Interest Rates: Empirical Patterns and Modeling Issues

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Interest Rates Empirical Patterns and Modeling
Issues
Casualty Actuarial SocietySeminar on Dynamic
Financial Analysis

• Kevin C. Ahlgrim, ASA, MAAA, Ph.D.
• Department of Finance
• University of Illinois at Urbana-Champaign
• June, 2000

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Overview of Presentation

• Beginner, intermediate, and advanced topics
• Historical interest rate movements
• General and specific interest rate models
• Choosing interest rate model parameters
• Concluding remarks

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Need for Interest Rate Models

• Many financial instruments have cash flows
  related to interest rates, either directly or
  indirectly
• Interest rate caps are options that payoff when
  interest rates increase
• Insurance prices depend on level of inflation
• We need a process that will simulate future
  interest rates to help value these instruments

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Interest Rate Models

• Definition
• Mathematical representation of interest rate
  movements
• Generate future term structures (or yield curves
  or forward curves)
• Short, long rate
• Slope
• Curvature

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A First Look at Interest Rate Models

• Historical term structures provide some
  information on potential range of term structure
  movements
• Shape of term structure
• Relationships between rates of different
  maturities
• Caution History is not necessarily an accurate
  predictor of the future

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Shapes of the Yield Curve
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Shapes of the Yield Curve
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How Do Curves Shift?

• Litterman and Scheinkmann (1991) investigated the
  factors that affect yield movements
• Over 95 of yield changes are explained by a
  combination of three different factors
• Level
• Steepness
• Curvature

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Level Shifts

• Rates of maturities shift by approximately the
  same amount
• Also called a parallel shift

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Steepness Shifts

• Short rates move in one direction, but the longer
  rates move in the other direction
• Changes the slope of the yield curve

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Curvature Shifts

• Shape of curve is altered
• Short and long rates move in one direction,
  intermediate rates move in the other

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Characteristics of Historical Interest Rate
Movements

• Rule out negative interest rates
• Higher volatility in short-term rates, lower
  volatility in long-term rates
• Mean reversion (weak)
• Correlation between rates closer together is
  higher than between rates far apart
• Volatility of rates is related to level of the
  rate

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Summary Statistics for Historical Rates
(1953-2001)
http//www.federalreserve.gov/releases/H15/data.ht
m
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Understanding a General Term Structure Model (p.1)

• Continuous time
• Applications will be discrete time (dt ? Dt)
• Change in interest rate
• a(it,t) is the expected change over the next
  instant
• Also called the drift

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Understanding a General Term Structure Model (p.2)

• dBt is a random draw from a standard normal
  distribution
• s(it,t) is the magnitude of the randomness
• Also called volatility or diffusion
• Alternative models depend on the definition of
  it, and form of a(it,t) and s(it,t)

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Vasicek Model (GE)

• Short-rate tends toward q
• Mean reversion affected by size of k
• Volatility is constant
• Negative interest rates are possible
• Yield curve driven by short-term rate
• Perfect correlation of yields for all maturities

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Determination of Yields

• Recall the comparison of rollover strategy with
  buy-and-hold

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Cox, Ingersoll, Ross Model (GE)

• Mean reversion toward a long-term rate
• Volatility is (weakly) related to the level of
  the interest rate
• Negative interest rates are ruled out
• Again, perfect correlation among yields of all
  maturities

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Summary Statistics for CIR Model
Notes Number of simulations 10,000, k
0.2339, q 0.0808, s 0.0854
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Choosing Appropriate Parameters

• Interest rate model should be reasonable based on
  existing market conditions
• Two approaches
• Estimate volatility, mean reversion, etc. based
  on historical movements (or future expectations)
• Match traded security values (called calibration)
  Determine the best fit parameters

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Heath, Jarrow, Morton Model (Arbitrage Free)

• Specifies process for entire term structure by
  including an equation for each forward rate
• Fewer restrictions on term structure movements
• Drift and volatility can have many forms

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Concluding remarks

• Interest rates are not constant
• A variety of models exist to help value interest
  rate dependent claims
• Appropriately pick parameters that reflect
  current environment or view
• Three types of interest rates YTM, spot, and
  forward rates
• Trade-off accuracy vs. complexity
• Model should reflect the application
• Investment banking or strategic planning

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Spot Rates vs. YTM

• Spot rate is the interest rate that applies to
  single cash flows
• Depends on maturity
• Yield/yield-to-maturity is the discount rate used
  to discount all cash flows of a coupon bond
• Unique YTM equates present value of cash flows
  with market price of bond

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Two Ways to Price Bonds

• Example 2-year, 8 annual coupon bond

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801000

• Using spot rates

• Using YTM

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Forward Rates

• Market consensus of future interest rates based
  on relationship between spot rates
• Also known as implied forward rates
• Any future period has a forward rate

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Forward Rate Example
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• Consider two strategies over 2-year horizon
• (Buy-and-hold) Invest at 2-year spot rate
• (Rollover) Invest for 1-year, then reinvest
  proceeds at end of year

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Some Documentation

• Feel free to contact me at ahlgrim_at_uiuc.edu
• This presentation is available at
  http//www.cba.uiuc.edu/ahlgrim/academic.htm
• For references, download the 1999 CAS Forum paper
  by Ahlgrim, DArcy, Gorvett
• www.casact.org/pubs/forum/99sforum/99sf001.pdf