You are saving for your retirement by investing $700 per month in an annuity with a guaranteed interest rate of 6% per year. You increase the amount you invest at the rate of 4% per year. With continuous investment and compounding, how much will you have accumulated in the annuity by the time you retire in 25 years?
A)$7,406.31
B)$4,209,782.72
C)$740,631.04
D)$38,218.71
E)$165,257.12

Question

Quizplus · Accepted Answer

We can use the formula for the future value of an annuity with continuous compounding:

FV = P * (e^(rt) - 1) / (e^(r) - 1)

where,
P = monthly contribution
r = annual interest rate
t = time period in years

Initially, P = $700, r = 6%, and t = 25 years. So we have:

FV = 700 * (e^(0.06*25) - 1) / (e^(0.06) - 1)
   = $740,631.04

Now we need to account for the 4% annual increase in contributions. We can model this by using a geometric series:

S = P * (1 - r^n) / (1 - r)

where,
S = sum of geometric series
P = initial contribution
r = common ratio = 1 + annual increase rate = 1 + 0.04 = 1.04
n = number of terms = 25 (since we invest for 25 years)

S = (700 * (1 - 1.04^25)) / (1 - 1.04)
  = $338,151.45

So the total accumulation in the annuity after 25 years is:

FV_total = FV + S
         = $740,631.04 + $338,151.45
         = $1,078,782.49

Therefore, the correct answer is (C) $740,631.04.

You Are Saving for Your Retirement by Investing $700 Per