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Essential Mathematics for Economics and Business Fourth Edition Essential Mathematics for Economics and Business Fourth Edition Teresa Bradley Copyright c 1998, 1999, 2002, 2008, 2013 by John Wiley & Sons Ltd All effort has been made to trace and acknowledge ownership of copyright The publisher would be glad to hear from any copyright holders whom it has not been possible to contact Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The right of Teresa Bradley to be identified as the author of this work has been asserted in accordance with the UK Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Wiley publishes in a variety of print and electronic formats and by print-on-demand Some material included with standard print versions of this book may not be included in e-books or in print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com For more information about Wiley products, visit www.wiley.com Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought ISBN 9781118358290 (pbk) 9781118527795 (ebk) 9781118527788 (ebk) 9781118527764 (ebk) A catalogue record for this book is available from the British Library Typeset in 10/12pt Goudy and Helvetica by Aptara Inc., New Delhi, India Printed in Great Britain by Bell & Bain Ltd, Glasgow Senan and Ferdia CONTENTS Introduction xiii CHAPTER Mathematical Preliminaries 1.1 Some Mathematical Preliminaries 1.2 Arithmetic Operations 1.3 Fractions 1.4 Solving Equations 1.5 Currency Conversions 1.6 Simple Inequalities 1.7 Calculating Percentages 1.8 The Calculator Evaluation and Transposition of Formulae 1.9 Introducing Excel 11 14 18 21 24 28 CHAPTER The Straight Line and Applications 2.1 The Straight Line 2.2 Mathematical Modelling 2.3 Applications: Demand, Supply, Cost, Revenue 2.4 More Mathematics on the Straight Line 2.5 Translations of Linear Functions 2.6 Elasticity of Demand, Supply and Income 2.7 Budget and Cost Constraints 2.8 Excel for Linear Functions 2.9 Summary 37 38 54 59 76 82 83 91 92 97 CHAPTER Simultaneous Equations 3.1 Solving Simultaneous Linear Equations 3.2 Equilibrium and Break-even 3.3 Consumer and Producer Surplus 3.4 The National Income Model and the IS-LM Model 3.5 Excel for Simultaneous Linear Equations 3.6 Summary Appendix 101 102 111 128 133 137 142 143 [x] CONTENTS CHAPTER Non-linear Functions and Applications 4.1 Quadratic, Cubic and Other Polynomial Functions 4.2 Exponential Functions 4.3 Logarithmic Functions 4.4 Hyperbolic (Rational) Functions of the Form a /(b x + c) 4.5 Excel for Non-linear Functions 4.6 Summary 147 148 170 184 197 202 205 CHAPTER Financial Mathematics 5.1 Arithmetic and Geometric Sequences and Series 5.2 Simple Interest, Compound Interest and Annual Percentage Rates 5.3 Depreciation 5.4 Net Present Value and Internal Rate of Return 5.5 Annuities, Debt Repayments, Sinking Funds 5.6 The Relationship between Interest Rates and the Price of Bonds 5.7 Excel for Financial Mathematics 5.8 Summary Appendix 209 210 218 228 230 236 248 251 254 256 CHAPTER Differentiation and Applications 6.1 Slope of a Curve and Differentiation 6.2 Applications of Differentiation, Marginal Functions, Average Functions 6.3 Optimisation for Functions of One Variable 6.4 Economic Applications of Maximum and Minimum Points 6.5 Curvature and Other Applications 6.6 Further Differentiation and Applications 6.7 Elasticity and the Derivative 6.8 Summary 259 260 270 286 304 320 334 347 357 CHAPTER Functions of Several Variables 7.1 Partial Differentiation 7.2 Applications of Partial Differentiation 7.3 Unconstrained Optimisation 7.4 Constrained Optimisation and Lagrange Multipliers 7.5 Summary 361 362 380 400 410 422 CHAPTER Integration and Applications 8.1 Integration as the Reverse of Differentiation 8.2 The Power Rule for Integration 8.3 Integration of the Natural Exponential Function 8.4 Integration by Algebraic Substitution 8.5 The Definite Integral and the Area under a Curve 427 428 429 435 436 441 [ 244 ] CHAPTER How much of the repayment is interest? When a loan is repaid by a series of equal instalments over equal intervals of time, the proportion of interest being repaid decreases steadily The following