In this section we present some fundamental concepts in term structure theory, such as the discount function, the spot rate and spot yield, and the forward rate. While these initially may appear to be esoteric in nature, they are in fact closely interrelated quantities that directly repre- sent the term structure, or act to influence the course of future interest rates in an arbitrage-free environment. In this section these concepts are shown to be incorporated into the different expressions that describe the various qualitative term structure theories, such as the expectation, preferred-habitat, and liquidity-preference hypotheses. The continuous- time term structure model discussed later in this chapter evolved from the eclectic compilation of earlier theories.
Discount Function
The discount function incorporates market yield-curve information to express the present value of a future dollar as a function of the term to its receipt. As such, the discount function is a valid expression of the term structure of interest rates by virtue of the price/yield relationship.
Since the discount function is used to quantify the value of a future dol- lar, the discount function also provides a direct means to value a coupon paying bond since the coupon and principal payments are simply scalar multiples of a single dollar. As a result, the discount function can be used as a reference check for other quantitative term structure models.
Quantitative term-structure models ultimately deal with the analysis of pure discount bonds. (Discount bonds, or zero-coupon bonds, are the simplest types of bonds to analyze as there is only the repayment of par at maturity. Further, all other bonds can be built from a series of dis-
5-Audley/Chin-TermStructModel Page 99 Thursday, August 29, 2002 10:00 AM
count bonds and options on discount bonds.) As a consequence of mod- eling the yield movements of discount bonds, term structure models describe their price movements since the price/yield relationship allows the term structure to be analyzed in terms of either price or yield.
This relationship is addressed further later in this chapter, in which the term structure model is expressed in terms of price as a function of rate and time.
If it is assumed that the discount bond pays one dollar at maturity, then the present value of the bond is some decimal fraction less than one. For a set of discount bonds of increasing maturities, there is the corresponding set of present values starting from approximately 0.999 and decreasing thereafter. This set of present values is called the “dis- count function,” and is shown in Exhibit 5.1.
The discount function is the term-to-maturity relationship of the present value of a future unit of cash flow. More formally, for a cash flow, CF, received after a term, T, from today, t, the present value, PV, of that cash flow is discounted, d, from the future value CF as expressed by the relation
(1) where
EXHIBIT 5.1 Discount Function
PV(t,T) = present value of the cash flow at t
d(t,T) = discount at t for a cash flow received T after t CF(t,T) = cash flow received at t + T
PV t T( , ) = d t T( , )×CF t T( , )
Term Structure Modeling 101
As we are able to generate the discount function, d, for all terms-to- maturity, T, this can be a valid representation of the term structure of interest rates. Indeed, the discount function reflects the Treasury term structure when the discount function exactly reprices the current-coupon Treasury issues.
Deriving the Discount Function for On-the-Run Treasuries
More generally, let P(t,i) be the set of closing prices on (date) t for the set of current-coupon Treasury bonds (where the index, i, associates a specific issue)
Each of these instruments has its own time series of cash flows, each with its own individual term-to-maturity. For the Treasury bills, the cash flows and associated terms-to-maturity are
and for the periodic instruments,
where the term to each of the cash flows, T(i,j), is specific to the instrument.
The index j is the sequence of the cash flow in the time series for security i.
The present value of a coupon paying instrument is simply the sum of the discounted present values of the cash flows that make up the cou- pon payments and the payment of principal. Accordingly, for the dis- count function to model the Treasury term structure (i.e., the market sector defined by the on-the-run Treasury reference set), the following equations must be simultaneously satisfied. In this way, the discount function will reprice the current-coupon Treasury issues.
P(t,3-month): price of the 3-month (13-week) bill, at time t P(t,6-month): price of the 6-month (26-week) bill, at time t P(t,2-year): price of the 2-year note, at time t
. . . .
P(t,30-year): price of the 30-year bond, at time t
3-month bill: CF(t,T(3-month,1)) 6-month bill: CF(t,T(6-month,1))
2-year note: CF(t,T(2-year,1)), CF(t,T(2-year,2)), CF(t,T(2-year,3)), CF(t,T(2-year,4))
. . . .
