10. FixedIncome Securities. Basic Concepts


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1 0. FixedIncome Securities Fixedincome securities (FIS) are bonds that have no default risk and their payments are fully determined in advance. Sometimes corporate bonds that do not necessarily have certain future payments are also called xedincome securities. Nominal bonds: Fixed coupon payments, i.e., xed in nominal terms Indexed bonds: Coupon payments indexed to in ation, i.e., xed in real terms In principle xed income securities are as any other securities, but there are some special features:. FIS markets are developed separately from security markets: Own institutional structure, terminology and (academic) study traditions 2. Markets extremely large 3. FISs have a special place in nancial theory: no cash ow uncertainty, so that their price vary only as discount rates vary (with e.g. stocks, also expected future cash ows (dividends) change as discount rates change). Nominal bonds carry information about nominal discount rates, and indexed bonds about real discount rate. 4. Many other assets can be seen as combinations of FISs and derivative security; e.g. a callable bond is a FIS minus a put option. 8 Basic Concepts Zero coupon or discount bonds make a single payment at a date in the future known as the maturity date. The size of this payment is the face value of the bond. The length of time to the maturity date is the maturity of the bond. Coupon bonds make coupon payments of a given fraction of the face value at equally spaced dates up to and including the maturity date, when the face value is also paid. Note: Coupon bonds can be though as packages of discount bonds, one corresponding to each coupon payment and one to the  nal coupon payment together with the repayment of principal. (STRIPS, Separated Trading of Registered Interest and Principal Securities.) 82
2 Yieldtomaturityon a bond is that discount whichequatesthepresentvalueofthebond's payments to its price. For example the yield to maturity on a threeyear bond with annual interest payment of $00, a principal payment of $ 000, and present price $900 is the rate Y that equates the present value of the three years cash ows on bond with its present price 900 = 00 +Y ( + Y ) 2 ( + Y ) 3 : So that Y =4:3%. Discount Bonds Suppose that P nt is the time t price of a discount bond that makes a single payment of $ at time t + n. Thentheyieldtomaturity is obtained from P nt = ( + Y nt ) n; so that Turning to log or continuously compounded variables, we obtain y nt = n p nt: The term structure of interest rates is the set of yields to maturity at a given time, on bonds of di erent maturities. The yield spread s nt = y nt y t is the di erence between the yield on an nperiod bond andtheyieldonaoneperiodbond,andisa measure of the shape of the term structure. The yield curve is a plot of the term structure, that is the plot of Y nt or y nt against n on some particular date t. +Y nt = P n nt : 83 84
3 HoldingPeriod Returns: The holdingperiod return on a bond is the return over some holding period less than the bond's maturity. Let R n;t+ denote the oneperiod holdingperiod return on an nperiod bond purchased at time t andsoldattimet +. The bond will be an (n )periodbondwhenitissold at sale price P n ;t+, and the holding period return is +R n;t+ = P n ;t+ = P nt In logs r n;t+ = p n ;t+ p nt ( + Y nt ) n ( + Y n ;t+ ) n : = ny nt (n )y n ;t+ = y nt (n )(y n ;t+ y nt ): Wecanalsowrite: p nt = r n;t+ + p n ;t+, i.e., today's price is related to tomorrow's price and return over the next period. Solving forwardweobtain(notethatp 0t =sothat p 0t =logp 0t =0) p nt = or in terms of the yield n X r n i;t++i i=0 n X y nt = r n i;t++i : n i=0 I.e., the average per period logreturn. Forward Rates: Bonds of di erent maturities can be combined to guarantee an interest rate on a xedincome investment to be made in the future; the interest rate on this investment is called a forward rate
4 The forward rate is de ned as the return of the time t + n investment P nt =P n+;t : Example. To guarantee at time t an interest rate on oneperiod investment to be made at time t+n, an investor can proceed as follows: ² Suppose the desired future investment will pay $ at time t + n +. ² Buy one (n + )period bond which costs P n+;t at time t and pays $ at time t + n +. But one wants to transfer the cost of this investment from time t to time t + n. To do this { Sell P n+;t =P nt nperiod bonds to nance the investment (and hence transferring time t of P n+;t to time t+n). This produces the desired cash ow P nt (P n+;t =P nt )=P n+;t at time t, exactly enough to o set the negative time t cash ow from the rst transaction. { Pay at time t + n the cash ow of P n+;t =P nt, which is in fact the cost of investment made at t + n for one period. ( + F nt )= = ( + Y n+;t) n+ P n+;t =P nt ( + Y nt ) n : In logarithms f nt = p nt p n+;t = (n +)y n+;t ny nt = y nt +(n +)(y n+;t y nt ); where y nt =log(+y nt ). We observe: ² f nt is positive whenever discount bond prices fall with maturity. ² f nt is above both the nperiod and (n + )period discount bond yields when the (n +)period yield is above the nperiod yield (yield curve is upward sloping) 87 88
