10. Fixed-Income Securities. Basic Concepts

10. Fixed-Income Securities

Fixed-income securities (FIS) are bonds that have no default risk and their payments are fully deter- mined in advance. Sometimes corporate bonds that do not necessarily have certain future payments are also called ¯xed-income securities.

Nominal bonds: Fixed coupon payments, i.e., ¯xed in nominal terms

Indexed bonds: Coupon payments indexed to in°a- tion, i.e., ¯xed in real terms

In principle ¯xed income securities are as any other securities, but there are some special features:

1. FIS markets are developed separately from secu- rity markets: Own institutional structure, termi- nology 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 ex- pected future cash °ows (dividends) change as discount rates change). Nominal bonds carry in- formation about nominal discount rates, and in- dexed 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.

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 pack- ages of discount bonds, one corresponding to each coupon payment and one to the ¯- nal coupon payment together with the re- payment of principal. (STRIPS, Separated Trading of Registered Interest and Principal Securities.)

Yield to maturity on a bond is that discount which equates the present value of the bond's payments to its price. For example the yield to maturity on a three-year bond with an- nual interest payment of $100, a principal payment of $1 000, 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 = 100

1 + Y + 100

(1 + Y )² + 100 + 1000 (1 + Y )³ : So that Y = 14:3%.

Discount Bonds

Suppose that P_nt is the time t price of a dis- count bond that makes a single payment of

$1 at time t + n. Then the yield to maturity is obtained from

P_nt = 1

(1 + Y_nt)ⁿ; so that

1 + Y_nt = P^¡

1

ntn:

Turning to log or continuously compounded variables, we obtain

y_nt =¡1 np_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¡ y1t is the di®er- ence between the yield on an n-period bond and the yield on a one-period bond, and is a measure of the shape of the term structure.

The yield curve is a plot of the term struc- ture, that is the plot of Y_nt or y_nt against n on some particular date t.

Holding-Period Returns: The holding-period return on a bond is the return over some holding period less than the bond's maturity.

Let R_n;t+1 denote the one-period holding- period return on an n-period bond purchased at time t and sold at time t + 1. The bond will be an (n¡ 1)-period bond when it is sold at sale price P_n¡1;t+1, and the holding period return is

1 + R_n;t+1 = P_n¡1;t+1

P_nt = (1 + Y_nt)ⁿ (1 + Y_n¡1;t+1)^n¡1: In logs

r_n;t+1 = p_n¡1;t+1¡ pnt

= ny_nt¡ (n ¡ 1)yn¡1;t+1

= y_nt¡ (n ¡ 1)(yn¡1;t+1¡ ynt):

We can also write: p_nt = ¡rn;t+1+ p_n¡1;t+1, i.e., today's price is related to tomorrow's price and return over the next period. Solving forward we obtain (note that P_0t = 1 so that p_0t = log P_0t = 0)

p_nt = ¡

nX¡1 i=0

r_n¡i;t+1+i or in terms of the yield

y_nt = 1 n

nX¡1 i=0

r_n¡i;t+1+i:

I.e., the average per period log-return.

Forward Rates: Bonds of di®erent maturi- ties can be combined to guarantee an inter- est rate on a ¯xed-income investment to be made in the future; the interest rate on this investment is called a forward rate.

Example. To guarantee at time t an interest rate on one-period investment to be made at time t+n, an investor can proceed as follows:

² Suppose the desired future investment will pay $1 at time t + n + 1.

² Buy one (n + 1)-period bond which costs P^n+1;t at time t and pays $1 at time t + n + 1. But one wants to transfer the cost of this investment from time t to time t + n. To do this

{ Sell Pn+1;t=Pnt n-period bonds to ¯nance the investment (and hence transferring time t of Pn+1;tto time t+n). This produces the desired cash °ow Pnt(Pn+1;t=Pnt) = Pn+1;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 Pn+1;t=Pnt, which is in fact the cost of investment made at t + n for one period.

The forward rate is de¯ned as the return of the time t + n investment P_nt=P_n+1;t:

(1 + F_nt) = 1

P_n+1;t=P_nt = (1 + Y_n+1;t)ⁿ⁺¹ (1 + Y_nt)ⁿ : In logarithms

f_nt = p_nt¡ pn+1;t

= (n + 1)y_n+1;t¡ nynt

= y_nt+ (n + 1)(y_n+1;t¡ ynt);

where y_nt = log(1 + Y_nt).

We observe:

² fnt is positive whenever discount bond prices fall with maturity.

² fnt is above both the n-period and (n + 1)-period discount bond yields when the (n + 1)-period yield is above the n-period yield (yield curve is upward sloping)

In summary, we have the interpretation: The yield to maturity is the average cost of bor- rowing 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

1 + Y_cnt+ C

(1 + Y_cnt)²+¢ ¢ ¢+ 1 + C (1 + Y_cnt)ⁿ

² When Pcnt = 1 the bond is said to selling at par, and Y_cnt = C.

