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

By Arjun K. Gupta, Tamas Varga

to Actuarial arithmetic via A. ok. Gupta Bowling eco-friendly kingdom collage, Bowling eco-friendly, Ohio, U. S. A. and T. Varga nationwide Pension coverage Fund. Budapest, Hungary SPRINGER-SCIENCE+BUSINESS MEDIA, B. V. A C. I. P. Catalogue checklist for this publication is accessible 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 revealed on acid-free paper All Rights Reserved © 2002 Springer Science+Business Media Dordrecht initially released via Kluwer educational Publishers in 2002 No a part of the cloth safe by way of this copyright become aware of should be reproduced or used in any shape or whatsoever, digital or mechanical, together with photocopying, recording or through any details garage and retrieval method, with no 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 services of Mortality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eighty four 2. three. Mortality Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety eight bankruptcy three. existence INSURANCES AND ANNUITIES . . . . . . . . . . . . . . . . . . . . . 112 three. 1. Stochastic money Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 three. 2. natural Endowments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a hundred thirty three. three. existence Insurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 three. four. Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 three. five. lifestyles Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 bankruptcy four. charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 four. 1. internet charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 four. 2. Gross rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Vll bankruptcy five. RESERVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 five. 1. web top rate Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 five. 2. Mortality revenue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 five. three. transformed Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 solutions TO ODD-NuMBERED difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Since this is the last payment, the outstanding capital should be zero. 27 = O. ) We can summarize the payments in the following payment schedule. 21. ) If we are only interested in the capital repayment and the interest content of one payment we do not need to fill in a whole table. First we need to find the outstanding capital just after the previous payment. This can be obtained either as the accumulated value of the past cash flow or as the present value of the future cash flow. The interest on this outstanding capital will be the interest content of the next payment we are focusing on, and the difference between the actual payment and the interest payment is the capital repayment.

An amount of $300 is deposited at a bank on January 1 of each year from 1981 to 1989. What is the accumulation on December 31, 1989? Use a 3% annual rate of interest. Solution: The term of this annuity-due is 9 years (1989 - 1981 + 1). 03, I om . 4639. 17. Next, assume an annuity-due is purchased whose payments start in year m + 1 and continue until year m + n. So there are no payments made in the first m years. This is called a deferred annuity-due. 2, we obtain (6) and hence I" m a n1 vm - v m+n d (7) There is also another way of evaluating a deferred annuity.

A) Based on a compound interest satisfying (2), what is the accumulated value on June 1, 1989? If the accumulation is withdrawn and redeposited immediately afterwards, how large will the account grow by January 1, 1990? b) Answer the same questions as in part (a), but this time use a simple interest. 5. A sum of $3000 is deposited on March 1, 1988. 5% in 1992 and 1993, and the a compound interest satisfies (2) in each calendar year, determine the accumulation on March 1, 1993. In the remaining problems of this section, we assume a compound interest rate satisfying (2).

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