5 In summary, we have the interpretation: The yield to maturity is the average cost of borrowing for n periods, while the forward rate is the marginal cost of extending the time period of the loan. Coupon Bonds Let C denote the coupon rate per period (i.e. per period paid coupon price divided by the principal value of the bond), then the yield to maturity Y cnt is obtained as the discount rate which equates the present value of the bond's payments equal to its price at time t P cnt = C C +C + +Y cnt ( + Y cnt ) 2+ + ( + Y cnt ) n Duration and Immunization For discount bonds maturity is the length of time that a bondholder has invested money. For a coupon bond maturity is an imperfect measure of this length of time because much of the investment is paid back as coupons before the maturity date. Abettermeasureis 0 D cnt = nx P cnt i= ( + Y cnt ) i + n ( + Y cnt ) n A : Called Macaulay's duration ² When P cnt = the bond is said to selling at par, andy cnt = C. ² When maturity n is in nite, the bond is called consol or perpetuity, and Y ct = C=P ct. 89 Macaulay, F. (938). Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yield, and Stock Prices in the United States Since 856. National Bureau of Economic Research, New York 90
6 ² If C =0thenD cnt = n the maturity. ² If C>0thenD cnt <n. ² ForaparbondP cnt =, Y cnt = C and D cnt = ( + Y cnt) n ( + Y cnt ) ² For a consol bond with Y ct = C=P ct, D ct = +Y ct Y ct : Furthermore, we observe that D cnt = dp cnt +Y cnt d(+y cnt ) P cnt = dp cnt =P cnt d(+y cnt )=(+Y cnt ) ; i.e. the (negative) elasticity of a coupon bond's price with respect to its gross yield ( + Y cnt ). 9 Modi ed duration D cnt = dp cnt +Y cnt dy cnt P cnt measures the proportional sensitivity of a bond's price to a small absolute change in its yield. Example. If modi ed duration is 0, an increase of one basis point in the yield (say from 3.00% to 3.0%) will cause a 0 basis point (0.0%) drop in the bond price. Immunization was originally de ned as a process to make business immune to general change in interest rate. Nowadays it is de ned as a technique to eliminate sensitivity to shifts in the term structure by matching duration of the assets to the duration of the liabilities. For example one may want to match zero coupon liabilities, such as pension liabilities, to coupon paying Treasury. The problem here is that the Bond portfolio includes short and long term bonds, whose yield curves are not the same. Consequently, term structure of interest is a key issue in the immunization. 92
7 Convexity = d2 P cnt dy 2 cnt P cnt µ = P C P n i(i+) cnt i= (+Y cnt ) i+2 + n(n+) (+Y cnt ) n+2 ; which indicates, for example, how the modi ed duration changes as yield changes. It canbealsousedinasecondordertaylorapproximation of the price impact of a change in yield: dp cn P cn ¼ dp cn dy cn sycn 2 Pcn dy cn + 2 d2 P cn P cn (dy cn ) 2 = (mod dur)dy cn + 2 (conv)(dy cn) 2 : A Loglinear Model for Coupon Bonds: Duration can be used to nd approximate linear relationships between log coupon bond yields, holding period returns, and forward rates that are analogous to the exact relationships for zerocoupon bonds (see earlier). Using a similar approach as with the stock return, we can write r c;n;t+ ¼ k + ½p c;n ;t+ +( ½)c p cnt ; 93 where and ½ = + exp(c p) k = log ½ ( ½)log(=½ ): For a par selling bond ½ ==( + C) =(+ Y cnt ). Using the approximation and solving forward, we obtain p cnt = n X i=0 ½ i h k +( ½)c r c;n i;t++i i : A similar approximation of the log yield to maturity y cnt produces p cnt ¼ P n i=0 ½i [k +( ½)c y cnt ] = ½n ½ [k +( ½)c y cnt] 94
8 Using these two expression of p cnt gives y cnt ¼ ½n ½ n X i=0 ½ i r c;n i;t++i : Thus there is an approximate equality between the log yield to maturity on coupon bond and a weighted average of the returns onthebondwhenitisheldtomaturity. From the above formula we also see that D cnt ¼ ½n ½ = ( + Y cnt) n ( + Y cnt ) : Thus (an approximate analogy for a zerocoupon bond) r c;n;t+ ¼ D cnt y cnt (D cnt )y c;n ;t+ : Finally a similar analysis for an nperiodahead period forward rate implicit in the couponbearing term structure is f nt ¼ D c;n+y c;n+;t D cn y cnt D c;n+ D cn : Estimating the ZeroCoupon Term Structure Suppose we know the prices of discount bonds P ;P 2 ;:::;P n maturing at each coupon date, that is the coupon term structure. Then the price of a coupon bond is P cn = P C + P 2 C + + P n ( + C): Similarly if a complete coupon term structure that is, the prices of coupon bonds P c ;P c2 ; :::;P cn maturing at each coupon date is available, then the zero coupon terms structure can be found applying iteratively the above coupon bond price: P c = P ( + C), so P = P c =( + C), and generally P n = P cn P n C P C +C 95 96
