Question

THE SETUP AND KEY INPUTS:                                                                   

Your best friend James just celebrated his 25th birthday and wants to start saving for his anticipated retirement. James plans to retire in 35 years and believes that he will have 25 good years of retirement (he has looked at the life expectancy tables and is playing the odds here) and believes that if he can withdraw $130,000 at the end of each year, he can enjoy his retirement. In this problem, assume that the full 25 years of retirement payments are made, and the account has a zero balance at the end.

Assume that a reasonable rate of interest for James for all scenarios presented below is 8% per year. This is an annual rate, review each individual question for more specifics on compounding periods per year.

Because James is planning ahead, the first withdrawal will not take place until one year after he retires. he wants to make equal annual deposits into his account for his retirement fund.                   

Hints: Picture the problem like this: We invest for retirement over our working lives and then we withdraw set amounts each year during retirement. Note that there are two different periods in the following timeline, N1 years for investing and a different N2 for the retirement period. In all parts of the problem the annual interest rate is 8%, whether you are looking at the investment or the retirement time periods.

|---------------------------------------------------------------------|---------------------------------------------------|

|                                Investment period, investing X$ every period                    Retirement period, receiving Y$ and having $0 at the end      |

For each question, add blank lines as needed to provide your solution.

A. If he starts making these deposits in one year and makes his last deposit on the day he retires, what amount must James deposit annually to be able to make the desired withdrawals at retirement?

A1) First: Amount James needs to have saved as of his retirement (3 pts):

A2) The amount James must save each year (beginning at the end of the first year) to fund his retirement is (3 pts):    

A3) If James decides to make monthly deposits for 35 years to reach his same retirement goal, how much must James start depositing one month from today (3 pts)?                                                                

B. If James decides he isn’t earning enough money yet and wants to wait several years before starting his investment deposits. Assume that instead of starting immediately (that is, the end of year 1), James waits for 10 years (first deposit at the end of year 10) leaving 10 fewer years (than the original plan of 35 years) to grow his retirement nest egg, what amount must he deposit annually to be able to make the desired withdrawals at retirement (4 pts)?           

                                               

C. Suppose your friend has just inherited a large sum of money. Rather than making equal annual payments for 35 years, he has decided to make one lump sum deposit today to cover his retirement needs. What amount does he have to deposit today? (3 pts)   

Answer 1

Answer A1

Total no. of years after retirement =25 years

Since first payment is made at the beginning of year, there will be further 24 withdrawals of $130000 each.

Hence PV of these withdrawal at the day he retires will be =130000+ PV of annuity of $130000 of 24 years at 8%

Hence PV of annuity =A*(1-(1+r)^n)/r

=130000*(1-(1+8%)^-24)/8%

=130000*(1-(1.08)^-24)/0.08

=130000*(1-0.1577)/0.08

=130000*0.8423/0.08

=1368738.58

Hence total PV at the day of retirement required = 130000+1368738.58=$1498738.58

Answer A2

If James s saving each year he will make 35 deposits of say Amount A at 8%, hence the Future value of the deposits =$1498738.58

Hence FV = A*((1+r)^n-1)/r

or, 1498738.58=A*((1+8%)^35-1)/8%

or, 1498738.58=A*((1.08)^35-1)/0.08

or, 1498738.58=A*(14.7853-1)/0.08

or, 1498738.58=A*13.7853/0.08

or, A=1498738.58*0.08/13.7853

or, A = 8697.57

Hence the annual deposits should be $8697.57

Answer A3

If deposits are monthly there will be 12*35 = 420 payments and interest rate will be 8%/12 = 0..67%

Hence Monthly deposit will be calculated as:

1498738.58=A*((1+0.67%)^420-1)/0.67%

or, 1498738.58=A*((1.0067)^420-1)/0.0067

or, 1498738.58=A*(16.5207-1)/0.0067

or, 1498738.58=A*15.5207/0.0067

or, A=1498738.58*0.0067/15.5207

or, A = 6469.77

Hence the monthly deposits should be $6469.77

Answer B

Since first payment is made at the end of year 10 instead of year 1 he missed 9 payments hence total no of payments will be 35-9=26

Hence FV = A*((!+r)^n-1)/r

or, 1498738.58=A*((1+8%)^26-1)/8%

or, 1498738.58=A*((1.08)^26-1)/0.08

or, 1498738.58=A*(7.3963-1)/0.08

or, 1498738.58=A*6.3963/0.08

or, A=1498738.58*0.08/6.3963

or, A = 18744.91

Hence the annual deposits should be $18744.91

Answer C

If lumpsum amount is deposited today it will grow for 35 years

Hence the PV of 1498738.58 should be deposited

Hence PV = 1498738.58/(1+8%)^35 =1498738.58/1.08^35= 1498738.58/14.7853 = $101366.50

Hence $101366.50 should be deposited