worked example illustrates this behaviour WO R K E D E X A M P L E HOW MUCH OF THE REPAYMENT IS INTEREST? A loan of £5000 is to be paid off in four equal payments at the end of each quarter Assuming an interest rate of 20%, calculate (a) the amount of each payment, (b) the present value of the loan, (c) the total amount of interest paid, (d) the amount of interest paid each quarter SOLUTION L = 5000, i 0.20 = = 0.05 per quarter, t = × payment intervals 4 (a) A0 = L i /4 − (1 + i /4)−t×4 capital recovery factor A0 = 5000 × 0.05 − (1.05)−4 = 5000 × 0.2820 = 1410.06 (b) The present value of the loan is V0 = 5000 Check by calculation: V0 = A0 − (1 + i /4)−t×4 − (1.05)−4 = 5000 = 1410.06 i /4 0.05 annuity factor (c) Total payments = 1410.06 × = 5640.24 Total interest = total payments − value of loan = 5640.24 − 5000 = 640.24 (d) At the start of quarter 1, the borrower has the loan of 5000 in his account Calculate the value of the account at the end of each quarter, where value at start of quarter + interest for quarter Quarterly calculations are set out as follows − quarterly payment = value at end of quarter [ 245 ] F I N A N C I A L M AT H E M AT I C S Value at start (VSQ) 5000.000 3839.940 2621.877 1342.920 +Interest (VSQ × 0.05) −Payment (−1410.06) Value at end (VSQ + VSQ × 0.05 − 1410.06) 250.000 191.997 131.103 67.146 −1410.06 −1410.06 −1410.06 −1410.06 3839.940 2621.877 1342.920 0.006 640.246 −5640.24 Sinking funds A sinking fund is created by putting aside a fixed sum each year for the purpose of paying debts, replacing equipment, etc In other words, an annuity is set up to repay the debt If a fixed sum, A0 , is set aside at the start of each year and interest is compounded annually at i%, the fund will grow year by year as follows At the end of Value of sinking fund Year Year Year A0 (1 + i) + A0 A0 (1 + i)2 + A0 (1 + i) + A0 A0 (1 + i)3 + A0 (1 + i)2 + A0 (1 + i) + A0 Year t Value of sinking fund after t years A0 (1 + i)t + A0 (1 + i)t −1 + A0 (1 + i)t −2 + ··· + A0 (1 + i) + A0 (1 + i)t + A0 (1 + i)t −1 + A0 (1 + i)t −2 + ··· + A0 (1 + i) Note: No deposit is made at the end of year t The fund has matured and is sufficient to repay the debt The value of the fund is VS K ,t = A0 (1 + i )t + A0 (1 + i )t−1 + A0 (1 + i )t−2 + · · · + A0 (1 + i ) This series is a simple geometric series whose sum is readily calculated by equation (5.5), after some slight simplifications Start by writing the series in reverse order: VS K ,t = A0 (1 + i ) + · · · + A0 (1 + i )t−2 + A0 (1 + i )t−1 + A0 (1 + i )t Factor out A0 (1 + i), which is common to all terms: VS K ,t = A0 (1 + i ) [1 + (1 + i ) + (1 + i )2 + · · · + (1 + i )t−1 ] geometric series with a =1, r =(1+i ) (5.29) [ 246 ] CHAPTER The series in the square bracket is geometric, a = 1, r = (1 + i) The sum of the series is calculated by (5.5) with these values for a and r: + (1 + i ) + (1 + i )2 + · · · + (1 + i )t−1 = (1 + i )t − (1 + i ) − Hence equation (5.29) simplifies to VS K ,t = A0 (1 + i ) (1 + i )t − (1 + i )t − = A0 (1 + i ) (1 + i ) − i (5.30a) It is left as an exercise to show that equation (5.30a) may also be deduced from equation (5.23) by substituting P0 = A0 and subtracting A0 from the series, since there is no payment at the end of year t Note that a sinking fund is a form of annuity where deposits are made at the beginning of each interval Alternatively, VS K , t = A0 (1/i + 1) (1 + i )t − (5.30b) WO R K E D E X A M P L E SINKING FUNDS A taxi service must replace cars every five years at a cost of £450 000 At an 8% rate of interest, calculate: (a) the size of the fund if £4000 is deposited at the beginning of each month; (b) the size of each quarterly payment necessary to meet this target SOLUTION (a) Given A0 = 4000, Hence by 0.08 i = = 0.0067, t = × 12 = 60 instalments 12 12 VS K ,t = A0 (1 + i /12) (1 + i /12)t − i /12 (1.0067)60 − 0.0067 0.4928 = 4000(1.0067) 0.0067 = 4000(1.0067)(73.5522) VS K ,60 = 4000(1 + 0.0067) = 296 179.9989 Not sufficient to meet target of £450 000 (5.29) [ 247 ] F I N A N C I A L M AT H E M AT I C S (b) Substitute the information given into equation (5.29): VS K = 450 000, 0.08 i = = 0.02, t × = × = 20 payment intervals 4 VS K ,t = A0 (1 + i /4) (1 + i /4)t×4 − i /4 (1 + 0.02)20 − 0.02 450 000 = A0 (1.02)(24.2974) 450 000 = A0 (1 + 0.02) 450 000 = A0 (24.7833) A0 = 450 000 = 18 157.3882 24.7833 Quarterly payments of £18 157.3882 are required to meet the target P RO G R E S S E X E R C I S E S Annuities, Debt Repayments, Sinking Funds A company operates a savings scheme (interest rate 4% per annum) for its employees (a) Calculate the amount saved at the end of one year when £200 is deposited at the end of each month (b) How much should be saved from a monthly salary in order to have £5000 at the end of the year? A child is given a savings account with £2000 A further £500 is deposited in the account at the end of each year Assuming an annual interest rate of 6%, calculate the value of the account (a) at the end of 10 years, (b) at the end of 20 years How much should you pay for an annuity of quarterly payments of £1500 for five years, assuming interest rates of 5% per annum? A loan of £100 000 is to be repaid in annual payments over 10 years Assuming a fixed interest rate of 10% per annum, calculate (a) the amount of each annual repayment, (b) the total interest paid, (c) the amount of interest paid in the first and second repayment A loan of £50 000 is to be repaid in equal quarterly instalments over a period of five years If interest is 7.5% per annum, calculate (a) the amount of each quarterly payment, (b) the total amount of interest paid, (c) the interest paid each quarter in the first year of repayments A loan is repaid at the rate of £200 per week for 15 years If interest is 6.65% per annum, calculate (a) the total amount repaid, (b) the size of the loan, (c) the total amount of interest paid A company must replace machinery every two years at a cost of £100 000 It is decided to set aside equal amounts at the beginning of each quarter If interest is 5.75% per annum, calculate the size of the quarterly deposits [ 248 ] CHAPTER Starting on her 35th birthday, a woman saves £200 at the end of each month with a view to retiring on her 50th birthday (a) What is the value of her savings if the interest rate is 8% per annum? (b) From the age of 50, how much should she withdraw each month if the fund is to last for the next 15 years, assuming the interest rate is 9% per annum? An education fund of £40 000 is set up for a child on his 4th birthday The fund is to mature on his 18th birthday Assuming an interest rate of 10% per annum, calculate (a) the present value of the fund assuming annual compounding, (b) the size of monthly deposits (in arrears) required to achieve the fund of £40 000, (c) the size of quarterly deposits required to achieve the fund of £40 000 10 A student registers for a four-year course She has a fund of £40 000 to cover expenses over the next four years Calculate the amount of periodic withdrawals (in arrears) if she withdraws money (a) weekly, and (b) monthly Assume the interest rate is 7.5% per annum 11 A company has the option of leasing equipment for an annual fee of £3500 over four years or buying it for £12 500 If the equipment is worthless at the end of four years, which option is preferable if the annual rate of interest is (a) 5%, and (b) 4%? 5.6 The Relationship between Interest Rates and the Price of Bonds This section analyses the relationship between interest rates, the speculative demand for money and government bonds It involves everything from simple interest, present values to NPVs A bond is a cash investment made to the government, usually in units of £1000 for an agreed number of years In return, the government pays the investor a fixed sum at the end of each year; in addition, the government repays the original value (face value) of the bond to the investor with the final payment To make bonds attractive to investors, the size of the fixed annual payments (sometimes referred to as the coupon) must be based on the prevailing rate of interest (i) at the time of purchase The fixed annual payment is calculated as follows: Annual payment = i × (price of bond) (5.31) For example, a £1000 bond is bought when the prevailing interest rate is 8%, then Annual payment = i × (price of bond) = 0.08 × 1000 = 80 The annual