9 Sometimes, however, the terms structure may be morethancomplete in the sense that at least one coupon bond matures on each coupon date and several coupon bond mature on some coupon dates. The prices are likely di erent in these multiple cases. One possibility is to determine a single price by compromising with a regression model P ci n i = P C i + P 2 C i + + P ni ( + C i )+u i ; i =;:::;I,whereC i isthecouponontheith bond and n i is the maturity of the ith bond. The coe±cients are discount bond prices P j, j =;:::;N,whereN =maxn i is the longest coupon bond maturity. OLS can be applied provided that the term structure is complete and I N. Interpreting the Term Structure of Interest Rates Theories of the term structure. Pure expectation hypothesis: For zero coupon bonds E t [R n;t+ ]=r t, for all maturities n, where r t is the riskfree rate. 2. Expectation hypothesis: E t [R n;t+ ] r t = c a constant for all maturities n. 3. Liquidity preference hypothesis: E t [R n;t+ ] r t = T (n) where T (n) >T (n ) >. 4. Time varying risk: E t [R n;t+ ] r t = T (n; z t ), where T is some function of n and set of variables z t. 5. Etc. In practice the term structure, however, is incomplete and other methods must be applied, e.g. spline
10 Here we consider only to some extend the expectation hypotheses. Expectation Hypotheses Pure expectation hypothesis (PEH): Expected excess returns on longterm over shortterm bonds are zero. Expectation Hypothesis (EH): Expected excess returns are constants over time. The rst form PEH equates the one period expected returns on oneperiod and nperiod bonds. The oneperiod return on a oneperiod bond, + Y t,isknown,so +Y t = E t [ + R n;t+ ] = (+Y nt ) n E t h( + Y n ;t+ ) (n )i : A second form of PEH equates the nperiod expected returns on oneperiod and nperiod bonds: ( + Y nt ) n = E t h ( + Yt ) ( + Y ;t+n ) i : From this implies +F n ;t = ( + Y nt) n ( + Y n ;t ) n = E t[+y ;t+n ]: Also it holds that ( + Y nt ) n =(+Y t )E t h ( + Yn ;t+ ) n i : This is inconsistent with the rst form whenever interest rates are random, because then generally " # E t ( + Y n ;t+ ) n 6= h E t ( + Yn ;t+ ) n i 99 00
11 Implications of the Log PEH First Secondly Finally y t = E t [r n;t+ ]: y nt = n n X i=0 E t [y ;t+i ]: f n ;t = E t [y ;t+n ]; which implies furthermore that f nt = E t [y ;t+n ] = i E t he t+ [y ;t+n ] = E t [f n ;t+ ] i.e., f n;t is a martingale. The expectation hypothesis is more general than the PEH allowing di erences in expected returns on bonds of di erent maturities. These di erences are sometimes called term premia. In PEH term premia are zero and in EH they are constant through time. Yield Spreads and Interest Rate Forecasts The yield spread between nperiod and oneperiod yield is s nt = y nt y t. Because we can write y nt = n nx r n i;t++i i= " nx s nt = n E t (y;t+i y t )+(r n+ i;t+i y ;t+i ) # i= " nx = n E t (n i) y;t+i +(r n+ i;t+i y ;t+i ) # i= 0 02
12 That is the yield spread equals a weighted average expected future interest rate changes and an unweighted average of expected future excess returns on long bonds. If the changes in interest rate ( y ;t+i )arestationary and the excess returns r n+ i;t+i y ;t+i are stationary then the yield spread is cointegrated. According to EH E t [r n+ i;t+i y ;t+i ] are constants. This implies that the yield spread is the optimal forecaster of the change in the longbond yield over the life of the short bond, and the optimal forecaster of changes in short rates over the life of the long bond. Recalling that r n;t+ = y nt (n )(y n ;t+ y nt )andy t = E t [r n;t+ ], we obtain under the EH and after some algebra n s nt = E t [y n ;t+ y nt ] and 2 3 n X s nt = E t 4 ( i=n) y 5 ;t+i : i= 03 The former equation shows that when the yield spread is high, the long rate is expected to rise. A high yield spread gives the long bond a yield advantage that must be o set by an anticipated capital loss. The latter equation shows that when the yield spread is high, short rates are expected to rise. An econometric model for testing the former is µ sn t y n ;t+ y n;t = n + n + ² n;t : n An econometric model for testing the latter claim is where s n;t = ¹ n + n s nt + ² nt s n n;t = X ( i=n) y ;t+i i= is the ex post value of the shortrate changes. The expectation hypothesis implies that = for all n. 04
C(t) (1 + y) 4. t=1. For the 4 year bond considered above, assume that the price today is 900$. The yield to maturity will then be the y that solves
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