Analysis

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ANA:4:11/13 - 1 -

Quantitative Analysis: Forecasting

Learning Objectives

1. Discuss loss data that should be developed, why it should be developed, and how to adjust loss data for frequency and severity development. (p. 2)

2. Calculate a loss frequency and total dollar ultimate loss forecast with confidence intervals using fully adjusted loss data. (p. 16)

3. Describe types of triangulation and calculate frequency development factors, severity development factors, and dollar payout ratios. (p. 23)

4. Identify when it is appropriate to use industry loss data rather than the organization’s loss data to calculate development factors. (p. 30)

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Learning Objective #1: Discuss loss data that should be developed, why it should be developed, and how to adjust loss data for frequency and severity development.

I. Adjusting Loss Data

A. Loss data that should be developed and why

1. Individual case losses – loss by loss

2. Frequency development (true IBNR) – year-to-year

3. Overall development – because of the “long tail” or lengthy payout period (includes dollar severity and frequency development; often referred to as broad or bulk IBNR)

4. Payout development (payout patterns) – it is important to calculate the worth of actual dollar payment outflows over time; focus is on “cash out the door”

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B. Loss data – note any obvious frequency and severity patterns from the below loss data

C. Linear regression – best way to capture trends, e.g., line fitting; although the following illustration uses five data points, linear regression is more effective if greater than 30 data points are used

as of 12/31/X5

Year # Claims Total Loss $ # Employees

X1 156 125,986 494 X2 115 469,091 535 X3 148 386,550 543 X4 192 291,555 552 X5 138 357,171 565

100

150

200

250

'X1 'X2 'X3 'X4 'X5

Ye a rs

N u

m b

e r

o f

C la

im s

X1 X2 X3 X4 X5

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D. Historic loss analysis

1. Property – losses are generally reported in the year they are incurred and are not expected to change from the historic values

2. Liability – IBNR (incurred but not reported) losses are common and take time to be fully reported, e.g., workers compensation, asbestos exposure. Those losses incurred in the most recent years may not yet be fully reported.

Assume the frequency development factors are provided as follows:

Year # Claims

(a)

Frequency Development

Factor (b)

Developed Frequency

(a x b)

X1 156 1.00 156 X2 115 1.00 115 X3 148 1.03 152 X4 192 1.09 209 X5 138 1.45 200

Mean 166

Note: Just because it may appear to be a trend, there is no trend in developed losses as shown below; therefore, regression will not improve the forecast.

0 5 0

1 0 0

1 5 0

2 0 0

2 5 0

' X 1 ' X 2 ' X 3 ' X 4 ' X 5

Y e a r s

N u

m b

e r

o f

D e v e lo

p e d

C la

im s

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E. Key exposures that have predictive value

1. Miles driven represent the exposure base for trucking claims

2. Hospital beds represent the exposure base for hospital claims

3. Payroll represents the exposure base for workers compensation claims

4. Revenue represents the exposure base for general liability claims

5. Number of employees is normally considered a key predictor of claims; as the number of employees changes, the number of reported claims should likewise be expected to change

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II. Adjusting Loss Data: Frequency Development

A. If exposures are constant and remain the same for the following year, exposure adjustments can be ignored. The mean and standard deviation can be calculated for the number of developed claims and a confidence interval can be constructed.

Year

Developed Frequency

(pg. 4)

(a)

Mean (pg. 4)

(b)

Deviation (a – b)

(c)

Squared Deviation

(c x c)

(d) X1 156 166 -10 100 X2 115 166 -51 2,601 X3 152 166 -14 196 X4 209 166 43 1,849 X5 200 166 34 1,156

Total 832 5,902

Developed frequency mean = 832/5 = 166 average expected ultimate claims per year

To calculate the standard deviation:

475,1 4

902,5

)15(

902,52 

 s 41.38475,1 S

Using the Empirical Rule:

Interval Approx. Probability 1 SD = ± 38 claims* 128 – 204 claims 68.2% 2 SD = ± 76 claims 90 – 242 claims 95.4% 3 SD = ± 114 claims

52 – 280 claims

99.7%

* 38.41 is rounded down to 38

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B. If exposures are not expected to remain constant, exposure history should be used to improve forecast of claims for the following year

1. The average incident rate (claims per exposure) can be calculated if the history of exposures and fully developed claim frequencies are available.

