Floating rate bonds and term structure of interest ratesl

In this lesson, we looked at pricing a contract when the future payouts depend on a floating rate.  The rate that will be paid out will not be revealed until that period occurs.

We used the theorem of linear pricing to do this: in linear pricing, given that the price of cash flow A is c(a) and the price of cash flow B is c(b), then the price of the contract that pays c = c(a) + c(b) = p(a) + p(b).   This just means that the price of the contract is the sum of the prices of all the individuals cash flows.

Then we moved on to floating interest rates.  We assume no knowledge of the interest rate in period k before period k begins.  Before the period k begins, we could model the interest rates as random quantities.



The cash flow of a floating rate bond that pays coupons can be modeled as follows:

According to the theorem of linear pricing, we can split the price of the bond into separate prices and then sum them.  We can model them as follows:

where the final coupon payment is based on r(0), the interest rate at the purchase of the bond.

Thus, the total price is:





Using this bond, we aim to build a portfolio that a has a deterministic cash flow, i.e. one that does not depend on the random quantities of the floating interest rate.
To do this, we will need to have an intermediate purchase and sale based on the floating bond rate.  We can construct the portfolio according to the table below:

We can see that in the beginning (period 0) we: buy the contract at price p(k), borrow alpha at rate r(0) from period 0 to period k-1, and loan alpha from period 0 to period k.

In period k-1, we: repay the loan from above at price alpha*(1+r(0))^(k-1) and we also take a loan in the same amount to finance that repayment.

In period k, we: receive financial payment from the bond of r(k-1)*F, repay the loan from period (k-1) of amount [alpha*(1+r(0))^(k-1)](1+r(k-1)), and receive our loan from period 0 plus interest for the amount alpha*(1+r(0))^k

We can see that the cash flow in the final period is:

Using this information, we can set this equation equal to zero and solve for alpha.  We then set alpha.

Using this alpha, we have changed the portfolio from being random to being deterministic. Finally, using the formula for total price above, we can price the portfolio.

The price of the floating rate bond is equal to the face value of the bond.


Now, we move onto term structure of interest rates.
Term structure - interest rate depends on length or duration of the loan i.e. the number of periods in each loan.  The longer the loan, the higher the interest rate.  This is due to multiple reasons:
1. Investors prefer liquidity so in order to induce loans for longer periods, investors must be enticed by higher rates.
2. Investors expect rates to either go up or go down in the future so the rate they receive today depends on how they expect rates to change in the future.
3. People offered different loans at different times must be offered different rates based on market segmentation.

We can model these rates with a simple change to our previous model of return on investment by changing the rate r to r(t), where t denotes the length of the loan/bond.  Thus we make the rate dependent on the length of the loan/bond.  In this case we use s(t) to denote the spot interest rate.

We can use current bond prices and rates on government bonds to infer the spot market rates.

We can also find the future rate between two periods, u and v, in the future using bond rates.  The forward rate can be modeled as follows:

where u is the beginning period of the loan and v is the ending period of the loan.

Another way to model the relationship is as follows:

This simply states that the spot rate between period 0 and t is equal to the product of the future rates for every year between 0 and t.