Finance - Series - Annuities

The questions on this page focus on:

1. Finding n for a pre-determined total.

Finding n for a pre-determined total.	1. Sophie opened a savings account at her local Bank. The nice people down there offered her an interest rate of 4.8% p.a. on her investments compounded monthly at the end of each month. Sophie contributed $300 from her salary at the beginning of each month. (i) Show that the value of her first contribution after 5 months was $306.05. (ii) Sophie has a savings goal of $20,000. If she continues to invest $300 per month, how many months will it take her to reach her goal? Answer.59 months.
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Missing payments.	4. Amy has joined a Superannuation fund and her employer is paying $600 per month into her account (at the end of each month as is usual). The fund pays 9% p.a. interest. After 4 years, Amy took extended maternity leave for 2 years. No employer contributions were made during that time. She then resumed work and her contributions continued at the previous level (there were no wage rises) for the next 11 years (that is until she had been employed for 20 years. (i) How much was in Amy's superannuation account when she began her leave (i.e. after 4 years). (ii) Show that she had $41,291.14 in her account after she returned to work. (iii) Show if she had not taken leave, she would have had $57,004.22 in her account at the end of the 6 years. (iv) Find the difference between the amounts in her account after 20 years between her taking and not taking leave. Answer.(i) $34,513.43. (iv) Difference = $64,411.90
	5. During a significant fall in employment due to a widespread virus infection, contributors were allowed to withdraw $10,000 from their superannuation accounts. Jamie had been contributing $100 at the beginning of each week to his fund for the past 5 years. His fund pays interest at the rate of 0.077% p.w. (4% p.a.). At the end of the 5 years, he withdraws $10,000 immediately but then cannot contribute to his fund for another 12 months because of being unemployed. He does receive compound interest on his investment. After that year, he resumes his weekly contributions for the next 19 years. (i) How much did Jamie have in his superannuation fund at the end of the first 5 years of contributions. (ii) How much would Jamie have in his fund at the end of 25 years if the virus outbreak did not occur and so his contributions would have continued unchanged and he made no withdrawal at the end of the 5th year? (iii) How much did Jamie have in his fund at the end of the 6th year after his withdrawal and one year of no contributions? (iv) How much did Jamie have in his fund at the end of the interrupted 25 year period? (v) How much change in the total of Jamie's retirement fund was effected by the withdrawal and lack of contributions during the 6th year? Answer.(i) $34,513.43. (ii) $223,542.59. (iii) Difference = $13,079.46. (iv) $176,052.37. (v) $47,490.22
Increasing interest rate or deposit.	7. Ben is planning to take a special holiday in three years time and estimates it will cost $8,000 for himself and his partner. To achieve that target, he intends to save a fixed amount ($A) at the beginning of each month in an investment account paying 6% interest compounded monthly. (i) Show that the value of Ben's investment savings ($S) at the end of 3 months can be expressed as S_n = A×1.005(1 + 1.005 + 1.005²). (ii) Find, to the nearest dollar, the amount of Ben's monthly contribution to realise his goal. (iii) At the end of two years, the interest rate increased to 9%. If Ben maintained his level of contribution, how much additional money would he have for some special dinners and extra sightseeing? Answer.(ii) $203. (iii) An extra $233.
	8. Matilda's goal is to travel to Vanuatu, Chile, Antartica and Portugal. She estimates her trip will cost $15,000. She therefore invests all of her savings on 3 January 2018 by depositing $4,890 in an account paying 6% p.a. compounding monthly. She plans to travel immediately after her deposit has been made for three years - say starting about 7 January 2021. To ensure she has enough money for her travels, Matilda makes an additional monthly deposit, starting at the same time, of $T. (i) Show that the balance in Matilda's account after two months can be written as: A₂ = 1.005{4,890 × 1.005 + 200T(1.005² - 1)} (ii) Hence, or otherwise, determine the value of Matilda's additional monthly investment $T she needs to make if she is to reach her travel goal. Answer.(ii) T = $231.41.