The Term Structure, Zero-Coupon, and Spot Curves 

The term structure is actually a function, which we will denote by  , that relates time to the interest rate prevailing in the market. For each  , is the interest rate per period (a year) paid for a dollar invested for years, from time zero to time . We keep our units of time measured in years and report the interest rate based on semiannual compounding. Hence, is the rate such that a dollar invested from time zero to time will grow to be 

at time . This function is referred to as the term structure of interest rates, and for each the rate is called a spot rate.  

These are the rates paid on an investment over a period that starts now (on the spot) and ends at some future time . The graph of this function is sometimes referred to as the spot curve or the zero-coupon curve.  In what follows, we justify this name followed by a graphical representation of the term structure which will be explored shortly afterward.

Certain bonds pay only one payment at the maturity time of the bond. These bonds are referred to as zero-coupon bonds, since they make no coupon payments. Furthermore, zero-coupon bonds are sold at a discount. The interest rate offered by such bonds is implicit in the difference between the face value of the bond and its market price.  Namely, if the price of a zero-coupon bond maturing at time , years from now, is and its face value is $100, the interest it pays is the solution to the equation 

. 

(3.3) 

Hence will be given by  

>	solve({P=100*(1+r/2)^(-2*t)},{r});

 

>

 

where we assume again that is measured in years and compounding is done semiannually. 

The zero-coupon bonds are bonds which repay the principal at some time and pay nothing at any other time. The zero-coupon bonds, therefore, are the elementary building blocks of coupon bonds as the latter can be interpreted as a portfolio of zero-coupon bonds.   

Consider a bond that pays a coupon of   at times ,..., and matures at time where it pays . This bond may be replicated by a portfolio of zero-coupon bonds, each with a face value of $1. A portfolio composed of   units of zero-coupon bonds maturing at times ,...,, and of a zero-coupon bond which matures at time , has the same cash flow as the coupon-paying bond.  Zero-coupon bonds each with a face value of $1 is equivalent to of a zero coupon bond with a face value of $100. Hence, the above argument can be done utilizing zero coupon bonds with $100 of face value. 

If the principal (face value) of a zero-coupon bond maturing at time is $1, the price of the bond is the discount factor for time . Hence, the spot rates extracted from coupon-paying bonds or those extracted from zero-coupon bonds which are the building blocks of the coupon bonds, will be the same. Thus, the term structure of interest rates is sometimes referred to as the zero-coupon curve.

Let us consider an example in order to demonstrate this idea. We will start with a market of coupon bonds and estimate the discount factors. Then we will replace the bonds in this market by zero-coupon bonds with face values of $1. The prices we will assign to these zero-coupon bonds will be the discount factors obtained in the first market. We then re-estimate the discount factors. Of course, the discount factors in these two markets should be the same.  
 

Consider the market specified in NarbitB below, where executing NarbitB confirms that the no-arbitrage condition is satisfied, and the discount factors and the function Vdis are solved for. 

>	NarbitB([[105,0,0],[5,105,0],[4,4,104]],[94,97,89]);

 

>

 

The discount factors are specified above, but of course, the discount factor for time one will be the value of a zero-coupon bond maturing at time one and paying $1. Hence the discount factors for times one, two, and three are given by, respectively. 

>	Vdis([1,0,0]),Vdis([0,1,0]),Vdis([0,0,1]);

 

>

 

Let us now re-run NarbitB on a market composed of only these three zero-coupon bonds. The prices are specified above and applied to the corresponding zero-coupon bond.  

>	NarbitB([[1,0,0],[0,1,0],[0,0,1]],[94/105, 1943/2205, 180577/229320]);

 

>

 

We indeed confirm that the resultant market and discount factors are the same as those in the first market.  This should come as no surprise to the reader.  The bond market specified by the coupon bonds and the one specified by the zero coupon bonds are in fact the same markets.  Two markets are the same if every (risk free) cash flow in these two markets has exactly the same price. That this is the case in our two markets, is evidenced from the fact that the discount factors in these two markets are the same. The price of every (risk free) cash flow in each of these markets, , and is the value of the linear function 

.  

Consequently these two markets are equivalent. Hence, the zero-coupon curve, or the discount factor function, is like the DNA of the market.  

There is another way of viewing equivalent markets. The initial securities defining a market are called primary securities. Consider the market where the primary securities were the coupon bonds. If a primary security is removed from the market (the market is assumed to be complete and not including redundant bonds) and another security (replicated by the primary securities) is added in its place, an equivalent market is obtained.  Let us demonstrate this below.   

The following structure calculates a portfolio of the three primary bonds that replicate the cash flow 0,1,0 . It also calculates the price of this portfolio. 

>	solve({105*B1+5*B2+4*B3=0,0*B1+105*B2+4*B3=1,0*B1+0*B2+104*B3=0,Price=B1*94+B2*97+B3*89});

 

>

 

Notice that the cash flow 0,1,0 can be generated by a combination of bond 1 and bond 2, specifically a short position in bond 1 and a long position in bond 2. Not surprisingly the price of this portfolio is exactly the value of the second discount factor.  If we now replace one of the primary bonds, bond 1 or bond 2, with this new portfolio we will again obtain an equivalent market to our original market. This is demonstrated below by solving, utilizing NarbitB, for the discount factors in this market and noticing that they are the same as the original discount factors. 

