Question

NON-EXCEL help on Growing Annuity Problem

In this problem, we will assume all cash flows occur at the BEGINNING OF THE PERIOD (Annuity Due). Therefore, you need to set the calculator to BEGIN. You may want to review the Lecture Video on Growing Annuity for help in working the exam.

Problem:
Assume you are 32 years old and plan to retire in 35 years at age 67. You are currently earning $75,000/year and expect average annual salary increases of 4.0%/year over the next 35 years. You have $0 saved for retirement.

You are trying to determine how much money to save (invest) each year in your 401(k) Plan to fund your retirement in order to pay yourself 70% of your final salary each year (that increases with inflation). [Remember this is an Annuity Due, so your first annual investment is made in Year 0….and your final payment is in Year 34.] You plan to maintain an investment as a percent of your salary, which simply means your payment into the 401(k) will also increase by 4.0% per year as your salary increases 4.0% each year.

You believe that you can earn 8.0%/year over the next 35 years while saving for retirement.

Once you retire, you have a life expectancy of 25 years. You plan to be more conservative in your investments and expect to earn only 5.0%/year on your investments over the 25 years while in retirement. You also want to maintain your purchasing power by increasing your “annual retirement pay” by the expected inflation rate of 3.0% each year. [Remember, your first withdrawal will be made in Year 0 of retirement (i.e., Year 35 on the timeline.] Assume that after you withdraw the 25th payment, you will have $0 left in the account.

Instructions:
- SET CALCULATOR TO “BEGIN”!
- Set up and show your Timeline with payments. You MUST “show your work” in order to get partial credit.
- For this Exam it would be better to handwork the problem and save as a pdf before submitting.
- BOX in all your ANSWERS!
- You may round all answers to the dollar throughout the problem.
- Carry your adjusted interest rates out to 6 decimal places to reduce rounding errors.

Questions: [NOTE: there are obviously many steps to get to these answers…scoring is below.]
1. What is your final year’s salary?
What is 70% of your salary? [Round to the dollar.]

2. How much will you need in your 401(k) at retirement in order to “pay yourself” 70% of your final year’s salary,
AND have that payment to yourself increase at 3.0% each year to maintain purchasing power?

3. How much do you need to contribute each year (i.e., save, invest) over the next 35 years in order to fund your retirement. Remember, payment into the 401(k) is increasing by 4.0% each year.

Scoring (Do not use Excel):

1. Readability of your Exam. Is it organized, do you show all work, are your answers BOXED in, etc. (10 points)
2. Correctly work problem as an Annuity Due (10 points)
3. Calculate the FV of salary and 70% of salary as first withdrawal from your 401(k) in retirement. (15 points)
4. Do you show a Timeline and is it labeled properly? (15 points)
5. Calculate adjusted interest rate for Retirement Annuity section. (10 points)
6. Calculate the PVA for the Retirement Annuity. (10 points)
7. Calculate the adjusted interest rate for the saving/investing section of problem. (10 points)
8. Calculate the adjusted FVA correctly that is used to determine amount to save each year. (10 points)
9. Calculate the amount you need to save each year; where payment increases as salary increases. (10 points)

Answer 1

1.Final year’s salary

75000*(1+0.04)^35=

295957

70% of salary=

295957*70%=

207170

2. Total Money needed in 401(k) at retirement in order to “pay yourself” 70% of your final year’s salary

AND have that payment to yourself increase at 3.0% each year to maintain purchasing power

is the PV of growing annuity due

whose PV is to be found out

Using PV of growing annuity-due formula, as below

PV of G/A=(Pmt./(r-g))*(1-((1+g)/(1+r))^n)*(1+r)

where, pmt.= annual payment into 401(k)

r=reqd. return 5%

g= growth/inflation rate

n= no.of years

Applying the values,in the formula,

PV at Yr. 35 end/36 beginning=(207170/(5%-3%))*(1-(1.03/1.05)^25)*(1.05)

4151549

3.Amt. needed to be contributed each year (i.e., save, invest) over the next 35 years in order to fund your retirement. Remember, payment into the 401(k) is increasing by 4.0% each year.

ie. Future Value of annuity due   of 35 nos. year-beginning payments into 401(k)

with required return r=8% &

growth rate of annual savings , g =4%

we have the future value of 35 -beginning -of- the -year savings as $ 4151549

& we need to find the annual savings, pmt.=?

Using the Future Value of Growing annuity, which is as below:

FV(GA)=Pmt.*(((1+r)^n-(1+g)^n)/(r-g))*(1+r)

Applying the values,in the formula,

4151549=Pmt.*(((1+0.08)^35-(1+0.04)^35)/(0.08-0.04))*(1+0.08)

Solving the above,we get the annual savings needed to be

14186