Finance - Fund withdrawal

The questions on this page focus on:

Number of withdrawals.	1. Jenny wants to help the school she taught at for many years and so establishes a fund with a single investment of $1,200. She wishes to have the school award a prize to the female student in Year 12 who best combines academic and sporting qualities. The prize will be valued at $80. The fund attacts a concessional interest rate of 5% per year compounding annually. The first occasion on which the prize will be awarded is one year after the fund is established. (i) Calculate the balance in the fund at the end of the second year. (ii) Let $Y_n be the amount in the fund at the end of n years (and after the n^th prize has been awarded). Show that . (iii) At the end of the 12^th year (and after the 12^th prize has been awarded), the school decides to increase the value of the prize to $120. For how many more years will the school be able to award Jenny's prize? Answers.(i) $1,180. After 12 years, balance is $881.66. Calculation = 9.38 which rounds down to 9 more years.
	2. Tony has set up a retirement fund and after 10 years he has accumulated $67,000. Due to an accident, he is no longer able to work and so he can make no further contributions to his fund. He leaves the money in the fund to accumulate interest at 8% p.a. compounded monthly. He does however have to withdraw $1,000 at the end of each month for living expenses. (i) Show that at the end of two months, Tony will have $67,686.80 in the fund. (ii) Develop an expression to describe the amount Tony still has invested in the fund at the end of n months. (iii) Hence determine how long the fund will be able to continue to pay Tony his monthly allowance. Answers.(iii) 89.2 months - so 89 months.
Withdrawal amount.	3. Chris invests $150,000 at 8% per annum compounded quarterly. He intends to withdraw $M at the end of each quarter to undertake home renovations that he estimates will take him 4 years (16 quarters) to complete. Let $A_n be the amount remaining in the account after n quarters. (i) Write an expression for the amount remaining in Chris' account after his second withdrawal. (ii) After 4 years, Chris will have spent all the money in his account (but will have a really nice home). Show that each quarterly withdrawal is approximately $11,047.52.
	4. Yoko has recently retired and has a lump sum payout of $100,000 for valuable service rendered over the years. She invests it into one of the best retirement funds which pays interest at the rate of 12% p.a. She intends to withdraw $Y each month from her fund. (i) Develop an equation showing how much Yoko has left in the fund $B after n months. (ii) Yoko decides she wants to have no balance in the fund after 15 years. How much $Y should she withdraw each month? Answers.(ii) Y = $1,200.17
Changing the rate of withdrawal by a % amount.	5. Phyllis deposits $300,000 into a special account at her bank. She argues for a higher rate of interest than is usually given by any of the Big Four banks and reaches an agreement for a 4% p.a. compound interest rate with an annual compounding deposit. Phyllis intends to transfer an amount at the end of each year to a different account so that she can meet her expenditures easily. Her first withdrawal is $A. As her costs increase, she plans to transfer an extra 5% of her previous amount each year. (i) Show that the balance $B₂ in her investment account after her second withdrawal can be expressed as B₂ = 300000 (1.04)² - A(1.04 + 1.05). (ii) Write an expression for the amount in her investment account at the end of the third year. (iii) Show that if Phyllis still has funds in her investment account, then .
	6. An important Tennis Club with some excellent representatives established a Prize Fund for the annual Awards night. A deposit of $15,000 was invested in an account one year before the next annual Awards night. The committee agreed to one strategy which entailed withdrawing $650 from the fund every year to purchase the annual prizes. The money in the Fund was invested at 4% p.a. compounded annually with the prize money withdrawn immediately after the interest was paid. (i) Show that, with this strategy, the fund would contain $14,898 after the second Awards night. (ii) Show that the amount remaining in the Prize Fund after the n^th Award night could then be expressed as (iii) Find the expected balance in the Fund after the 20th Awards night (express your answer to the nearest dollar. (iv) Find the maximum number of Awards night which could be funded under this arrangement. (v) An alternative strategy considered by the committee was to increase the amount withdrawn from the fund each year by 2%. If that strategy was adopted, what would then be the expected balance in the fund after the 20th Awards night. Answers.(i) End Y2 = $14,898. (iii) $13,511. (iv) 65 years. (v) $9,949.
Amount to invest.	7. Matt invests $P at 8% p.a. compounded annually. He plans to withdraw $3,000 at the end of each year for the next six years to cover special expenses during his double degree at University. (i) Write down an expression for the amount $A₁ remaining in the account after Matt's first $3,000 is withdrawn. (ii) Calculate the amount $P (to the nearest $10) that Matt needs to invest (on behalf of his parents) if the account is to have a $0 balance at the end of the six years. Answers.(i) $A₁ = P × 1.08 - $3,000. (ii) $28,160 (nearest $10).
	8. Harry invests $A into an investment account at 4% compounded annually. He intends to withdraw $3,000 at the beginning of each year to help defray his costs for his paramedical training. (i) Explain why the expression B₁ = (A - 3000)×1.04. describes the balance in Harry's account at the end of his first year of studies. (ii) Calculate the amount $A Harry needs to deposit in his investment account if he has a zero balance in his account after 4 years (answer to the nearest $50). Answer.(ii) A = $11,350 (to nearest $50).
	9. An Alumni from a well known school established a fund with $2,500 to provide an annual prize of $150 for the best student in the school in Financial Mathematics. The fund paid 6% p.a. interest compounded annually. The prize money for any year was withdrawn from the fund immediately after the annual interest was paid. (i) What is the balance of funds after the third annual prize had been withdrawn from the fund. (ii) What conclusion can you make about the fund and explain why is it so? Answer.(ii) A = $2,500. (ii) Prize awarded in perpetuity.