__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