>	NarbitB([[0,1,0],[5,105,0],[4,4,104]],[1943/2205,97,89]);

 

>

 

Yield and Spot Curves
 

The financial press reports a measure of a bond's return, which is called a yield. The yield of a bond maturing in years, is defined as the rate that solves the equation 

, 

(3.4) 

where is the coupon of the bond, is its face value, and is its price. Therefore the yield of a bond is a rate such that if payments from the bond obtained in years are discounted to its present value by , the price of the bond equals its present value.  Readers who are already familiar with the concept of internal rate of return will immediately recognize that the yield is the same.  From equation (3.3) it follows that the yield of a zero-coupon bond, maturing at time   and having a face value of $1, is the spot rate for time  .  Hence, if the market is populated only with such zero-coupon bonds, the concept of a yield would be equivalent to that of a spot rate. Let us calculate the yield for some zero-coupon bonds using the examples above.   

Consider the bond with the semiannual cash flow of  ().  Since, for this bond,   and , equation (3.4) reduces to equation (3.3).  Hence we can solve for the yield of this bond by solving the equation: 

There might be multiple solutions to this equation, and not all solutions must be positive numbers or even real numbers. Let us solve for y numerically (note we use 'fsolve' instead of 'solve') 

>	fsolve(Vdis([0,0,1])=(1+y/2)^(-3));

 

>

 

Our structure below can solve the equation symbiotically and chooses those solutions which are both real and positive. 

>	#evalf(map(proc (x) if type(x,'realcons') and evalf(0 <= x) then x else NULL fi end,\

 

>	[solve(Vdis([0,0,1])=(1+y/2)^(-3))]));

 

If no positive solution exists, a NULL will be returned. 

To compare and to confirm that this is the same result as that obtained from NarbitB, we can calculate the interest rate which spans the time interval .  This is done below.   

>	(1+%/2)^3-1;(1/Vdis([0,0,1.0]))-1;

 

We confirm that the results are those reported by NarbitB, albeit 1.5 years is three units of half a year and thus reported as time three in NarbitB. The yield of a zero-coupon bond coincides with the spot rate for the length of time corresponding to the maturity of the bond.   

A different result is obtained for the case of yields of coupon-paying bonds.  We examine now the yield of the bond maturing in a year with a cash flow of () and a current price of $97.  To solve for the yield of this bond, equation (3.4) is solved, using values of   and , i.e  

>	fsolve(5/(1+y/2)+105/(1+y/2)^2=97);

 

>

 

>	#evalf(map(proc (x) if type(x,'realcons') and evalf(0 <= x) then x else NULL fi end,\

 

>	[solve(5/(1+y/2)+105/(1+y/2)^2=97)]));

 

>

 

This time if we solve for the one year spot interest rate, based on this yield, we obtain 

>	(1+%/2)^2-1;(1/Vdis([0,1.0,0]))-1;

 

while, on the other hand, the result from NarbitB is 0.1348.  This is not due to a roundoff error.   

This example serves to demonstrate a few key points.  The yields of two bonds which mature at the same time may be different even though their maturity dates coincide.  It also highlights that the yield measure suffers from some deficiencies.  The price of a bond is the discounted value of its future cash flow. This is a consequence of the no-arbitrage condition.  Hence, the contribution of each future payment to the price of the bond is the present value of that payment.  

The price of a sure dollar to be obtained in a future time period is independent of the bond from which it is to be obtained.  The bonds, as explained above, can all be constructed from the same basic building blocks:  one dollar obtained at time .  These building blocks must therefore also have the same value in the current time period, regardless of the bonds they are used in. The yield concept, however, is a result of using different discount factors for payments obtained from different bonds at the same time.   

As was demonstrated by the example, a dollar obtained from the coupon bond with a maturity of one-year ( obtaining in one year) was discounted using the yield of the bond, which was different from the one year spot rate.  There is no economic rationale for using different discount factors for a dollar which has the same risk characteristics just because it is obtained from a different financial instrument, i.e., a zero-coupon versus a coupon-paying bond.  Hence, the calculation of present values using yields (unless it is a zero-coupon bond) is an incorrect procedure. 

One should think about the yield to maturity of a bond as some sort of "average" or mixture of returns on the bond.  We know that instead of in equation (3.4) we should have used  , where is the spot rate for years.  Solving for the yield as is done in equation (3.4), we are actually constraining the spot rate to be the same for all time periods (referred to as a flat term structure of interest rates).  The is really some sort of a "mixture" of the different spot rates .   

We demonstrate this below by solving for , in terms of and .  Given a bond which matures in one year, pays a coupon of  ,  and has a face value of , we display the first of the solutions produced (this is the meaning of the "[1]" below) . 

>	assume (r0.5>0,r1>0,c>0,FC>0);

 

>

 

>	simplify(solve((c/2)/(1+r0.5/2)+(FC+(c/2))/(1+r1/2)^2=(c/2)/(1+y/2) +(FC+(c/2))/(1+y/2)^2,y)[1]);

 

>

 

The next section elaborates on the term structure of interest rates and on its estimation.