Are Actuaries Systematically or Systemically Wrong (or not)?

Transcription

1 Are Actuaries Systematically or Systemically Wrong (or not)? This draft: February 2016 Abstract: Insurance reserving is a complicated matter. Actuaries estimate claims incurred today that will need to be paid over a number of years. Over time, as more information about these claims is acquired actuaries are able to adjust the insurer s reserves to reflect such information about incurred losses. Changes in reserves as they develop can be seen as errors in the initial reserve. We study the distribution of these errors. Under the assumption that errors are randomly distributed, we should find no dependence between errors across claim years (the year where the loss is finally paid), accident years (the year where the loss is incurred), or reserves of the same age (stage of development). We develop a model of reserve development that allows for the testing of correlations across estimation and accident years, insurers, and lines of business to determine if the errors are indications of manipulation or are more systemic in nature. JEL Classification: G22. Keywords: Insurance Reserves, Loss Development, Insurance Pricing. Proposal for 2016 American Risk and Insurance Association Annual Meeting 2

2 1 Introduction Nobody makes mistakes on purpose, especially not when money is on the line. Mistakes can be the result of individuals not being aware of their limited ability, having a lack of expertise or information, or running out of time to make a clear choice. Alternatively, mistakes can just be bad luck! When faced with the possibility of having to pay losses in the future, managers of insurance companies ask their actuaries to use their experience and knowledge to evaluate the current value of the claims that will need to be paid with respect to the losses that the insurers policyholders have incurred today. Despite the high level of uncertainty associated with distant cash flows, the assessment of incurred losses is not made lightly. Still, the assessment of future claims and the appropriate current reserves entail a great deal of uncertainty. Fortunately for actuaries though, new information about future claims to be paid reaches their desk continuously so that they become more and better informed about the actual potential claims that will need to be paid in the future. In other words, with new information comes new assessments. When examining new assessments, prior assessments could appear to simply be unforcastable mistakes. Alternatively the mis-estimation of initial reserves may not be mistakes at all. Rather, the reserves may have been set purposefully higher or lower than the true assessment of future losses. Managers may alter reserves to improve their apparent capital position, avoid taxes, improve ratings, and increase their own bonuses, among other incentives. To that end, the penultimate error between final paid losses and initial reserve estimates that is seen in the reserving practice may have been purposefully made. In this paper, we examine how reserving errors are correlated across accident years, claim years,or reserves of the same age. Our hypothesis is that shocks to losses (the new information that reaches the actuaries office) are randomly distributed such that the yearly incurred loss update for a given accident year should be independent of the yearly incurred loss update for another given accident year. If the information about the shocks to incurred losses occurs randomly in each estimated year, then we should see no dependence between the shocks that affect a given accident year and the shocks that affect another accident year. Evidence of correlation in reserve errors would provide evidence of reserve manipulation, or of systemic risk in the insurance industry. 3

