## Saturday, 15 June 2013

### 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 = ert
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
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