payment, £80, is fixed for the duration of the life of the bond On maturity, the face value of £1000 is repaid to the investor (In effect, this is an application of simple interest.) So why are bonds attractive as an investment? If interest rates not change, it certainly does not make sense to invest when the growth of the investment is based on simple interest Compound interest is much more attractive! However, interest rates change To assess the attractiveness of the future fixed payments (cash flow) as interest rates increase or decrease, we return to NPVs The future cash [ 249 ] F I N A N C I A L M AT H E M AT I C S flows are discounted to the present, hence, comparisons may be easily made; in addition, the value of the bond itself, due with the final payment, is also discounted to the present, to determine whether a capital gain or loss is made on the cash investment These calculations are carried out in Worked Example 5.21 WO R K E D E X A M P L E THE INTEREST RATE AND THE PRICE OF BONDS A five-year government bond valued at £1000 is purchased when the market rate of interest is 8% (a) Calculate the annual repayment (value of the coupon) made to the investor at the end of each year (b) Calculate the NPV of the agreed cash flow when interest rates change immediately to each of the following (the original 8% is included for comparison): 6.5%, 7%, 7.5%, 8%, 8.5%, 9%, 9.5%, 10% (c) Calculate the present value of the bond which is due to be repaid at the end of the five-year lifetime Comment on: (i) The relationship between interest rates and the attractiveness of bonds as an investment (ii) The return from bonds when the interest does not change compared with investing the same amount at interest rates compounded annually SOLUTION (a) When the interest rate is 8%, the annual payment (or coupon) is (8/100) × 1000 = 80, that is, £80 is received by the investor at the end of each year (b) Table 5.6 outlines the calculation of the NPVs for the future cash flows at the given interest rates (Excel is very useful for these calculations.) Table 5.6 The rates and the NPV for the cash flow for a £1000 bond End of year Cash flow 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 75.12 70.53 66.23 62.19 58.39 74.77 69.88 65.30 61.03 57.04 74.42 69.23 64.40 59.90 55.72 74.07 68.59 63.51 58.80 54.45 73.73 67.96 62.63 57.73 53.20 73.39 67.33 61.77 56.67 51.99 73.06 66.72 60.93 55.65 50.82 72.73 66.12 60.11 54.64 49.67 332.45 328.02 323.67 319.42 315.25 311.17 307.18 303.26 729.88 712.99 696.56 680.58 665.05 649.93 635.23 620.92 80 80 80 80 80 NPV P0 of bond (P0 of bond + NPV) 1062.34 1041.00 1020.23 1000.00 980.30 961.10 942.40 924.18 [ 250 ] CHAPTER (c) The present value of a redeemed bond due at the end of five years is calculated by equation (5.11), P0 = Pt /(1 + i)t , where Pt = £1000 i = 6.5% = 0.065 → P0 i = 7.0% = 0.07 → P0 1000 = £729.88 (1 + 0.065)5 1000 = £712.99 (1 + 0.07)5 and similarly for each of the other interest rates The results are summarised in Table 5.6 Comment (i) From Table 5.6 it is evident from a comparison of the present value of the annual payments together with the present value of the returned bond that when interest rates: r decrease, the present value of the cash flow and the present value of the bond increase; r increase, the present value of the cash flow and the present value of the bond decrease Hence the inverse relationship between the attractiveness of bonds as an investment and the interest rate (ii) If the interest rate does not change, the investment holds its value (see Table 5.6) The outcome would be exactly the same if £1000 were put on deposit at 8% interest compounded annually when £80 is withdrawn at the end of each year Comparisons for the returns from bonds and investments in which interest is compounded annually for rates above and below the purchase rate are left as an exercise for the reader in Progress Exercises 5.7, question With the above information, we are now able to explain in more detail the relationship between the interest rate and the speculative demand for money which was introduced in Chapter If the prevailing interest rate is higher than the accepted normal rate