3 2 Motivation and Literature Review The majority of a property and casualty insurer s liabilities are associated with not-yet paid losses and loss adjustment expenses. Clearly, the estimation (or more importantly, the misestimation) of such future losses can have a non-trivial impact on the insurer s financial health since the loss reserves will affect the size of the insurer s ability to pay future claims, its shareholder and/or policyholder surplus, its tax liability, and the pricing of its policies (and even more so in states where premium rates increases are subject to regulatory oversight). The mis-estimation of incurred liabilities can therefore attract unwanted regulatory attention, and even result in an insurer being placed into receivership or liquidated. Accounting standards, as applied to insurance companies (known in the United States as the Statement of Statutory Accounting Principles), demand that the insurer s management team assess correctly the liability that are recorded in the insurer s financial statements. The management team s best estimate often needs to be updated as new information flows to the market and to the insurer. These updates should normally be the result of random errors if the best estimate principle is applied. However, the errors may be systematically positive if actuaries are conservative or pessimistic (i.e. over-reserve) about the future, or may even be systematically negative if actuaries are naturally aggressive and optimistic (i.e. under-reserve) about the future. In any case, the pattern of updates in the insurer s assessment of incurred liability is not a trivial concern as pessimistic actuaries end up reducing the insurer s current tax liability and apply upward pressure to premiums, whereas optimistic actuaries end up increasing the insurer s risk of regulatory scrutiny and potential insolvency. Motivation for this paper also comes from the well developed literature utilizing reserve errors as measures of reserve manipulation. Though earlier papers examined changes in reserves, Smith (1980) and Weiss (1985) were some of the first papers to use loss development as a measure of managerial manipulation. Using a five year development period (which has become the standard) Weiss (1985) shows evidence of firms using reserving practices to smooth underwriting results. Other studies including Grace (1990) and Beaver, McNichols, and Nelson (2003) followed and showed similar results using the now more common five year development of loss reserves. 1 In addition to incentives to smooth income, other research has shown reserving practices to be associated with avoidance of regulatory scrutiny (e.g. Petroni, 1992; Gaver and Paterson, 2004; 1 Beaver, McNichols, and Nelson (2003) also use assets to scale the development. Other scalars are used (Weiss (1985), for example uses the initial loss), though most studies show results robust to scaling. 4

4 Grace and Leverty, 2012) and executive compensation (e.g. Eckles and Halek, 2010; Eckles et al., 2011). Research has also shown that the presence of certain auditors and actuarial firms can help mitigate reserve manipulation (e.g. Petroni and Beasley, 1996; Gaver and Paterson, 2001, 2007). The papers cited above all utilize a reserve error that considers a five year development period. The development outside of those years are ignored. Few papers have examined the methodological importance of the reserve error measure used. Using the standard measure of reserve error (now, popularly called the KFS error), Kazenski, Feldhaus, and Schneider (1992) examines the development period and finds that longer development horizons should be used to improve measurement precision. 2 As we develop our model and methodology below, we note that our methodology proposes using more of the reserve development triangle in an effort to maximize the use of available information. 3 Model of reserving update 3.1 Dependency of errors and independence of perturbations Suppose that in year τ actuaries are asked to evaluate an insurer i s incurred losses for year t τ, and for line of business j, denoted as L i,j t,τ. Using all the information available at time τ for losses incurred in line j for year t for insurer i, Ξ i,j t,τ, actuaries are therefore asked to measure [ ] E. Reserves are then put aside tax free in year τ to pay for losses that have been incurred L i,j t Ξ i,j t,τ in year t. The logic behind the accumulation of reserves is that they are the insurer s best estimate, given the amount of information they have, of the losses they have indeed incurred in the accident year. The reserves should therefore be the best estimate in year τ of the losses incurred in year t that insurer i will end up paying for line j losses in more or less near future. When an insurer measures its expected losses for accident year t in evaluation year τ (that [ ] is, E L i,j t Ξ i,j t,τ ), some proportion of the total loss has already been paid. We will posit that this [ ] paid loss amount is equal to E LP i,j t Ξ i,j t,τ. For losses that have not been paid, some have been [ ] reported to the insurer already (say an amount E ), but some have not. The insurer LR i,j t knows, however, that these losses have been incurred, even if they have not yet been reported to [ ] the insurer. We will label these incurred but not reported losses as E LQ i,j t Ξ i,j t,τ. Of course, [ ] [ ] [ ] [ ] E = E + E + E. L i,j t Ξ i,j t,τ LP i,j t Ξ i,j t,τ LQ i,j t Ξ i,j t,τ LR i,j t In terms of the uncertainty about these three loss values, they can be ranked from the most 2 Weiss (1985) is commonly attributed to utilizing a different error construction. However, the error in Weiss (1985) is based upon that in Smith (1980) which is the same as the KFS error developed over one year. In order to avoid confusion, we will use KFS to refer to the standard error calculation. Ξ i,j t,τ Ξ i,j t,τ 5

