Happy Birthday! You turned 25 today. You intend to retire at age 65 and want to

Question

Happy Birthday! You turned 25 today. You intend to retire at age 65 and want to be able to receive a 25-year, $80,000 beginning of the year annuity with the first payment to be received on your 65th birthday. You would like to save enough money over the next 40 years to achieve your objective; that is, you want to accumulate the necessary funds by your 65th birthday.   You expect your investments to earn on average 8% per year until age 65, and 5% thereafter. What equal, annual amount must you save at the end of each month for the next 40 years until you are age 65 to achieve your objective? Assume you currently have saved $15,000 toward this goal.

Explanation / Answer

Ans:

If amount already collected $15000 and if I invest that money for 40 years @8% , I get after 40 years

=15000(1.08)40

=$ 325,868

So the remaining amount has to be collected.

The annuity is $80,000

Formaula for Annuity fund :

P= r(PV)/ 1-(1+r)-n

Where P = Annuity payment received =$80,000

n=No of period =25 yrs

PV = Fund value of Annuity to be invested

R=rate of interest =5%

Therefore :

80,000=0.05(PV)/1-(1.05)-25

Or, 80,000=0.05PV/(1-.295303)

Or, 56,376=0.05PV

Or, PV=1,127,516

So the Annuity fund required after 40 years is $1,127,516

Future value of $15000 already accumulated = $325,868

Remaining amount to be accumulated : = $801,648

Formula for future value of Annuity :

FV= A [ (1+k)n-1/k]

FV = Future anuuity value

A = periodical (monthly) investement

K=interest rate=8/12=0.67% per month

N=periods=480 months

Therefore ,

801,648=A[{(1.0067)480-1}/.0067]

Or, 801,648=A(23.66)/0.0067

Or, A=801,648*.0067/23.66

Or, A = 227

Therefore , monthly amount to be collected for 40 years for the required annuity is $227

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