As an example, employees are used below as the exposure base.

Year

Developed Frequency

(pg. 4) (a)

# Employees (pg. 3)

(b)

Incident Rate (a / b)

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2. Confidence intervals can be constructed for the incident rate given the mean incident rate of 0.309 and the given standard deviation of the incident rate of 0.064.

68% C.I. 0.309 ± 0.064 for an incident rate interval of 0.245 to 0.373

95% C.I. 0.309 ± 0.128 for an incident rate interval of 0.181 to 0.437

99% C.I. 0.309 ± 0.192 for an incident rate interval of 0.117 to 0.501

Note: • Mean developed frequency = 832/5 = 166 • Mean incident rate = 1.544/5 = 0.309 • Standard deviation (given) = 0.064 • Budgeted X6 employees = 589

Year

Developed Frequency

(pg. 4) (a)

# Employees (pg. 3)

(b)

Incident Rate

(a / b) (c)

X1 156 494 0.316 X2 115 535 0.215 X3 152 543 0.280 X4 209 552 0.379 X5 200 565 0.354

Total 832 1.544 X6 589 (bud) 0.309

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C. From the incident rate confidence intervals, confidence intervals for the number of claims around the expected number of claims for X6 can also be constructed with the following factors:

Budgeted # employees (589, page 8) x mean incident rate (0.309, page 8) = expected number of claims (182)

589(0.309) = 182

68% C.I.

(0.245)(589) to (0.373)(589) = 144 claims to 220 claims

95% C.I.

(0.181)(589) to (0.437)(589) = 107 claims to 257 claims

99% C. I.

(0.117)(589) to (0.501)(589) = 69 claims to 295 claims

These forecasts use fully developed frequency and exposure-based data.

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D. Eyeball technique (sometimes called ocular regression) – if qualitative data is available to make a better judgment, the mean may not be used; the eyeball technique can be used to estimate the next period’s incident rate if regression capabilities are not available.

There is no clear trend over time in the graph supporting a need for regression; however, to be sure that the mean is a good estimate, various spreadsheets and statistical programs are available to perform regression, e.g., Excel.

0.150

0.200

0.250

0.300

0.350

0.400

'X1 'X2 'X3 'X4 'X5

Ye ars

In c id

e n

c e R

a te

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If regression is used, the expected incident rate for X6 could be calculated as follows using the formula*:

y = m(x) + c

y (X6 incident rate) = 0.024(6) + 0.237 = 0.381

If the budgeted number of employees for X6 is 589 and the X6 incident rate is 0.381, the accidents forecasted for X6 would be calculated as follows:

589(0.381) = 224

But, from the graph trending, r2 = 0.347 is too low to be reliable. Therefore, calculate forecasted claims from mean in confidence intervals.

*from section 3, pg. 26

SUMMARY OUTPUT for Incident Rates

Regression Statistics

Multiple R 0.589080346 R Square 0.347015655 Adjusted R Square 0.129354206 Standard Error 0.060121252

Observations 5

ANOVA

df SS MS F Significance F

Regression 1 0.005762668 0.005762668 1.594290845 0.295936044 Residual 3 0.010843695 0.003614565

Total 4 0.016606363

Coefficients Standard Error t Stat P-value Lower 95%

Intercept 0.236638242 0.063055701 3.752844535 0.033051743 0.035966672

X Variable 1 0.024005557 0.019012009 1.262652306 0.295936044 -0.036499198

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III. Adjusting Loss Data: Severity Development

A. Severity loss forecast – it is important to have severity per loss forecast calculations in addition to the frequency per loss forecast

1. Severity should be developed for IBNR. Since it changes for IBNR, development is required for the more recent years that have not reached full development.

