The Bootstrap Method
It is convenient to be able to determine a set of discount rates for zero coupon bonds from current coupon bonds. Recall, for example, the immunization problem of the previous chapter for which the liability had duration 10 years but there were no matching zero coupon bonds available to hold against that liability. Bootstrapping is a method that extracts discount rates for zero coupon bonds using observed prices on existing coupon bonds. I will demonstrate the method and follow it with an application.
Assume all bonds have a face value M of $100. The following table lists some hypothetical bond information on maturity, coupon, and price.
Bootstrap Table A.
We will use continuous discounting, which means that the continuously discounted rate rc is determined as before, that is, as
Taking natural logarithms extracts the continuously compounded rate of return, viz.,
The first bond is a 90-day maturity zero coupon bond that returns $1.5 on the $98.5 price for a discrete annual return of 4∗$1.5/$98.5 = 6.09 percent. Converting to continuous time, this rate is 4∗ln(1 + .0609/4) = 6.04 percent. We do likewise for the six-month and the one-year bonds, which have discrete annual returns of 6.19 percent and 7.53 percent, respectively. Their continuously compounded analogs are 6.14 percent and 7.46 percent, respectively. The 1.5-year bond pays a $2 semiannual coupon and has market price equal to $94.
This is easy to solve and gives rc = 8.19 percent, which is the yield on a 1.5-year zero bond. Likewise, for the two-year bond paying a semiannual coupon of $4 with market price $98, we solve
This returns a two-year zero yield, equal to 9 percent.
Bootstrap Table B.
The last column in the table shows the continuous zero coupon yield. Using this method we can bootstrap an entire set of zero coupon yields across the term structure.
A yield curve play is a strategy that tries to exploit the forward rate structure. If, for example, we believe that short rates will remain low and the term structure is rising, then we could conceivably borrow by paying lower short yields and investing long, rolling over our short-term loans over time. Using the preceding table and forecasting that rates on 90-day bonds will stay near 6 percent, then we borrow at this rate and invest in two-year bonds yielding 9 percent. Every quarter, we could roll over our loans, effectively refinancing at the new 90-day yield. As long as short rates stay low, this is a good bet. But, as we saw in the previous chapter, this play does not come without its risk—the Fed could raise rates, for example, which would not only increase the cost of short-term borrowing but also depress the value of the bonds we invested in (as their yields begin to rise). The classic case of the yield curve play going wrong is Orange County, California in the mid-1990s.