² When maturity n is in¯nite, the bond is called consol or perpetuity, and Y_c1t = C=P_c1t.

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.

A better measure is D_cnt = 1

P_cnt

0

@C

Xn i=1

i

(1 + Y_cnt)ⁱ + n (1 + Y_cnt)ⁿ

1 A: Called Macaulay's duration^¤

¤Macaulay, F. (1938). Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yield, and Stock Prices in the United States Since 1856. National Bureau of Economic Research, New York

² If C = 0 then Dcnt = n the maturity.

² If C > 0 then Dcnt < n.

² For a par bond Pcnt = 1, Y_cnt = C and D_cnt = 1¡ (1 + Ycnt)^¡n

1¡ (1 + Ycnt)^¡1

² For a consol bond with Yc1t = C=P_c1t, D_c1t = 1 + Y_c1t

Y_c1t :

Furthermore, we observe that D_cnt = ¡_d(1+Y^dPcnt_cnt) ^1+Y_Pcnt^cnt

= ¡_d(1+Y^dP_cnt^cnt_)=(1+Y^=Pcnt _cnt);

i.e. the (negative) elasticity of a coupon bond's price with respect to its gross yield (1 + Y_cnt).

Modi¯ed duration D_cnt

1 + Y_cnt = ¡dP_cnt dY_cnt

1 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 10, an in- crease of one basis point in the yield (say from 3.00% to 3.01%) will cause a 10 basis point (0.10%) drop in the bond price.

Immunization was originally de¯ned as a pro- cess 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.

Convexity = ^d2Pcnt

dY_cnt² 1 P_cnt

= _P¹

cnt

µ

C^Pn_i=1 ⁱ⁽ⁱ⁺¹⁾

(1+Ycnt)ⁱ⁺² + ⁿ⁽ⁿ⁺¹⁾

(1+Ycnt)ⁿ⁺²

¶

; which indicates, for example, how the mod- i¯ed duration changes as yield changes. It can be also used in a second-order Taylor ap- proximation of the price impact of a change in yield:

dPcn

Pcn ¼ ^dP_dY^cn_cnP¹_cndY_cn+ ¹₂ ^d2Pcn

sY_cn² 1

Pcn(dY_cn)²

= ¡(mod dur)dYcn+ ¹₂(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 rela- tionships for zero-coupon bonds (see earlier).

Using a similar approach as with the stock return, we can write

r_c;n;t+1 ¼ k + ½pc;n¡1;t+1+ (1¡ ½)c ¡ pcnt;

where

½ = 1

1 + exp(c¡ p) and

k = ¡ log ½ ¡ (1 ¡ ½) log(1=½ ¡ 1):

For a par selling bond ½ = 1=(1 + C) = (1 + Y_cnt)^¡1.

Using the approximation and solving forward, we obtain

p_cnt =

nX¡1 i=0

½^ihk + (1¡ ½)c ¡ rc;n¡i;t+1+i i:

A similar approximation of the log yield to maturity y_cnt produces

p_cnt ¼ ^Pn¡1_i=0 ½ⁱ[k + (1¡ ½)c ¡ ycnt]

= ¹₁^¡½n

¡½ [k + (1¡ ½)c ¡ ycnt]

Using these two expression of p_cnt gives y_cnt ¼ 1¡ ½ⁿ

1¡ ½

nX¡1 i=0

½ⁱr_c;n¡i;t+1+i:

Thus there is an approximate equality be- tween the log yield to maturity on coupon bond and a weighted average of the returns on the bond when it is held to maturity.

From the above formula we also see that D_cnt ¼ 1¡ ½ⁿ

1¡ ½ = 1¡ (1 + Ycnt)^¡n 1¡ (1 + Ycnt)^¡1:

Thus (an approximate analogy for a zero- coupon bond)

r_c;n;t+1 ¼ Dcnty_cnt ¡ (Dcnt ¡ 1)yc;n¡1;t+1: Finally a similar analysis for an n-period-ahead 1-period forward rate implicit in the coupon- bearing term structure is

f_nt ¼ D_c;n+1y_c;n+1;t¡ D^cny_cnt D_c;n+1 ¡ D^cn :

Estimating the Zero-Coupon Term Structure Suppose we know the prices of discount bonds P₁; P₂; : : : ; P_n maturing at each coupon date, that is the coupon term structure. Then the price of a coupon bond is

P_cn = P₁C + P₂C +¢ ¢ ¢ + Pn(1 + C):

Similarly if a complete coupon term structure|

that is, the prices of coupon bonds P_c1; P_c2; : : : ; P_cn maturing at each coupon date|is available, then the zero coupon terms struc- ture can be found applying iteratively the above coupon bond price: P_c1 = P₁(1 + C), so P₁ = P_c1=(1 + C), and generally

P_n = P_cn ¡ Pn¡1C ¡ ¢ ¢ ¢ ¡ P1C 1 + C

Sometimes, however, the terms structure may be more-than-complete 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_cini = P₁C_i + P₂C_i +¢ ¢ ¢ + Pn_i(1 + C_i) + u_i; i = 1; : : : ; I, where C_i is the coupon on the ith bond and n_i is the maturity of the ith bond.

