3. The Term Structure, its Estimation, and Smoothing 69
3.3.2 Forward Rate: A Practical Approach
A forward rate (not to be confused with a forward agreement or with a forward price) is not only a theoretical concept but an implied rate to prevail in the future. It is actually possible to borrow at this rate. It is the rate at which an investor can secure, or commit to, a loan now (at time zero) which will be taken at some future time and be repaid at a later time . Hence, in a sense, the forward rate resembles a forward contract and perhaps its name originates from the similarity.
Coincide an agreement made now for delivery of a certain amount of money, at time t and repayment of $ plus interest at time t. The interest rate , is agreed upon now at time . Thus the amount paid back at time will be where we denote by the interest rate paid over the time interval . (For simplicity we suppress the 0 subindex and use the notion instead of ). Let us demonstrate the above with an example.
3.3.2 Forward Rate: A Practical Approach
A forward rate (not to be confused with a forward agreement or with a for- ward price) is not only a theoretical concept but an implied rate to prevail in the future. It is actually possible to borrow at this rate. It is the rate at which an investor can secure, or commit to, a loan now (at time zero) which will be taken at some future timet1and be repaid at a later timet2. Hence, in a sense, the forward rate resembles a forward contract and perhaps its name originates from the similarity.
Coincide an agreement made now for delivery of a certain amount of money, $x at time t1 and repayment of $x plus interest at time t2. The interest rater0(t1,t2), is agreed upon now at time 0. Thus the amount paid back at timet2 will be $x(1+r0(t1,t2))where we denote byr0(t1,t2)the interest rate paid over the time interval[t1,t2]. (For simplicity we suppress the 0 subindex and use the notion r(t1,t2) instead of r0(t1,t2)). Let us demonstrate the above with an example.
Consider a bond market with three periods and three bonds as specified inNarbitB:
> NarbitB([[105,0,0],[10,110,0],[8,8,108]],[94,97,85]);
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,..]
First we would like to check if it is possible to have a certain self-financing portfolio in this market. The reader will recall that a self- financing portfolio is one with a zero cash flow at time zero. We are in- terested in a self-financed portfolio with a cash inflow of $1 at time two followed by a cash outflow at time three. If such a portfolio exists in this market, we would like to know what the cash outflow will be at time three.
Since there are no arbitrage opportunities in this market, as was just confirmed, every self-financing portfolio which has a zero cash flow at time one and a cash inflow of $1 at time two, should have a cash outflow at time three. To this end, we solve the system of equations (3.10) where B1, B2, andB3 stand for the holdings of bonds one, two, and three in the portfolio, respectively. The cash outflow at time three is denoted by RF and is also one of the variables for which we seek a solution.
94B1+97B2+85B3=0
105B1+10B2+8B3=0 110B2+8B3=1
108B3=RF. (3.10)
This system of equations in (3.10) is solved below.
> solve({B1*94+B2*97+B3*85=0,B1*105+B2*10+B3*8=0,
B1*0+B2*110+B3*8=1,B1*0+B2*0+B3*108=RF},{B1,B2,B3,RF});
B1=− 37
412535,B2= 8173
825070,B3=− 1849
165014,RF=−99846 82507
Hence, we see that if we hold a long position in bond two of 8173 825070 units, a short position in bond one of− 37
412535units, and a short position in bond three of− 1849
165014 units, the cost of this portfolio is zero. It produces a cash inflow of $1 at time two and a cash outflow of −99846
82507 dollars at time three. The portfolio is purchased at time zero, at which time no cash changes hands. At time two, the buyer receives $1 and then repays the loan with 99846
82507 dollars at time three. The buyer is essentially “buying”
this portfolio at time zero to secure a loan which will be in effect from time two to time three. Let us see what interest rate the borrower who purchases this portfolio is paying.
We refer to the transaction above as a synthetic loan since it is the same sort of cash flow as a loan, but it is not a loan in the conventional sense.
The cash flows are those of a loan which is agreed upon today but which will be transacted at time two. The interest rate implicit in this loan is the value ofrwhich solves the equation
1+r= 99846 82507.
(3.11) Hence, the implicit rate is
> solve((1+r)=99846/82507);
17339 82507
Thus, given the current market conditions, the investor can secure, at time zero, a loan which will be transacted at time two and repaid at time three with an interest rate of 17339
82507. This rate is referred to as theforward rate from time two to time three. In the same manner, the forward rate from time one to time two can be calculated. This is left as an exercise for the reader.
The above argument demonstrates the concept of a forward rate by the replication argument. Let us see how the same concept will be explained utilizing the discount factor valuation approach. Consider the cash flow generated above. It costs zero to generate, produces $1 at time two, and
−$(1+r)at time three. We would, thus, like to find the numerical value ofr such that the value of the cash flow [0,1,−1−r] will be zero at time zero. That is, valuing from the point of view of time zero, what should a person return at time three for a dollar he or she receives at time two. This can be calculated easily by the Vdis function. We want to solve for the value ofrsuch thatVdis([0,1,−1−r]) =0. This is submitted to Maple.
> solve(Vdis([0,1,-1-r])=0);
17339 82507
Indeed, the same value for the forward rate is obtained. The concept of a forward rate is thus the rate, r(t1,t2), at which one can secure at time zero a loan from time 0<t1 to some time t2, t1 <t2. If the market is not complete, we cannot be sure that these types of loans can actually be executed by purchasing certain portfolio combinations. Nevertheless, the concept of a forward rate has been extended for everyt1andt2making use of the continuous approximation of the discount factors.
We can then offer yet another explanation for forward rates to justify equation (3.9). Given an approximation of the continuously compounded interest rater(t)spanning the time interval[0,t], proceed in the following manner. Consider an investor who borrows a dollar at time zero which will be returned at timet2. In other words the investor receivese(−r(t2)t2)dollars at time zero. This amount is immediately invested until time t1, where t1<t2. Hence, at timet1the investor will havee(r(t1)t1)e(−r(t2)t2)dollars and will have to give back one dollar at time t1. The interest rate implicit in such a loan, reported as a continuously compounded rate, is the r(t1,t2)
that solves
e(r(t1)t1)e(−r(t2)t2)e(r(t1,t2)(t2−t1))=1. (3.12) Multiplying (3.12) bye(r(t2)t2)results in equation (3.10), and hence the same solution for the forward rater(t1,t2)is obtained.