10. Fixed-Income Securities
Fixed-income 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 ¯xed-income 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
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.
securities, but there are some special features:
1. 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.
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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.)
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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 annual 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
100
100 + 1000
100
+
+
:
1+Y
(1 + Y )2
(1 + Y )3
So that Y = 14:3%.
900 =
Discount Bonds
Suppose that Pnt is the time t price of a discount bond that makes a single payment of
$1 at time t + n. Then the yield to maturity
is obtained from
1
;
Pnt =
(1 + Ynt)n
so that
1 + Ynt =
Turning to log or continuously compounded
variables, we obtain
1
ynt = ¡ pnt:
n
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 snt = ynt ¡ y1t is the di®erence 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 structure, that is the plot of Ynt or ynt against n
on some particular date t.
¡1
Pnt n :
83
84
Holding-Period Returns: The holding-period
return on a bond is the return over some
holding period less than the bond's maturity.
Let Rn;t+1 denote the one-period holdingperiod 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 Pn¡1;t+1, and the holding period
return is
1 + Rn;t+1 =
Pn¡1;t+1
Pnt
(1 + Ynt)n
=
:
(1 + Yn¡1;t+1)n¡1
We can also write: pnt = ¡rn;t+1 + pn¡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 P0t = 1 so that
p0t = log P0t = 0)
pnt = ¡
n¡1
X
rn¡i;t+1+i
i=0
or in terms of the yield
ynt =
X
1 n¡1
rn¡i;t+1+i:
n i=0
I.e., the average per period log-return.
In logs
rn;t+1 = pn¡1;t+1 ¡ pnt
= nynt ¡ (n ¡ 1)yn¡1;t+1
= ynt ¡ (n ¡ 1)(yn¡1;t+1 ¡ ynt):
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Forward Rates: Bonds of di®erent maturities can be combined to guarantee an interest rate on a ¯xed-income investment to be
made in the future; the interest rate on this
investment is called a forward rate.
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The forward rate is de¯ned as the return of
the time t + n investment Pnt=Pn+1;t:
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 Pn+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;t to 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.
87
(1 + Fnt) =
1
Pn+1;t=Pnt
=
(1 + Yn+1;t )n+1
(1 + Ynt)n
:
In logarithms
fnt = pnt ¡ pn+1;t
= (n + 1)yn+1;t ¡ nynt
= ynt + (n + 1)(yn+1;t ¡ ynt);
where ynt = log(1 + Ynt).
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)
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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 Ycnt is obtained as the discount
rate which equates the present value of the
bond's payments equal to its price at time t
C
C
1+C
+
+¢ ¢ ¢+
Pcnt =
2
1 + Ycnt (1 + Ycnt )
(1 + Ycnt )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.
A better measure is
Dcnt =
0
1
n
X
1 @
n
i
A:
C
+
i
n
Pcnt
(1 + Ycnt )
i=1 (1 + Ycnt )
Called Macaulay's duration¤
² When Pcnt = 1 the bond is said to selling
at par, and Ycnt = C.
² When maturity n is in¯nite, the bond is
called consol or perpetuity, and Yc1t =
C=Pc1t .
89
¤ 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
90
² If C = 0 then Dcnt = n the maturity.
² If C > 0 then Dcnt < n.
² For a par bond Pcnt = 1, Ycnt = C and
Dcnt =
1 ¡ (1 + Ycnt )¡n
1 ¡ (1 + Ycnt )¡1
² For a consol bond with Yc1t = C=Pc1t,
Dc1t =
1 + Yc1t
:
Yc1t
Furthermore, we observe that
1+Ycnt
dPcnt
Dcnt = ¡ d(1+Y
cnt) Pcnt
dP
=P
cnt
= ¡ d(1+Y cnt)=(1+Y
;
cnt
cnt)
i.e. the (negative) elasticity of a coupon
bond's price with respect to its gross yield
(1 + Ycnt ).
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Modi¯ed duration
Dcnt
dPcnt 1
=¡
1 + Ycnt
dYcnt Pcnt
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 increase 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 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
2
Convexity = d P2cnt P1
=
1
Pcnt
µ
dYcnt
where
cnt
P
n(n+1)
i(i+1)
C n
i=1 (1+Ycnt)i+2 + (1+Ycnt)n+2
¶
;
which indicates, for example, how the modi¯ed duration changes as yield changes. It
can be also used in a second-order Taylor approximation of the price impact of a change
in yield:
dPcn
Pcn
½=
1 d2Pcn 1 (dY )2
cn 1
¼ dP
cn
2 Pcn
dYcn Pcn dYcn + 2 sYcn
2
= ¡(mod dur)dYcn + 1
2 (conv)(dYcn) :
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 zero-coupon bonds (see earlier).
Using a similar approach as with the stock
return, we can write
and
1
1 + exp(c ¡ p)
k = ¡ log ½ ¡ (1 ¡ ½) log(1=½ ¡ 1):
For a par selling bond ½ = 1=(1 + C) = (1 +
Ycnt)¡1.
