Boor's paper mentions several variants of the Harwayne method. Know particularly the method used in Table 2 on page 335. To illustrate the method, let us consider the following practice problem.

You are setting workers' compensation rates for construction workers in New York. Your company writes only in New York, New Jersey, and Pennsylvania, and writes only construction risks and manufacturing risks. You have the following historical data: State Class Payroll ($00,000) Indemnity Benefits ($000) New York Construction 200 600 Manufacturing 500 800 New Jersey Construction 400 2000 Manufacturing 2000 4000 Pennsylvania Construction 100 150 Manufacturing 100 100 Payroll is in hundreds of thousands of dollars, and indemnity benefits are in thousands of dollars. You will file a pure premium which is a credibility weighted average of the "indicated" pure premium for New York construction workers and the "credibility complement" for this indicated pure premium. What is the "indicated" pure premium for New York construction workers? What is the "credibility complement" for this indicated pure premium, using Harwayne's method?

The Indicated Pure Premium

The indicated pure premium is the historical benefit costs divided by the payroll, or $600,000 ÷ $20,000,000 = $3.00 per $100 of payroll. This examination question, since it is based on the Boor reading, not the Workers' Compensation Ratemaking study note, does not ask about partial pure premiums, or the pure premium for serious losses versus the pure premium for non-serious losses.

How much credibility should be assigned to this indicated pure premium? The Boor paper does not discuss the amount of credibility to be assigned to the figure indicated from historical experience versus the amount of credibility to be assigned to the complement. Rather, the Boor paper discusses how to determine the complement. Some candidates are confused by this, so remember: Boor's examples do not show the final rate indication. They show only the derivation of the amount which is assigned the complement of credibility.

State Benefit Levels

Harwayne's method at first seem complex. To make the method clear, let us first solve this problem incorrectly, without first adjusting New Jersey and Pennsylvania to New York's benefit level.

New Jersey's indicated pure premium for construction workers is

Similarly, Pennsylvania's indicated pure premium for construction workers is

Thus, the overall (countrywide) pure premium, excluding New York, is an average of the New Jersey and Pennsylvania pure premiums. If we were to take a pure average, we would get ($5.00 + $1.50) ÷ 2 = $3.25. Since New Jersey has four times the exposure as Pennsylvania, we might want to take a weighted average, using the exposures as weights, or

Adjusting the Benefit Levels

What is wrong with these answers? Well, these other states might have differing benefit levels than New York has. Looking at both construction and manufacturing workers, we see that New Jersey provides higher average benefits than New York does, and Pennsylvania provides lower average benefits than New York does. The benefit levels may differ (i) because of statutory differences between jurisdictions (e.g., one state's compensation rate may be 65% of pre-injury wage while another states's may be 70% of pre-injury wage) or (ii) because of differences in the claim environment (e.g., some states have high levels of attorney involvement in workers' compensation claims). [The benefit levels shown here are illustrative only, and do not correspond to actual New York, New Jersey, and Pennsylvania statutes.]

We adjust New Jersey's and Pennsylvania's benefits payments to the New York benefit level. To do this, we first complete the spreadsheet to show the totals for all classes combined in these three states.

Harwayne's Method
NEAS study aid: Practice Problem
State	Class	Payroll	Benefit	Rate	Reweighted	Reduction Factor	Adjusted Rate
New York	Construction	$200	$600	3.000			3.325
	Manufacturing	$500	$800	1.600			1.417
	Total	$700	$1,400	2.000
New Jersey	Construction	$400	$2,000	5.000			3.500
	Manufacturing	$2,000	$4,000	2.000			1.400
	Total	$2,400	$6,000	2.500	2.857	0.700
Pennsylvania	Construction	$100	$150	1.500			2.625
	Manufacturing	$100	$100	1.000			1.750
	Total	$200	$250	1.250	1.143	1.750

We have completed the worksheet just as Boor does on page 335 of his paper. Boor's exhibit is confusing, because the numbers he shows in the worksheet are not the numbers that he uses in the Harwayne method. So let us proceed slowly, showing what figures we actually use.