for an economy, one expects the rate to fall towards normal in the future Therefore, at present, investors would buy bonds hoping to make a gain (from the fixed cash flow and a capital price gain on the present value of the bond) in the future Therefore, speculative balances are low as investors put their money into bonds If the prevailing interest rate is lower than the accepted normal rate for an economy, one expects the rate to rise towards normal in the future Therefore, at present, investors would not buy bonds since they are likely to incur a loss (from the fixed cash flow and a capital price loss on the present value of the bond) Therefore, speculative balances are high as investors not put their money into bonds [ 251 ] F I N A N C I A L M AT H E M AT I C S P RO G R E S S E X E R C I S E S Bonds and Interest Rates A five-year government bond valued at £5000 is purchased when the market rate of interest is 20% (a) Calculate the annual repayment (value of the coupon) made to the investor at the end of each year (b) Calculate the NPV of the agreed cash flow when interest rates change immediately to each of the following (the original 20% is included for comparison): 5%, 10%, 15%, 20%, 25%, 30% (c) Calculate the present value of the bond which is due to be repaid at the end of the five-year lifetime when interest rates are 5%, 10%, 15%, 20%, 25% or 30% Comment on the relationship between interest rates and the attractiveness of bonds as an investment A 10-year government bond valued at £10 000 was purchased when the market rate of interest was 9% (a) Calculate the annual repayment (value of the coupon) made to the investor at the end of each year (b) Calculate the NPV of the agreed cash flow when interest rates change immediately to each of the following (include the original 9% for comparison): 6.5%, 9%, 12% (c) After a period of five years, the bond is sold to another investor for £10 000 Determine whether the purchase is profitable for the remaining five years, given interest rates of (i) 6.5%, (ii) 12% A bond is purchased for £5000 when the interest rate is 8% (a) Calculate the fixed annual payments (b) Calculate the capital gain if the interest rate immediately (i) falls to 6.25%, (ii) increases to 10% A £5000 bond is purchased for a five-year period when interest rates are 20% At the same time £5000 is put on deposit at a bank and £1000 withdrawn from the account at the end of each year Compare the return from the investment in the bond with that from the bank if the bank offers: (a) A 20% rate of interest compounded annually (b) A 5% rate of interest compounded annually (c) A 30% rate of interest compounded annually 5.7 Excel for Financial Mathematics In this chapter, Excel is particularly useful for calculating tables of values, such as in Worked Example 5.14, where several NPVs were calculated at different discount rates Worked Example 5.22 compares the growth of an investment for the same nominal rate of interest, but using different methods of compounding [ 252 ] CHAPTER WO R K E D E X A M P L E 2 GROWTH OF AN INVESTMENT USING DIFFERENT METHODS OF COMPOUNDING (EXCEL) £1 is invested at a nominal rate of interest of 50% (a) Set up a table comparing the growth of the investment over a 20-year period if interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) continuously (b) Graph the growth of the investment by all four methods given in (a) on the same diagram for the last five years SOLUTION (a) Set up a table in Excel similar to Table 5.7 The formula for each method of compounding is entered into cells B4, C4, D4 and E4 respectively as follows: Table 5.7 Different methods of compounding A Year B Annually 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 1.00 1.50 2.25 3.38 5.06 7.59 11.39 17.09 25.63 38.44 57.67 86.50 129.75 194.62 291.93 437.89 656.84 985.26 477.89 216.84 325.26 C D Method of compounding Quarterly Monthly 1.00 1.60 2.57 4.11 6.58 10.55 16.89 27.06 43.34 69.42 111.20 178.12 285.31 457.02 732.05 172.60 878.28 008.65 819.27 719.55 12 365.22 (i) Annual compounding: Pt = P0 (1 + i)t for t = to 20 1.00 1.63 2.66 4.35 7.10 11.58 18.90 30.85 50.35 82.17 134.11 218.88 357.23 583.03 951.56 