• Document: Mathematics of Interest Rates and Finance Gary C. Guthrie Larry D. Lemon First Edition
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Mathematics of Interest Rates & Finance Guthrie Lemon 1e Mathematics of Interest Rates and Finance Gary C. Guthrie Larry D. Lemon First Edition ISBN 978-1-29203-983-1 9 781292 039831 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 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, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. ISBN 10: 1-292-03983-3 ISBN 10: 1-269-37450-8 ISBN 13: 978-1-292-03983-1 ISBN 13: 978-1-269-37450-7 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America Compound Interest Example 9.2 Find the percentage rate of change for two funds that are both growing at an instantaneous rate of change of $5000 per month when one fund contains $50,000 and the other contains $250,000.  The fund containing $50,000 that is growing at a rate of $5000 per month has a percentage rate of change of 10% per dollar invested. The fund containing $250,000 that is growing at a rate of $5000 per month has a percentage rate of change of only 2% per dollar invested. Using the force of interest gave a much more perceptive view of the change in each fund. Simple Interest Rates of Change We can now apply each of the first three rate of change concepts (16) to (19) to simple interest.  The average rate of change for simple interest: a (t ) − a(t − 1) (1 + it ) − (1 + i (t − 1)) Start with . Substitute to give = i. t − (t − 1) t − (t − 1) The numerator is simply the interest earned by $1 at simple interest for 1 year, while the denominator represents an interval of 1 year and t ≥ 1 . d  The instantaneous rate of change for simple interest: a' (t ) = [1 + it ] = i dt The average rate of change and the instantaneous rate of change (slope) are the same because the simple interest accumulation function is a straight line with slope i.  The percentage change for simple interest expressed as a decimal: a (t ) − a (t − 1) (1 + it ) − (1 + i (t − 1)) i it = = = (This gives i for t = 1.) a (t − 1) 1 + i (t − 1) 1 + i (t − 1) Remember that this percentage change expressed as a decimal is also the effective rate since it expresses the ratio of the interest earned during a unit interval to the principal at the beginning of that interval. Note that as t increases the effective rate is decreasing asymptotically to zero. In a practical sense, this is not a serious concern since simple interest is seldom used for times beyond two or three years. Example 9.3 Find the effective rates of 6% per annum simple interest for 1, 2, and 3 years. .06  i1 = = .06 = 6% 1 + (.06)(0) 173 Compound Interest .06 i2 = = .0566 = 5.66% 1 + (.06)(1) .06 i3 = = .05357 = 5.357% 1 + (.06)(2) We migh

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