2. The severity per loss requires both frequency and dollar loss development. For now, the development factors for dollar severity are provided as follows:

Year

Total

Incurred $ (pg. 3)

Development Factor (given)

Ultimate Total Loss $

(a  b)

Developed Frequency

(pg. 4)

Average Severity $

(c / d)

(a) (b) (c) (d) (e) X1 125,986 1.00 125,986 156 808 X2 469,091 1.11 520,691 115 4,528 X3 386,550 1.30 502,515 152 3,306 X4 291,555 1.57 457,741 209 2,190 X5 357,171 2.51 896,499 200 4,482

3. The average severity could be calculated at this point, BUT the dollar amounts cover different time periods. The purchasing power of a dollar in X1 is not the same as the purchasing power of a dollar in X5. Inflation adjustment of the dollars over time is needed before the average severity per loss is calculated.

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4. The inflation index should represent price changes affecting the purchasing power of the claims. For example, if the claims represent medical care costs, an index of changes in medical care prices is needed.

5. Assume prices increased 12% per year over the period of collected claims. The older prices would need to be adjusted more than the current ones. If the base year is X6 (index = 1.0), then X5 dollars would need to be adjusted by 1.12 to have equivalent purchasing power in X6 dollars. Dollars in X4 would need to be adjusted by a factor of 12% per year for two years (1.12 x 1.12) = 1.254.

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Note: Indexed average severity was calculated using an Excel spreadsheet and unrounded numbers. Any discrepancies are due to rounding.

6. The dollar amounts in the last column represent severity measured in prices of X6.

The mean average severity from developed and inflation adjusted data =

020,5$746,2$645,4$127,7$423,1$  = $4,192

7. A price index is used to adjust dollar severity for

inflation unless prices linked to the losses are constant over time. All severity dollars in the loss history are now in X6 prices. The severity per loss can be calculated by using either the average or regression. In the example, the average was best.

Year

Average

Severity $ (pg. 12)

(a)

Inflation Index

(12%)

(b)

Indexed Average

Severity $ (a x b)

(c)

X1 808 1.762 1,423 X2 4,528 1.574 7,127 X3 3,306 1.405 4,645 X4 2,190 1.254 2,746 X5 4,482 1.120 5,020

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8. A reasonable estimate for severity – from the graph, trending by regression will not be useful (actual r2 = 0.040 if a regression is run and that is TOO LOW to be reliable). The mean average severity (from fully developed and exposure adjusted losses) = $4,192.

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Learning Objective #2: Calculate a loss frequency and total dollar ultimate loss forecast with confidence intervals using fully adjusted loss data.

IV. Critical Steps to Loss Data Adjustment

A. Calculating loss frequency

1. If loss data includes IBNR claims, frequency development factors should be used to develop loss frequency

2. If exposures leading to claims are not constant over time, loss frequency should be adjusted for the exposure base resulting in an incident rate per year

3. An expected incident rate can be calculated using the mean or regression; a confidence interval estimate can also be constructed for the incident rate, such as the following, using information from page 10:

a. 95% confidence interval: 0.309 ± 2 (0.064)

b. 0.309 ± 0.128 for an incident rate interval of 0.181 to 0.437

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4. If the number of exposures is known for the next year (589 from page 8), use the following formula to estimate the number of claims for the next year:

(expected incident rate) x (exposures expected next year) = estimated number of claims next year

(0.181)(589) to (0.437)(589) = 107 to 257 claims

Multiply the upper and lower end of the incident rate confidence interval by the expected number of exposures for the following year.

Note: The concept of outliers is always a concern when considering measures of central tendency and dispersion. However, there is no concern over frequency outliers as one particular year of adverse losses may be unexpected but is not an outlier. Outliers are a greater concern in severity predictions because of the effect of catastrophic losses. Severity outliers that would normally give rise to more variance in total losses and lead to larger confidence intervals can be managed with the selection of retentions and excess insurance.

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B. Calculating total dollar loss

1. If loss data includes IBNR claims, severity development factors are used to develop total dollar losses. The developed severity per loss is calculated by dividing developed total dollars by the developed number of claims over the loss history.

