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Financial Risk Management of Insurance Enterprises


Financial Risk Management of Insurance Enterprises Interest Rate Models ... (Brennan-Schwartz) Short term rate and volatility parameter (Longstaff-Schwartz) ... – PowerPoint PPT presentation

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Title: Financial Risk Management of Insurance Enterprises

Financial Risk Management of Insurance Enterprises
  • Interest Rate Models

Interest Rate Models
  • Classifications of Interest Rate Models
  • Term Structure of Interest Rate Shapes
  • Historical Interest Rate Movements
  • Parameterizing Interest Rate Models

Classifications of Interest Rate Models
  • Discrete vs. Continuous
  • Single Factor vs. Multiple Factors
  • General Equilbrium vs. Arbitrage Free

Discrete Models
  • Discrete models have interest rates change only
    at specified intervals
  • Typical interval is monthly
  • Daily, quarterly or annually also feasible
  • Discrete models can be illustrated by a lattice

Continuous Models
  • Interest rates change continuously and smoothly
    (no jumps or discontinuities)
  • Mathematically tractable
  • Accumulated value ert
  • Example
  • 1 million invested for 1 year at r 5
  • Accumulated value 1 million x e.05 1,051,271

Single Factor Models
  • Single factor is the short term interest rate for
    discrete models
  • Single factor is the instantaneous short term
    rate for continuous time models
  • Entire term structure is based on the short term
  • For every short term interest rate there is one,
    and only one, corresponding term structure

Multiple Factor Models
  • Variety of alternative choices for additional
  • Short term real interest rate and inflation (CIR)
  • Short term rate and long term rate
  • Short term rate and volatility parameter
  • Short term rate and mean reverting drift

General Equilibrium Models
  • Start with assumptions about economic variables
  • Derive a process for the short term interest rate
  • Based on expectations of investors in the economy
  • Term structure of interest rates is an output of
  • Does not generate the current term structure
  • Limited usefulness for pricing interest rate
    contingent securities
  • More useful for capturing time series variation
    in interest rates
  • Often provides closed form solutions for interest
    rate movements and prices of securities

Arbitrage Free Models
  • Designed to be exactly consistent with current
    term structure of interest rates
  • Current term structure is an input
  • Useful for valuing interest rate contingent
  • Requires frequent recalibration to use model over
    any length of time
  • Difficult to use for time series modeling

Which Type of Model is Best?
  • There is no single ideal term structure model
    useful for all purposes
  • Single factor models are simpler to use, but may
    not be as accurate as multiple factor models
  • General equilibrium models are useful for
    modeling term structure behavior over time
  • Arbitrage free models are useful for pricing
    interest rate contingent securities
  • How the model will be used determines which
    interest rate model would be most appropriate

Term Structure Shapes
  • Normal upward sloping
  • Inverted
  • Level
  • Humped

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

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

Steepness Shifts
  • Short rates move more (or less) than longer term
    interest rates
  • Changes the slope of the yield curve

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

Parameterizing the Yield Curve
  • Level 6 month yield
  • Steepness (or slope) 10 year yield 6 month
  • Curvature 6 month yield 10 year yield
  • 2 x 2 year yield
  • Based on Brandt and Chapman (2002)

Characteristics of Historical Interest Rate
  • 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

Table 1 Summary Statistics for Historical
Rates April 1953-July 1998
Run Graph Show of Interest Rates
  • Go to
  • http//
  • Download Graph Show
  • Click on Historical (4/53-5/99)
  • Click on Start Graph Show
  • You may want to shorten the time interval to
    speed up the process
  • Note how interest rates have moved over the last
    46 years
  • Pay attention to the level of interest rates, the
    shape of the yield curve and the volatility over
  • Alternative source for the yield curve movements
  • http//

Current Interest Rates
  • Yields
  • Spot rates
  • Implied forward rates

(No Transcript)
  • U. S. Government stopped issuing 30 year bonds in
    October, 2001
  • Reduced supply of long term bonds has increased
    their price, and reduced their yields
  • Effect has distorted the yield curve

Parameterizing Interest Rate Models
  • Vasicek
  • Cox-Ingersoll-Ross (CIR)
  • Heath-Jarrow-Morton (HJM)

Heath-Jarrow-Morton model
  • 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

Table 2 Summary Statistics for Vasicek Model
Notes Number of simulations 10,000, ?
0.1779, 0.0866, ? 0.0200
Table 3 Summary Statistics for CIR Model
Notes Number of simulations 10,000, ?
0.2339, 0.0808, ? 0.0854
Table 4 Summary Statistics for HJM Model
Notes Number of simulations 100, ? 0.0485,
? 0.5
Concluding remarks
  • Interest rates are not constant
  • Interest rate models are used to predict interest
    rate movements
  • Historical information useful to determine type
    of fluctuations
  • Shapes of term structure
  • Volatility
  • Mean reversion speed
  • Long run mean levels
  • Dont assume best model is the one that best fits
    past movements
  • Pick parameters that reflect current environment
    or view
  • Recognize parameter error
  • Analogy to a rabbit