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7-1 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Chapter 7 Annuities Introductory Mathematics & Statistics

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7-2 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Learning Objectives Understand and apply annuities Distinguish between future and present values of annuities Solve problems involving the future value of an annuity Calculate the present value of an annuity Calculate the periodic payment of a present value annuity (amortisation) Calculate the periodic payment of a future value annuity (sinking fund)

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7-3 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.1 Introduction annuityAn annuity is a series of equal payments, often made under contract, paid at equal intervals (e.g. quarterly or monthly) Definitions 1. The rent period (or payment period) is the interval of time between the payments of an annuity 2. The term of an annuity is the time from the beginning of the first payment period to the end of the last payment period

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7-4 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.1 Introduction (cont…) The two main types of annuities are simple and general A simple annuity is where interest is compounded at the same times as the annuity payments A general annuity is where interest is compounded at times that are either greater or smaller than when the annuity payments are made

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7-5 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.1 Introduction (cont…) Simple annuities can be classified into four types 1. An ordinary annuity is one where the payments are made at the end of each period 2. An annuity due is one where the payments are made at the beginning of each period 3. A deferred annuity is one where the payments do not commence until a period of time has elapsed 4. A perpetuity is an annuity in which the payments continue indefinitely

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7-6 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.1 Introduction (cont…) Ordinary annuities have the following aspects 1. The future value of an annuity: this is the value of the annuity at the end of its term 2. The present value of an annuity: this is the value of the annuity at its beginning 3. Amortisation: this is the making of periodic payments to repay a debt (including the principal and interest) 4. Sinking fund: this is a fund into which the periodic payments necessary to realise a given sum of money in the future are made

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7-7 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.2 Future value of an annuity The future value (or accumulated value) of an annuity is the amount due at the end of the term That is, it is the sum of all the periodic payments made and interest accrued up to and including the final payment period Where: S = future value of the annuity R = amount of the annuity payment made per period i = interest rate per payment period n = total number of payments

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7-8 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.2 Future value of an annuity (cont…) The formula can also be written as Where The value of s n i (or s n at i %) may be thought of as the future value at an interest rate of i% of $1 paid at the end of each period for n periods.

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7-9 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.2 Future value of an annuity (cont…) Example A customer deposits $250 every 3 months into a building society account that pays interest at a rate of 8% per annum convertible quarterly. How much money will be in the account at the end of 10 years? Solution

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7-10 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.2 Future value of an annuity (cont…) Solution (cont…) Hence, the customer will have $15 100.50 in the account at the end of 10 years.

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7-11 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.3 Present value of an annuity The present value (or discounted value) of an annuity is its value at the beginning of the initial rent period. That is, it is the sum of the compound present values of all the payments one period before the initial payment. Where: A = present value R = annuity payment per period i = interest rate per period n = number of payments

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7-12 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.3 Present value of an annuity (cont…) Another way of writing the formula Where The value of a n i (or a n at i %) may be thought of as the present value at an interest rate of i % per period of $1 paid at the end of each period for n periods

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7-13 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.3 Present value of an annuity (cont…) Example Melinda took out a loan from a credit union in order to purchase a home computer. She was to repay the loan in monthly instalments of $120 for 5 years. Calculate the present value of these repayments if the interest rate was 9%, convertible monthly Solution From Table 3:

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7-14 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.3 Present value of an annuity (cont…) Solution (cont…) That is, the repayments are worth $5780.80 at the beginning of the loan

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7-15 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.4 Amortisation When an individual or business pays a debt (including interest) by making periodic payments at regular intervals, the debt is said to be amortised. An amortisation problem involves finding the periodic payment that will discharge a debt. In particular, as each periodic payment is made, this amount covers part of the principal and interest on the balance of the principal. In turn, there is a reduction in the remaining principal such that at the end of the term of the annuity the debt is extinguished.

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7-16 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.4 Amortisation (cont…) Example A family undertakes a mortgage of $40 000 from a bank in order to buy its new home. The bank charges interest at a rate of 12% per annum, compounded quarterly over 20 years. What quarterly payments will the family have to make on this loan? Solution From Table 3:

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7-17 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.4 Amortisation (cont…) Solution (cont…) Hence, the family must repay the loan at the rate of $1324.47 per quarter. Note that the family will, in fact, be paying back to the bank a total of $1324.47 × 80 = $105 957.60, considerably more than the original $40 000 loan!

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7-18 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.5 Sinking funds A sinking fund is a fund into which periodic payments are made so as to accumulate a nominated amount of money at the end of a specified period It is assumed that each deposit earns compound interest until the end of the period Where: S = future value R = annuity payment per period i = interest rate per period n = number of payments or

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7-19 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.5 Sinking funds (cont…) Example Connie is planning to spend her Christmas holidays in 2 years time in the United States. She estimates that she will need an amount of $6000 to pay for her airfares, accommodation and other expenses. She is going to save her money in an account that attracts 10% interest per annum, compounded quarterly. How much will Connie have to deposit into her account at the end of each quarter, to have the desired amount at the end of 2 years?

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7-20 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 7.5 Sinking funds (cont…) Solution So If Connie deposits $686.80 every 3 months over the 2 years, she will have the required $6000 at the end of that time

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7-21 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Summary We looked at understanding and applying annuities We found the difference between future and present value of annuities We solved problems involving the future value of an annuity We calculated the present value of an annuity We calculated the periodic payment of a present value annuity (amortisation) We calculated the periodic payment of a future value annuity (sinking fund)

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