Download An Introduction to Actuarial Mathematics by Arjun K. Gupta, Tamas Varga PDF

By Arjun K. Gupta, Tamas Varga

to Actuarial arithmetic via A. okay. Gupta Bowling eco-friendly kingdom college, Bowling eco-friendly, Ohio, U. S. A. and T. Varga nationwide Pension assurance Fund. Budapest, Hungary SPRINGER-SCIENCE+BUSINESS MEDIA, B. V. A C. I. P. Catalogue checklist for this booklet is out there from the Library of Congress. ISBN 978-90-481-5949-9 ISBN 978-94-017-0711-4 (eBook) DOI 10. 1007/978-94-017-0711-4 published on acid-free paper All Rights Reserved © 2002 Springer Science+Business Media Dordrecht initially released by means of Kluwer educational Publishers in 2002 No a part of the cloth secure by way of this copyright discover can be reproduced or used in any shape or in any way, digital or mechanical, together with photocopying, recording or through any info garage and retrieval procedure, with out written permission from the copyright proprietor. To Alka, Mita, and Nisha AKG To Terezia and Julianna television desk OF CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix bankruptcy 1. monetary arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1. Compound curiosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 2. current price. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1. three. Annuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . forty eight bankruptcy 2. MORTALITy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eighty 2. 1 Survival Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eighty 2. 2. Actuarial features of Mortality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eighty four 2. three. Mortality Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety eight bankruptcy three. lifestyles INSURANCES AND ANNUITIES . . . . . . . . . . . . . . . . . . . . . 112 three. 1. Stochastic money Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 three. 2. natural Endowments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one hundred thirty three. three. existence Insurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 three. four. Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 three. five. existence Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 bankruptcy four. rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 four. 1. web charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 four. 2. Gross charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Vll bankruptcy five. RESERVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 five. 1. web top class Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 five. 2. Mortality revenue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 five. three. transformed Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 solutions TO ODD-NuMBERED difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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These names are used traditionally, although they do not seem to be very logical. The first payment of an "annuity-immediate" is not made immediately at the beginning of the first payment period, rather, it is due at the end of it. An annuity whose payments are equal is called a level annuity. We will study annuities whose payments are $1, since any other level annuity can be obtained from this by a simple multiplication. First we examine annuities that make payments once a year. They are called yearly annuities.

To to v , e (19) If (12) is satisfied, (19) can also be expressed as A Vt Furthermore, if p(t) =r e = fee 8{te-t) to p(t)dt. (20) then A Vte =r (1 + i) t -to 8 e - 1 (21) 42 CHAPTER 1 If a cash flow contains both discrete and continuous elements, its accumulated value is the sum of the accumulated values of the discrete and the continuous parts. If we take t~ > t e, then (22) Moreover, we can find a simple relationship between present values and accumulated values. Since we are working with compound interest, (1) can be expressed as Then, using (18) we get Similarly, for continuous cash flows, (10) and (19) imply that Thus, (23) is always true.

Determine the accumulation of $2500 if it is invested for a) one day. b) one week. c) one year. 12. A sum of $2000 is invested at a 7% annual rate of interest for four years. a) How much interest is paid at the end of year four? b) Determine the interest payments if they are made at the end of each year. c) If the interest is paid monthly, find its monthly amount. 13. The interest on a $500 deposit is paid continuously for 3 years. Assume the annual rate of interest is 7%. a) Determine the annual rate of the interest payment.

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