2. The final estimate of next year’s total dollar losses equals the expected number of losses next year times the developed and inflation adjusted dollar severity per loss. The confidence interval for total dollar losses is obtained by taking the confidence interval for the number of claims and multiplying the number of claims at each end of the interval times dollar severity.

The following formula can be used to calculate the total dollar loss:

Number of Losses (pg. 11)

Severity $ (pg. 14) Total Loss $

Low 107 x 4,192 = 448,544 Midpoint 182 x 4,192 = 762,944 High 257 x 4,192 = 1,077,344 95% interval: $448,544 to $1,077,344

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Summary of Developed and Adjusted Data

Year Developed Frequency

(pg. 4)

Incident Rate

(pg. 7)

Developed Total $ (pg. 12)

Average Severity $

(pg. 12)

Trended and Developed Average

Severity $ (pg. 14)

X1 156 0.316 125,986 808 1,423 X2 115 0.215 520,691 4,528 7,127 X3 152 0.280 502,515 3,306 4,645 X4 209 0.379 457,741 2,190 2,746 X5 200 0.354 896,499 4,482 5,020

Avg. 0.309 4,192 Std. Dev.

0.064

Based on 589 exposures, 589 x 0.309 average incident rate = 182 claims

 95% confidence interval on incident rate = 0.309 – 2(0.064) to 0.309 + 2(0.064) = 0.181 to 0.437

 95% confidence interval on number of claims = (0.181)(589) to 0.437(589) = 107 to 257  95% confidence interval on total dollar claims = (107)($4,192) to (257)($4,192) = $448,544 to $1,077,344

Predicted losses = (182)($4,192) = $762,944

MIDPOINT ESTIMATE IS ESTABLISHED AND 95% CONFIDENCE INTERVAL FOR FULLY DEVELOPED, EXPOSURE BASED, AND

INFLATION ADJUSTED TOTAL DOLLAR LOSS.

Note: As severity of individual claims is not distributed normally, it is not actually statistically valid to choose low and high intervals of equal size; this is shown only to illustrate the desirability of considering intervals.

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C. Loss rates rather than incident rates from fully developed data

1. Loss rates are a measure of dollars; incident rates are a measure of frequency

2. If the exposure base is revenue rather than employees, losses per revenue can be calculated

3. In the example below, it is assumed both revenue and costs are trending at the same rate

Year Revenue $ Loss $ ÷ Revenue $ Loss Rate

X1 27,400,000 300,000 ÷ 27,400,000 0.0109

X2 30,700,000 320,000 ÷ 30,700,000 0.0104

X3 32,600,000 440,000 ÷ 32,600,000 0.0135

X4 34,100,000 290,000 ÷ 34,100,000 0.0085

X5 35,800,000 340,000 ÷ 35,800,000 0.0095

Total 0.0528

Avg. 0.0528/5=.0106

Average loss rate = 0.0106 (rounded up) Standard deviation = 0.00188

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D. Total dollar loss forecast based on the loss rate

1. If losses tend to be related to revenue, a forecast of losses from the loss rate and a projection of next year’s revenues can be calculated

Example: Assume next year’s revenue is expected to be $36,500,000 and no trend in loss rates is expected (the mean can be used rather than a regression estimate)

Estimated losses = (0.0106)($36,500,000) = $386,900

95% confidence interval on the loss rate = 0.0106 - 2(0.00188) to 0.0106 + 2(0.00188) = 0.00684 to 0.01436

95% confidence interval on losses = (0.00684)($36,500,000) to (0.01436)($36,500,000) = $249,660 to $524,140

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2. Inflation adjustments

a. If the inflation rate for costs trends at the same rate as the price changes for revenue, an adjustment is not necessary (both the numerator and denominator of the ratio have the same trend in this case and inflation cancels out)

b. If the inflation rate for revenue is not the same as the inflation rate for costs, the relevant indexes to adjust both revenue and costs into the same purchasing power would be needed

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Learning Objective #3: Describe the types of triangulation and calculate frequency development factors, severity development factors, and dollar payout ratios.

