Question

Sarah has just finished her studies, and has started her career by accepting a position at an online learning company. Being a forward-thinking young woman, she is already looking ahead to her retirement, and wants to start putting money away now. She wants to retire 30 years from now, and wants a lump sum of R3 million at that point. To allow her to plan effectively, and to keep things simple, she wants to put away the same amount for the full period, at the end of each year. Using the tables, and showing your workings clearly, answer each of the following questions: 2.1 Assuming that she receives an average growth rate of 9% p.a. for the period, how much does Sarah need to put away at the end of each year? Start writing here: 2.2 If she were to invest a lump sum now instead, to reach R3 million in 30 years, which also grew at 9% p.a., compounded annually, how big would this lump sum need to be? Start writing here:

Answer 1

2.1 To accumulate a sum of R3 million , the sum of future value of all her annual deposits (A) should should be equal to R3 million

Her first installment at the end of 1st year will earn interest for 29 years,

2nd installment will earn interest for 28 years and so on.

So the sum of future value of her installments is

i.e. A*(1+0.09)^29 +A*(1+0.09)^28 + .... + A*(1+0.09) + A = R3 million

As the Left hand side represents the future value of an annuity of A for 30 years at 9%

=> A * FVIFA (9%, 30 years) = R3 million

=> A * 136.3 = R3 million (From Table B corresponding to 9% and 30 years)

=> A = R3 million/136.30

= R 22010.27 per year

So, Sarah needs to put away R22010.27 at the end of each year

NOTE: As the table gives approximate value, the actual amount is R 22009.05 per year

2.2 To accumulate R3 million by depositing a lump sum amount (say A) the future value of A should be equal to R3 million

So,

A*(1+0.09)^30 = R3 million

or A *FVIF(9%, 30 years) = R3 million

=> A * 13.27 = R3 million (From Table A corresponding to 9% and 30 years)

=> A = R 226073.85

So, Sarah needs to put away R226073.85 today to reach her goal

NOTE: As the table gives approximate value, the actual amount is R 226113.41

2.3 The present value of all the retirement amounts (A) withdrawn annually would be equal to R3 million at the time of retirement (Assuming that the 1st amount is withdrawn at end of year 1)

So, A/1.12 + A/1.12^2+ ... + A/1.12^20 = R 3 million

As the Left hand side represents the present value of an annuity of A for 20 years at 12%

=> A * PVIFA(12%, 20 years) = R3 million

=> A * 7.469= R3 million (From Table D corresponding to 12% and 20 years)

=> A = R 401660.20

So, Sarah will receive R401660.20 at the end of each year after retirement

NOTE: As the table gives approximate value, the actual amount is R 401636.34 per year

2.4 The worth of the 5th installment of R401660.20 received at the end of 35 years today is given by the present value of the amount discounted by inflation rate of 6% for 35 years

So, value today = R401660.20 /1.06^35

= R401660.20 * PVIF( 6%, 35 years)

=R401660.20*0.130      (From Table C corresponding to 6% and 35 years)

= R52215.83

So, Value of 5th installment today is  R52215.83

NOTE: As the table gives approximate value, the actual amount is R52254.98