5 certain to the most uncertain. For instance losses paid are not volatile because they have been paid [ ] already and are therefore no longer subject to randomness, we can then write that E = LP i,j t LP i,j t,τ. The most uncertain losses are the incurred but not reported losses since insurers do not have as much information about the potential claim as for losses that have been reported but not paid. In the following year, the actuaries acquire more information such that their assessment of L t [ ] [ ] possibly changes. In other words, E L i,j t Ξ i,j t,τ+1 may not be equal to E L i,j t Ξ i,j t,τ. This difference [ ] [ ] E E can be seen as the one-year reserving error associated with the loss L i,j t Ξ i,j t,τ+1 L i,j t Ξ i,j t,τ incurred in year t for insurer i in line j We shall label this error ε i,j t,τ+1. If actuaries are evaluating losses in an unbiased manner, then given the information available at time τ we should have [ ] E ε i,j t,τ+1 Ξi,j t,τ = 0 for all i, j, τ and t. The error from period τ to period τ + 1 cannot be due to a misestimation of losses paid up until period τ, LP i,j t,τ. This does not mean that losses paid do not change from year to year because losses paid are expected to rise with each passing year so that LP i,j i,j t,τ+1 LPt,τ 0. In other words, the measurement of losses paid cannot possibly be an error, but at the same time the evolution in the losses paid is not perfectly known at time t. Let us measure this difference as δ P i,j t,τ Ξ i,j t,τ 0. The misestimation of the reserve error can only come from the last two sources: A misestimation of losses reported and not paid LR i,j t,τ, and a misestimation of incurred but not reported losses LQi,j t,τ. Let us represent the two sources of errors as perturbations η R i,j t,τ reserving, we should then observe that [ ] E ε i,j t,τ+1 Ξi,j t,τ [ = E δ P i,j t,τ+1 Ξi,j t,τ ] }{{} changes in losses paid [ + E η Q i,j t,τ+1 Ξi,j t,τ ] and η Q i,j t,τ. If there is no bias in [ + E η R i,j t,τ+1 Ξi,j t,τ } {{ } changes in reserves ] = 0 Put differently, the combined changes in the Q and R reserves should be equal to the change in the losses paid. The question that arises now is how do perturbations to losses come to be? In other words, how are δ P i,j t,τ, ηq i,j t,τ and η R i,j t,τ distributed across accident years t, evaluation years τ, and across companies i and lines of business j? If perturbations are purely caused by idiosyncratic shocks, we should not observe any dependence. In other words, our presumption will be that [ ] E ε i,j,j t,τ εi t,τ = 0 t t, τ τ, i i, j j. 6

6 3.2 Using the reserve development triangle In statutory reports, insurers report reserve triangles as shown below where each cell represents the reserves set aside for losses occurring in one year, evaluated in another. For example, a78 represents the reserves set aside in 2008 for accident year 2007: 3 Insurer Reserve Triangle Evaluation year (τ) Accident year (t) a00 a01 a02 a03 a04 a05 a06 a07 a08 a a11 a12 a13 a14 a15 a16 a17 a18 a a22 a23 a24 a25 a26 a27 a28 a a33 a34 a35 a36 a37 a38 a a44 a45 a46 a47 a48 a a55 a56 a57 a58 a a66 a67 a68 a a77 a78 a a88 a a99 We can then represent the one year development (errors) for accident year/evaluation year as: Evaluation year (τ) Accident year (t) a01-a00 a02-a01 a03-a02 a04-a03 a05-a04 a06-a05 a07-a06 a08-a07 a09-a a12-a11 a13-a12 a14-a13 a15-a14 a16-a15 a17-a16 a18-a17 a19-a a23-a22 a24-a23 a25-a24 a26-a25 a27-a26 a28-a27 a29-a a34-a33 a35-a34 a36-a35 a37-a36 a38-a37 a39-a a45-a44 a46-a45 a47-a46 a48-a47 a49-a a56-a55 a57-a56 a58-a57 a59-a a67-a66 a68-a67 a69-a a78-a77 a79-a a89-a Of course it is impossible to calculate the reserve error for losses incurred in accident year 2009 since there has been no development. We can then can re-write this matrix using the terminology presented earlier in the section as: 3 Year 2009 is used as a sample baseline year. 7

