 # Time Structure Models

## Equilibrium Term Structure Models

Equilibrium term structure models are built on theories about the economy. They explain the stochastic process that describes the dynamics of the yield curve (term structure). They are primarily based on macroeconomic variables. That includes inflation, unemployment, economic growth, GDP, and balance of payments.

### Properties of the Equilibrium Term Structure Models

Equilibrium term structure models have three main properties:

• They can be one-factor or multifactor models: While one-factor models work under the assumption that there is only one unique macroeconomic variable that affects the term structure of interest rates, multifactor models assume that interest rates are a function of multiple macroeconomic variables. Good examples of one-factor models are the Vasicek model and the CIR model. They assume a single factor, short-term interest rate(r), can explain all the interest rate movements.
• The behavior of the underlying factors is based on assumptions: For example, the short rate (r) can be assumed to show a mean-reverting behavior or even exhibit “jumps.”
• In contrast with arbitrage-free term structure models, equilibrium term structure models are generally more sparing with regard to the number of parameters that ought to be estimated.

## The Vasicek Model

The Vasicek model is a single-factor model that assumes the movement of interest rates can be modeled based on a single random) factor. It was introduced in 1977 by Oldřich Vašíček.

The model employs the following equation to model the short-term rate of interest, $$r$$:

$$dr=\alpha\left(\mu-{r}\right)dt+\sigma dw$$

The first part on the right-hand side of the equation is the deterministic part, while the second part is the stochastic part.

Where:

• $${\alpha}$$ determines the speed of the mean reversion, i.e., the speed at which the interest rate returns to its long-term mean level ($${\mu}$$). A higher $$\alpha$$ implies faster reversion.
• $${\mu}$$ is the long-term mean level of the interest rate, calculated based on historical data.
• dr is an infinitely small increment in the short-term interest rate
• $$dt$$ is an infinitely small increment in time
• $${\sigma}$$ is the volatility. It is a positive number, to ensure that r is mean reverting to the constant value $$\mu$$
• $$dw$$ is a stochastic process that models risk.
• $${\alpha} > 0$$, $$\mu > 0$$ and, $$\sigma$$ are constants

The stochastic part is normally distributed with a mean of 0 and standard deviation $$\sigma\sqrt t$$.

The Vasicek model assumes that interest rates do not increase or decrease to extreme levels but rather exhibit mean reversion. An increase in the short rate leads to a negative drift, pulling the rate back to $${\mu}$$. Similarly, a decrease in the short rate leads to a positive drift, pulling the rate back to $${\mu}$$.

Ideally, interest rates should be positive. Otherwise, it becomes difficult to find depositors because no one would want to pay in order to keep their money in the bank, rather than generating a profit. With negative interests, banks themselves would have to pay to keep their excess reserves at the central bank.

However, the Vasicek model allows interest rates to be negative.

Finally, the Vasicek model allows the yield curve to be upward-sloping, downward-sloping, or humped based on the selected parameter values.

## The Cox-Ingersoll-Ross (CIR) Model

Under the CIR model, interest rate movements can be explained in terms of an individual’s investment preferences and consumption and the risk-return payoff of the productive processes of the economy.

The model is given by:

$$\text{dr}=\alpha\left(\mu-{r}\right)dt+\sigma\sqrt{r}dw$$

The first part on the right-hand side of the equation is the deterministic part, while the second part is the stochastic part.

We have seen that the Vasicek model allows the short-term interest rate to be negative. However, this assumption is not ideal.

The CIR model incorporates $$\sqrt{r}$$ in the volatility coefficient, which guarantees that interest rates remain strictly positive.

Note to candidates: The main difference between the models is that the CIR model assumes that interest rate volatility increases with an increase in interest rates. In contrast, the Vasicek model assumes that interest rate volatility is constant.

## Arbitrage-Free Models

Arbitrage-free models are less theoretical models that employ the realized (actual) yield curve to estimate future interest rates.

These models are based on the assumption that the market term structure is correct, and thus no arbitrage opportunities exist.

## Ho–Lee Model

The Ho-Loo model is an arbitrage-free model set to market data and uses a binomial lattice approach to generate a distribution of possible future interest rates.

It is expressed as:

$$dr=\theta dt+\sigma dw$$

Where $$\theta$$ is the drift parameter.

The resulting interest rates can be used to determine the prices of zero-coupon bonds and the spot curve.

## Question

Which of the following is the least appropriate feature of the Vasicek model?

1. The short-term interest rate exhibits mean reversion.
2. It assumes that interest rate volatility increases with increases in the level of interest rates.
3. It allows the short-term interest rate to be negative.

#### Solution

Vasicek model assumes that interest rate volatility is constant, whereas the CIR model assumes that interest rate volatility increases with increases in interest rates.

Reading 29: The Arbitrage-Free Valuation Framework

LOS 29 (i) Describe time structure models and how they are used.

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