Multiple Interest Rate Factors
Key Rates & Perturbations
We assume that
where are the 2-year, 5-year, 10-year and 30-year par coupon rates.
For some security with deterministic cash flows, the first-order approximation can be expressed as
The quantities
can be thought of as the DV01s corresponding to the i-th key rate.
The quantities
can be thought of as the durations corresponding to the i-th key rate.
The second-order approximation takes the form
with multiple factors, duration is described by a vector and convexity is described by a matrix.
Key Rate Perturbations
Let’s assume that a change of one basis point in each individual key rate leads to a change in the par-coupon yield curve of basis points where the function are as shown in the graphs below (Note that all the impacts in the graphs are our assumptions.)
Par Coupon yields and Discount Factors
Note that we can derive the discount factors from the par-coupon yields.
Recall that the par coupon yield corresponding to maturity is given by
starting with a par-coupon yield curve , the discount factors can be computed recursively by
Construct a Hedge using Key Rates
Example
Generalization of the Method
Suppose we have sold a security whose price is given by
and we wish to hedge the sale of this security using a portfolio of basic securities. (Typically we will want securities in the hedging portfolio). If the price of the hedging portfolio is given by
then we choose the amounts of each of the securities in the hedging portfolio so that
This is analogous to matching all of the key rate DV01s. (note that for corresponding to the original yield curve - the perturbation is zero).
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