A Random Walk Model for Paid Loss Development Daniel D. Heyer

Why revisit a topic that has already been visited… a lot!. Typical reserving methods focus upon projecting the expected ultimate loss, not the range of possible results. Need a “range of results” model for...  Loss Portfolio Transfer / Commutation  Reserve Liability Securitization  Managing Reserving and Investment Policy Typical reserving methods focus upon projecting the expected ultimate loss, not the range of possible results. Need a “range of results” model for...  Loss Portfolio Transfer / Commutation  Reserve Liability Securitization  Managing Reserving and Investment Policy

Let’s start someplace familiar… Exposure Growth Development Pattern

You know what comes next… QUESTION: Why, when everyone says that Link Ratios are evil, do they work so well?

Roger Hayne took us further. Assume that A2A factors are time-stable (no report/accident year trends). Assume that A2A factors are Lognormal distributed. Estimate the  and  for each development interval (each A2A factor gets its own). Use properties of compounded Lognormal variables to estimate the ultimate loss distribution. Assume that A2A factors are time-stable (no report/accident year trends). Assume that A2A factors are Lognormal distributed. Estimate the  and  for each development interval (each A2A factor gets its own). Use properties of compounded Lognormal variables to estimate the ultimate loss distribution.

Why get more complicated? Traditional approaches have enough parameters to fit curves to Grandma’s lace apron (projecting noise). Traditional approaches ignore the underlying stochastic process so it is difficult to make stochastic statements about results. No data for estimating tail parameters. May be difficult to adjust model for different development intervals. Traditional approaches have enough parameters to fit curves to Grandma’s lace apron (projecting noise). Traditional approaches ignore the underlying stochastic process so it is difficult to make stochastic statements about results. No data for estimating tail parameters. May be difficult to adjust model for different development intervals.

So what are we going to do? Identify a functional form for incremental loss payments (e.g. incremental payments decay exponentially). Move to continuous time so we are modeling the payments made during each “small” interval rather than over large intervals like years. Split process into expected development and volatility around that development. Identify a functional form for incremental loss payments (e.g. incremental payments decay exponentially). Move to continuous time so we are modeling the payments made during each “small” interval rather than over large intervals like years. Split process into expected development and volatility around that development.

The Model Assumptions: A2A factors for each interval dt are lognormal and time-stable We can get development factors for a larger period [ t 1, t 2 ] by multiplying all of, the A2A factors between t 1 and t 2. This is equivalent to adding the logs of the A2A factors. Remember: we’re modeling in continuous- time so adding means integrating. Assumptions: A2A factors for each interval dt are lognormal and time-stable We can get development factors for a larger period [ t 1, t 2 ] by multiplying all of, the A2A factors between t 1 and t 2. This is equivalent to adding the logs of the A2A factors. Remember: we’re modeling in continuous- time so adding means integrating.

Testing Time-Stability of Development

Testing Log-normality of Factors

Analysis Objective We need to find an instantaneous log-A2A function a(s) that we can integrate to get the log-A2A factor between any two times. (like Wiser’s model) These factors should “replicate” the observed experience. We need to find an instantaneous log-A2A function a(s) that we can integrate to get the log-A2A factor between any two times. (like Wiser’s model) These factors should “replicate” the observed experience.

What is this function? For various theoretical reasons (discussed in the paper) we find that A2A factors are Lognormal distributed with…  For various theoretical reasons (discussed in the paper) we find that A2A factors are Lognormal distributed with…  Expected Development Volatility Removes “bias”

From Model to Method First find a volatility function v(t) that integrates to yield the observed variance of the log-A2A factors. Then find a growth function g(t) that together with the volatility function v(t) integrates to yield the observed mean of the log-A2A factors. Thanks for the advice, but how can I find these functions? First find a volatility function v(t) that integrates to yield the observed variance of the log-A2A factors. Then find a growth function g(t) that together with the volatility function v(t) integrates to yield the observed mean of the log-A2A factors. Thanks for the advice, but how can I find these functions?

Make a Picture – What is our intuition?

