How much money invested at 7% compounded continuously for 18 years will yield $25,000?

Question

Using your answer from the above, how much will that initial investment earn if we have it invested at 4.5%, compounded quarterly, for 36 years?

What is the difference in yield between the two? Which investment has a higher return?

Please explain how you got to the final answers.

Answer 1

The equation for continuous compounding is: P(t) = P₀e^rt

P(t) = $25,000

P₀ = initial investment

r = 0.07

t = 18 years

using the eqn and solving for P₀ :

P₀ = 25,000 / [e^{(0.07 * 18)}]

P₀ = $7091.35

______________________

If P₀ = $7091.35 and r = 0.045 and t = 36 years, what is P(t)

plug new values into the eqn.

P(t) = 7091.35 * e^(0.045*36)

P(t) = $35,833.24

Difference in yield: $35,833.24 - $25,000 = $10,833.24

The second investment has a higher return.

Hope this helps!

Answer 2

Continuous compounding:

P(t) = P₀·e^rt

1. P(t) is the amount of money after t years = $25,000
2. P₀ is the starting amount of money (what we're trying to find)
3. r = interest rate in decimal form = 0.07
4. t = years = 18

P(t) = P₀·e^rt

$25,000 = P₀·e^(0.07)(18)

$25,000 = P₀·e^1.26

$25,000/e^1.26 = P₀

Use your calculator to compute the answer.

The quarterly compounding formula:

P(t) = P₀·(1 + r/n)^nt

1. P(t) = amount of money at year t
2. P₀ = starting amount computed from continuous compounding above
3. r = rate as a decimal = 0,045
4. n = number of compounding per year = quarterly = 4
5. t = years = 36

Plug in the numbers and use your calculator to get the answer. Compare this answer to the $25,000 from the continuous compounding. Which is greater?