108156.pdf

10. Variable Interest Rates 231
    
    
    • Buy a bond maturing at timeNand write a fractionB(n, N)ofabond
      maturing atn. (Here we use the assumption that the future bond prices are
      known today.) This givesB(0,n)B(n, N)−B(0,N) dollars now.
    
    
    • At timensettle the written bonds, raising the required sum ofB(n, N)by
      issuing a single unit bond maturing atN.
    
    
    • At timeNclose the position, retaining the initial profit.
    
    
    
    

The reverse inequalityB(0,N)>B(0,n)B(n, N) can be dealt with in a similar
manner, by adopting the opposite strategy.

Employing the representation of bond prices in terms of yields, we have

B(n, N)=

B(0,N)

B(0,n)

=eτny(0,n)−τNy(0,N).

This would mean that all bonds prices (and so the whole term structure) are
determined by the initial term structure. However, it is clear that one cannot
expect this to hold in real bond markets. In particular, this relation is not
supported by historical data.
This shows that assuming deterministic bond prices would go too far in
reducing the complexity of the model. We have no choice but to allow the future
term structure to be random, only the initial term structure being known with
certainty. In what follows, future bond prices will be random, as will be the
quantities determined by them.

10.2.1 Forward Rates .....................................

We begin with an example showing how to secure in advance the interest rate
for a deposit to be made or a loan to be taken at some future time.

Example 10.10

Suppose that the business plan of your company will require taking a loan of
$100,000 one year from now in order to purchase new equipment. You expect to
have the means to repay the loan after another year. You would like to arrange
the loan today at a fixed interest rate, rather than to gamble on future rates.
Suppose that the spot rates arey(0,1) = 8% andy(0,2) = 9% (withτ=1).
You buy 1,000 one-year bonds with $100 face value, paying 100,000e−8%∼=
92 , 311 .63 dollars. This sum is borrowed for 2 years at 9%. After one year
you will receive the $100,000 from the bonds, and after two years you can
settle the loan with interest, the total amount to pay being 92, 311 .63e^2 ×9%∼=
110 , 517 .09 dollars. Thus, the interest rate on the constructed future loan will