INTEREST RATE MODELS

INTEREST RATE MODELS

Classifications of Interest Rate Models

·         Discrete vs. Continuous

·         Single Factor vs. Multiple Factors

·         General Equilbrium vs. Arbitrage Free

Discrete Models

·         Discrete models have interest rates change only at specified intervals

·         Typical interval is monthly,daily, quarterly or annually also feasible

·         Discrete models can be illustrated by a lattice approach

Continuous Models

Interest rates change continuously and smoothly (no jumps or discontinuities)

Mathematically tractable

Accumulated value = e^rt

Example

            $1 million invested for 1 year at r = 5%

            Accumulated value = 1 million x e^.05 = 1,051,271

Single Factor Models

Single factor is the short term interest rate for discrete models

Single factor is the instantaneous short term rate for continuous time models

Entire term structure is based on the short term rate

For every short term interest rate there is one, and only one, corresponding term structure

Multiple Factor Models

Variety of alternative choices for additional factors

Short term real interest rate and inflation (CIR)

Short term rate and long term rate (Brennan-Schwartz)

Short term rate and volatility parameter (Longstaff-Schwartz)

Short term rate and mean reverting drift (Hull-White)

General Equilibrium Models

Start with assumptions about economic variables

Derive a process for the short term interest rate

Based on expectations of investors in the economy

Term structure of interest rates is an output of model

Does not generate the current term structure

Limited usefulness for pricing interest rate contingent securities

More useful for capturing time series variation in interest rates

Often provides closed form solutions for interest rate movements and prices of securities

Arbitrage Free Models

Designed to be exactly consistent with current term structure of interest rates

Current term structure is an input

Useful for valuing interest rate contingent securities

Requires frequent recalibration to use model over any length of time

Difficult to use for time series modeling

Which Type of Model is Best?

There is no single ideal term structure model useful for all purposes

Single factor models are simpler to use, but may not be as accurate as multiple factor models

General equilibrium models are useful for modeling term structure behavior over time

Arbitrage free models are useful for pricing interest rate contingent securities

How the model will be used determines which interest rate model would be most appropriate

Term Structure Shapes

Normal upward sloping

Inverted

Level

Humped

Litterman and Scheinkmann (1991) investigated the factors that affect yield movements

Over 95% of yield changes are explained by a combination of three different factors

• Level
• Steepness
• Curvature

Level Shifts

Rates of maturities shift by approximately the same amount

Also called a parallel shift

Steepness Shifts

Short rates move more (or less) than longer term interest rates

Changes the slope of the yield curve

Curvature Shifts

Shape of curve is altered

Short and long rates move in one direction, intermediate rates move in the other