future value (A) for the second investment:
A ≈ $40,000 * 2.10997
A ≈ $84,398.80
Comparing the future values of the two investments, we can see that the investment with 6.5% compounded continuously will earn more interest in 10 years.