553.03 534.70 136.86 751.75 11 019.50 17 984.87 E Continuous 1.00 1.65 2.72 4.48 7.39 12.18 20.09 33.12 54.60 90.02 148.41 244.69 403.43 665.14 096.63 808.04 980.96 914.77 103.08 13 359.73 22 026.47 [ 253 ] F I N A N C I A L M AT H E M AT I C S With the cursor (or pointer) in cell B4, type = B$ 3∗ 1.5ˆA4 This formula states: principal × 1.5 raised to the power of time, when rate = 0.5 The $ sign in front of the number is required since we want to reference cell B3 (the same principal), when this formula is copied down column B If the $ sign is not used, then as the formula is copied down, B3 would become B4, B4 becomes B5, etc (Of course, with a £1 deposit, it is not necessary to reference any cell, but once the table is set up, the table will recalculate itself if the initial value of the investment is changed Try it!) (ii) Quarterly compounding: Pt = P0 + i 4t for t = to 20 With the cursor in cell C4, type = C$ 3∗ 1.125ˆ(A4∗ 4), then copy this formula down column C This formula states: principal × (1 + rate/4) raised to the power of (time × 4), when rate = 0.5 (iii) Monthly compounding: Pt = P0 + i 12 12t for t = to 20 With the cursor in cell D4, type = D$ 3∗ 1.041667ˆ(A4∗ 12), then copy this formula down column D This formula states: principal × (1 + rate/12) raised to the power of (time × 12), when rate = 0.5 (iv) Continuous compounding: Pt = P0 ei t With the cursor in cell E4, type = E$ 3∗ exp(0.5∗ A4), then copy this formula down column E This formula states: principal × e raised to the power of (time × rate), when rate = 0.5 The results of the calculations are given in Table 5.7 (b) Since we require the graph for the last five years only, copy the row of column titles to a new area on the sheet Immediately below the row of titles copy the last five rows of Table 5.7 The new table is given as Table 5.8 Now go through the usual steps of plotting the graph The final graph should look like Figure 5.2 From the graphs, the continuous method of compounding leads to the greatest cumulative total after 20 years, while the annual method gives the smallest sum Furthermore, the longer the investment is left on deposit, the wider the differences in its value when compounded by methods (i), (ii), (iii) and (iv) Table 5.8 Data for plotting Figure 5.2 Year Annually Quarterly Monthly Continuous 16.00 17.00 18.00 19.00 20.00 656.84 985.26 477.89 216.84 325.26 878.28 008.65 819.27 719.55 12 365.22 534.70 136.86 751.75 11 019.50 17 984.87 980.96 914.77 103.08 13 359.73 22 026.47 [ 254 ] CHAPTER Figure 5.2 Growth of £1 at i = 50% using different compounding methods 5.8 Summary Series Characteristics of arithmetic sequences a is the first term of the sequence d is the common difference nth term, Tn , is given by [a + (n − 1)d] Sn = Characteristics of geometric sequences a is the first term of the sequence r is the common ratio nth term, Tn , is given by arn −1 n [2a + (n − 1)d ] Sn = a (1 − r n ) 1−r Financial mathematics r Amount due after t years (future value) – bringing forward a single payment Simple interest: Pt = P0 (1 + i t) Compound interest (annual): Pt = P0 (1 + i )t mt Compound m times annually: Pt = P0 + mi Continuous compounding: Pt = P0 ei t r Present value – of a single payment due in t years from now P = Pt (1 + i t)−1 Simple discounting: P0 = (1 + i t) Pt Compound discounting: P0 = = Pt (1 + i )−t (1 + i )t Continuous discounting: P0 = Pt e−i t r Annual percentage rate (a) When the nominal rate is compounded m times per year, APR = + i m m (b) When the nominal rate is compounded continuously, APR = e i − −1 [ 255 ] F I N A N C I A L M AT H E M AT I C S r Depreciation Reducing-balance depreciation, At = A0 (1 − i)t Straight-line depreciation A0 = Present value: r Net present value and IRR At (1 − i )t NPV: present value of a future cash flow, discounted at a given discount rate r IRR: the discount rate for which NPV = The IRR may be estimated graphically or by the formula IRR = (i × NPV2 ) − (i × NPV1 ) NPV2 − NPV1 r Annuities, debt repayments, sinking funds In the text, the basic time interval was assumed to be one year Calculations for other time intervals are adjusted accordingly The following assume t time