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Answers to Selected Exercises 335Chapter 22.1 (a) 12.5779 (b) 4.5460 (c) 30.9822 (d) 22.0185 (e) 202.2 (a) 1.9505 (b) 4.1757 (c) 10.66242.5 (a) $6,910.67 (b) $3,387.58 (c) $6,432.772.6 $4,716.352.7 14.47%2.8 $612.972.9 (a) 5.4920 (b) 6.16, No[ ]1+i (1 + i)22.10 $1,000 −i (1 + i) 4 − 12.11 $74.902.12 $7,351.402.13 $92.912.14 $9.00212.16 $808.612.17 2.081 years2.18 $50,811.182.19 $528.592.22 ¨s m⌉ at the interest rate i k ,wherei k =(1+i) k − 12.23d (12)r (52)2.24 44 months2.26 $97.812.27 (a) i(Ia) n⌉ (b) The principal is n
336 APPENDIX B2.28 $2,000(¨s (4)7⌉ − ¨s(4) 2⌉ )2.29 $1,000(¨s 12⌉ +¨s 8⌉ +¨s 4⌉ )2.302(a)t +22.31 $1,789.31(b) 1.56672.32 n −n(n − 1)d22.34 (a) a 60⌉s 4⌉(b) s 23⌉ − s 5⌉s 3⌉2.36 $21,154.842.37 (Ia) n⌉ + v n (Da) n⌉2.38 (a) $138.87 (b) $370.30[ ]n(n +1) n(n + 1)(2n +1)2.39+i262.40 $688.1232.41 $20,748.232.43 $177.59602.44 $297.26 at t =18or $315.37 at t =20or $299.05 at t =18.22.45 352.46 $57.832.47 (a) 5.7905% (b) 6.6563% (c) 4.9494% (d) 4.4778%2.48 2ln(1+t)2.50 (a) $1,361.23 (b) $10,429.65 (c) $1,553.932.51 (a) n =10, $1,369.73(b) $2,630.27 at t =11; $4,000 at t =12, ··· , 28; $44.95 at t = 29[ (y ) ]− xk [ ]2.53 x 2 1− 1xy − 2x
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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
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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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Page 314 and 315: Interest Rates and Financial Securi
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Page 348 and 349: APPENDIXAReview of Mathematics and
Page 350 and 351: Review of Mathematics and Statistic
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Page 354 and 355: APPENDIXBAnswers to Selected Exerci
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Page 360 and 361: Answers to Selected Exercises 3394.
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Page 370 and 371: IndexAaccrued interest, 187, 198-20
Page 372 and 373: Index 351RATE, 63, 70XIRR, 109-111,
Page 374: Index 353dollar-weighted, 105, 112,
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