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

Solution to part a

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

Rate = 12% pa i.e. 0.12/12 = 0.01 per month

So r = 0.01

Investment Annuity, A = $600 p.m

Payment periods, n = 20 yrs *12 months = 240 months

FV = 600*[(1+0.01)240 - 1] / 0.01

FV = 600*[1.01240 - 1] / 0.01

FV = 600*[10.8926 - 1] / 0.01

FV = $ 593,556

Future value of the savings after last deposit = $593,556

Solution to part b:

There would be 4 bonus deposits of $1100 at the end of every 5 years for 20 years

Future value of a single deposit = PV * (1+r)n

Thefuture values of the 4 deposits at the end of 20 years would be:

1st deposit would become = $1100 * (1+0.01)15*12 =1100 * 1.01180 = 6595.38

2nd deposit would become = $1100 * (1+0.01)10*12 =1100 * 1.01120 = 3630.43

3rd deposit would become = $1100 * (1+0.01)5*12 =1100 * 1.0160 = 1998.37

4th deposit would become = $1100 * (1+0.01)0 =1100 * 1 = 1100 = 1100

Total FV = FV of the annuity of $600 pm + FV of the 4 bonus deposits

= 593556 + 6595.38 + 3630.43 + 1998.37 + 1100

= $ 606,880.18

Solution to part c

Effective annual rate is 12%

Effective quarterly rate, r = 12% /4 = 3% or 0.03

Solution to part d

Let the initial deposit be denoted by I and the annuity of quarterly deposits by A

The future value of the initial deposit plus the future value of the deposit annuity equals $500,000

So we have the equation:

FV of initial depost + FV of annuity = 500,000

{I * (1+r)n} + {A* [(1+r)n - 1] / r} = 500,000

Number of deposit periods, n = 4 quarters * 20 years = 80

and r = 0.03

{I * 1.0380} + {A * (1.0380 - 1) / 0.03} = 500,000

(10.6409*I) + (9.6409*A / 0.03) = 500,000

10.64I + 321.36A = 500,000

Neither is there any ratio provided between the initial deposit & annuity nor does the problem specify how much we can deposit initially.

So there is no definate answer; however we will make some assumptions here.

Assuming no initial deposit, we have 321.36A = 500,000

So A would be 1555.89

We will round this to 1,500 and take it as the value of A

Substituting this in the above equation:

10.64I + 321.36*1500 = 500000

10.64I = 500000 -482040

I = 17960 / 10.64

I = $ 1687.97

Thus initial deposit is $1687.97 and quarterly deposit is $1500