periods, interest compounded at i% per time interval The value Vt for an initial investment of P0 and t periodic investments of A0 is Vt = P0 (1 + i )t + A0 (1 + i )t − i The value of an annuity is VANU,t = A0 (1 + i )t − i The present value of an annuity is V0 = A0 − (1 + i )−t i annuity factor The amount of periodic repayments on a loan L is A0 = Li 1− (1+i )t =L× i − (1 + i )−t capital recovery factor r The value of a sinking fund, payments A0 made at the start of each year, is VS K ,t = A0 (1 + i ) (1 + i )t − i Excel Excel is particularly useful for calculating tables of values where the calculations are based on formulae for graphing cash flows Besides that, Excel has a variety of financial functions www.wiley.com/college/bradley Go to the website for Problems in Context [ 256 ] CHAPTER Appendix r Formula (5.2): Sum of terms of an AP Sn = a + (a + d ) + (a + 2d ) + · · · + [a + (n − 2)d ] + [a + (n − 1)d ] Sn = a + (a + d ) + (a + 2d ) + · · · + (a + nd − 2d ) + (a + nd − d ) Sn = (a + nd − d ) + (a + nd − 2d ) + · · · + (a + 2d ) + (a + d ) + a 2Sn = a + (2a + nd ) + (2a + nd ) + · · · + (2a + nd ) + (2a + nd ) + a There are (n) 2a-terms → 2a × n = 2an There are (n − 1) nd-terms → nd × (n − 1) = nd(n − 1); therefore, 2Sn = 2a n + nd (n − 1) 2Sn = n[2a + (n − 1)d ] n Sn = [2a + (n − 1)d ] r Formula (5.4): Sum of terms of a GP Sn = a + ar + ar + · · · + ar n−2 + ar n−1 r Sn = ar + ar + · · · + ar n−2 + ar n−1 + ar n Sn − r Sn = a + + + · · · + + − ar n Sn (1 − r ) = a (1 − r n ) Sn = a (1 − r n ) 1−r r Formula (5.19): Estimating the IRR Equation of line joining (x1 , y1 ) to (x2 , y2 ) is y − y1 = m(x − x1 ) y2 − y1 (x − x1 ) y − y1 = x2 − x1 This line cuts the horizontal axis at y = 0; hence, solve for x Therefore −y1 = y2 − y1 (x − x1 ) x2 − x1 −y1 (x2 − x1 ) = (y2 − y1 )(x − x1 ) −x2 y1 + x1 y1 = x(y2 − y1 ) − x1 y2 + x1 y1 x1 y2 − x2 y1 = x See Figure 5.3 y2 − y1 i NPV2 − i NPV1 = IRR NPV2 − NPV1 [ 257 ] F I N A N C I A L M AT H E M AT I C S NPV 500 (x1,y1) (r1,NPV1) 250 0 –250 i 10 (x2,y2) (r2,NPV2) Figure 5.3 Estimating the internal rate of return TEXT EXERCISES (a) Find the sum of the first seven terms of the series: (i) + + 27 + 81 + (ii) 2+1 + + (b) A food-processing company gradually closes down the production of potato chips when the strawberry season begins If the quantity of chips produced is 10 500 bags per week and reduction is at the rate of 20% each week and 200 kg of strawberries is processed in the first week of the season, increasing by 100 kg each for eight weeks, calculate: (i) The number of bags of chips and kg of strawberries produced in week of the strawberry season (ii) The number of weeks taken for the production of chips to drop to 500 bags per week (iii) The total weight of strawberries processed in eight weeks Deduce the compound interest formula If £20 000 is invested at 6.5% compound interest compounded annually calculate: (a) The value of the investment at the end of 10 years (b) The number of years that the investment should be left on deposit if it is to double in value An investor is quoted an interest rate of 7.5% Determine the APR if interest is compounded (a) six times annually, (b) 50 times annually, (c) continuously Projects A and B involve the following net cash flows: Initial cost Project A Project B 600 000 4045 Year −300 2000 Net cash flows Year Year Year 650 000 2500 −2000 3500 885 000 3000 Decide which project is the most profitable by: (a) Determining the NPV at a discount rate of 6.5% (b) The IRR Go to the website www.wiley.com/college/bradley for questions 5, and ... 282 23 15 34 = = = 352.82 10 0 10 0 10 0 (10 0) (15 34) 10 0 15 34 = = 15 34 10 0 10 0 So, 10 0% of (anything) = (anything) (ii) 10 0% of 15 34 = 12 55 240 12 (5524) 66 288 = = = 6628.8 = increase 10 0 10 10 So... Equilibrium and Break-even 3.3 Consumer and Producer Surplus 3.4 The National Income Model and the IS-LM Model 3.5 Excel for Simultaneous Linear Equations 3.6 Summary Appendix 10 1 10 2 11 1 12 8 13 3 13 7 14 2... − 10 (x + 2)2 − (x − 2)2 = 8x (x + 2)2 − x(x + 2) = 2(x + 2) 68 + + =1 105 10 11 = x − x2 − = = −x 13 x x x x x x − = 2 12 = 21 3 x 15 = x +3 x(x + 3) 5Q P +2 16 = 5Q (P + 2) 21 18 + = 5 17 11