7 Accident year (t) Insurer Yearly Errors Evaluation year (τ) ε i,j 0,1 ε i,j 0,2 ε i,j 0,3 ε i,j 0,4 ε i,j 0,5 ε i,j 0,6 ε i,j 0,7 ε i,j 0,8 ε i,j 0, ε i,j 1,2 ε i,j 1,3 ε i,j 1,4 ε i,j 1,5 ε i,j 1,6 ε i,j 1,7 ε i,j 1,8 ε i,j 1, ε i,j 2,3 ε i,j 2,4 ε i,j 2,5 ε i,j 2,6 ε i,j 2,7 ε i,j 2,8 ε i,j 2, ε i,j 3,4 ε i,j 3,5 ε i,j 3,6 ε i,j 3,7 ε i,j 3,8 ε i,j 3, ε i,j 4,5 ε i,j 4,6 ε i,j 4,7 ε i,j 4,8 ε i,j 4, ε i,j 5,6 ε i,j 5,7 ε i,j 5,8 ε i,j 5, ε i,j 6,7 ε i,j 6,8 ε i,j 6, ε i,j 7,8 ε i,j 7, ε i,j 8, Each entry is an actual dollar amount. Each ε i,j t,τ value is drawn drawn from a distribution that is, presumably, centered at zero. If the errors are truly random, they should not be correlated with one another (outside of systematic changes in reserving practices/needs) Data We utilize annual statement data for property and liability insurers from the National Association of Insurance Commissioners (NAIC). Data for the reserve errors come from Schedule P. Our data spans 1991 to This allows us to calculate one year errors for years between 1990 and 2011 and five year errors between 1986 to We consider insurer groups and unaffiliated firms and exclude firms with incurred losses less than $500, with total assets less than $5,000,000, and with a total reserve error greater than 100% of assets. We also remove the firm from final analysis if any one year error is greater than 100% of assets. This leaves a sample of 26,517 observations. Table 1 shows the descriptive statistics, by year, of the one year error for losses incurred one year ago scaled by total assets. The data represents the most recent year of reserve development. For example, in reporting year 2012, the First Year Error of 0.27% is the average change in reserves for accident year 2011 (i.e. the reserves set in 2011 minus the reserves set in 2012) for the 2011 accident year. [Insert Table 1 about here] 4 Again, 2009 is used as a sample baseline year. In our analysis ε 8,9 will refer to the most recent one year error, not that reported in 2009 specifically. 8