V. Triangulation

A. Types of triangulation using three different types of data

1. Frequency – history of reported number of claims

2. Severity – history of reported dollar losses

3. Payout – history of actual dollar payouts

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B. Frequency

The chart below illustrates the raw data as of 12/31/X5 made available from the organization’s internal risk management information system or from a third-party provider used to calculate the frequency development factors previously used.

Note: The frequency development factors used in previous examples were provided. The above chart shows how these factors were calculated from raw data.

Year # Emps # Claims

(pg. 4) (a)

Frequency Development Factor

(pg. 4) (b)

Developed Frequency

(a  b)

X1 494 156 1.00 156 X2 535 115 1.00 115 X3 543 148 1.03 152

X4 552 192 1.09 209

X5 565 138 1.45 200

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Frequency to Date Claims as of 12/31/X5

Months from Inception

Year 12 24 36 48 60

X1 116 146 154 156 156

X2 77 106 110 115

X3 100 136 148

X4 144 192

X5 138

Age-to-Age Development Year 12 – 24 24 – 36 36 – 48 48 – 60 X1 1.26 1.05 1.01 1.00 X2 1.38 1.04 1.05 X3 1.36 1.09 X4 1.33 X5

Totals 5.33 3.18 2.06 1.00 Averages 1.33 1.06 1.03 1.00

Year Age to Ultimate Development 1 to 5 1.33 1.06 1.03 1.00 = 1.45 2 to 5 1.06 1.03 1.00 = 1.09 3 to 5 1.03 1.00 = 1.03 4 to 5 1.00 = 1.00

5 and beyond 1.00

* Any adding error is due to rounding

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C. Severity

1. Differentiate from frequency development

2. Both number of claims and dollar claims must be developed; for now, the severity development factors are provided

Note: Development factors and ultimate total loss dollars were calculated using unrounded numbers and Excel. Any discrepancies are due to rounding.

Year

Total Incurred $

(pg. 12) (a)

Development Factors (pg. 12)

(b)

Ultimate Total $ Losses

(a x b) (c)

X1 125,986 1.00 125,986 X2 469,091 1.11 520,691 X3 386,550 1.30 502,515 X4 291,555 1.57 457,741 X5 357,171 2.51 896,499

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Severity Development Factors

Incurred Claims $ (in thousands)

Months from Inception Year 12 24 36 48 60 X1 47 76 98 113 126 X2 215 323 394 469 X3 219 346 387 X4 173 292 X5 357

Note: The factor for 5 and beyond is 1.00, based on the assumption that the claims have reached their ultimate value. Since claims may be reopened, an IBNR factor can be used to reflect future values of claims, including those closed claims that are reopened.

Year Age to Ultimate Development 1 to 5 1.60 1.21 1.17 1.12 = 2.54 2 to 5 1.21 1.17 1.12 = 1.59 3 to 5 1.17 1.12 = 1.31 4 to 5 1.12 = 1.12 5 and beyond 1.00

Age-to-Age Development (months)

Year 12-24 24-36 36-48 48-60 X1 1.62 1.29 1.15 1.12 X2 1.50 1.22 1.19 X3 1.58 1.12 X4 1.69 X5

Totals 6.39 3.63 2.34 1.12 Averages 1.60 1.21 1.17 1.11

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D. Payout ratios

1. 100% of all actual dollars from all policy years are used when determining payout patterns using triangulation.

Paid Claims $ (in thousands) Months from Inception

Year 12 24 36 48 60 X1 15 41 47 67 75 X2 140 234 304 469 X3 93 186 205 X4 63 141 X5 143

Age-to-Age Development Year 12-24 24-36 36-48 48-60 X1 2.73 1.15 1.43 1.12 X2 1.67 1.30 1.54 X3 2.00 1.10 X4 2.24 X5

Totals 8.64 3.55 2.97 1.12 Averages 2.16 1.18 1.48 1.12

Year Age to Ultimate Development Dev.

