Let us generalize our model so that we can assume that the term structure uncertainty is governed by a more generalized process, not only by a par- allel shift. Consider the following market, where we estimate the discount function and try to extrapolate it to 6 years.
Note the ‘3’ inNarbitB([[105,0,0], [10,110,0], [8,8,108]], [94,97,85], 3, Dis, 1,6), stands for the number of approximating functions used to esti- mate the discount factor. ‘Dis’ is chosen by the user as a name for the discount factor function. ‘1’ is for producing the graph of the discount factor function and if a 0 is entered for this parameter no graph will be produced. Finally ‘6’ is the upper bound of the domain along which the discount factor function is estimated. The presumption of this procedure is that the time intervals are equal. That is the time interval between the cur- rent time (on which the price of the bonds are given) and the first coupon payment is equal to the time between the first and second coupon payment, etc.
> NarbitB([[105,0,0],[10,110,0],[8,8,108]],[94,97,85],
3,Dis,1,6);
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,‘Dis’, (.)
Let us first determine if the discount factor function is a perfect fit, as it does not seem so from the graph:
> SumAbsDiv;
135043 171680
Since the value is not zero we try to increase the number of the approx- imating functions from 3 to 7.
> NarbitB([[105,0,0],[10,110,0],[8,8,108]],[94,97,85], 7,Dis,1,6);
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, ‘Dis’, (.)
> SumAbsDiv;
0
Having obtained a good discount functionDis(t), we can define a func- tion for the term structure, i.e., forR(t). Assuming continuous compound- ing means thatDis(t) =e−R(t)t and consequently that the term structure is the functionRsuch that−ln(Dis(t))
t =R(t).HenceRis defined below :
> R:=unapply(evalf(-ln(Dis(t))/t),t);
R:=t→ −1t (1.ln(piecewise(0.≤tandt≤6.,
−0.1058861591t−0.02051130324t2+0.03426317753t3
−0.01285192791t4−0.0006447858262t5+0.001030316421t6
−0.0001698574536t7+0.000008634794668t8,Float(unde f ined) It is obvious that the function is not a line parallel to the x-axis, hence we are not dealing with a market in which the term structure is flat. For example the term structure for time 2 is
> R(2);
0.1113012862 While for time 5 it is
> R(5);
0.1858648871
To see a plot of the term structure we can issue the following command:
> plot(R(t),t=0..6,y=0..0.22);
Nevertheless, it is easy to show that if we continue to work with a mea- sure of duration, then as before, it will immunize us against changes in the
term structure of a parallel nature. These changes are of the type exem- plified as follows — they cause a shift in the term structure curve, up or down, by the same amount.
> plot([R(t)+0.05,R(t),R(t)-0.05],t=0..6, color=[red,green,blue]);
In the above figure the solid graph represents the current term structure of interest rate, R, and two possible parallel changes; an up shift of the curve by 0.05 and a down shift by −0.05. It is however not realistic to assume that the term structure will be changed by a parallel shift as it means that long term rates are changed by the same amount of the short term rates.
We thus further generalize the situation and assume that the term structure can change not only by parallel shifts, but allows forms that will affect its curvature.
Recall that under certain conditions every continuous function can be (uniformly) approximated as closely as desired by a polynomial function (the Weierstrass approximation theorem). We thus allow the change to the term structure to be an addition of a polynomial, and not only a constant as in the case of a parallel shift, and consider an addition of the form
n
∑
i=1
αit(i−1),
(4.26)
where the alphas are random coefficients. Note that wheni=1 the change is of a parallel shift and the duration we defined can be used to immunize the portfolio for such a case. In general, however, we assume that if the current term structure isR(t)it will be changed to
R(t)+
n
∑
i=1
αit(i−1).
(4.27) We first define, below, the discount factor functionDisGCas a function of the arrayα= [α1, . . . ,αn]andt, that is induced by such a change i.e,
DisGC([α1, . . .,αn],t) =e−
R(t)+∑n
i=1
αit(i−1)
(4.28)
> DisGC:=(t,alpha)->exp(-(R(t)+sum(alpha[i]*tˆ(i-1), i=1..nops(alpha)))*t);
DisGC := (t,α)→e
− R(t)+
nops(α)
∑
i=1
αiti−1
! t
There are two arguments to this function,tthe time variable and an ar- ray of alpha. Hence if one would like to consider a change to the term struc- ture that is of the form
3
∑
i=1
αit(i−1)where the alphas are [0.05,0.0,0.005] the new term structure will be
> DisGC(t,[0.05,0.0,0.005]);
e−(R(t)+0.05+0.005t2)t
If the array [0.05,.0,0.005] would be replaced by [0,0,0] then one would get the current discount factor function. Hence to plot the current discount factor function and the one that would result after such a change, i.e., an addition of 05+0t1+0.005t2 to the current term structure, one would execute:
> plot([DisGC(t,[0,0,0]),DisGC(t,[0.05,.0,0.005])], t=0..6,colour=[green,red]);
To plot the current term structure and the one that would result after such a change, i.e., an addition of 05+0t1+0.005t2 to the current term structure, one would execute:
> plot([-ln(DisGC(t,[0,0,0]))/t, -ln(DisGC(t,[0.05,.0,0.005]))/t], t=0..6,colour=[green,red]);
Note that indeed the changes we considered are not all feasible as we
cannot have a non-decreasing discount factor function. However, our im- munization strategy will not be adversely influenced by such an assumption (although it is not as efficient as it could have been).
Generalizing the possible changes to the term structure, of course, has its influence on the immunization strategy. When only a parallel shift in the term structure was considered, there was only one measure of dura- tion,
n
∑
t=1
w(t)t. In the generalized model described above there are many measures of durations that are defined similarly.
The next additional measure of duration is one that takes care of changes to the term structure that are induced byα2, an addition of a linear function. This measure is
n
∑
t=1
w(t)t2 or the convexity as defined by equa- tion (4.10). Consequently, if the current term structure isR(t)and after the change the term structure will be given in the form of R(t) +
2
∑
i=1
αit(i−1) immunization will require that the first order duration,
n
∑
t=1
w(t)t, of the as- set and the liability will be the same and also the second order duration
n
∑
t=1
w(t)t2(referred to as convexity) of the asset and the liability will be the same.
Allowing a more realistic change to the term, i.e., considering a largern in
n
∑
i=1
αit(i−1), requires more duration constraints to be satisfied. In general one must satisfy nduration constraints, of the form
n
∑
t=1
w(t)tn that insure that thenth order duration of the asset and the liability are the same.
This might not always be possible and second best solutions are some- times utilized. When short sales are allowed, more degrees of freedom are introduced and it might be possible to meet the duration constraints that are not feasible with long positions only.
To help us in the calculation we have defined the following function that calculates the kth order duration of a bond with a coupon rate of c where the discount function isd and the maturity of the bond isnunits of time (measured in years); GenDur(c,d,n,k). For simplicity this function
assumes that coupons are paid annually. Thus the first order duration of a bond with a coupon rate of 10%, a discount factor function named Dis and maturity of three years is given by executing:
> GenDur(0.10,Dis,3,1);
2.711204142
The table below demonstrates the value of the different duration order.
A B C D E F G
5 4 3 2 1
Durations of Diffrent Orders