8 We see errors ranging from an under-reserving of approximately 0.2% in 2001 to an over reserving of approximately 0.94% in Table 2, then, shows the one year error for an accident year five years in the past. That is, the 0.05% mean in reporting year 2012 is the one year change in the reserve amount for accident year In other words, it is calculated as the reserve amount for accident year 2007 reported in 2011 minus the reserve amount for accident year 2007 reported in Not surprisingly, the one year errors that are five years old are much smaller as these losses would have had more time to develop and a larger percentage of those reserves have moved from estimated to paid losses. Here the errors range from an under-reserving of 0.04% in 2002 to over-reserving of 0.085% in [Insert Table 2 about here] Finally, Table 3 shows the descriptive statistics, again by year, of five year reserve errors as a function of the accident year (the KFS errors). The average errors presented in Table 3 are consistent with other studies on insurer reserving practices. 4 Preliminary Results [Insert Table 3 about here] We first examine the degree to which the errors are normally distributed. If the errors are indeed random, they should appear to be normally distributed. Table 1, in addition to the descriptive statistics of the first year error distribution, shows the goodness of fit (of a normal distribution) for the one year errors. Column six shows the goodness of fit with a normal distribution with a mean of zero and column seven shows the goodness of fit allowing the normal distribution to be centered around the sample mean. The goodness of fit is measured by the Akaike information criterion (AIC). The AIC estimates the quality of the distribution being fit compared to the other distributions. It does not provide a critical value to determine if a distribution fits. Rather, it provides a relative measure to compare distributions. A lower AIC score indicates a better fit. Not surprisingly, allowing the normal distribution to fit the sample mean consistently improves the goodness of fit. As shown, the fits vary across the years. Similarly, Table 2 shows the fit for the one year errors from accident years five years in the past. The errors are much better fits with the normal distribution. Taken together, these results suggest that the errors in the latter years (where losses have evolved more and are less uncertain) are more random (may have less manipulation); 9

9 whereas, the one year errors of relatively new losses (where the losses are more uncertain) appear to be less random (may have more manipulation). Figures 1 through 6 provide some more detail on the overall fit of the normal distribution to selected years one year errors. Figure 1 shows the distribution for the first year error (the change from 2004 to 2005 in the estimates of the 2004 accident year reserves) in 2005, the year with the highest over-reserving in the sample. Here we see that the distribution of errors does not appear to follow a normal distribution (normal probability density function shown in red, Normal (0,0.02)). The sample distribution has a mean substantially above one (nearly 1% of assets) and is positively skewed with more companies having adjusted reserves down from 2004 (over reserved in 2004). The sample distribution is also more peaked than a normal distribution. The sample has more than 90% of the observations between and while the normal distribution would only have 60% of its observations in that area. Figure 2 shows that there are three other distributions that fit better than the normal distribution. The Laplace, loglogistic, and logistic distributions all fit better than the normal distribution even when allowing the normal distribution to be centered about the sample mean. Similarly, for the year with the lowest average error (2001), Figures 3 and 4 also suggest the normal distribution provides a, relatively, poor fit. The sample distribution is skewed again (this time negatively) and the normal distribution underestimates the observations near the mean, overestimates errors as you move away from the mean, but then underestimates observations out in the left-tail. Once again, there are other distributions (Laplace, logistic) that fit better than the normal distribution. Figures 5 and 6 show the fit of the normal distribution to two years of one year errors that are for accident years that occurred 5 years before the reporting year (e.g. the change in reserves from 2008 to 2009 for the 2004 accident year). These sample distributions are much narrower, given that significantly more of the reserve is already paid. While not normal, the normal distribution does fit the data better (see the AIC values in Table 2). If we consider the standard five year errors, we see different results. Figures 7 through 10 show the five year KFS error for the same reporting year error as above. Though the normal distribution is still not the best fit, the distributions appear to be more normal. These results suggest that by examining the five year errors, manipulation can potentially be smoothed out over time resulting in a lack of identification of manipulation in the intervening years.this leads to our concern that 10

