## Berquist-Sherman Paid Claim Development Adjustment

### Description

The B-S Paid Claim Development Adjustment methods adjusts paid claims based on the underlying relation between paid and closed claims.

### Usage

BS.paid.adj(Triangle.rep.counts = NULL, Triangle.closed, Triangle.paid,
ult.counts = NULL, regression.type = "exponential")


### Arguments

 Triangle.rep.counts cumulative reported claim counts triangle. Assume columns are the development period, use transpose otherwise. A (mxn)-matrix C_{ik} which is filled for k \leq n+1-i; i=1,\ldots,m; m\geq n , see qpaid . Triangle.closed cumulative closed claim counts triangle. Assume columns are the development period, use transpose otherwise. Triangle.paid cumulative paid claims triangle. Assume columns are the development period, use transpose otherwise. ult.counts vector of ultimate claim counts. regression.type Default = "exponential". Type of regression used in the model, it can take 'exponential' or 'linear'. See also 'Details'

### Details

The importance of recognizing the impact of shifts in the rate of settlement of claims upon historical paid loss data can materially affect the ultimate projections.

This functions adjusts the paid claims based on the numerical method described in the B-S paper.

Berquist and Sherman presented a technique to adjust the paid claim development method for changes in settlement rates. The first step of the paid claims adjustment is to determine the disposal rates by accident year and maturity.

The disposal rate is defined as as the cumulative closed claim counts for each accident year-maturity age cell divided by the selected ultimate claim count for the particular accident year.

If ultimate claim counts have been provided, they will be used to calulate the disposal rates, otherwise ultimate claim counts will be estimated from the cumulative reported claim counts triangle with a standard development method.

The disposal rates along the latest diagonal will be selected as the basis for adjusting the closed claim count triangle, The selected disposal rate for each maturity are multiplied by the ultimate number of claims to determine the adjusted triangle of closed claim counts.

Berquist and Sherman then use regression analysis to identify a mathematical formula that approximates the relationship between the cumulative number of closed claims (X) and cumulative paid claims (Y). The algorithm gives the possibility, through the choice of the 'regression.type' field, to fit an exponential model, Y = a*e^(bX), or a linear model, Y = a+b*X.

The relation is estimated based on unadjusted closed claim counts and unadjusted paid claims. Once the regression coefficients are estimated, they will be used to adjust paid claims based on such coefficients and the adjusted closed claim counts triangle.

### Author(s)

Marco De Virgilis devirgilis.marco@gmail.com

### References

Berquist, J.R. and Sherman, R.E., Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach, Proceedings of the Casualty Actuarial Society, LXIV, 1977, pp.123-184.

See also qpaid for dealing with non-square triangles, inflateTriangle to inflate a triangle based on an inflation rate,

### Examples

# Adjust the Triangle of Paid Claims based on Reported Claim Counts

adj_paid <- BS.paid.adj( Triangle.rep.counts = AutoBI$AutoBIReportedCounts, Triangle.closed = AutoBI$AutoBIClosed,
Triangle.paid = AutoBI$AutoBIPaid, regression.type = 'exponential' ) # Calculate the IBNR from the standard unadjusted Paid Triangle std_ibnr <- summary(MackChainLadder(AutoBI$AutoBIPaid))$Totals[4, 1] # Calculate the IBNR from the adjusted Paid Triangle adj_ibnr <- summary(MackChainLadder(adj_paid))$Totals[4, 1]

# Compare the two