Finance - Series - Annuities

The questions on this page focus on:

1. Determining a total investment amount.

Determining a total investment amount.	1. Emily joins a superannuation fund on her 18th birthday. She deposits $2,000 into her fund on that day. The fund pays 11% p.a. Each half-year, Emily deposits another $2,000 into her fund which compounds her investment each 6 months. She continues to contribute the same amount until the day before she is 55 years of age. How much will be in her fund at that stage? Answer.$1,978,152.86.
	2. Alex establishes an annuity fund and makes quarterly contributions of $500 at the end of each quarter. His fund pays 8% p.a. interest compounding quarterly. How much does Alex have in his fund at the end of 10 years? Answer. $30,200.99.
	3. David decides to salary sacrifice $200 per month to deposit into his Superannuation Fund which pays 10% pa. If he continues with that monthly deposit pattern for the next 20 years, how much will David have in his fund due just to that contribution (to the nearest $100)? Answer. $153,100.
	4. Lindsay invested $1,000 per year for 8 years into an annuity account paying 4% p.a.compound interest. She makes her deposit at the end of each year. How much did Lindsay have in her annuity account at the end of the 8 years? Answer. $9,214.22.
Finding the required contribution.	5. Thirty years ago, a kind lady decided she wanted to leave a substantial sum of money for her grandchildren to share equally. She arranged an investment fund which paid 6% p.a. compound interest and made an initial contribution of $10,000. She has made annual contributions of $M to the fund at the beginning of each year since (starting in Year 2). (i) How much will the original deposit amount to at the end of 30 years? (ii) Show that the amount in the fund at the end of the second year can be expressed as A₁ = 10,000 (1.06)² + M×1.06 (iii) Show that the amount in the fund at the end of three years can be expressed as A₃ = 10000 (1.06)³ + M×1.06 (1 + 1.06). (iv) If the total amount in the fund at the end of 30 years was $100,000, what was the annual amount (to the nearest $5) that Granny contributed? (v) What was the total amount that Granny contributed? Answer. (i) $57,434.91 (iv) $545 (v) $25,805.
	6. Evie and Mimi are sisters and they work in the same department. Each year on 1 January, the company gives them a bonus for the efforts over the previous year of $5,000 (they are good workers). They receive their first bonuses in 2020. Every year, Mimi invests her bonus in superannuation at 9% p.a. compounding interest. Every year, Evie spends her bonus on an overseas surfing trip. (i) Find an expression for the amount that will be in Mimi's superannuation account on 31 December 2030 (just before her next bonus is credited). (ii) Show Mimi will have about $100,700 (to the nearest $10) in her account on 1 January 2031 after the bonus has been paid. Evie decides on at the end of 2030 that, after 11 years of world surfing holidays, she will start saving for her retirement. She naturally wants the same amount as her sister at the end of 2050. Evie's superannuation fund also pays 9% p.a. compounding interest. (iii) How much will Evie need to save with annual deposits at the beginning of each year to have the same total amount as Mimi will have at the end of 2050 (including the amounts Mimi invested in the first 10 years). (iv) How much more will Evie have to invest over her reduced years of saving than Mimi will have invested over all the years since the beginning of 2020? Answer. (i) $95,703.60 (ii) $100,700 (iii) about $14,620 p.a. (iv) Evie spends $137,366.80 more.
	7. Charlotte wants to have a special holiday at the end of her school studies to recover from "the big exams" at the end of next year. She feels she needs $3,000 and has 20 months in which to save. She can access a special account at the bank which will pay her 4% p.a. compound interest. She has a part time job and wants to deposit into her "Holiday Saver" account each 2 weeks when she is paid. How much must Charlotte contribute each fortnight to reach her goal? Answer. Charlotte needs to contribute at least $33 per fortnight.