10 we are leaving information on the table regarding studying only one measure of reserve errors, i.e. the KFS error. Suppose insurer A purposely over-reserved in 2004 for accident year 2004 by 10%. The expected losses are $1,000,000 but the insurer sets aside $1,100,000 to reduce its tax liability in At some point that over-reserving needs to be undone. If insurer A does that in a systematic manner, for example reducing the over-reserving by 50% each year, the reserve estimate would look like the example below: Evaluation year (τ) Accident year (t) a00 a01 a02 a03 a04 a05 a06 a07 a08 a a11 a12 a13 a14 a15 a16 a17 a18 a a22 a23 a24 a25 a26 a27 a28 a a33 a34 a35 a36 a37 a38 a ,100,000 1,050,000 1,025,000 1,012,500 1,006,250 1,003, a55 a56 a57 a58 a a66 a67 a68 a a77 a78 a a88 a a99 The resulting pattern of error calculations would be: Accident year (t) Evaluation year (τ) ε i,j 0,1 ε i,j 0,2 ε i,j 0,3 ε i,j 0,4 ε i,j 0,5 ε i,j 0,6 ε i,j 0,7 ε i,j 0,8 ε i,j 0, ε i,j 1,2 ε i,j 1,3 ε i,j 1,4 ε i,j 1,5 ε i,j 1,6 ε i,j 1,7 ε i,j 1,8 ε i,j 1, ε i,j 2,3 ε i,j 2,4 ε i,j 2,5 ε i,j 2,6 ε i,j 2,7 ε i,j 2,8 ε i,j 2, ε i,j 3,4 ε i,j 3,5 ε i,j 3,6 ε i,j 3,7 ε i,j 3,8 ε i,j 3, ,000 25,000 12,500 6,250 3, ε i,j 5,6 ε i,j 5,7 ε i,j 5,8 ε i,j 5, ε i,j 6,7 ε i,j 6,8 ε i,j 6, ε i,j 7,8 ε i,j 7, ε i,j 8, In other words, the error term each year would be positively related. This would be the case for any systematic undoing of intentional over/under-reserving. Using one year errors, we can test for a relationship among the errors from the same accident year through time (the row of the loss triangle). If errors are random, there should be no relationship. If the relationships are generally positive, it could indicate systematic over/under-reserving. 11

11 In addition to studying the relationships of reserve errors on the row, we can look at the relationships of reserve angles on the diagonals. The diagonals from upper left to lower right represent the same age reserve errors from different accident years. For example, ε 1,3 ; ε 2,4 ; and ε 3,5 (from the above example) are the one year errors from the 2003, 2004, 2005 reporting years for the accident year two years previous (2001, 2002, 2003 respectively). This reserve development from 12 to 24 months old losses may be positively related if the insurer is using the chain ladder method of reserving. Examining this relationship may indicate that any systematic errors are being induced by the reserving methods, not any manipulation of reserves. Finally, we can also study the relationship among the error terms in the same column, that is the same reporting year. For example, a change to the 2008 accident year reserves in 2009 may be similar to the change to the one year change in the 2006 accident year reserves from 2008 to This could occur if there were a systematic change to all insurer losses, regardless of the accident year in This could happen in rare occasions, but it should not happen every year. There should not be systematic changes to all insurer loss reserves frequently. The reserves represent different losses from different accident years and likely different stages of the settlement process. As an initial test, we estimate the relationship between the one year reserve errors in the current reporting year with the one year reserve errors for other reserving years. In addition to the reserve errors, we also include year control variables as well as a control for firm size. Because the error terms in each model are presumably not independent, we estimate the models utilizing the seemingly unrelated regression model framework. The triangle below shows how our regression models are structured. For example, to examine the error relationships to ε 4,9 (the dependent variable) we are looking at the one year errors along the row (in blue), the one year errors along the diagonal (in yellow) and the one year error in the column (in red), while controlling for the reporting year and size of the organization. 12

