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9Interest Rates andFinancial SecuritiesIn this chapter we outline the determination of rates of interest froman economic view point. We discuss the yields on various financialsecurities, including Treasury bonds. We examine the role of the governmentin setting interest rate targets and policies. Our discussionwill focus mainly on the U.S. economy, which is the largest in theworld and whose policies have major impact on other economies.
292 CHAPTER 9Learning Objectives• Determination of the level of interest rates• Yields and rates of interest of various financial securities• The role of the Central Bank• The Federal Reserve System of the U.S.• Macroeconomic management and interest rates• Interest rates in an open economy9.1 Interest Rate DeterminationGiven an accumulation function, the present value of a single payment or stream ofpayments can be computed using (2.17). When the term structure is flat, Sections2.1 through 2.3 provide formulas for the computation of the present values of variousannuities. The calculation of the future value of an annuity, however, requiresassumption about how future payments accumulate in value. Under a fixed interestrate environment with a flat term structure this problem is easy to resolve, as allpayments earn the same effective rate of interest i over any period. In practice, interestrates vary over time and we must be careful with our assumption about howfuture payments accumulate in value. In Section 1.7 we consider the assumptionof future payments earning the current spot rates (i.e., the current term structure remainsunchanged over time), whereas in Section 3.3 we introduce the assumptionthat future payments earn the (implicit) forward rates of interest. It must be emphasizedthat these assumptions are made ex ante. As interest rates are in practice timevarying, both assumptions will generally be wrong ex post. It is thus important tounderstand how interest rates are determined in practice.Depending on the type of financial security, the rate of interest (yield) earnedmay be explicit (e.g., a savings account) or imputed from the security prices (e.g.,Treasury bonds). In Chapter 7, we describe methods to determine the spot ratesof interest (and hence the term structure) from Treasury securities. Note that theterm structure is estimated from the market prices of Treasury securities and is notdirectly observable. As the prices of Treasury securities are dependent on theirsupply and demand, the Central Bank can influence their prices by varying theirsupply. To the extent that interest rates affect the demand of capital investment and
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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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Page 262 and 263: Bond Yields and the Term Structure
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Page 266 and 267: 8Bond ManagementThe rate of interes
Page 268 and 269: Bond Management 247Using PV(C t ) a
Page 270 and 271: Bond Management 249again is less th
Page 272 and 273: Bond Management 251To use the modif
Page 274 and 275: Bond Management 253Example 8.5: A 1
Page 276 and 277: Bond Management 255Figure 8.2:Bond-
Page 278 and 279: Bond Management 257Rule 3: Other th
Page 280 and 281: Bond Management 259the duration D P
Page 282 and 283: Bond Management 261Example 8.8: A c
Page 284 and 285: Bond Management 263Figure 8.4:Illus
Page 286 and 287: Bond Management 265Example 8.10: A
Page 288 and 289: Bond Management 267of the asset and
Page 290 and 291: Bond Management 269Figure 8.5:Illus
Page 292 and 293: Bond Management 271Solving the abov
Page 294 and 295: Bond Management 273First, in classi
Page 296 and 297: Bond Management 275wherePV(C t )=C
Page 298 and 299: Bond Management 277Example 8.14: Co
Page 300 and 301: Bond Management 2795. Immunization
Page 302 and 303: Bond Management 281Time toCoupon ra
Page 304 and 305: Bond Management 2838.15 Using Exerc
Page 306 and 307: Bond Management 2858.22 Revisit Exe
Page 308 and 309: Bond Management 2878.29 Suppose you
Page 310 and 311: Bond Management 289(c) Calculate th
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Page 326 and 327: 10Stochastic Interest RatesIn previ
Page 328 and 329: Stochastic Interest Rates 307Table
Page 330 and 331: Stochastic Interest Rates 309ands 5
Page 332 and 333: Stochastic Interest Rates 311Next,
Page 334 and 335: Stochastic Interest Rates 313Theref
Page 336 and 337: Stochastic Interest Rates 315which
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Page 342 and 343: Stochastic Interest Rates 32110.3 W
Page 344 and 345: Stochastic Interest Rates 32310.13
Page 346 and 347: Stochastic Interest Rates 32510.24
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Page 350 and 351: Review of Mathematics and Statistic
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Page 354 and 355: APPENDIXBAnswers to Selected Exerci
Page 356 and 357: Answers to Selected Exercises 335Ch
Page 358 and 359: Answers to Selected Exercises 337Ch
Page 360 and 361: Answers to Selected Exercises 3394.
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Answers to Selected Exercises 3415.
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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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