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1 Annuities and Loan Repayments H General Maths HSC NAME: 1
2 HSC CAPACITY MATRIX GENERAL MATHEMATICS TOPIC: Financial Mathematics 5 Annuities & loan repayments 3 weeks AM1 rn1 CONTENT CAPACITY BREAKDOWN! DONE GOT ON MY WORKING HELP!!!! IT!!!! IT!!!!! WAY! ON IT! 1. Deep understanding of the term annuity Compound interest W/S 2. Calculation of the future value (A) of an annuity (or Skillsheet 8.1 the contribution per period), using ( + ) Ex 8A all even questions S/S task Annuity calc r = 3. Calculation of the present value (N) of an annuity (or the contribution per period), using N = M n ( 1+ r ) 1 A or N = r ( 1+ r ) n ( 1+ r ) n 4. Using tables to solve problems involving annuities 5. Use the present value formula for annuities to calculate loan instalments, and hence the total amount paid over the term of a loan Ex 8B all even questions S/S task Future and present value tables Ex 8C all even questions Ex 8D all even questions 6. Investigate various processes for repayment of loans 7. Calculate the fees and charges for different options for borrowing money in order to make a purchase. Types of loans task Your say! What was the most important thing you learned? What was something new you learnt? 2
3 What part(s) of this topic will you need to work on? 3
4 QUICK REVISION - COMPOUND INTEREST Compound Interest formula is it on the Formulae sheet? eg Calculate the value of each of the following investments. (a) $5000 invested at 7% p.a. for 4 years with interest compounded annually. (b) $ invested at 9% p.a. for 5 years with interest compounded sixmonthly. (c) $ invested at 6.8% p.a. for 4 years with interest compounded quarterly. 4
5 INVESTIGATING ANNUITIES But firstly, what is an annuity? Let s investigate: eg Laura invests $5000 at the end of each year at 8% p.a. for 4 years with interest compounded annually. (a) Sketch a timeline. (b) Find the amount to which the first $5000 will grow. (c) If Laura invests another $5000 at the end of the second year, to what amount will this $5000 grow? (d) A third $5000 is invested after another year. To what amount will this $5000 grow in one year? (e) How much will Laura make on her investment? (f) How much altogether? 5
6 A QUICKER METHOD PERHAPS? S The value of an annuity is calculated by adding the value of each amount contributed as a separate compound interest investment. The formula below will calculate the value of an annuity: This formula is given to you! FUTURE VALUE OF ANNUITY FORMULA A = M ( 1+ r ) r n 1 where M is the amount of each periodical contribution; r is the interest rate per period expressed as a decimal; n is the number of deposits. S The amount of each contribution to annuity to reach a certain goal can be calculated using the formula: M = Ar n ( 1+ r ) 1 This formula is NOT given to you can you see how to change the above formula into this one? eg Lance invests $1000 into a fund at the end of each year at 8%pa interest, for 10 years. Calculate the future value of the annuity after 10 years. S 6
7 eg Belinda invests $500 in a fund every six month at 9.5% pa interest, compounding six-monthly for 15 years. Calculate the future value of the annuity after 15 years. eg Luke wants to save $ in the next five years. The best interest rate that he can obtain is 7.5% pa, with interest compounded annually. Calculate the amount of each annual contribution that Luke must make. 7
8 PRESENT VALUE OF AN ANNUITY DEFINITION: The present value of an annuity is the single sum that can be invested under the same terms as annuity and will produce the same financial outcome. The present value of an annuity can be calculated using the formula: A N = 1+ r ( )n To compare an annuity with a single sum investment, you need the present value of the annuity! when you know the future value of an annuity. If you know the amount of each contribution of the annuity, you can calculate the present value using the formula N = M n ( 1+ r ) 1 ( ) n r1+ r, where M is the contribution period; r is the interest rate per period (as a decimal); and n is the number of interest periods. Investments can be compared using the present value formula. The investment with the greater present value will produce the greater financial outcome over time. 8
9 eg Scott has an annuity that has a future value of $ on his retirement in 32 years. The annuity is invested at 6.5%pa with interest compounded annually. Calculate the present value of Scott s annuity. eg Tom has an annuity to which he contributes $1100 annually at 7.5%pa interest, compounded annually. The annuity will mature in 25 years. Calculate the present value of the annuity. 9
10 eg Which of the following investments would give the greater financial return? INVESTMENT A: An annuity of $100 deposited per month for 20 years at 12% pa interest, compounding six monthly; INVESTMENT B: A single deposit of $ invested for 20 years at 12% pa with interest compounding six monthly. 10
11 USING TABLES A table such as the one below can be used to find the value of an annuity by multiplying the amount of the annuity by the future value of $1. eg Use the table to calculate the value of an annuity into which $8 000 is deposited at the end of each year at 7% pa interest, compounded annually for 11 years. 11
12 eg Phoebe invests $850 per year in an annuity at 6% pa for 9 years, with interest compounded annually. Use the table to calculate the present value of Phoebe s annuity. 12
13 LOAN REPAYMENTS When a loan is taken out and is repaid in equal monthly instalments, the pattern of repayments works similar to an annuity. Each month interest compounds on the balance owing on the loan and then a repayment is made. Consider the present value (N) formula: n N = M nr ( 1+ r ) 1 ( ) r1+ To calculate the amount of each monthly repayment, we need to make M the subject of this formula. In this formula, M is the amount of each repayment, N is the amount borrowed, r is the interest rate per repayment period as a decimal and n is the number of repayments to be made. Z By considering the amount borrowed in a loan as the PRESENT VALUE OF AN ANNUITY, you can apply the present value formula to calculate the amount of each repayment. Z The total cost of the loan can be calculated by multiplying the amount of each repayment by the number of repayments to be made; Z The length of time that it will take to repay a loan can be calculated by using GUESS AND REFINE! 13
14 eg Calculate the monthly repayments on a loan of $ to be repaid in monthly instalments over 14 years at an interest rate of 12.5% pa. eg Calculate the total cost of repaying a $ home loan at 9% pa in equal monthly repayments over a 25 year term. You can calculate the total cost of the repayments by multiplying the amount of each repayment by the number of repayments. 14
15 eg A $ home loan is taken out over a 25 year term at an interest rate of 12% pa reducible interest. The minimum monthly repayment on the loan is $ How long will take the loan to be repaid at $1200 per month? Guess and refine! e! BLURK!!!! 15
16 2011 HSC MC Past HSC questions Q23 16
17 Q HSC Q25 17
18 2009 HSC MC 18
19 Q26 19
20 Q27 20
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