3. The Term Structure, its Estimation, and Smoothing 69
3.3.1 Forward Rate: A Classical Approach
Let r(t) be the interest rate which prevails in the market from time zero to time t. For simplicity, we assume that the rates are quoted based on continuous compounding and that time is measured in years. Hence, a dollar invested at time zero for one year will grow into er(1)ã1and a dollar invested at time zero for two years will grow into er(2)ã2.
There is another way to invest money from time zero to time two. The dollar could be invested from time zero to time one, and then invested from time one to time two at the rate that will prevail in the market at that time.
At the current time this rate is unknown, as it is a random variable. Under this investment strategy, a dollar invested from time zero to time one will be worth er(1) at time one. This amount is then invested for one further time period at the rate prevailing in the market at that time. If the dollar had been invested for two time periods at the prevailing rate of interest covering the interval from time zero to time two, it would have grown to a value of er(2)ã2.
Let us denote byx1(1,2)the (as yet unknown) interest rate from time one to time two. The subindex 1 ofxmeans that this is the interest rate that will prevail at time one and the (1,2) means that this rate is spanning the time interval[1,2]. Hence, the dollar at time zero amounts to er(1)at time one, which grows to er(1)ex1(1,2)at the end of two time periods. The value ofx1(1,2)which solves
er(1)ex(1,2)=er(2)ã2 (3.8)
is theforward ratefrom time one to time two, as of time zero. As has been our custom before, we omit the index for zero and use two indexes in order to denote the span of time for which this forward rate is applicable. That is, we user(1,2) instead ofr0(1,2)in the same manner we user(t)instead ofr0(t)when the span is[0,t].
The intuitive meaning of the forward rate is that it is a rate implicit in the spot rate which will prevail in the market from time one to time two.
If an investor perceives thatr(2)is not high enough to induce the investor to pursue a two-period investment horizon, then that investor will prefer to use a rollover strategy. The investor invests for one period, from time zero to time one, and then “rolls over” the investment for another period, from time one to time two. From the definition of a forward rate it is apparent thatr(1,2)is the rate that makes our “rollover” strategy equivalent to the
“buy and hold” one. The rates prevailing in the market are those which induce an equilibrium: r(1,2)is the value which makes these two strate- gies equivalent. Hence, these rates convey information about the market’s anticipation of future rates which will prevail from time one to time two.
This discussion was based on an example. One can extract the forward
rate as of time zero for any future time periodt1tot2(t1<t2). A dollar in- vested from time zero to timet1will grow to be er(t1)t1 dollars. If the dollar is invested from time zero to timet2, it will grow to er(t2)t2 andr(t1,t2)will be the solution to
er(t1)t1er(t1,t2)(t2−t1)=er(t2)t2. (3.9) Thusr(t1,t2)denoted byr12 below will be given by:
> solve({exp(rt1*t1)*exp(r12*(t2-t1))=exp(rt)*t2}, {r12});
r12=−ln(t2) +rt−rt1 t1
−t2+t1
In our example we defined the continuously compounded term structure of interest rates as the functionTs(t). It is redefined here again in the steps below:
> NarbitB([[105,0,0],[10,110,0],[8,8,108]],[94,97,85], 5,ConApp2):
The no-arbitrage condition is satisfied.
The discount factor for time,1,is given by, 94
The interest rate spanning the time interval,[0,1],is given by,105 0.1170
The discount factor for time,2,is given by, 1849
The interest rate spanning the time interval,[0,2],is given by,2310 0.2493
The discount factor for time,3, is given by, 82507 124740
The interest rate spanning the time interval,[0,3],is given by, 0.5119 The function Vdis ([c1,c2,..]), values the cashflow [c1,c2,..]
The continuous discount factor is given by the function, ‘ConApp2’, (.) In our example we defined the continuously compounded term structure of interest rates as the functionTs(t).
> Ts:=unapply(-ln(ConApp2(t))/t,t);
We can also now define the functionFr(t1,t2), based on equation (3.10), as the forward rate from timet1to timet2.
> Fr:=(t1,t2)->(Ts(t2)*t2-Ts(t1)*t1)/(t2-t1);
Fr := (t1,t2)→ Ts(t2)t2−Ts(t1)t1 t2−t1
The functionFrshould always have the second argumentt2greater than the first argumentt1.
Armed with the function so defined, we can plot it together with the term structure and see how the term structure is related to the forward rate. In the following figure, the forward rate (the graph with the lower y-intercept) is graphed together with the term structure (the graph with the higher y-intercept), where the time domain is 0.5 to 3 and the forward rate is graphed from time 0.5 to any timetin the interval[0.5,3].
> plot([Ts(t),Fr(.5,t)],t=0.5..3,color=[green,red], thickness=2,labels=[Time,Rate],title=‘The Forward Rate and the Term Structure‘,titlefont=[TIMES,BOLD,10]);
We can also display the forward rate from timet1tot2 where we lett1 range from 0 to 3 and t2 fromt1 to 3. This is displayed in the following figure. Note that the graph of the forward rate in the preceding figure is a slice of the graph in the following one, fort1=0.5.
> plot3d(Fr(t1,t2),t1=0..3,t2=t1..3,axes=normal, labels=[‘Initial Time‘,‘End
Time‘,Rate],orientation=[-162,56], title=‘The Forward Rate from Time t1 to t2: 0<t1<3,
t1<t2<3‘,axes=framed,titlefont=[TIMES,BOLD,10]);