Using Directional Bit Sequences to Reveal
the Property-Liability Underwriting Cycle 
as an Algorithmic Process

Joseph D. Haley
St. Cloud State University
jhaley@stcloudstate.edu

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The P-L Underwriting Cycle 1930-2000

(Legended[, (LineLegend["U Margin"])])             ({{Aggregate profits characterized by irregular fluctuations
}, {Hard Market: Rising Prices and Strict Underwriting
}, {Soft Market: Falling Prices and Liberal Underwriting
}, {Theoretically, Profits should behave more randomly}})

Important Features of the Simulation Model

Only pure premium is modeled. (Equilibrium is profit margin of zero.)

Agent-based: Individual insurers follow pricing rule separately

Each company has its own target capital structure

For example, 40% equity and 60% debt (as expected losses)

40/60 structure: $2,000,000 equity will support $3,000,000 in expected losses

Insurance is mandatory and the industry faces a downward sloping demand curve

The Simple Pricing Rule

This year’s premium is based on

Expected Losses, L (a constant, ≈$3,123)

Last year's Surplus (a.k.a. Equity), S

Last year's Size of Risk Pool, n

The Simple Pricing Rule is .....

(Subscript[Premium, t]) = L+([(r/((1-r)))]·L-((Subscript[S, (t-1)])/(Subscript[n, (t-1)])))               ...where r=(Surplus/Assets)

Premiums are collected at the beginning of the year, whereas Losses are an end-of-the-year cash flow. So there is a two-year lag.

(Subscript[S, (t-1)])  →   (Subscript[P, t])     →   (Subscript[L, (t+1)])

Losses are simulated using the SkewNormalDistribution[] function

Simulation X Results of 75 Companies for 200 years (Quick Look)

Starting in equilibrium,the simulated weighted average underwriting margin.

The weighted average loss per insured.

Directional Bit Sequences

4-tuple directional bit sequences were determined for the actual underwriting margin data (1930-2000) and the Simulation X data. 

1=up, 0=down. Point 1 on the graph is (1,1,1,1), while Point 16 is (0,0,0,0).

Probabilities

Correlation = .9053
The bit sequences clearly appear to have similar structures. Let’s look at more runs of the simulation.....

4 Other Simulation Runs

({{
                  Correlation = .9011, 
                 Correlation = .8623}, {
                 Correlation = .9006, 
                 Correlation = .8914}})

‘Shape’ of Simulation X Underwriting Time Series

What about the other four simulations displayed?

More Distribution Shapes for the Simulated Underwriting Margin

Several different shapes are possible.

Some Initial Conclusions

The directional bit sequences reveal a common structure between the actual and simulated time series.

The common structure shows the underwriting cycle following an algorithmic (or rule-based) process.

Continuously seeking equilibrium is the only consistent feature of the alogorithmic process. This is indicated by the zero mean.

The usual notion of a continuous probability distribution being the basis of the data generating process can be relaxed.

The actual underwriting cycle we observe is one of many, many possiblities.

Simulated Companies Differ from Industry

Correlation = .6970                                                                                  Correlation = .0041
             
               Correlation = -.2899                                                                              Correlation = .3530

Microeconomic decisions cannot be deduced from macroeconomic data.

Evidence of Emergence?

A Look at Volatility

({{({{□, t = 1-125, t=126-200}, {Average Correlation
Across 75 Companies, .11257, .5278}, {Standard Deviation of 
Aggregate Underwriting Margin, .776%, 2.22%}}), 
Volatility increases significantly around t=125.}, {Simulated Aggregate
Underwriting Margin
(not Simulation X), }})

Fewer Companies Result in Greater Volatility

Nine additional simulations for each pretend industry of size 15, 35, and 55 companies were run. The relationship between the overall standard deviation of underwriting margins and average correlation of underwriting margins across companies shows that an industry with fewer companies tends to be more volatile.

A CDF of the Simulation

Conclusions (Again)

The P-L Underwriting Cycle can be modeled as an algorithmic process.

While equilibrium has a gravity-like pull on economic activity, movement towards equilibrium is not uniform (or symmetrical) through time. Some equilibrium deviations result in faster adjustment than others.

Direction of changes can be analyzed separately from magnitude of changes. There is no consistent DGP.

With each company following the pricing rule, and the industry operating along the demand curve, a type of emergence occurs.

Company activity is more correlated during periods of volatility.

An industry with fewer companies is prone to more volatility.