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)

Properties of a Wiener Process 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 σ 2

Adjusting the Mean of a Brownian Motion Suppose our stochastic process x is expected to change in value by: 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 σ 2 dt

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

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

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

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 Variance Term Vasicek Model There are considerably different approaches to the variance term Vasicek model assumes constant volatility 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

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

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