Question

You plan to start regular savings for retirement. You have been offered 3 different savings plans by a financial institution.

Option 1: You will deposit $600 at the end of each month for the next 20 years. The nominal interest rate is 12% per annum compounded monthly.
a) Calculate the future value of your savings immediately after the last deposit. (1 mark)

b) To help you, your parents will deposit a bonus of $1100 into your savings account at the end of every 5 years, in additional to your deposits in part a). Calculate the future value of your savings immediately after the last deposit.

Option 2: This savings plan requires you to make your first deposit immediately. You will make regular quarterly deposits for the next 20 years. Your savings goal for retirement is $500,000 (at the end of the 20 years). The effective annual rate is 12%.
c) Calculate the effective quarterly interest rate. (1 mark)

d) Calculate the size of the required quarterly deposit.

Option 3: You will make regular deposits for the next 20 years. Specifically, you will make regular semi-annual deposits of $6,000 for the next 15 years. Then you will stop saving for a year. After that, you will make regular deposits of $10,000 every 2 years for the remaining 4 years. The first deposit is made 6 months from now. The effective annual rate for the first 16 years (starting today) is 10% and the nominal interest rate in subsequent years is 6% per annum compounded daily.
e) Calculate the future value of your savings immediately after the last deposit.

f) How many deposits it will take for the balance to first exceed $130,000? (1 mark)

Answer 1

OPTION 1

a) Monthly interest rate = 12%/12 =1% =0.01

No. of deposits = 20*12 = 240

Future value of savings = 600/0.01*(1.01^240-1) = $593553.22

b) If parents deposit $1100 at the end of every 5 years (total 4 deposits)

Future value of savings = 600/0.01*(1.01^240-1) + 1100*1.01^180+1100*1.01^120+1100*1.01^60+1100

= $606877.39

OPTION 2

c) Effective quarterly rate r is given by

(1+r/4)^4= 1+ 0.12

=> r = 0.114949 or 11.4949%p.a. or 2.8737% per quarter

d) No. of deposits = 20*4 = 80

The quarterly deposits (A) is given by equating future value of deposits (Annuity) to target of $500000

A/0.028737* (1.028737^80-1)= 500000

=> A* 300.8731 = 500000

=> A = $1661.83

So, size of quarterly deposit is $1661.83

OPTION 3

There will be 30 deposits of $6000 every 6 months and two deposits of $10000 each at the end of 18th year and 20th year

e) Interest rate for 6 months during first 16 years = 1.1^0.5-1=0.048809

No of payments = 15*2 = 30

So, Future value of payments by the end of 15th year = 6000/0.048809*(1.048809^30-1)

= $390574.45

Future value after 16 years = 390574.45*1.048809^2= $429631.89

Now,

effective interest rate per day = 6%/365 = 0.000164

So, value of savings after 20 years = 429631.89*1.000164^(365*4) + 10000*1.000164^(365*2)+10000

=$567433.27

f)As the future value after 15 years is $390574.45 , to reach $130000 , it will take less than 15 years

Let it take n deposits to first exceed $130000

So, 6000/0.048809*(1.048809^n-1) > 130000

1.048809^n-1> 1.057525

=>1.048809^n > 2.057525

Taking natural log of both sides

n> ln (2.057525)/ln(1.048809)

n> 15.14

Actually balance will exceed $130000 sometimes after 15 deposits but before 16th deposit.  

So, deposits required = 15 (balance will be more than $130000 before the 16th deposit)