Financial Risk Management of Insurance Enterprises

Stochastic Interest Rate Models

Today

• How do we model interest rates?– What is a stochastic process?– What interest rate models exist?

Stochastic Processes• A stochastic process is an elaborate term for a

random variable

• Future values of the process is unknown

• We want to model some stochastic process– Future interest rates can be viewed as a stochastic process

• Basic stochastic processes:– Random walk– Wiener process– Brownian motion

Application of Stochastic Processes

• As we have seen, the cash flows of insurers can be dependent on the level of interest rates

• To help determine the range of potential outcomes to an insurer, we want to model interest rates

• Models of interest rates involve assuming some distribution of future interest rates

• This distribution is defined by the stochastic process

Random Walk• A college student leaves a bar late Saturday night

• He doesn’t know where home is and supports himself from the light posts down Green Street

• He can only move from one light post to the next– Unfortunately, when he gets to the new light, he forgets

where he came from

• On average, where does this college student wake up Sunday morning?– Right back where he started

Features of a Random Walk

• Memory loss– History reveals no information about the future

• Expected change in value is zero– Over any length of time, the best predictor of future

value is the current value– This feature is termed a martingale

• Variance increases with time– As more time passes, there is potential for being

farther from the initial value

Brownian Motion

• A Brownian Motion is the limit of the discrete case random walk– This is a continuous time process

• The simplest form of Brownian Motion is a Wiener process (dz)

dz dt~

~

Where is the stochastic term responsible for

changing the value of z. It is assumed to have

a standard normal distribution

Properties of a Wiener Process

E

Var

~

~

~

So, over some finite period (0,T), we are summing the

terms. Thus, the change in the variable has a normal

distribution

z(T) - z(0) ~ N(0,T)

0

1

• Thus, a Wiener process is a continuous time representation of the discrete time random walk

• A continuous time, stochastic process is also referred to as a diffusion process

Generalizing Pure Brownian Motions

• For most applications, the assumptions of a Wiener process (dz) do not fit with the modeled stochastic process

• For example, if we want to model stock prices– Expected future value is not the current value– The variance of the process is not 1

• Depending on the nature of the process, we can adjust the Wiener process to fit our needs

Adjusting the Variance of a Brownian Motion

• Suppose we want to model the stochastic process x, which has variance σ2

dx dz

E dx E dz E dz

Var dx Var dz Var dz

02 2

Adjusting the Mean of a Brownian Motion

• Suppose our stochastic process x is expected to change in value by:

dx dt dz • The μdt term is deterministic, i.e. has no randomness

– This term is called the “drift” of the stochastic process

• Now, the stochastic process predicts a change with a mean of μdt and a variance of σ2dt

Notes to Fabozzi

• Fabozzi’s definition of a standard Wiener processes includes the adjustments to the mean and variance

• Later he states the standard Wiener process has variance 1

• Bottom line: beware of the book’s terminology

Understanding Stochastic Processes

• Let’s interpret the following expression:

• First, recall that we are modeling the stochastic process x– Think of x as a stock price or level of interest rates

• The equation states that the change in variable x is composed of two parts:– A drift term which is non-random

– A stochastic or random term that has variance σ 2

– Both terms are proportional to the time interval

dx dt dz

Complications to the Process

• In general, there is no reason to believe that the drift and variance terms are constant

• An Ito process generalizes a Brownian motion by allowing the drift and variance to be functions of the level of the variable and time

dzx,tdttxdx )( ),(

Modeling Interest Rates

• In an earlier lecture, we described different approaches to interest rate modeling

• A one-factor model, as its name suggests, describes the term structure with one variable– Typically, this one-factor is the short term rate and all longer term

rates are related to short term rates

• Two-factor models have two variables driving the level of interest rates– Typically, one factor is the short-term rate and the other is a long-

term rate

One Factor Models

• There are various types of one factor interest rate models

• The short term interest rate is the underlying stochastic process considered

• No model is “perfect” in empirical studies

• Most models use a specific form of an Ito process

dr r t dt dz

r

(r, t)

Where now indicates the level of interest rates

( , )

The Drift Term

• A constant drift term does not make economic sense

• Most models assume mean reversion– There is a long-run average interest rate– Interest rates are drawn to this average rate

the long - run average rate of interest

the speed of mean - reversion

( , ) ( )

r t r r

r

The Variance TermVasicek Model

• There are considerably different approaches to the variance term

• Vasicek model assumes constant volatility

( , )r t • Assumption is the volatility is independent of the level of interest

rates and time

• Potential problem is the interest rates can become negative

Variance Term - Dothan

• Dothan model suggests that the volatility of interest rates is related to level of the rate

• This approach has intuitive appeal because empirically, as interest rates increase, their volatility does increase

( , )r t r

Variance Term - CIR

• CIR stands for Cox, Ingersoll, and Ross

• This model extends the Dothan model but the volatility is not as extreme

• The complete formulation of this model becomes a mean-reverting, square-root diffusion process

( , )r t r

dr r r dt rdz ( )

Applications• Use of these models requires estimating the

long-run average interest rate and the volatility– Also the strength of mean reversion

• Might use historical data to develop estimates

• Then, use a random number generator to take draws from the standard normal distribution

Next Time...

• More on interest rate models