Initialization 

>	with(FixIncFun):

 

>

 

The term structure is a function from the time domain to the interest rate domain i.e., the argument of the function is a time and the value of the function is the spot interest rate at that time. A realistic market requires the knowledge of the discount factors and of the spot rates, not only for times one, two, and three, as in our examples, but for every possible in some interval . Sometimes the value of is required even for a time period which extends beyond the maximum maturity, time three in our example. The common practice in the marketplace is to use the knowledge of the spot rate of interest for the discrete points (1, 2, and 3 in our examples) and generate from these an estimation of a discount factor or a spot rate for each in a certain time interval. This process is referred to as smoothing.  

In the scope of this book, we will give only an overview of these techniques.  The Appendix will explore these ideas further.  Since spot rates are recovered from discount factors, and vice versa, the smoothing can be done in terms of either spot rates or discount factors. Our discussion of smoothing will be with respect to discount factors. Equation (3.2) showed how one determines the discount factor from the spot rate and vice versa. 

The general form of smoothing techniques is based on the assumption that the discount factor for time , is a function which can be approximated by some structure. Usually, the estimation of the discount factor, denoted below by is of the form 

,                                       

 (3.5) 

where the are certain functions and the , for i=1,...,K are numerical coefficients. Hence utilizing (3.5) implicitly assumes that the discount factor function can be written in such a form. The different smoothing techniques are based on choices of the approximating functions .   

The first step is to choose the set of    functions ,..., . Next, an optimization problem is solved in order to uncover the values of the for i=0,...,K, such that the present value of the cash flow from bond , where is the payment from bond i at time t, is as close as possible to , the market price of bond i.  

To this end, one must choose how to measure the distance between the present value, calculated based on , and the market price of the bond. Here again, methods differ from one another. Some selection criteria are based on the sum of squared differences and some on the sum of the absolute value of the differences.   

The procedure NarbitB is also capable of producing the smoothing. It uses the criteria of absolute values for measuring the distance. It is possible to think of, and display, the smoothing process as being done in two phases. In the first phase the discount factors for the discrete payment dates of the bonds are identified. In the second phase an optimization problem is solved. It uncovers the values of the for i=0,...,K, such that the value of the continuous curve is as close as possible to the value of the discount factors at the payment dates. Whenever this is possible, NarbitB will demonstrate it with a graph. The reader will find more information about these techniques and the specification of the functions   in the Appendix. 

Let us now examine an example of smoothing the discount factors. For simplicity we assume a market in which coupons are paid annually. Hence in NarbitB and its output, the time unit is a year. In order to accomplish this, we need to run the procedure NarbitB as before, but with the addition of two parameters.  We need to specify the number of approximating functions and the name which will be assigned to our new (smooth) discount factors function. This function is termed the continuous approximation of the discount factors. 

Suppose we would like to have six approximating functions (i.e., K=5), and we would like to call the continuous approximation ConApp.  We thus need to run NarbitB as demonstrated below. The output of the procedure also generates a graph.  This is illustrated in the Figure which appears directly following the output. (You can add a zero as a parameter at the end to suppress the graph of the discount factor, i.e., execute 'NarbitB([[105,0,0],[10,110,0],[8,8,108]],[94,97,85],6,ConApp,0);' instead of the below) 

>	NarbitB([[105,0,0],[10,110,0],[8,8,108]],[94,97,85],6,ConApp);

 

>

 

>	solve(d(t) = (1/(1+(1/2)*r(t)))^(2*t),r(t));

 

>	subs(d(t)=ConApp2(t),%):

 

>

 

>	r:=unapply(%,t):

 

>	simplify(r(t));

 

>

r(1);

 

>

r(2):

 

We can now plot this function and see how the spot rate in this market behaves. 

>	plot(r(t),t=0..3,labels=[Time,`Rate`],thickness=2,title=`A Continuous Approximation of a Term Structure`,titlefont=[TIMES,BOLD,10]);

 

>

 

The fact that the rates are reported on an annual basis allows us to compare the rates for different maturities. We see that the rates are decreasing as time increases and then, at about 1.5 years, the rates increase. Different shapes of the term structure are possible.  There are a few theories which explain the shape of the term structure of interest rates.  These theories are summarized in the Appendix.  

Footnotes