Jun 20

Practical toolkit for term structure estimation (Paper & XLS software)

by David Harper, CFA, FRM, CIPM

Here is a fantastic primer on the term structure by the folks at fixedincomerisk.com. For the FRM candidate, this is chock-full of useful stuff (if you would like to reinforce some of the Tuckman). Also, some really clear math in case you'd like more practice with exponential function:

Building blocks: bond prices, spot rates, forward rates. You need to know how spot rate curve embeds a forward curve.
Day count basis: "The Actual/Actual basis is used for Treasury bonds, the Actual/360 basis is used for U.S. Treasury bills and other money market instruments, and the 30/360 basis is used for U.S. corporate and municipal bonds."
Yield to maturity (YTM): "â€¦is a complex weighted average of zero-coupon rates. The size and timing of the coupon payments influence the yield to maturity, and this effect is called the coupon effect. In general, the coupon effect will make two bonds with identical maturities but with different coupon rates or payment frequencies have different yield to maturities if the zero-coupon yield curve is non-flat. The coupon effect makes the term structure of yields on coupon bonds lower (higher) than the term structure of zero-coupon rates, when the latter is sloping upward (downward)."
Shape off term structure (inverted, humped, normal and steep): "The steep shape of the term structure typically occurs at the trough of a business cycle, when after many interest rate reductions by the central bank, the economy seems poised for a recovery in the future. The inverted shape of the term structure typically occurs at the peak of a business cycle, when after many interest rate increases by the central bank, the economic boom or a bubble may be followed by a recession or a depression"

They say a good term structure estimation method should meet four criteria:

Ensures a suitable fitting of the data.
Estimated zero-coupon rates and the forward rates remain positive over the entire maturity spectrum.
Estimated discount functions, and the term structures of zero-coupon rates and forward rates are continuous and smooth.
Allows asymptotic shapes for the term structures of zero-coupon rates and forward rates at the long end of the maturity spectrum.

Then they review "the three most commonly used" estimation approaches:

Bootstrapping
Polynomial/exponential spline (I have an 8 min screencast example of this here)
Nelson & Siegel (Exponential functional form). An elegant approach that uses a single exponential function over the entire maturity.