Tail-Functions The observed variance appears to be smoothly decreasing so v(t) should be smoothly decreasing, too. Similar conclusion is reached about g(t). These functions are sometimes called “Tail Functions” Actuaries have lots of tail functions at their disposal: Scaled PDFs for the Exponential, Pareto, etc. The observed variance appears to be smoothly decreasing so v(t) should be smoothly decreasing, too. Similar conclusion is reached about g(t). These functions are sometimes called “Tail Functions” Actuaries have lots of tail functions at their disposal: Scaled PDFs for the Exponential, Pareto, etc.

Matching the Observed and Fitted A2A Factors INTEGRATE

The Fitted Functions The fitted functions were of the form… (Pareto PDF) For t -years, the fitted parameters {K,c,a} were…  {0.329, 2.099, 0.114} for v(t)  {0.913, 0.951, 1.337} for g(t) The fitted functions were of the form… (Pareto PDF) For t -years, the fitted parameters {K,c,a} were…  {0.329, 2.099, 0.114} for v(t)  {0.913, 0.951, 1.337} for g(t)

The Fitted Functions The Pareto CDF was chosen to keep the example simple. Should look at decay rate of tail and select appropriate candidates. (Get out those Real Analysis books!) An A2A or A2U factor is obtained by computing the  and  parameters for the lognormal distribution. Then compute expected value, confidence intervals, etc. for the factor. Nice theory. Does it work? The Pareto CDF was chosen to keep the example simple. Should look at decay rate of tail and select appropriate candidates. (Get out those Real Analysis books!) An A2A or A2U factor is obtained by computing the  and  parameters for the lognormal distribution. Then compute expected value, confidence intervals, etc. for the factor. Nice theory. Does it work?

WHY?

Similar Answers. So What? For each development age, we can estimate reserves using the computed A2U factor and losses paid-to-date. Random Walk model allows us to compute the distribution of the A2U factor (Lognormal). Then we know the distribution of our possible reserve outcomes! For each development age, we can estimate reserves using the computed A2U factor and losses paid-to-date. Random Walk model allows us to compute the distribution of the A2U factor (Lognormal). Then we know the distribution of our possible reserve outcomes!

Required Reserve and 90% Probability Interval (Recall Khury’s Reserve Radius?) Expected A2U: % PI for A2U:[1.037, 1.053]

Discounted Reserves ( Discounting is Trickier – See the paper. )

What can I do with these distributions? Consider reserve adequacy on a probabilistic basis. (Gee, thanks!) Evaluate the marginal adverse/favorable development risk from accident year, line of business, etc. Better valuation of LPT, commutation, etc. (via distribution of discounted reserves). Find other use for assets supporting full-value reserves in excess of the 99 th %-ile of the discounted reserve (securitization opportunity). Consider reserve adequacy on a probabilistic basis. (Gee, thanks!) Evaluate the marginal adverse/favorable development risk from accident year, line of business, etc. Better valuation of LPT, commutation, etc. (via distribution of discounted reserves). Find other use for assets supporting full-value reserves in excess of the 99 th %-ile of the discounted reserve (securitization opportunity).

How does this method compare to commercially available Lognormal-type models? It’s free. Has more parameters (used to reflect decreasing payment and volatility rates) but fewer than traditional loss development methods. Does not adjust for accident/report year trends. Correlation of incremental payments arises through expected development structure. Can explicitly include other sources of “correlation” via multi-factor model (e.g. judicial/legislative shocks, economic factors, etc.) It’s free. Has more parameters (used to reflect decreasing payment and volatility rates) but fewer than traditional loss development methods. Does not adjust for accident/report year trends. Correlation of incremental payments arises through expected development structure. Can explicitly include other sources of “correlation” via multi-factor model (e.g. judicial/legislative shocks, economic factors, etc.)

Where can I learn more? Basic Theory Steele, “ Stochastic Calculus and Financial Applications ” Intermediate Theory Oksendall, “ Stochastic Differential Equations ” Advanced Theory Musiela and Rutkowski, “ Martingale Methods in Financial Modelling ” Practical Application Tavella and Randall, “ Pricing Financial Instruments ” Basic Theory Steele, “ Stochastic Calculus and Financial Applications ” Intermediate Theory Oksendall, “ Stochastic Differential Equations ” Advanced Theory Musiela and Rutkowski, “ Martingale Methods in Financial Modelling ” Practical Application Tavella and Randall, “ Pricing Financial Instruments ”