By Arjun K. Gupta, Tamas Varga

to Actuarial arithmetic via A. okay. Gupta Bowling eco-friendly kingdom collage, 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 list for this ebook 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 through Kluwer educational Publishers in 2002 No a part of the fabric safe by way of this copyright observe might be reproduced or used in any shape or in any respect, digital or mechanical, together with photocopying, recording or by means of any details 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 worth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. existence INSURANCES AND ANNUITIES . . . . . . . . . . . . . . . . . . . . . 112 three. 1. Stochastic funds Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 three. 2. natural Endowments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one hundred thirty three. three. lifestyles Insurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 three. four. Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 three. five. existence 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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An Introduction to Actuarial Mathematics

To Actuarial arithmetic through 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 ebook is on the market from the Library of Congress. ISBN 978-90-481-5949-9 ISBN 978-94-017-0711-4 (eBook) DOI 10.

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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. b) Find the total amount of the interest payment.

17) It is also useful to keep in mind that since d = 1- v we get d 1 . V V ' -=--1=1 that is (18) Using the results for the annuity-due and applying (16), (17), and (18) we get 1- v n a nl =-i-' (19) 1 = i a nl + v n . (20) and The values of a nl are tabulated in Appendix 1 for selected values of i and n. CHAPTER 1 56 Formula (20) can be obtained by general reasoning, which is very similar to the explanation given to (3). The only difference is that the interest payments are made at the end of each year.

There are also annuities-due whose payments form a geometric sequence. That is, the payment is A in the first year, bA in the second year, ... , bn-1A in year n, where b is positive. Let us define j as 54 CHAPTER 1 (13) j=b-l. The number j can be negative but j? -1 is always true. The expression 100j gives the percentage change in annual payments. Now, the present value of this annuity-due is Note that this is equivalent to valuing a level annuity-due of A per annum at an annual rate of interest i*, where 1 1 + i* = v* = v .

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