Using the approximation and solving forward,
we obtain
pcnt =
n¡1
X
i=0
h
i
½i k + (1 ¡ ½)c ¡ rc;n¡i;t+1+i :
A similar approximation of the log yield to
maturity ycnt produces
pcnt ¼
Pn¡1 i
i=0 ½ [k + (1 ¡ ½)c ¡ ycnt]
n
= 1¡½
1¡½ [k + (1 ¡ ½)c ¡ ycnt ]
rc;n;t+1 ¼ k + ½pc;n¡1;t+1 + (1 ¡ ½)c ¡ pcnt ;
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94
Using these two expression of pcnt gives
X
1 ¡ ½n n¡1
ycnt ¼
½irc;n¡i;t+1+i:
1 ¡ ½ i=0
Estimating the Zero-Coupon Term Structure
Thus there is an approximate equality between 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
Dcnt ¼
1 ¡ ½n
1 ¡ (1 + Ycnt )¡n
:
=
1¡½
1 ¡ (1 + Ycnt)¡1
Thus (an approximate analogy for a zerocoupon bond)
rc;n;t+1 ¼ Dcntycnt ¡ (Dcnt ¡ 1)yc;n¡1;t+1:
Finally a similar analysis for an n-period-ahead
1-period forward rate implicit in the couponbearing term structure is
fnt ¼
Dc;n+1yc;n+1;t ¡ Dcnycnt
Dc;n+1 ¡ Dcn
Suppose we know the prices of discount bonds
P1; P2 ; : : : ; Pn maturing at each coupon date,
that is the coupon term structure. Then the
price of a coupon bond is
Pcn = P1C + P2C + ¢ ¢ ¢ + Pn(1 + C):
Similarly if a complete coupon term structure|
that is, the prices of coupon bonds Pc1; Pc2;
: : : ; Pcn maturing at each coupon date|is
available, then the zero coupon terms structure can be found applying iteratively the
above coupon bond price: Pc1 = P1 (1 + C),
so P1 = Pc1=(1 + C), and generally
Pn =
Pcn ¡ Pn¡1C ¡ ¢ ¢ ¢ ¡ P1C
1+C
:
95
96
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
Pcini = P1Ci + P2 Ci + ¢ ¢ ¢ + Pni (1 + Ci) + ui;
i = 1; : : : ; I, where Ci is the coupon on the ith
bond and ni is the maturity of the ith bond.
The coe±cients are discount bond prices Pj ,
j = 1; : : : ; N , where N = max ni 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
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] ¡ rt = c a constant for all maturities n.
3. Liquidity preference hypothesis: Et[Rn;t+1] ¡ rt =
T (n) where T (n) > T (n¡1) > ¢ ¢ ¢.
4. Time varying risk: Et [Rn;t+1 ]¡rt = T (n; zt), where
T is some function of n and set of variables zt .
5. Etc.
In practice the term structure, however, is
incomplete and other methods must be applied, e.g. spline.
97
98
Here we consider only to some extend the
expectation hypotheses.
A second form of PEH equates the n-period
expected returns on one-period and n-period
bonds:
Expectation Hypotheses
h
Pure expectation hypothesis (PEH): Expected
excess returns on long-term over short-term
bonds are zero.
From this implies
1+Fn¡1;t =
Expectation Hypothesis (EH): Expected excess returns are constants over time.
(1 + Ynt)n
= Et [1+Y1;t+n¡1]:
(1 + Yn¡1;t)n¡1
Also it holds that
(1 + Ynt
The ¯rst form PEH equates the one period
expected returns on one-period and n-period
bonds. The one-period return on a oneperiod bond, 1 + Y1t, is known, so
)n
h
= (1 + Y1t)Et (1 + Yn¡1;t+1
)n¡1
1 + Y1t = Et [1 + Rn;t+1]
i
"
1
#
:
1
h
i
=
6
n¡1
(1 + Yn¡1;t+1 )n¡1
Et (1 + Yn¡1;t+1)
= (1 + Ynt)nEt (1 + Yn¡1;t+1)¡(n¡1) :
99
i
This is inconsistent with the ¯rst form whenever interest rates are random, because then
generally
Et
h
i
(1 + Ynt)n = Et (1 + Y1t ) ¢ ¢ ¢ (1 + Y1;t+n¡1) :
100
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.
Implications of the Log PEH
First
y1t = Et [rn;t+1]:
Secondly
Yield Spreads and Interest Rate Forecasts
X
1 n¡1
Et [y1;t+i]:
ynt =
n i=0
The yield spread between n-period and oneperiod yield is snt = ynt ¡ yt. Because
Finally
n
1 X
ynt =
rn¡i;t+1+i
n i=1
fn¡1;t = Et[y1;t+n¡1];
which implies furthermore that
we can write
fnt = Et [y1;t+n]
h
i
= Et Et+1[y1;t+n ]
snt
= Et [fn¡1;t+1]
#
" n
X£
¤
1
= Et
(y1;t+i ¡ y1t) + (rn+1¡i;t+i ¡ y1;t+i)
n
i=1
#
" n
X£
¤
1
= Et
(n ¡ i)¢y1;t+i + (rn+1¡i;t+i ¡ y1;t+i)
n
i=1
i.e., fn;t is a martingale.
101
102
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 (¢y1;t+i) are stationary and the excess returns rn+1¡i;t+i ¡ y1;t+i
are stationary then the yield spread is cointegrated.
According to EH Et[rn+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 rn;t+1 = ynt ¡(n¡1)(yn¡1;t+1 ¡
ynt) and y1t = Et [rn;t+1 ], we obtain under the
EH and after some algebra
and
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
+ ²n;t:
yn¡1;t+1 ¡ yn;t = ®n + ¯n
n¡1
An econometric model for testing the latter
claim is
s¤n;t = ¹n + °nsnt + ²nt
where
1
snt = Et [yn¡1;t+1 ¡ ynt]
n¡1
s¤n;t =
n¡1
X
i=1
2
3
n¡1
X
(1 ¡ i=n)¢y1;t+i5 :
snt = Et 4
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.
103
104