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Financial Risk Management of Insurance Enterprises
Stochastic Interest Rate Models
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Today
• How do we model interest rates?– What is a stochastic process?– What interest rate models exist?
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 )( ),(
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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
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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
( , )
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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
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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
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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
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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 ( )
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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
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Next Time...
• More on interest rate models