Fixed-Income Securities

IN 1 ItlSCIt.Mtt.Rand the next we turn our attention to the bond markets. West udy bonds thai have no call provisions or default risk, so that their payments are fully specified in advance. Such bonds deserve the name: fixed-income securities that is often used more loosely to describe bonds whose future payments are in fact uncertain, hi the US markets, almost all true lixcd-incotue securities are issued by the US Treasury. Conventional Treasury securities make fixed payments in nominal lerins, but in early 1990 the Treasury announced plans lo issue indexed bonds whose nominal payments are indexed lo nidation so thai their payments are fixed in real terms.1

Many of the ideas discussed in earlier chapters can be applied lo fixed-income securities as well aslo any other asset. But there are several reasons to devote special attention lo fixed-income securities, first, the fixed-income markets have developed separately from die equity markets. They have their own institutional structure and their own terminology. Likewise the academic study of fixed-income securities has its own traditions. Second, the markets for Treasury securities are extremely large regardless of whether size is measured by quantities outstanding or quantities traded. Third, fixed-income securities have a special place iu finance theory because they have-no cash-How uncertainty, so their prices vary only as discount rates vary. By studying fixed-income securities we can explore the effects of changing discount rales without having to face the complications introduced by changing expectations of future cash Hows. The prices of conventional Treasury securities carry information about nominal discount rates, while the prices of indexed securities carry information about real discount rales, finally, many oilier assets can be seen as combinations of lixcd-incoiue securities and derivative securities; a callable bond, for example, is a fixed-income security less a put option.

.Such bonds have already been issued hv I lie I > k, C.an.uti.in. and sevci al other govci mucins. Sec Campbell and Shiller (I.l.Ki) lor a review.

that there are 253 trading days in a year and that market prices have no volatility when markets arc closed, i.e., weekends, holidays.

9.4.2 Provide a 95% confidence interval for /7(0) and an estimate of the number of simulations needed to yield a price estimate that is within $.05 of the true price.

9.4.3 I low docs this price compare with the price given by the Colduiau-Sosin-Gatto formula? Can you explain the discrepancy? Which price-would you use to decide whether to accept or reject Cl.Ms proposal?

The literature on fixed-income securities is vast.2 We break it into two main parts. First, in ibis chapter we introduce basic concepts and discuss empirical work on linear lime-scries models оГ bond yields. This work is only loosely motivated by theory and has the practical aim of exploring the forecasting power of the term structure of interest rates. In Chapter 11 we turn to more ambitious, fully specified term-structure models that can be used to price interest-rate derivative securities.

10.1 Basic Concepts

In principle a fixed-income security can promise a stream of future payments of any form, but there are two classic cases.

Zero-coujmn bonds, also called discount bonds, make a single payment at a date in the future known as the maturity date. The size of this payment is l\\c face value ol the bond. The length of time to the maturity date is the maturity of the bond. US Treasury bills (Treasury obligations with maturity at issue of up to 12 months) take this form.

Coupon bonds make coupon payments of a given fraction of face value at equally spaced dales up lo and including the maturity date, when the lace value is also paid. US Treasury notes and bonds (Treasury obligations with maturity at issue above 12 months) take this form. Coupon payments on Treasury notes and bonds arc made every six months, but the coupon rales Tor these instruments are normally quoted at an annual rate; thus a 7% Treasury bond actually pays 3.5% of face value every six months up to and including maturity.1

Coupon bonds can be thought of as packages of discount bonds, one corresponding to each coupon payment and one corresponding to the final coupon payment together with the repayment of principal. This is not merely an academic concept, as the principal and interest components оГ USTreasury bonds have been traded separately under the Treasurys STRIPS (Separate Trading of Registered Interest and Principal Securities) program since 1085, and the prices ol such Treasury strips al all maturities have been reported daily in the Wall Street Journal since 1980.

Fortunately it lias increased jM quality since F.d Капе judgement: Il is generally agreed thai, ceteris paribus, die fertility of a field is roughly proportional to the<tianlily of manure thai has been dumped upon it in the recent pasi. Ily this standard, the term strut lute of intet est rates has become ... an extraordinarily fertile lieltl indeed (Kane I .I7()). See Melino (Н1.ЧК) or Shiller (lU.IO) foi excellent recent surveys, and Sundaresan (10%) for a book-length treatment.

See a textbook such as Fabo i and Fabo/yi (ЮНГ,) or Fabn/zi (Itltlli) for limber details on ihe markers lor US Treasury securities.