Factor

% Ultimate Payout

(1/Dev. Factor)

Payout Pattern

1 to 5 2.16 1.18 1.48 1.12 = 4.22 24% 24% 2 to 5 1.18 1.48 1.12 = 1.96 51% 27% 3 to 5 1.48 1.12 = 1.66 60% 9% 4 to 5 1.12 = 1.12 89% 29% 5 and beyond 1.00 100% 11%

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2. The last column in the previous table is the percentage of ultimate payouts made at the end of a given year. Percentages to ultimate payout ratios are calculated as: 1 / development factor for payouts For example, at the end of the first year, 1 / 4.22 = 24% of the ultimate dollar payouts were made.

3. The payout for each year can also be determined.

For example, in Year 2, 51% - 24% = 27% of the ultimate payouts were made. The 51% represents the percentage of total payouts made in the first two years, so that means 24% of the total payouts were made in the first year and 27% of the total payouts were made in the second year.

Dev. Factor (% Ult. Payout) Payout Pattern 4.22 24% 24% 1.96 51% 27% 1.66 60% 9% 1.12 89% 29% 1.00 100% 11%

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Learning Objective #4: Identify when it is appropriate to use industry loss data rather than the organization’s loss data to calculate development factors.

VI. Industry Data versus Organizational Data

A. Organizational data is preferred over industry data when the data is complete, consistent, relevant and has integrity (as described in section 2).

1. Organizational data issues

a. Insufficient volume of quality data

b. Bad data (garbage in, garbage out, or GIGO)

c. Qualitative factors must be considered, e.g., changes in the legal climate, coverage, nature of operations, risk control and safety programs, and claims handling procedures

2. Industry data issues

a. Industry may be too broad for specific organization

b. Data may be unreliable

c. Classifications may not fit the organization

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B. Sources of industry factors

1. Rating bureaus (ISO, NCCI, etc.)

2. Consultants (actuaries, accounting firms)

3. Other (Conning & Co., A.M. Best)

C. Role of actuaries and consultants

1. Perform simulations – an alternative way to develop ranges, e.g., Monte Carlo models

2. Provide broader expertise and experience

3. Compare and contrast approaches

4. Maintain development factors and other data useful for comparison purposes

5. Outside opinions can lend credibility

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VII. Summary of Development and Trending Methodology

A. Determine actual losses (frequency and severity) from loss data

B. Develop number of IBNR claims to calculate frequency development

C. If exposures are constant, forecast frequency of claims from history of developed claims using the mean, eyeball technique, or the construction of a confidence interval

D. If exposures change, use history of exposures and history of developed number of claims to create a history of incident rates. Use expected number of exposures next year and the expected incident rate (mean or regression) to predict next year’s number of claims. A confidence interval can be constructed for the incident rate (must have the expected incident rate and the standard deviation of the incident rate).

E. Construct an estimate of next year’s number of claims and a confidence interval using the expected incident rate and the incident rate confidence interval

F. Develop the dollar loss history for severity

G. Create an average dollar severity history by dividing total developed dollar losses by exposures for each year

H. Use the relevant inflation index to adjust the history of severity for changes in purchasing power. Calculate the severity per loss mean or use regression.

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I. Construct the estimated losses and confidence interval for total developed losses. The estimated losses is the number of fully developed losses next year times the fully developed and inflation adjusted severity per loss. The confidence interval is the confidence interval from the number of losses times the average inflation adjusted severity per loss.

J. Using the history of actual payouts, calculate the payout ratios

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Review of Learning Objectives

1. Discuss loss data that should be developed, why it should be developed, and how to adjust loss data for frequency and severity development. (p. 2)

2. Calculate a loss frequency and total dollar ultimate loss forecast with confidence intervals using fully adjusted loss data. (p. 16)

3. Describe types of triangulation and calculate frequency development factors, severity development factors, and dollar payout ratios. (p. 23)

4. Identify when it is appropriate to use industry loss data rather than the organization’s loss data to calculate development factors. (p. 30)