30-year bond: CF(t,T(30-year,1)), CF(t,T(30-year,2)), . . ., CF(t,T(30- year,60)),
P(t,3-month) = d(t,T(3-month,1))×CF(t,T(3-month,1)) P(t,6-month) = d(t,T(6-month,1))× CF(t,T(6-month,1))
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The last cash flow of each series consists of the principal payment and, for the notes and bond, one coupon payment. The solution to these simultaneous equations furnishes many distinct points of term in which the discount function is defined; the long bond alone may have as many as 60 term points. Depending on the circumstances surrounding each auction, there may be as many as over 90 distinct points of term defin- ing the discount function.
As with the earlier “connect-the-dots” model for the yield curve, in which the yield points were connected to generate intermediate values for the term structure, similar ideas can be used to accommodate the cash flows that do not fall on one of the terms, T(i,j), enumerated above. In fact, interpolation techniques using spline functions may be applied to create a continuous discount-function curve.1
The discount function forms the basis for the development of a term structure model, as will be developed further in later sections. As the discount function is an expression of the term structure based on price, there is no ambiguity of compounding periodicity, as with yield based term structure models. The discount function simply expresses the non- dimensional, fractional, present value of a unit cash flow to be received after some term. The term may be specified in a unit of time (e.g., years, months, or days) or in periods, in which the period length is a unit of time.
Spot-Yield Curve
With the assumption of a compounding convention (usually semian- nual), the discount function can be used to derive the equivalent Trea- sury zero-coupon structure—sometimes referred to as the spot-yield curve. In this case, the spot-yield curve is an equivalent term structure representation based on yield that provides a view of the term structure
P(t,2-year) = d(t,T(2-year,1)) ×CF(t,T(2-year,1)) +d(t,T(2-year,2)) ×CF(t,T(2-year,2)) +d(t,T(2-year,3)) ×CF(t,T(2-year,3)) +d(t,T(2-year,4)) ×CF(t,T(2-year,4))
. . . . . .
P(t,30-year)) = d(t,T(30-year,1)) ×CF(t,T(30-year,1)) +d(t,T(30-year,2)) ×CF(t,T(30-year,2)) . . .
+d(t,T(30-year,60)) ×CF(t,T(30-year,60))
1See Oldrich A. Vasicek and H. Gifford Fong, “Term Structure Modeling Exponen- tial Spline,” Journal of Finance (May 1982), pp. 339–348.
Term Structure Modeling 103
that is more familiar to readers. The equivalence between these two forms of the term structure is used later in this chapter.
Thespot yield,R, is related to the discount function, d, through the price/yield relation. By definition of the internal rate of return (IRR), the present value at t, PV(t,n), of a cash flow received n periods in the future,CF(t,n), has the IRR (or spot yield), R(t,n), through the relation
(2)
We use the discrete notion of integer periods, with each period of length P, to keep the math simple at this point.
Comparing equations (2) and (1) provides the relation between the spot yield and the discount function
(3)
where
The spot-yield curve is just the set of spot yields for all terms-to- maturity. In contrast, the spot rate is simply the one-period rate prevail- ing on t for repayment one period later. In the above notation, the spot rate is denoted R(t,1).
We can generalize the earlier comment about coupon paying bonds in terms of the set of spot yields. The present value of a coupon paying instrument is simply the sum of the discounted (present value) of the cash flows that make up the coupon payments and the payment of prin- cipal. The analogy to equation (2) for a coupon paying bond using spot yields is
(2a)
Similarly, the analogy to equation (1) for a coupon paying bond using the discount function is given by
PV(t,n) = d(t,1)×CF(t,1) + d(t,2)×CF(t,2) + . . . + d(t,n)×CF(t,n) (1a) d(t,n) = discount of a cash flow received n periods after t
R(t,n) = n-period spot yield on t PV t n( , ) CF t n( , )
1+R t n( , )
[ ]n
---
=
d t n( , ) 1 1+R t n( , )
[ ]n
---
=
PV t n( , ) CF t( ,1) 1+R t( ,1)
[ ]
--- CF t( ,2) 1+R t( ,2)
[ ]2
--- … CF t n( , )
1+R t n( , )
[ ]n
---
+ + +
=
5-Audley/Chin-TermStructModel Page 103 Thursday, August 29, 2002 10:00 AM
Implied Forward Rate
A consequence of the discount function, spot yield, and spot rate is the immediate relation to the (implied) forward rates. The implied forward rate is the spot rate embodied in today’s yield curve for some period in the future. The forward rate generally is regarded as an indication of future spot rates in an arbitrage-free economy. In the absence of arbitrage and uncertainty, the future spot rate, by definition, is equal to the forward rate.
