The following article is reprinted from the July, 1996 issue
of On the Edge,
the Interactive Data Fixed Income Analytics bimonthly newsletter.
Treasury Curves in BondEdge
Teri Geske
Senior Vice President, Product Development
When we speak about the analytical
models, valuations and simulations in BondEdge, we often use the term "Treasury
Curve" as the starting point for the discussion. It is important to recognize that
various forms of the treasury curve are used in different analyses, depending upon the
task at hand. We thought it would be a good idea to review the concepts involved when the
term Treasury Curve is used, and to clarify how these concepts are applied throughout the
system.
The treasury curve displayed in Security Valuation and in Specified Scenario is a Yield
Curve which shows the Yield-to-Maturity of the Treasuries corresponding to nine maturity
points. The nine points which are displayed are either the "On-the-run" yields,
i.e. the yields of the most recently issued bonds at each maturity point, or are taken
from a fitted Treasury curve constructed from the population of Treasury bonds (certain
screening criteria are used to eliminate outlying instruments). By default, we display the
yields from the fitted Treasury curve, but you may switch to the On-the-run yields via the
system Defaults screen (under Utility - Miscellaneous - Defaults). Note that switching to
the On-the-run curve affects Specified Scenario and Security Valuation calculations only.
It would be inappropriate to use the yield-to-maturity of coupon-bearing Treasuries as
discount rates when pricing securities; to do so would cause cashflows which occur in the
same period to be discounted at different rates. Instead, we compute a series of
zero-coupon "spot rates" which correspond to each point on the yield curve, and
use those spot rates when discounting the cashflows which occur in a given period. The
spot rates are derived using a common technique which recognizes that a coupon-bearing
bond may be viewed as a portfolio of zero-coupon bonds comprised of coupon payments and
the principal payment at the maturity date. By "stripping" the coupon payments
out of the price of each Treasury in the yield curve, we are left with present value of
the principal payment at the maturity date of each treasury. This allows us to solve for
the discount rates, i.e. the spot rates which correspond to a zero-coupon bonds maturing
at the nine maturity points of interest.
These spot rates form the Spot Curve which is the basis for a number of computations in
BondEdge. When we solve for a security's option-adjusted spread (which is then used to
compute Effective Duration and Convexity), this OAS is computed relative to the Spot
Curve. When you change the Yield Curve in Specified Scenario, the ending Yield Curve is
translated into a Spot Curve, which is used to value each security as of the horizon date
using the OAS derived from the beginning Spot Curve. The Spot Curve is a primary input to
the corporate bond option model and the Monte Carlo-Representative Pathsİ algorithm used
to value mortgage pass-throughs, CMOs and ARMs. In the COMPARE system, we compute Key Rate
Durations by shocking each spot rate independently and observing the change in price due
to independent shocks to each spot rate.
The Spot Curve is the basis for computing the (Implied) Forward Curve. Whereas the Spot
Curve allows us to discount a set of future cashflows to today, the Forward Curve is used
to discount a single future cash flow to some nearer future date. The Forward Curve is
derived from the Spot Curve by observing that today's interest rates imply the interest
rate between two future dates. Consider the following risk-free zero-coupon bonds: the
first matures in 6 months with a yield of 5.00%, the second matures in one year with a
yield of 5.10%. We can choose between two investment strategies (i) buy the one year bond
and earn 5.10%, or (ii) buy the six-month bond and roll over into another six month
investment. The 6-month rate six months from now is implied by the fact that these two
strategies are expected to offer the same return. We solve for the interest rate which
satisfies the following equation: (1.0255)2 = (1.025)(1+X/2), where X is the
implied 6-month rate, six months from today (5.20% in this example).
Forward rates are used in valuing securities with embedded options (e.g. calls, puts,
prepayments, reset/lifetime caps and floors, etc.), whose expected future cashflows depend
upon the interest rate environment in the future. Although we usually speak in terms of
the short-term (e.g. 6-month) forward rate, when valuing mortgage pass-throughs and CMOs
we also compute 10-year forward rates, since the 10-year rate determines the refinance
incentive component of the Prepayment Model and we need to know the refinance incentive
which will exist in the future.
We hope this review has helped to clarify how BondEdge uses different types of Treasury
curves. For an in-depth discussion of term structure modeling, please contact your Interactive Data Fixed Income Analytics
representative to request a copy of "The H.J. M. Paradigm,
Theory and Application".
