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
• Department of Finance
• University of Illinois at Urbana-Champaign
• July 18, 2000

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Overview of Presentation

• Beginner, intermediate, and advanced topics
• Define several types of interest rates
• Discussion of issues to consider when selecting
  an interest rate model
• Look at several types of interest rate models
• Comment on importance to insurance
• Implementation issues
• Concluding remarks

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Spot Rates

• Spot rate is the interest rate that applies to
  single cash flows
• Depends on maturity
• Term structure (or spot rate curve) illustrates
  level of spot rates by maturity

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Yield to Maturity (YTM)

• 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
• Internal Rate of Return (IRR)
• Yield curve is the graph of bond yield by maturity

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Building the Term Structure

• Determine the spot rates (0rn)

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Example Spot Rates
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Example Yields vs. Spot Rates

• YTM is the one discount rate that equates the PV
  of cash flows with market price

• Note that the YTM (8.9) is an average of the
  individual spot rates

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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 Example

• In the previous example, the one-year spot rate
  was 8 and the two-year rate was 8.3
• Consider two investment strategies for a two year
  horizon
• 1. (Rollover) Buy the one-year zero coupon bond
  and reinvest all the proceeds at maturity
• 2. (Buy and hold) Buy a two-year zero coupon
  Treasury

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Forward Example (p.2)

• The strategies are identical if

• f is called the implied forward rate since it is
  derived from the relationship between the one-
  and two-year rates
• If investor chooses rollover strategy, she is
  making a gamble that future spot rate will exceed
  8.6

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Similarities and Differences

• Set of yields, spot rates, and forward rates
  provides equivalent information
• They are all interest rates
• Which interest rate to model depends on use
• Yields useful in capital markets (bonds, swaps)
• Spots useful for present value applications
• Forwards have some mathematical benefits

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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

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

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Summary Statistics for Historical Rates
(1953-1999)
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Alternative Interest Rate Models

• General equilibrium vs. arbitrage free
• One vs. multiple stochastic factors
• Choice of model should reflect the application
• Investment banking or strategic planning

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General Equilibrium vs. Arbitrage Free

• GE models part of larger economic model where
  bond prices (and yields) are derived from the
  expected movement of short-term rate
• Bond, option prices can have analytic solutions
• - Model does not fit existing term structure
• Arbitrage free models match existing term
  structure
• Better for pricing options (Hull (2000) and
  Jegadeesh (1998))
• - May be more difficult to use

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One Factor vs. Multiple Factors

• One-factor models (typically) model short-term
  interest rate movements
• Simpler to use
• - Movements in entire yield curve are constrained
• Two-factor models have two stochastic variables
• Allow for increased variety of yield curves
• - Numerical methodology may be quite technical

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Advantages() and Disadvantages(-) of Alternative
Interest Rate Models
General Equilibrium
Arbitrage Free
One-Factor
Two-Factor
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Understanding a General Term Structure Model (p.1)

• Continuous time
• Applications will be discrete time
• 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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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
• Simplest case is where volatility is constant
• Ho-Lee model

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Applications of Interest Rate Models to Insurance

• Model asset portfolio
• Bond values
• Prepayments on mortgage obligations, CMOs
• Model liabilities
• Inflation affects claims
• Underwriting cycle

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Real vs. Nominal Interest (p.1)

• Interest rates are comprised of two factors
• The real rate of interest is the underlying price
  of exchanging money across time
• The nominal rate of interest has an additional
  component that adjusts for (expected) inflation
• Nominal rate compensates investors for time value
  of money and inflation

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Real vs. Nominal Interest p.2

• Nominal interst is i, real interest is r,
  inflation is p

• Generally, i r p is close enough
• Relating interest rates to inflation can be done
  through a regression
• Can be used to translate interest rates to line
  of business inflation

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Importance of Multiple Factors in Insurance
Valuation

• P-L insurance losses may extend over long periods
  of time
• But cash flows may be related to short-term
  interest rates
• P-L insurance costs are related to inflation
• Are multiple factors required for analyzing
  insurer interest rate risk?
• Are simple interest rate models adequate to
  capture insurer risk profile?
• Help insurers set capital and surplus

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Sample DFA Model Output
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Interest Rate Model Calibration

• Interest rate model should reflect existing
  market conditions
• Determine the best fit parameters for alternative
  interest rate models
• Model prices should approximate market prices

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Concluding remarks

• Several interest rates exist
• Interest rates are not constant
• A variety of models exist to help value interest
  rate dependent claims
• Pick parameters that reflect current environment
  or view

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Some Documentation

• Feel free to contact me at ahlgrim_at_uiuc.edu
• This presentation is available at
  http//www.students.uiuc.edu/ahlgrim/academic.htm
  
• For references, download the CAS Forum paper by
  Ahlgrim, DArcy, Gorvett