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6.7 Modern Term Structure Models

6.7.1 Modern Term Structure Models and their Use

  • Modern term structure models provide quantitatively precise descriptions of how interest rates evolve.
  • Interest rate models attempt to capture the statistical properties of interest rate movements.
  • Two major types of such models:
  • General equilibrium, including Vasicek and Cox-Ingersoll-Ross (CIR) models
  • Arbitrage-free models, including the Ho-Lee models
  • Equilibrium term structure models are models that seek to describe the dynamics of the term structure using fundamental economic variables that are assumed to affect interest rates.
  • They share the following characteristics:
  • They are one-factor or multifactor models
  • They make assumptions about the behavior of factors
  • They are, in general, more sparing with respect to the number of parameters that must be estimated compared with arbitrage-free term structure models
  • One-factor models assume that a single observable factor (sometimes called a "state variable") drives all yield curve movements. Both the Vasicek and CIR models assume a single factor, the short-term interest rate, r.
  • This approach is plausible because empirically, parallel shifts are often found to explain more than 90% of yield changes. In contrast, multifactor models may be able to model the curvature of a yield curve more accurately but at the cost of greater complexity.

6.2.1.1 General Equilibrium Models CIR MODEL The CIR model assumes that every individual has to make consumption and investment decisions with his or her limited capital.

The CIR model can explain interest rate movements in the following terms:

  • An individual's preferences for investment and consumption
  • The risks and returns of the productive processes of the economy ( ) sqrt where dr and dt = infinitely small increments of short-term interest rate and time, respectively; ( ) a deterministic part, where b = a long-run value of interest rate and a = a positive parameter; sqrt = a stochastic part, which models risk and follows the random normal distribution with a mean of zero; sqrt is the standard deviation factor. The model shows how the short-term interest rate is related to the risks facing the productive processes of the economy.

Assuming that an individual requires a term premium on the long-term rate, the model shows that the short-term rate can determine the entire term structure of interest rates and the valuation of interest rate-contingent claims.

For simplicity of presentation, the formula in the slide assumes that the term premium of the CIR model is equal to zero.

VASICEK MODEL The Vasicek model is similar to CIR model.

( )

It has the same drift term as the CIR model and thus tends toward mean reversions in the short-term rate Unlike the CIR model. Interest rates are calculated assuming that volatility remains constant over the period of analysis Because both the Vasicek model and the CIR model require the short-term rate to follow a certain process, the estimated yield curve may not match the observed yield curve.

But if the parameters of the models are believed to be correct, then investors can use these models to determine mispricings.

6.2.1.2 ARBITRAGE-FREE MODELS:

  • In arbitrage-free models, the analysis begins with the observed market prices of a reference set of financial instruments and the underlying assumption is that the reference set is correctly priced.
  • An assumed random process with a drift term and volatility factor is used for the generation of the yield curve.
  • These models are called "arbitrage-free" because the prices they generate match market prices.
  • Arbitrage-free models do not attempt to explain the observed yield curve.

Instead, these models take the yield curve as given. For this reason, they are sometimes labeled as partial equilibrium models.

  • The basic arbitrage-free concept can be used to solve much broader problems.

These models can be extended to value many bond types, allowing for a term structure of volatilities, uncertain changes in the shape of the yield curve, adjustments for the credit risk of a bond, and much more.

THE HO-LI MODEL The Ho-Lee model is expressed as .

  • The model can be calibrated to market data by inferring the form of the time- dependent drift term, , from market prices.

6.7.2 Measuring the bond's exposure to each of the factors driving the yield curve can

be measured and how these exposures can be used to manage yield curve risks.

Shaping risk – Defined as the sensitivity of a bond's price to the changing shape of the yield curve

  • For active bond management, a bond investor may want to base trades on a forecasted yield curve shape or may want to hedge the yield curve risk on a bond portfolio.

A yield curve factor model – defined as a model or a description of yield curve movements that can be considered realistic when compared with historical data

  • Litterman and Scheinkman (1991) decomposed yield curve movements into a combination of three independent movements, which they interpreted as level, steepness,and curvature.
  • The level movement refers to an upward or downward shift in the yield curve.
  • The steepness movement refers to a non-parallel shift in the yield curve when either short-term rates change more than long-term rates or long-term rates change more than short-term rates.
  • The curvature movement is a reference to movement in three segments of the yield curve: The short-term and long-term segments rise while the middle-term segment falls, or vice versa.

Factors Affecting the Shape of the Yield Curve

  • The method to determine the number of factors-and their economic interpretation- begins with a measurement of the change of
  • Then, the historical variance/covariance matrix of these interest rate movements is obtained.
  • The next step is to try to discover a number of independent factors that can explain the observed variance/covariance matrix.
  • The approach that focuses on identifying the factors that best explain historical variances is known as principal components analysis (PCA).
  • PCA creates a number of synthetic factors defined as (and calculated to be) statistically independent of each other; how these factors may be interpreted economically is a challenge to the researcher that can be addressed by relating movements in the factors (as we will call the principal components in this discussion) to movements in observable and easily understood variables.

Yield Curve Risk Management Yield curve risk can be managed on the basis of several measures of sensitivity to yield curve movements:

1. Effective duration, which measures the sensitivity of a bond's price to a small parallel shift in a benchmark yield curve.

2. Key rate duration, which measures a bond's sensitivity to a small change in a benchmark yield curve at a specific maturity segment.

  • Effective duration is used to measure the change in the level factor, whereas key rate duration is used to measure the change in steepness and curvature factors.

We can calculate a measure based on the decomposition of yield curve movements into parallel, steepness, and curvature movements:

whereDL, DS, andDC= sensitivities of portfolio value to small changes in the level, steepness, and curvature factors, respectively; is change in respective factors.

6.7.3 Maturity Structure of Yield Volatilities and Their Effect on Price Volatility

Quantifying interest rate volatilities is important for at least two reasons:

1. The values of bonds with embedded options crucially depend on the level of interest rate volatilities.

2. Risk management includes controlling the impact of interest rate volatilities on the instrument's price volatility.

The term structure of interest rate volatilities is a representation of the yield volatility of a zero- coupon bond for every maturity of security.

This volatility curve (or "vol") or volatility term structure measures yield curve risk.

Interest rate volatility is not the same for all interest rates along the yield curve.

The volatility term structure typically shows that short-term rates are more volatile than long- term rates.

Research indicates that short-term volatility is most strongly linked to uncertainty regarding monetary policy whereas long-term volatility is most strongly linked to uncertainty regarding the real economy and inflation.

Lesson Wrap-Up

This lesson should leave you able to explain the modern term structure models in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

  1. Explain modern term structure models and their use in your own words.
  2. Explain measuring the bond's exposure to each of the factors driving the yield curve can in your own words.
  3. Explain maturity structure of yield volatilities and their effect on price volatility in your own words.
  4. State one exam-style risk, valuation, or market implication of the modern term structure models.