In the arbitrage-free term structure model discussed later in this chapter, it can be shown that the future spot rate continuously converges toward the forward rate as the spot rate evolves over time.
Specifically, the one-period forward rate, F, can be determined from the spot yields as follows. Consider the one-period and two-period spot yields; the forward rate, F, may be found from
(1 + R(t,2))2 = (1 + R(t,1))× (1 + F(t,1,1)) (4) where
This relation follows from the no-arbitrage assumption intrinsic in the concept of forward rates. The calculation of the forward rate pre- sumes that an investment today for two periods provides the same return as a one-period investment today immediately rolled into another one-period investment one period from now. That is
(5)
(6)
By equating equations (5) and (6), equation (4) results.
Deriving Forward Rates from Spot Yields
Implied from the term structure, through the spot-yield curve, is a set of forward rates. These forward rates may be iteratively defined from the above and written as follows
(1 + R(t,n))n = (1 + R(t,n−1))n−1× (1 + F(t,1,n−1)) R(t,2) = two-period spot yield on t
R(t,1) = one-period spot rate on t
F(t,1,1) = one-period forward rate one-period from t
PV t( ) CF t( ,2) 1+R t( ,2)
[ ]2
---
=
CF t( ,2) 1+R t( ,1)
[ ]×[1+F t( , ,1 1)] ---
=
Term Structure Modeling 105
where in addition to the earlier notation, F(t,1,n−1) = one-period for- ward rate n−1 periods from t, and noting, through substitution, that
(7) which furnishes the first n− 1 one-period forward rates.
The relation between spot yield, spot rate and forward rates, equa- tion (7), can be combined with equation (2) to furnish a method for cal- culating the present value, at t, of a single n-period future cash flow based on a series of one-period forward rates
(8) Since the present value of a coupon paying security is simply the sum of the discounted present value of the cash flows that make up the coupon payments and the payment of principal [see equations (la) and (2a)], the analogy to equation (8) for determining the present value of a coupon paying bond is
(8a)
Equation (8a) may be used to define multi-period forward rates.
Deriving Forward Rates from the Discount Function
The discount function provides a direct method for generating forward rates. The one-period forward return n − 1 periods from t is obtained through the following
(9) 1+R t n( , )
( )n
1+R t( ,1)
( )×(1+F t( , ,1 1))×(1+F t( , ,1 2))×…×(1+F t( , ,1 n–1))
=
PV t n( , ) CF t n( , ) 1+R t( ,1)
[ ] …× ×[1+F t( , ,1 n–1)] ---
=
PV t n( , ) CF t( ,1) 1+R t( ,1)
[ ]
---
=
CF t( ,2) 1+R t( ,1)
[ ]×[1+F t( , ,1 1)] --- +
…
+ CF t n( , )
1+R t( ,1)
[ ] …× ×[1+F t( , ,1 n–1)] --- +
1+F t( , ,1 n–1) d t n( , –1) d t n( , ) ---
=
5-Audley/Chin-TermStructModel Page 105 Thursday, August 29, 2002 10:00 AM
Equation (9) may be derived from earlier equations, or from the fol- lowing argument that creates a synthetic forward position. For each unit of cash delivered n periods from today, t, we pay d(t,n). We take a long position in this zero. We also short d(t,n)/d(t,n−1) units of cash to be delivered n − 1 periods from t. For this we receive d(t,n−1) times d(t,n)/d(t,n−1), or simply d(t,n), units. There is no net change in our cash position today. After n − 1 periods we pay out d(t,n)/d(t,n−1) and aftern periods receive one unit of cash. Thus the forward price per unit, FP, to be paid n− 1 periods from now is
(9a)
where
The forward price then gives the forward one-period rate, n − 1 periods from t as
(9b)
Equating (9a) to (9b) results in Equation (9).