The coe±cients are discount bond prices P_j, j = 1; : : : ; N , where N = max n_i is the longest coupon bond maturity. OLS can be applied provided that the term structure is complete and I ¸ N.

In practice the term structure, however, is incomplete and other methods must be ap- plied, e.g. spline.

Interpreting the Term Structure of Interest Rates

Theories of the term structure

1. Pure expectation hypothesis: For zero coupon bonds Et[Rn;t+1] = rt, for all maturities n, where rt is the riskfree rate.

2. Expectation hypothesis: Et[Rn;t+1]¡ r^t= c a con- stant for all maturities n.

3. Liquidity preference hypothesis: Et[Rn;t+1]¡ r^t = T⁽ⁿ⁾ where T⁽ⁿ⁾ > T^(n¡1)> ¢ ¢ ¢.

4. Time varying risk: Et[Rn;t+1]¡r^t= T (n; zt), where T is some function of n and set of variables zt. 5. Etc.

Here we consider only to some extend the expectation hypotheses.

Expectation Hypotheses

Pure expectation hypothesis (PEH): Expected excess returns on long-term over short-term bonds are zero.

Expectation Hypothesis (EH): Expected ex- cess returns are constants over time.

The ¯rst form PEH equates the one period expected returns on one-period and n-period bonds. The one-period return on a one- period bond, 1 + Y_1t, is known, so

1 + Y_1t = E_t[1 + R_n;t+1]

= (1 + Y_nt)ⁿE_t^h(1 + Y_n¡1;t+1)^¡(n¡1)i:

A second form of PEH equates the n-period expected returns on one-period and n-period bonds:

(1 + Y_nt)ⁿ = E_t^h(1 + Y_1t)¢ ¢ ¢ (1 + Y1;t+n¡1)ⁱ: From this implies

1+F_n¡1;t = (1 + Y_nt)ⁿ

(1 + Y_n¡1;t)^n¡1 = E_t[1+Y_1;t+n¡1]:

Also it holds that

(1 + Y_nt)ⁿ = (1 + Y_1t)E_t^h(1 + Y_n¡1;t+1)^n¡1i: This is inconsistent with the ¯rst form when- ever interest rates are random, because then generally

E_t

"

1

(1 + Y_n¡1;t+1)^n¡1

#

6

= 1

E_t^h(1 + Y_n¡1;t+1)^n¡1i

Implications of the Log PEH First

y_1t = E_t[r_n;t+1]:

Secondly

y_nt = 1 n

nX¡1 i=0

E_t[y_1;t+i]:

Finally

f_n¡1;t = E_t[y_1;t+n¡1];

which implies furthermore that f_nt = E_t[y_1;t+n]

= E_t^hE_t+1[y_1;t+n]ⁱ

= E_t[f_n¡1;t+1] 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 n-period and one- period yield is s_nt = y_nt¡ yt. Because

y_nt = 1 n

Xn i=1

r_n¡i;t+1+i we can write

snt = 1 nEt

" _n X

i=1

£(y1;t+i¡ y^1t) + (rn+1¡i;t+i¡ y^1;t+i)¤#

= 1 nEt

" _n X

i=1

£(n¡ i)¢y^1;t+i+ (rn+1¡i;t+i¡ y^1;t+i)¤#

That is the yield spread equals a weighted av- erage expected future interest rate changes and an unweighted average of expected fu- ture excess returns on long bonds. If the changes in interest rate (¢y_1;t+i) are station- ary and the excess returns r_n+1¡i;t+i¡ y1;t+i

are stationary then the yield spread is coin- tegrated.

According to EH E_t[r_n+1¡i;t+i ¡ y1;t+i] are constants. This implies that the yield spread is the optimal forecaster of the change in the long-bond 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+1 = y_nt¡(n¡1)(yn¡1;t+1¡ y_nt) and y_1t = E_t[r_n;t+1], we obtain under the EH and after some algebra

1

n¡ 1s_nt = E_t[y_n¡1;t+1¡ ynt] and

s_nt = E_t

2 4

nX¡1 i=1

(1¡ i=n)¢y1;t+i 3 5:

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

y_n¡1;t+1¡ yn;t = ®_n + ¯_n

µ s_nt n¡ 1

¶

+ ²_n;t: An econometric model for testing the latter claim is

s^¤_n;t = ¹_n+ °_ns_nt+ ²_nt where

s^¤_n;t =

nX¡1 i=1

(1¡ i=n)¢y1;t+i

is the ex post value of the short-rate changes.

The expectation hypothesis implies that ° = 1 for all n.