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2AnnuitiesAn annuity is a series of payments made at equal intervals. There aremany practical examples of financial transactions involving annuities,such as• a car loan being repaid with equal monthly installments• a retiree purchasing an annuity from an insurance companyupon his retirement• a life insurance policy being purchased with monthly premiums• a bondholder receiving an annuity in the form of semiannualcoupon paymentsIn this chapter we consider annuities of level or varying amounts, thepayments of which are certain. Thus, we assume there is no defaultrisk in the case of a bond or an annuity purchased from an insurancecompany. We shall discuss the calculation of the present and futurevalues of these annuities. When there is uncertainty in the annuitypayments, as in the case of the default of a car loan, the paymentsare contingent upon some random events. Such annuities will not bediscussed in this book.
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Financial Mathematics forActuaries
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Financial Mathematics forActuaries
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To Bonnie, Nikki and Kevinfor their
Page 8 and 9: About the AuthorsWai-Sum Chan, PhD,
Page 10 and 11: Preface to the Second EditionA good
Page 12 and 13: ContentsAbout the AuthorsPreface to
Page 14 and 15: ContentsxiiiChapter 7 Bond Yields a
Page 16 and 17: List of Mathematical SymbolsSymbola
Page 18 and 19: List of Mathematical SymbolsxviiSym
Page 20 and 21: 1InterestAccumulation andTime Value
Page 22 and 23: Interest Accumulation and Time Valu
Page 48: Table 1.3:Summary of accumulated va
Page 57: Interest Accumulation and Time Valu
Page 61 and 62: Annuities 41Figure 2.1 illustrates
Page 63 and 64: Annuities 43Solution: The rate of i
Page 65 and 66: Annuities 45As an annuity-due of n
Page 67 and 68: Annuities 47Figure 2.3: Illustratio
Page 69 and 70: Annuities 49Also, it can be checked
Page 71 and 72: Annuities 512.5 Payment Periods, Co
Page 73 and 74: Annuities 53Note that (2.20) has a
Page 75 and 76: Annuities 55Example 2.11: Solve the
Page 77 and 78: Annuities 57Figure 2.8:Increasing a
Page 79 and 80: Annuities 59Second, we may consider
Page 81 and 82: Annuities 61n is an integer and 0 <
Page 83 and 84: Annuities 63Exhibit 2.1: Use of Exc
Page 85 and 86: Annuities 652.2 Let d (2) =2%. Eval
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Page 89 and 90: Annuities 692.37 A paused rainbow i
Page 91 and 92: Annuities 71(c) Find the amount of
Page 93 and 94: 3SpotRates, Forward Ratesand the Te
Page 95 and 96: Spot Rates, Forward Rates and the T
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Spot Rates, Forward Rates and the T
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4Rates of ReturnThe performance of
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Rates of Return 107Example 4.1: A p
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Rates of Return 109where y 4 is the
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Rates of Return 111Exhibit 4.3:Solu
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Rates of Return 113Figure 4.1: Cash
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Rates of Return 115Hence, from (4.5
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Rates of Return 117Calculate the ar
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Rates of Return 119denote it as R D
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Rates of Return 1214.4 Portfolio Re
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Rates of Return 123Solution: For (a
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Rates of Return 125To calculate the
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Rates of Return 127Finally, for (d)
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Rates of Return 129Figure 4.4: NPV
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Rates of Return 131When the project
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Rates of Return 133measure of the r
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Rates of Return 1354.7 Which of the
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Rates of Return 1374.19 A 7-year pr
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Rates of Return 139(a) Calculate th
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Rates of Return 141Assume all inves
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Rates of Return 143(a) Draw a graph
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Rates of Return 1454.50 Judy is con
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5Loans andCosts of BorrowingLoans a
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Loans and Costs of Borrowing 149can
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Loans and Costs of Borrowing 151Due
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Loans and Costs of Borrowing 153Tab
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Loans and Costs of Borrowing 155Exa
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Loans and Costs of Borrowing 157We
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Loans and Costs of Borrowing 159Exa
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Loans and Costs of Borrowing 161For
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Loans and Costs of Borrowing 1635.5
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Loans and Costs of Borrowing 165the
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Loans and Costs of Borrowing 167its
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Loans and Costs of Borrowing 169Not
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Table 5.5:Summary of reference rate
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Loans and Costs of Borrowing 175rem
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Loans and Costs of Borrowing 177Yea
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Loans and Costs of Borrowing 185let
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6Bondsand Bond PricingA bond is a c
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Bonds and Bond Pricing 189hand, bon
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Bonds and Bond Pricing 191Figure 6.
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Bonds and Bond Pricing 193Example 6
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Bonds and Bond Pricing 195Table 6.1
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Bonds and Bond Pricing 197Note that
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Bonds and Bond Pricing 199Figure 6.
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Bonds and Bond Pricing 201The Excel
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Bonds and Bond Pricing 203Assuming
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Bonds and Bond Pricing 205cash flow
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Bonds and Bond Pricing 2076. When t
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Bonds and Bond Pricing 209Half- Cou
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Bonds and Bond Pricing 2116.23 Show
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7Bond Yields andthe Term StructureW
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Bond Yields and the Term Structure
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8Bond ManagementThe rate of interes
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Bond Management 247Using PV(C t ) a
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Bond Management 249again is less th
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Bond Management 251To use the modif
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Bond Management 253Example 8.5: A 1
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Bond Management 255Figure 8.2:Bond-
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Bond Management 257Rule 3: Other th
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Bond Management 259the duration D P
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Bond Management 261Example 8.8: A c
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Bond Management 263Figure 8.4:Illus
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Bond Management 265Example 8.10: A
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Bond Management 267of the asset and
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Bond Management 269Figure 8.5:Illus
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Bond Management 271Solving the abov
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Bond Management 273First, in classi
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Bond Management 275wherePV(C t )=C
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Bond Management 277Example 8.14: Co
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Bond Management 2795. Immunization
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Bond Management 281Time toCoupon ra
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Bond Management 2838.15 Using Exerc
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Bond Management 2858.22 Revisit Exe
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Bond Management 2878.29 Suppose you
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Bond Management 289(c) Calculate th
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9Interest Rates andFinancial Securi
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Interest Rates and Financial Securi
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10Stochastic Interest RatesIn previ
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Stochastic Interest Rates 307Table
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Stochastic Interest Rates 309ands 5
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Stochastic Interest Rates 311Next,
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Stochastic Interest Rates 313Theref
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Stochastic Interest Rates 315which
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Stochastic Interest Rates 317There
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Stochastic Interest Rates 319We use
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Stochastic Interest Rates 32110.3 W
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Stochastic Interest Rates 32310.13
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Stochastic Interest Rates 32510.24
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APPENDIXAReview of Mathematics and
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Review of Mathematics and Statistic
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Review of Mathematics and Statistic
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APPENDIXBAnswers to Selected Exerci
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Answers to Selected Exercises 335Ch
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Answers to Selected Exercises 337Ch
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Answers to Selected Exercises 3394.
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Answers to Selected Exercises 34710
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IndexAaccrued interest, 187, 198-20
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Index 351RATE, 63, 70XIRR, 109-111,
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Index 353dollar-weighted, 105, 112,
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