The State Distribution

The overall benefit level in New York is correctly shown in the exhibit as $2.00 per $100 of payroll. This figure is determined as total benefits in New York divided by total payroll, or $1,400,000 ÷ $70,000,000 = $2.00 per $100 of payroll.

What about New Jersey? The worksheet shows the corresponding calculation, just as Boor's exhibit shows. Total benefits are $6,000,000, total payroll is $240,000,000, so the average benefit is $2.50 per $100 of payroll.

What is happening here? Well, the rate for construction workers is higher than the rate for manufacturing workers in both New York and in New Jersey (and in Pennsylvania as well), since workplace injuries are more frequent and more severe for construction workers than for manufacturing workers. New York has two and a half times as much manufacturing payroll as construction payroll, so the average rate is slightly closer to the manufacturing rate. New Jersey has five times as much manufacturing payroll as construction payroll, so the average rate is much closer to the manufacturing rate.

Harwayne's question is as follows:

If New Jersey had the New York distribution of exposures among its classes, what would be the New Jersey average rate?

This is the essential question, and its answer is the pivotal number for the Harwayne method. It is shown in the text of Boor's paper, not in his exhibit. We show it in the column captioned "reweighted."

For New Jersey, the indicated rates are $5.00 per $100 of payroll for construction and $2.00 per $100 of payroll for manufacturing. Using the New York distribution of payroll (not the New Jersey distribution of payroll), the average New Jersey rate is

Similarly, for Pennsylvania, the indicated rates are $1.50 per $100 of payroll for construction and $1.00 per $100 of payroll for manufacturing. Using the New York distribution of payroll (not the Pennsylvania distribution of payroll), the average Pennsylvania rate is

These figures, for Boor's example, are shown on the bottom of page 11, labeled PT and PU. Be careful as you review Boor's paper. The PT figure of 3.88 is not related to the 3.88 in the subtotal line for state T. [The similarity of the numbers is due to the similar exposure distributions in state S and state T in the paper.]

Reduction Factors

We now ask: "What is the relationship of New York benefit levels to New Jersey benefit levels?" Well, the average benefits levels are $2.00 in New York, $2.857 in New Jersey, and $1.143 in Pennsylvania, using the New York exposure distribution in each state. Therefore, the ratio of New York benefit levels to New Jersey benefit levels is 0.700 to 1, as shown in the column captioned "reduction factor" [2.000 ÷ 2.857 = 0.700.] Similarly, the ratio of New York benefit levels to Pennsylvania benefit levels is 1.750 to 1. [As before, 2.000 ÷ 1.143 = 1.750.] We call these "reduction factors," following the terminology of the Workers' Compensation Ratemaking study note. [There is no term for this in Boor's paper.]

We can now adjust the historical rate levels for construction workers in New Jersey and in Pennsylvania to the New York benefit levels.

• For New Jersey, the indicated rate is $5.00 per $100 of payroll. We multiply this by the reduction factor of 0.700 to get the "adjusted" New Jersey rate of $3.500 per $100 of payroll. The adjusted rate is shown in the column captioned "adjusted rate."
  
  

• For Pennsylvania, the indicated rate is $1.50 per $100 of payroll. We multiply this by the reduction factor of 1.750 to get the "adjusted" Pennsylvania rate of $2.625 per $100 of payroll.
  
  

Study Recommendation

"Isn't this a bit complex for an examination problem?" asks the Exam 5 candidate.

On the contrary. Boor spends four pages explaining this procedure, and he includes a complete example. The Workers' Compensation Ratemaking study note describes the same procedure, and it is a standard techniques both for workers' compensation ratemaking and for rate indications in other lines of business. The mathematics is trivial. However, it is simple to make mistakes, and most candidates will get it wrong. So memorize the procedure.