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Bonds and Bond Pricing 191Figure 6.1:Cash-flow pattern of a coupon-paying bond with n couponsBond pricePTime0 1 2 n − 1 n······Cash flowFr Fr Fr Fr+ CWe shall assume that the term structure is flat, so that cash flows at all times arediscounted at the same yield rate i. Thus, the fair price P of the bond is given bythe following basic price formulaP =(Fr)a n⌉ + Cv n , (6.1)where the interest and annuity functions are calculated at the yield rate i.Example 6.1: The following shows the results of a government bond auction:Type of Bond Government bondIssue Date February 15, 2010Maturity Date February 15, 2040Coupon Rate 4.500% payable semiannuallyYield Rate 4.530% convertible semiannuallyAssume that the redemption value of the bond is the same as the face value, whichis $100. Find the price of the bond.Solution: This is a 30-year bond and we can use the basic price formula (6.1)withF = 100,C = 100,r = 0.045 =0.0225,2i = 0.0453 =0.02265,2n = 60,
192 CHAPTER 6to calculate the price of the bond, which isP = (Fr)a n⌉+ Cv n= 2.25a 60⌉ 0.02265 + 100(1.02265)−60= $99.51.Example 6.2: The following shows the information of a government bond tradedin the secondary market:Type of Bond Government bondIssue Date June 15, 2005Date of Purchase June 15, 2009Maturity Date June 15, 2020Coupon Rate 4.2% payable semiannuallyYield Rate4.0% convertible semiannuallyAssume that the redemption value of the bond is the same as the face value, whichis $100. Find the purchase price of the bond immediately after its 8th couponpayment on June 15, 2009.Solution: This is a 15-year bond at issue but there are only 22 remaining couponpaymentperiods after its 8th coupon payment on June 15, 2009. We use the basicprice formula (6.1) withto obtainF = 100,C = 100,r = 0.042 =0.021,2i = 0.040 =0.020,2n = 22,P = (Fr)a n⌉+ Cv n= 2.1a 22⌉ 0.02 + 100(1.02)−22= $101.77.
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Financial Mathematics forActuaries
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Financial Mathematics forActuaries
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To Bonnie, Nikki and Kevinfor their
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About the AuthorsWai-Sum Chan, PhD,
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Preface to the Second EditionA good
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ContentsAbout the AuthorsPreface to
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ContentsxiiiChapter 7 Bond Yields a
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List of Mathematical SymbolsSymbola
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List of Mathematical SymbolsxviiSym
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1InterestAccumulation andTime Value
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Interest Accumulation and Time Valu
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Table 1.3:Summary of accumulated va
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2AnnuitiesAn annuity is a series of
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Annuities 41Figure 2.1 illustrates
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Annuities 43Solution: The rate of i
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Annuities 45As an annuity-due of n
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Annuities 47Figure 2.3: Illustratio
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Annuities 49Also, it can be checked
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Annuities 512.5 Payment Periods, Co
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Annuities 53Note that (2.20) has a
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Annuities 55Example 2.11: Solve the
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Annuities 57Figure 2.8:Increasing a
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Annuities 59Second, we may consider
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Annuities 61n is an integer and 0 <
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Annuities 63Exhibit 2.1: Use of Exc
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Annuities 652.2 Let d (2) =2%. Eval
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Annuities 672.23 Express s(52) n⌉
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Annuities 692.37 A paused rainbow i
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Annuities 71(c) Find the amount of
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3SpotRates, Forward Ratesand the Te
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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
Page 161 and 162: Rates of Return 141Assume all inves
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Page 165 and 166: Rates of Return 1454.50 Judy is con
Page 167 and 168: 5Loans andCosts of BorrowingLoans a
Page 169 and 170: Loans and Costs of Borrowing 149can
Page 171 and 172: Loans and Costs of Borrowing 151Due
Page 173 and 174: Loans and Costs of Borrowing 153Tab
Page 175 and 176: Loans and Costs of Borrowing 155Exa
Page 177 and 178: Loans and Costs of Borrowing 157We
Page 179 and 180: Loans and Costs of Borrowing 159Exa
Page 181 and 182: Loans and Costs of Borrowing 161For
Page 183 and 184: Loans and Costs of Borrowing 1635.5
Page 185 and 186: Loans and Costs of Borrowing 165the
Page 187 and 188: Loans and Costs of Borrowing 167its
Page 189 and 190: Loans and Costs of Borrowing 169Not
Page 191: Table 5.5:Summary of reference rate
Page 196 and 197: Loans and Costs of Borrowing 175rem
Page 198 and 199: Loans and Costs of Borrowing 177Yea
Page 206 and 207: Loans and Costs of Borrowing 185let
Page 208 and 209: 6Bondsand Bond PricingA bond is a c
Page 210 and 211: Bonds and Bond Pricing 189hand, bon
Page 214 and 215: Bonds and Bond Pricing 193Example 6
Page 216 and 217: Bonds and Bond Pricing 195Table 6.1
Page 218 and 219: Bonds and Bond Pricing 197Note that
Page 220 and 221: Bonds and Bond Pricing 199Figure 6.
Page 222 and 223: Bonds and Bond Pricing 201The Excel
Page 224 and 225: Bonds and Bond Pricing 203Assuming
Page 226 and 227: Bonds and Bond Pricing 205cash flow
Page 228 and 229: Bonds and Bond Pricing 2076. When t
Page 230 and 231: Bonds and Bond Pricing 209Half- Cou
Page 232 and 233: Bonds and Bond Pricing 2116.23 Show
Page 234 and 235: 7Bond Yields andthe Term StructureW
Page 236 and 237: Bond Yields and the Term Structure
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Bond Yields and the Term Structure
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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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