Term Structure in a Certain Economy
As discussed earlier, term structure models describe the evolution of inter- est rates over time. Often, future interest rates are expressed in terms of the future spot rate. If the future spot rate (or equivalently, the future rate of return on a bond) is known, the future term structure of interest rates may be found from the previously established inter-relationships between the spot rate and the discount function or spot yield. In fact, it is this relationship between the spot rate and the discount function that is used to motivate the formulation of the term structure model described later in this chapter as a function of the spot rate. As a precursor to a general- ized term structure theory, we first discuss the ramifications for a term structure in a certain economy.2
If the future course of interest rates is known with certainty, then arbitrage arguments demand that future spot rates be identical to future FP(t,1,n−1) = forward price of a one-period unit of cash n− 1 peri-
ods from now
2In this context, “certain” refers to an economy with a lack of randomness, in other words, a lack of uncertainty.
FP t( , ,1 n–1) d t n( , ) d t n( , –1) ---
=
FP t( , ,1 n–1) 1 1+F t( , ,1 n–1) ---
=
Term Structure Modeling 107
forward rates. In the notation presented in equation (7), this is equiva- lent to noting that
R(t + nP,1) = F(t,1,n) (10)
forn = 1, 2, 3, . . . and where P is the term of the period. If this condi- tion were violated, say, for example,
F(t,1,n) > R(t+nP,1)
then the same arbitrage argument may be made as before: If we buy the synthetic forward (this is a long position in a unit zero to be delivered n + 1 periods from today, t); and short d(t,n + 1)/d(t,n) units of cash to be delivered n periods from today, t, no cash changes hands today. How- ever, after n periods, we pay the forward price, FP,
to receive one unit of cash after n + 1 periods. Also, after n periods, at t +nP, we sell the one-period unit zero for a price of
We know we can do this since there is no uncertainty in the econ- omy. If, as assumed, F(t,1,n) > R(t + nP,1), then after n periods the long and short positions yield a positive net cash flow, or a riskless arbitrage, of
aftern periods with no uncertainty and with no net investment. Arbitra- guers will exploit the imbalance of the n-period forward rate with the spot rate n periods from now by continuing to buy the synthetic forward until demand outstrips supply. In this scenario, the synthetic forward price goes up, and the forward rate, F(t,1,n), goes down to R(t + nP,1)—
with predictable effect on d(t,n+1) and/or d(t,n). On the other hand, if F(t,1,n) < R(t + nP,1), we may reverse our positions and the same argu- ment carries through to show F(t,1,n) will increase to R(t + nP,1).
FP t( , ,1 n) 1 1+F t( , ,1 n) ---
=
1 1+R t( +nP,1) ---
1 1+R t( +nP,1)
--- 1 1+F t( , ,1 n) ---
– >0
5-Audley/Chin-TermStructModel Page 107 Thursday, August 29, 2002 10:00 AM
Using the no-arbitrage condition in a certain economy, equation (10), in the present value expression from the implied forward-rate expression, equation (8) (which always holds irrespective of assump- tions about the economy), we have,
(11)
This means that the certain return of holding an n + 1 period zero until maturity is the same as the total return on a series of one-period bonds over the same period. Later we will discuss the various forms of equation (11) from various qualitative term structure theories.
Given equation (11), we have, at time P (one period) later,
so we find that the single-period return on a long-term zero is
(12) Since the term-to-maturity was not specified, equation (12) must be true for zeros of any maturity. That is, the return realized on every dis- count bond over any period is equal to one plus the prevailing spot rate over that period. This will be expanded upon later in this chapter.
