Brenda is 25 years old today and she wishes to accumulate enough money over the
ID: 2697641 • Letter: B
Question
Brenda is 25 years old today and she wishes to accumulate enough money over the next 35 years to provide for a 20 year retirement annuity of $100,000 at the beginning of each year, starting with her 60th birthday. She can save $2,000 at the end of each of the next 10 years and $3,000 each year for the following 10 years. How much must she save each year at the end of years 21 through 35 to obtain her goal? Assume that the average rate of return over the entire period will be 10%.
Please show all steps and work.
Explanation / Answer
This problem requires 4 time value of money analyses: the retirement period and 3 different annuity periods prior to retirement.
The retirement period is 20 years long (n=20) with an interest rate of 10% (i=10). She wants to recieve $100,000 at the begining of each year (PMT= 100,000, annuity due). At the end of the retirement period there will be no money left (FV=0). So we're looking to solve for the present value, which will be the amount required at the begining of the retirement period.
Using a finacnicial calculator:
N=20 I/R= 10 FV= 0 PMT=100,000 and so PV= 936,492
Using the TVM tables:
PVAD= 100,000 (PMT) * 9.36492 (Table factor, N=20 i=10) = 936,492
This is the amount Brenda needs to finance her retirement.
Period 1:
Brenda can pay $2,000 (PMT) at the end of every year for 10 years (N=10). Same interest rate.
Using a financial calculator:
N=10 I/R=10 PV=0 PMT= 2000 and so FV=31,874.85
Using TVM tables:
Future Value of Ordinary Annuity= PMT * Table Factor (N=10, i=10)
FVA= 2000 * 15.9374 = 31974.8
Period 2:
Same as above but PMT= 3000. The present value of this period is the future value of the last period.
Using financial calculator:
N=10 I/R=10 PV=31,874.85 PMT=3000 and so FV= 130,487.43
Using tax table:
FV of the account= FV of lump sum + FV of annuity
FV= PV * Table 1 Factor (N=10, i=10) + PMT * Table 3 Factor (N=10, i=10)
FV= (31,874.85 * 2.59374) + (3000 * 15.9374)
FV= 130,487.27
Final Period (the one with the answer):
We've solved for 2 of our components already. The future value is the begining of the retirement period (936,492) and the present value is the future value of the previous period (130,487.43). The interest rate is the same but the period is 15 years.
Using financial calculator:
N=15 I/R=10 PV=130,487.43 FV=(-936,492) and so PMT=12,319.27
Using TVM tables:
FV of account= FV of lump sum + FV of annuity
FV= PV * Table 1 Factor (N=15, i=10) + PMT * Table 3 Factor (N=15, i=10)
936,492= (130,487.43 * 4.17725) + (PMT * 31.7725)
936,492= 545,078.62 + 31.7725(PMT)
391,413.38 = 31.7725(PMT)
PMT= 12,319.25
Brenda needs to save $12,319.25 at the end of each year in years 21-35 in order to finance her retirement.