Question 5.1: Valuing Monthly Cash Flows Suppose your bank account pays in...
Valuing Monthly Cash Flows
Suppose your bank account pays interest monthly with an effective annual rate of 6%. What amount of interest will you earn each month? If you have no money in the bank today, how much will you need to save at the end of each month to accumulate $100,000 in 10 years?
PLAN
We can use Eq. 5.1
Equivalent n-period Discount Rate =(1+r)^n-1 (5.1)
to convert the EAR to a monthly rate, answering the first question. The second question is a future value of an annuity question. It is asking how big a monthly annuity we would have to deposit in order to end up with $100,000 in 10 years. However, in order to solve this problem, we need to write the timeline in terms of monthly periods because our cash flows (deposits) will be monthly:
That is, we can view the savings plan as a monthly annuity with 10 × 12 = 120 monthly payments. We have the future value of the annuity ($100,000), the length of time (120 months), and we will have the monthly interest rate from the answer to the first part of the question. We can then use the future value of an annuity formula (Eq. 4.6)
FV(Annuity) = C\times \frac{1}{r} ((1+r)^N -1) (4.6)
to solve for the monthly deposit.
Execute
From Eq. 5.1, a 6% EAR is equivalent to earning (1.06)^{1/12} – 1 = 0.4868% per month. The exponent in this equation is 1/12 because the period is 1/12 th of a year (a month).
To determine the amount to save each month to reach the goal of $100,000 in 120 months, we must determine the amount C of the monthly payment that will have a future value of $100,000 in 120 months, given an interest rate of 0.4868% per month. Now that we have all of the inputs in terms of months (monthly payment, monthly interest rate, and total number of months), we use the future value of annuity formula from Chapter 4 to solve this problem:
FV (annuity)=C\times \frac{1}{r} [(1+r)^n-1]We solve for the payment C using the equivalent monthly interest rate r = 0.4868%, and n = 120 months:
C=\frac{FV (annuity)}{\frac{1}{r} [(1+r)^n-1]} =\frac{\$100,000}{\frac{1}{0.004868}[(1.004868)^{120}-1] } = \$615.4 per month
We can also compute this result using a financial calculator or spreadsheet:
Evaluate
Thus, if we save $615.47 per month and we earn interest monthly at an effective annual rate of 6%, we will have $100,000 in 10 years. Notice that the timing in the annuity formula must be consistent for all of the inputs. In this case, we had a monthly deposit, so we needed to convert our interest rate to a monthly interest rate and then use total number of months (120) instead of years.
