Luke S Tax Return

Annuities and loan repayments

In ‘Investing Money’ we calculated how much MT would require in ten years time for a deposit on her first home. In this unit we investigate the nature and mathematics of annuities, the processes by which they accrue and the ways of maximising their value as an investment.

An annuity is a form of investment involving a series of periodic equal contributions to an account, with interest compounding at the conclusion of each period.

- The future value (A) of an annuity (or the contribution per period), is found using:

where M = contribution per period, paid at the end of the period.

r = interest rate per period expressed as a decimal

n = the number of interest periods

- The present value (N) of an annuity (or the contribution per period), is found using:

- The minimum repayment (M) on a loan is found using:

where N = the amount borrowed.

In the activities that follow you will decide which of three annuities will allow MT Pockets to achieve her savings target. You will also examine the cost of borrowing and will investigate strategies to minimise both the interest paid and the term of the loan.

Go to annuities.xls (Excel 218kB)

Comparing annuities: Questions

MT Pockets has the goal of saving for a deposit on her first home. Our task is to help MT to decide which of three annuities will allow her to reach her savings target in ten years from now.

1.Annuity A pays7% pa, compounded annually. What is the value of the annuity after 10 years if $2000 is deposited at the end of each year?

2.Annuity B pays 5.5% pa, compounded half-yearly. What is the value of the annuity after 5 years if $500 is deposited at the end of every 6 months?

3.Annuity C pays 6.75% pa, compounded monthly. If $125 is deposited at the end of each month, what is the total interest earned at the end of 10 years?

4.If MT Pockets had the same total amount to invest each year in the three annuity funds, which annuity would givethe greatest return? Which annuity would give the least return?

5.In ‘Investing Money’ we calculated that in ten years time MT Pockets would need at least $51800 for a deposit on a home. Use the spreadsheet to determine the minimum amount that MT must invest each month in Annuity C in order to achieve her savings target. Give your answer to the nearest dollar.[Key Question]

6.Use the formula given on the introduction page to calculate the exact amount thatMT Pockets must invest each month in Annuity C in order to achieve her savings target. Give your answer to the nearest cent. Check you answer using the spreadsheet.

7.What is the minimum amount thatMT Pocketswould need to invest each half-year in Annuity B to produce a better financial result thanAnnuity C? Give your answer to the nearest ten dollars.

8.Suppose that MT Pockets could only afford to invest $1500 at the end of each year. What is the minimum interest rate that would need to be offered with Annuity A forMT to achieve her savings target? Give your answer to the nearest whole percentage.

9.The present value of an ordinary annuity is the single sum of money which, if invested today at the rate of compound interest which applies to the annuity, would produce the same financial result over the same period of time. Study the formulae in cells C21 and D21. Go to cell E21 and enter a formula that will calculate the present value of Annuity C.

10.What single sum would MT Pockets have to deposit today in an account paying 6.75% pa, compounded monthly, to reach her savings target ten years from now?

Comparing annuities: Answers

1.$27632.90

2.$5666.38

3.$6340.48

4.Annuity C would give the greatest return. Annuity B would give the least return.

5.If MT Pockets invested $304 each month, the value of the annuity at the end of ten years would be $51900 (to the nearest dollar).

6.A=$51800,

R=0.0675÷12=0.005625

n = 12x10=120

(to the nearest cent).

MT Pockets would need to invest $303.42 each month to achieve her savings target.

7.If MT Pockets invested $1980 each half-year, the value of the annuity after 10 years would be $51871 (to the nearest dollar).

8.An interest rate of 26% pa would allow MT Pockets to achieve her savings target. In ten years the annuity would accrue to $52417 (to the nearest cent).

9.The formula should read =E19/(1+D7/E7)^(E7*10)

10.MT Pockets would have to deposit $26424.76.

Home loan repayments: Questions

For most people, the largest loan they will take out will be for a home. The interest on a home loan is reducible, ie the interest is calculated on the amount of money owing at the time, rather than the amount borrowed. Our task is to investigate the cost of borrowing and the strategies to minimise both the interest paid and the term of the loan. Go to the ‘Home loan repayments’ spreadsheet.

1.Find the term of a loan of $300000 with an interest rate of 8.5% pa and a monthly repayment of $2500.

2.Find the final repayment required to reduce the balance of the loan in question (1) to zero. Calculate the total interest paid on the loan.

3.If the monthly repayment was doubled, would you expect the term of the loan in question (1) to be halved? Explain.

4.What would happen to the principal of the loan in question (1) if the regular payment was less than $2125? Explain.

5.In ‘Investing Money’ we estimated that in ten years time MT Pockets would require a loan of $466200 to purchase her dream home. If she wishes to pay-off the loan in 20 years, what would be the minimum monthly repayment (to the nearest hundred dollars) that she would need to make? (Assume an interest rate of 8% pa).

6.What would MT Pockets have to increase her loan repayments to if the interest rate was 9% pa for the term of the loan? (Give your answer to the nearest hundred dollars).

7.Use the formula given on the introduction page to calculate the minimum monthly repayment for the loan in question (6). Check your answer using the spreadsheet.

8.If MT Pockets could afford to repay no more than $3000 per month, what would be the minimum annual interest rate (to 1 decimal place) that would allow her to repay the loan within 20 years? What would be the total interest paid by MT?

9.If in months 10, 11 and 12 MT Pockets was able to make a repayment of $7000 instead of her regular payment of $3000, how would this affect the term of the loan in question (8)? How much interest would be saved?

10.What advice would you offer MT Pockets regarding repayments on a home loan?

Home loan repayments: Answers

1.The loan would be repaid after 269 months (or 22 years and 5 months).

2.A repayment of $1941.90 in the 269th month would reduce the balance of the loan to zero.

Total repayments = 268x$2500+$1941.90=$671941.90

Total interest =$671941.90-$300000 = $371941.90

3.No. The term of the loan would be reduced by 190 months to 79 months (6 years and 7 months), which is a reduction of more than 70%.

4.The principal would increase since the repayment is less than the interest that would be added to the principal each month. For example, if a monthly payment of $2124 was made, after 360 months (30 years) the principal would be $301638 (to the nearest dollar).

5.A monthly repayment of $3900 for 239 months and a payment of $3595.83 in the 240th month would reduce the balance of the loan to zero.

6.MT Pockets would have to increase her monthly repayment to $4200 per month for 239 months, leaving a final payment of $541.59 in the 240th month.

Therefore the minimum repayment would be $4194.52 (to the nearest cent). Entering this number into the spreadsheet gives a balance in the 241st month of $1.61. This results because of the rounding of the answer from the calculator.

8.An interest rate of 4.7% pa would allow her to pay off the loan in 20 years. The total interest paid = $719991.59 - $466200 = $253791.59

9.The loan would be repaid in 231 months. The total interest saved would be $719991.59–$703113.03=$16878.56

10.‘Sound advice’ could include the following:

Don’t borrow more than your income and lifestyle will allow you to repay.
Repay more than the minimum monthly repayment.
Make additional ‘one-off’ payments whenever possible.
Pay half the monthly repayment each fortnight.
‘Shop around’ for the best interest rate.

These measures will reduce the term of the loan and hence the total interest paid.