400 CHAPTER Mathematics of Finance Solution We seek the payment R for a deferred annuity with n ϭ payment periods, deferred for k ϭ 2(10) ϭ 20 periods, i ϭ 0.086 ϭ 0.043, and A(n,k) ϭ $50,000 A(n,k) ϭ R $50,000 ϭ R Ϫ (1 ϩ i )Ϫn (1 ϩ i )Ϫk i Ϫ (1.043)Ϫ8 (1.043)Ϫ20 0.043 $50,000 Ϸ R(2.865122499) $50,000 RϷ Ϸ $17,451.26 2.865122499 ■ In Example 8, note the effect of the deferral time The family receives 8($17,451.26) ϭ $139,610.08 from the original $50,000 investment ✓ CHECKPOINT ANSWERS (a) n ϭ 14 (b) i ϭ 0.01 Ϫ (1 ϩ i )Ϫn i Ϫ (1.01)Ϫ14 (d) An ϭ $2000 Ϸ $26,007.41 0.01 R Ϸ $37,757.17 A(20,due) Ϸ $85,230.28 (a) n ϭ 10 payment periods; k ϭ 20 deferral periods (b) i ϭ 0.03 Ϫ (1 ϩ i )Ϫn (c) A(n,k) ϭ R (1 ϩ i )Ϫk i (d) A(n,k) Ϸ $23,614.83 (c) An ϭ R | EXERCISES | 6.4 ORDINARY ANNUITIES In Problems 1–4, find the requested value (to the nearest dollar), and tell what each of the other values represents Ϫ (1 ϩ 0.04)Ϫ30 Find An: An ϭ 1300 0.04 Ϫ (1 ϩ 0.01)Ϫ120 Find An: An ϭ 2550 0.01 Ϫ (1 ϩ 0.005)Ϫ360 Find R: 135,000 ϭ R 0.005 Ϫ (1 ϩ 0.02)Ϫ20 Find R: 25,000 ϭ R 0.02 Find the present value of an annuity of $6000 paid at the end of each 6-month period for years if the interest rate is 8%, compounded semiannually Find the present value of an annuity that pays $3000 at the end of each 6-month period for years if the interest rate is 6%, compounded semiannually Suppose a state lottery prize of $5 million is to be paid in 20 payments of $250,000 each at the end of each of the next 20 years If money is worth 10%, compounded annually, what is the present value of the prize? How much is needed in an account that earns 8.4% compounded monthly in order to withdraw $1000 at the end of each month for 20 years? With a present value of $135,000, what is the size of the withdrawals that can be made at the end of each quarter for the next 10 years if money is worth 6.4%, compounded quarterly? 10 If $88,000 is invested in an annuity that earns 5.8%, compounded quarterly, what payments will it provide at the end of each quarter for the next 512 years? 11 A personal account earmarked as a retirement supplement contains $242,400 Suppose $200,000 is used to establish an annuity that earns 6%, compounded quarterly, and pays $4500 at the end of each quarter How long will it be until the account balance is $0? 12 A professional athlete invested $2.5 million of a bonus in an account that earns 6.8%, compounded semiannually If $120,000 is to be withdrawn at the end of each six months, how long will it be until the account balance is $0? 13 Suppose that a 25-year government bond has a maturity value of $1000 and a coupon rate of 6%, Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it SECTION 6.4 with coupons paid semiannually Find the market price of the bond if the yield rate is 5% compounded semiannually Is this bond selling at a discount or at a premium? 14 Suppose that a 10-year corporate bond has a maturity value of $25,000 and a coupon rate of 7%, with coupons paid semiannually Find the market price of the bond if the yield rate is 8% compounded semiannually Is this bond selling at a discount or at a premium? 15 The figure shows a graph that compares the present values of two ordinary annuities of $1000 annually, one at 8% compounded annually and one at 10% compounded annually (a) Determine which graph corresponds to the 8% rate and which to the 10% rate (b) Use the graph to estimate the difference between the present values of these annuities for 25 years (c) Write a sentence that explains this difference 19 20 21 22 $ Dollars 23 24 time Present Values of Annuities 401 (a) An annuity due that pays $1000 at the beginning of each year for 10 years (b) Taking $1000 now and establishing an ordinary annuity that pays $1000 at the end of each year for years Find the present value of an annuity due that pays $3000 at the beginning of each quarter for the next 7 years Assume that money is worth 5.8%, compounded quarterly Find the present value of an annuity due that pays $25,000 every months for the next 212 years if money is worth 6.2%, compounded semiannually What amount must be set aside now to generate payments of $50,000 at the beginning of each year for the next 12 years if money is worth 5.92%, compounded annually? Suppose an annuity will pay $15,000 at the beginning of each year for the next years How much money is needed to start this annuity if it earns 7.3%, compounded annually? A year-end bonus of $25,000 will generate how much money at the beginning of each month for the next year, if it can be invested at 6.48%, compounded monthly? A couple inherits $89,000 How much can this generate at the beginning of each month over the next 5 years, if money is worth 6.3%, compounded monthly? Years 16 The figure shows a graph that compares the present values of two ordinary annuities of $800 quarterly, one at 6% compounded quarterly and one at 9% compounded quarterly (a) Determine which graph corresponds to the 6% rate and which to the 9% rate (b) Use the graph to estimate the difference between the present values of these annuities for 25 years (100 quarters) (c) Write a sentence that explains this difference Dollars $ time Quarters ANNUITIES DUE 17 Explain the difference between an ordinary annuity and an annuity due 18 Is there any difference between the present values in parts (a) and (b)? Explain MISCELLANEOUS PROBLEMS FOR ORDINARY ANNUITIES AND ANNUITIES DUE In Problems 25–40, (a) decide whether the problem relates to an ordinary annuity or an annuity due, and then (b) solve the problem 25 An insurance settlement of $1.5 million must replace Trixie Eden’s income for the next 40 years What income will this settlement provide at the end of each month if it is invested in an annuity that earns 8.4%, compounded monthly? 26 A local library receives a bequest of $100,000 from a prominent local family How much will this provide at the beginning of each 3-month period for the next 221 years if money is worth 7.4% compounded quarterly? 27 A company wants to have $40,000 at the beginning of each 6-month period for the next 421 years If an annuity is set up for this purpose, how much must be invested now if the annuity earns 6.68%, compounded semiannually? 28 Is it more economical to buy an automobile for $29,000 cash or to pay $8000 down and $3000 at the end of each quarter for years, if money is worth 8% compounded quarterly? 29 Dr Jane Kodiak plans to sell her practice to an HMO The HMO will pay her $1.5 million now or will make a $500,000 partial payment now and make additional payments of $140,000 at the end of each year for the next 10 years If money is worth 6.5%, compounded Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... at the beginning of each 3-month period for the next 221 years if money is worth 7.4% compounded quarterly? 27 A company wants to have $40,000 at the beginning of each 6-month period for the next... years How much money is needed to start this annuity if it earns 7.3%, compounded annually? A year-end bonus of $25,000 will generate how much money at the beginning of each month for the next year,... compounded semiannually Is this bond selling at a discount or at a premium? 14 Suppose that a 10-year corporate bond has a maturity value of $25,000 and a coupon rate of 7%, with coupons paid