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LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach

LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach. Dave Clark American Re-Insurance 2003 Casualty Loss Reserve Seminar. LDF Curve-Fitting and Stochastic Reserving. Goals: 1. Describe loss emergence in a mathematical model to assist in estimating needed reserves

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LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach

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  1. LDF Curve-Fitting and Stochastic Reserving:A Maximum Likelihood Approach Dave Clark American Re-Insurance 2003 Casualty Loss Reserve Seminar

  2. LDF Curve-Fitting and Stochastic Reserving Goals: 1. Describe loss emergence in a mathematical model to assist in estimating needed reserves 2. Calculate the variability around the estimated reserves

  3. LDF Curve-Fitting and Stochastic Reserving

  4. LDF Curve-Fitting and Stochastic Reserving Growth Curve = Cumulative % of Ultimate G(t) = 1 / LDFt Inverse Power Curve: G(t|,) = 1 / [1+(/t)]

  5. LDF Curve-Fitting and Stochastic Reserving Why use a continuous curve? • Smoothing of Development Pattern • Interpolation & Extrapolation(including tail factor) • Handle irregular evaluation dates(e.g., latest diagonal less than 12 months from penultimate diagonal) • Avoid Over-Parameterization

  6. LDF Curve-Fitting and Stochastic Reserving Disadvantages of using a continuous curve: • Need curve-fitting engine (answers not in “real time”) • May not fit well unless the “right” curve form is used

  7. LDF Curve-Fitting and Stochastic Reserving Basic Model: • Convert loss development triangle to an incremental basis • For each “cell” of the triangle, we have ci,t =actual loss for AY i, between ages t and t-1 i,t =expected loss for AY i, between ages t and t-1

  8. LDF Curve-Fitting and Stochastic Reserving Two Methods for calculating the Expected Incremental Loss: 1. LDF Allows each accident year reserve to be estimated independently 2. Cape Cod Requires onlevel premium or other exposure base for each accident year

  9. LDF Curve-Fitting and Stochastic Reserving LDF Method: i,t =Ultimatei x [G(t|,) - G(t-1 |,)] n+2 Parameters: Ultimatei expected ultimate loss for accident year i  “scale” parameter of G(t)  “shape” parameter of G(t)

  10. LDF Curve-Fitting and Stochastic Reserving Cape Cod Method: i,t = Premiumi x ELR x [G(t|,) - G(t-1 |,)] 3 Parameters: ELR expected loss ratio for all years  “scale” parameter of G(t)  “shape” parameter of G(t) An onlevel Premiumi entry for each accident year must be supplied by the user

  11. LDF Curve-Fitting and Stochastic Reserving Why We Prefer the Cape Cod Method: • Provides the model with more information (an exposure base in addition to the triangle) • Requires estimation of fewer parameters • More stable estimate of immature year(s)

  12. LDF Curve-Fitting and Stochastic Reserving And now for the Stochastic part…

  13. LDF Curve-Fitting and Stochastic Reserving Assumptions: • The expected development in each cell, i,t is treated as the mean of a distribution. • Each cell has a different mean, but assumed to have the same ratio of Variance/Mean, 2.

  14. LDF Curve-Fitting and Stochastic Reserving Assumptions: • The distribution for each cell follows an Over-dispersed Poisson with a constant Variance/Mean ratio. • The model parameters are estimated using Maximum Likelihood Estimation (MLE).

  15. LDF Curve-Fitting and Stochastic Reserving What the heck is an Over-dispersed Poisson distribution? A discretized version of the aggregate loss amount, with the same shape as a standard Poisson - commonly used in Generalized Linear Models.

  16. LDF Curve-Fitting and Stochastic Reserving

  17. LDF Curve-Fitting and Stochastic Reserving Advantages of the Over-dispersed Poisson distribution: • Sum of ODP is also ODP • Can always match mean & variance • Reflects mass point at zero • Very convenient mathematics

  18. LDF Curve-Fitting and Stochastic Reserving Maximizing the Likelihood means solving for w and q that maximize the expression:

  19. LDF Curve-Fitting and Stochastic Reserving The Variance/Mean Ratio, 2, is estimated by:

  20. LDF Curve-Fitting and Stochastic Reserving Why use Maximum Likelihood Estimation (MLE)? • Familiar methods (LDF and Cape Cod) are exact MLE results • MLE provides estimate of the uncertainty in the parameters (“delta method” in Loss Models)

  21. LDF Curve-Fitting and Stochastic Reserving Comments & Limitations • Independent draws from Identical Distributions (the old “iid”) • Sources of Variance not included…

  22. LDF Curve-Fitting and Stochastic Reserving Comments & Limitations Sources of Variance: • Process • Parameter or estimation error • Mean • Variance • Model or specification error • “State of the World” risk These we can do!

  23. LDF Curve-Fitting and Stochastic Reserving Let’s look at an example…

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