10.1.1 Discount Bonds

We first define and illustrate basic bond market concepts for discount bonds. The yield to maturity on a bond is that discount rate which equates the present value of the bonds payments to its price. Thus if P , is the lime t price of a discount bond that makes a single payment of $1 al time / + n, and Y , is the bonds yield to maturity, we have

P , =---. (Ю.1.1)

(1 + K ,)

so the yield can be found from the price as

(1 + K, ) = /-(-). (Ю.1.2)

It is common in the empirical finance literature lo work with log or continuously compounded variables. This has the usual advanlage that it transforms the nonlinear equation (10.1.2) into a linear one. Using lowercase letters for logs the relationship between log yield and log price is

У , = -(-W (1°Л3)

The term structure of interest rates is the set of yields to maturity, at a given time, on bonds of different maturities. The yield spread S , = Y , - Y\t, or.in log terms .<; , = y , - y\ is the difference between the yield on ann-peripd bond and the yield on a one-period bond, a measure of the shape of the term structure. The yield curve is a plol of the term structure, that is, a plpt of Y, or y , against n on some particular date I. The solid line in Figure 10.1.1 shows the log zero-coupon yield curve for US Treasury securities t the end of January 1987.4 This particular yield curve rises at first, then falls at longer maturities so that it has a hump shape. This is nol unusual, although the yield curve is most commonly upward-sloping over the whole range of maturities. Sometimes the yield curve is inverted, sloping down over the whole range of maturities.

Holding-Period Returns \

The holding-fteriod return on a bond is the return over some holding period less than the bonds maturity. In order to economize on notation, we specialize at once to the case where the holding period is a single period.5 We

* This curve is not based on quoted strip prices, which arc readily available only for rccci years, but is estimated from the prices of coupon-bearing Treasury bonds. Figure 10.1.1 is clue to MrCiillorh and Kwon (HI.IX) and uses McC.ullochs (I.171, IУ7Г ) estimation method as discussed in section 10.1.3 below.

Shiller (IJJO) gives a much more comprehensive treatment, which requires more complicated notation.

It). Fixril-lnrmni Sri uritin

I llSl.ir I l.ll ItM M IS I;I II W.ll (I K.ltr

/си) Coupim YivM

in 15 20

M.uiuiiv in Years

Figure It). I. /11111.nii/inn Yield uml Im iiiiul-luitr tunn in January l>S7

define l( .,+ t as die one-period holding-period rcliirn on an (/-period bond purchased al time / and sold al lime / + I. Since the bond will be an ( n - I )-period bond when it is sold, die sale price is / -1.1+1 and the holding-period return is

(1 + /< ., и) =

(1 f V ,) (I + К, 1.-.1)

(lO.I.l)

The holding-period relurn in (10.1. I) is high illhe bond lias a high vielel when it is purchased at time /. and il it has a low yield when il is sold .11 time / + I (since .1 low yield tin responds lo a high price).

Moving to logs lor simplicity, die log holding-period relurn, i ,it r-logfl -I- /f ./ii).s

n.ii I = /11 tut /1,1 y.ii I D.V11 i.ni

--- V, ( - IX.Yi, i./.l -) () (ID.I.ri)

The last equality in (1(1.1. )) shows bow the holding-period return is determined bv the begiiiiiing-ol-peiiod yield (positively) and the change in ihe vield over ihe holding period (negatively). ,. : .

HI. I. Imsir Vdimrjih < ;l<l<l

l.cpt;tticiit (1(1.1.a) can he rearranged so that il relates die log bond price today lo the log price tomorrow and the relui 11 o\er the next period: I,. 1 - -..nl I / i.nl- (hie can solve litis tlillerence etiialiini loi ward, substituting niil Inline log bond prills null! ihe nullum dale is reached (;iihI inning dial 1I11- log price al in.ilinil\ eiit.ils /em) lo nlilain /> , = - 1 1,. 01 in lemis оГ the vield

y , = j 1 ( Ml. Hi)

I his Itjiialiou shows thai die log yield lo iii.tlinilv 011 .1 /eto-с (iiipun bond equals the aveiage log return per period il die bond is belli lo maturity.

hmxraiil I tales

Hi niils 1 ildillei cut 111.1l 111 ilies call he combined to guarantee an iiilcrcsl rate 01. a lixcd-iutnine investment to be made in the Itiluie: the interest rale on this investment is called a [inward rule.

lo guarantee ai time / an interest rale 011 a one-period invesinicul lobe in.ule .11 lime / I- 11. an investor can proceed as follows. Ihe desired future invesiincnl will pay $1 at time / + 11 -f- I so site first buvs one ( -I- I )-period bond, this costs / 11.1 at lime / and pays $ I at time I -I- 11 -1- 1. The investor wains to transfer the cost of this investment from time / to time / 4- 11; to do this she sells / , 1 , , /(-period bonds. This produces a positive cash How of I /.. \ .i/l i) - /.,+1.1 at time /, exactly enough to offset the negative tiine / cash (low from die lirsl transaction. The sale of -period bonds implies a negative cash How of / , 1., , at time / -f . This can he thought of as the cost ol die one-period investment to be made al lime / + . The cash Hows resulting from these transactions are illustrated in figure 10.2.

flu- forward rale is defined to be the return on the time /+ in vest men I ol /...: , , :

,., = = iill-. (1( .7,

(/.,. 1. Л.1) (I + >.,/)

In die in nal ion / , the first subscript refers 1 о the number of periods ahead lhat tin one-pel iocl investment is lo be made, and die second subscript relets in the dale ,u which ihe forward rate is sel. Al ihe cost of additional coutplexiiv in notation we could also define forward rales for niullipcriod iiiMsiiiiiiiis. bill wc do nol pursue ibis further here.

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