Alternatively, we can use our relation for the discount function in Equation (1), noting
and
and restate equation (12) in terms of the discount function
(12a) PV t n( , ) CF t n( , +1)
1+R t( ,1)
[ ]×[1+R t( +P,1)]×…×[1+R t( +nP,1)] ---
=
CF t n( , +1) 1+R t n( , +1)
[ ]n+1
---
=
PV t( +P,n) CF t n( , +1) 1+R t( +P,1)
[ ] …× ×[1+R t( +nP,1)] ---
=
PV t( +P) PV t( )
--- = 1+R t( ,1)
PV t( +P,n) = d t( +P,n)×CF t n( , +1)
PV t n( , ) = d t n( , +1)×CF t n( , +1)
d t( +P,n) d t n( , +1)
--- = 1+R t( ,1)
Term Structure Modeling 109
While these developments for the certain economy may appear triv- ial and obvious, they serve as a guide for modeling the term structure under uncertainty as well.
Term Structure in the Real World—Nothing Is Certain
In the real-world economy, the future course of interest rates contains uncertainty. In attempting to deal with uncertainty, however, it would not be inconceivable that a belief in the efficiency of the market would prompt one to use the term structure and the relation between forward rates and spot rates as indicators of expectation about the future. Indeed, market efficiency states that prices reflect all available information bear- ing on the valuation of the instrument. Equilibrium supply and demand for fixed-income instruments reflect a market cleared consensus of the economic future. As uncertainty represents a departure from this consen- sus, the expected equilibrium offers a natural starting point for analysis.
Expectations Hypothesis
The expectations theory of the term structure of interest rates offers a good starting point for dealing with an uncertain future. Actually, there is a whole family of expectations theories. Broadly, the expectations the- ory states that the expected one-period rate of return on an investment is the same, regardless of the maturity of the investment. That is, if the investment horizon is one year, it would make no difference to invest in a one-year instrument, a two-year instrument sold after one year, or two sequential six-month instruments.
The most common form of this statement uses equation (10) as the basis for the theory. This is referred to as the unbiased expectations hypothesis, which states that the expected future spot rate is equal to the forward rate, or
fork = 0, 1, . . ., n− 1, and where E[⋅] is the expectation operator.
Using this relation, we find from equation (8) that the present value in an economy characterized by unbiased expectations is
(13)
Therefore, the unbiased expectations hypothesis concludes that the guaranteed return from buying a (n + 1) period bond and holding it to
E R t[ ( +nP,1)] = F t( +kP, ,1 n k– )
PV t n( , )
CF t n( , +1) 1+R t( ,1)
[ ]×{1+E R t[ ( +P,1)]}×…×{1+E R t[ ( +nP,1)]} ---
=
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maturity is equivalent to the product of the expected returns from hold- ing one-period bonds using a strategy of rolling over a series of one- period bonds until maturity.
Alternatively, the return-to-maturity expectations hypothesis is based on equation (11). Here we find that present value in such an economy is
(14)
The return-to-maturity expectations hypothesis assumes that an investor would expect to earn the same return by rolling over a series of one-period bonds as buying an (n + 1)-period bond and holding it to maturity.
The last version of the expectations hypothesis that we will mention (there are others) is the local-expectations hypothesis (or risk-neutral hypothesis). This hypothesis is based on equation (12), or equivalently, the discount-function based equation (12a). Under this hypothesis, the expected rate of return over a single period is equal to the prevailing spot rate of interest. Applying these expressions recursively gives
(15)
Equations (13), (14), and (15) are clearly different in that the coeffi- cient of the cash flow, CF(t,n+1), received n + 1 periods in the future is a different expression in each case. Furthermore, by the principle from mathematical analysis known as Jensen’s inequality, only one of the expressions can be true if the future course of interest rates is uncertain.
In fact, in discrete time, we find that bond prices given by the unbi- ased and return-to-maturity hypotheses are equal but less than that given by the expectations hypothesis. Although the three hypotheses are different, in discrete time, any of these hypotheses is an acceptable description of equilibrium.
PV t n( , )
CF t n( , +1)
E{[1+R t( ,1)]×[1+R t( +P,1)]×…×[1+R t( +nP,1)]} ---
=
PV t( ) E PV t[ ( +P)] 1+R t( ,1)
[ ]
---
=
E PV t( +2P) 1+R t( +P,1)
[ ]×[1+R t( ,1)] ---
=
CF t n( , +1)
=
E 1
1+R t( ,1)
[ ]×[1+R t P( + ,1)] …× ×[1+R t nP( + ,1)] ---
×