12 Accident year (t) Evaluation year (τ) ε i,j 0,1 ε i,j 0,2 ε i,j 0,3 ε i,j 0,4 ε i,j 0,5 ε i,j 0,6 ε i,j 0,7 ε i,j 0,8 ε i,j 0, ε i,j 1,2 ε i,j 1,3 ε i,j 1,4 ε i,j 1,5 ε i,j 1,6 ε i,j 1,7 ε i,j 1,8 ε i,j 1, ε i,j 2,3 ε i,j 2,4 ε i,j 2,5 ε i,j 2,6 ε i,j 2,7 ε i,j 2,8 ε i,j 2, ε i,j 3,4 ε i,j 3,5 ε i,j 3,6 ε i,j 3,7 ε i,j 3,8 ε i,j 3, ε i,j 4,5 ε i,j 4,6 ε i,j 4,7 ε i,j 4,8 ε i,j 4, ε i,j 5,6 ε i,j 5,7 ε i,j 5,8 ε i,j 5, ε i,j 6,7 ε i,j 6,8 ε i,j 6, ε i,j 7,8 ε i,j 7, ε i,j 8, Tables 4 through 6 show the results when considering the scaled error (scaled by total assets). Table 4 shows the relationship between the dependent variable and the one year errors from the same accident year, but previous reporting years (i.e. the row). As can be seen, the relationship is generally positive (especially for younger errors) and significant. While statistically significant, the magnitude of the parameter estimates are fairly low, rarely getting above 0.05, and then declining as the time between the errors gets larger (further to the left in the triangle). [Insert Table 4 about here] Table 5 shows the relationship along the diagonal. Similar to Table 4, the relationship along the diagonal is generally positive and significant. The magnitude of the parameter estimates again decline as the errors move further up the diagonal. For younger errors (the top row), the relationship to the same age errors one year earlier is economically significant, with parameter estimates of 0.32, 0.18, and 0.21 respectively. [Insert Table 5 about here] Table 6 shows the relationship of errors in the same reporting year (the column). The dependent variable is highly positively related to the error term that appears just above it in the column. These two error terms are the two that are most manipulatable by actuaries. They are the two most recent accident years and will have the lowest ratio of paid losses to total reserves. This high positive correlation implies that any changes that are being made to the most recent reserves are also being made the reserves from the previous year. This would occur if a systematic changes to 13

13 reserves was needed, but should not be happening every year (as we control for reporting years in the analysis). The relationship seems to dissipate as the errors move up the column and appears to become more random. 5 Conclusion [Insert Table 6 about here] We first show that reserve errors developed over a shorter period are not normally distributed. If the errors were truly errors and not subject to manipulation, the errors would be normally distributed. Observing that errors in latter stages of development appear to be more normal gives further evidence that reserves with the most uncertainty (i.e. those that are most manipulatable) are subject to manipulation. Second, we show that the standard method of measuring reserve errors, the KFS error, appears to leave information on the table. That is, there is additional information we can glean from examining other development periods. References Beaver, W. H., M. F. McNichols, and K. K. Nelson, 2003, Management of the loss reserve accrual and the distribution of earnings in the property-casualty insurance industry, Journal of Accounting and Economics, 35: Eckles, D. L., and M. Halek, 2010, Insurer Reserve Error and Executive Compensation, The Journal of Risk and Insurance, 77: Eckles, D. L., M. Halek, E. He, D. W. Sommer, and R. Zhang, 2011, Earnings Smoothing, Executive Compensation, and Corporate Governance: Evidence From the Property & Liability Insurance Industry, The Journal of Risk and Insurance, 78: Gaver, J. J., and J. S. Paterson, 2001, The Association between External Monitoring and Earnings Management in the Property-Casualty Insurance Industry, Journal of Accounting Research, 39: Gaver, J. J., and J. S. Paterson, 2004, Do insurers manipulate loss reserves to mask solvency problems?, Journal of Accounting and Economics, 37:

14 Gaver, J. J., and J. S. Paterson, 2007, The influence of large clients on office-level auditor oversight: Evidence from the property-casualty insurance industry, Journal of Accounting and Economics, 43: Grace, E. V., 1990, Property-Liability Insurer Reserve Errors: A Theoretical and Empirical Analysis, The Journal of Risk and Insurance, 57: Grace, M. F., and J. T. Leverty, 2012, Property & Liability Insurer Reserve Error: Motive, Manipulation, or Mistake, The Journal of Risk and Insurance, 79: Kazenski, P. M., W. R. Feldhaus, and H. C. Schneider, 1992, Empirical Evidence for Alternative Loss Development Horizons and the Measurement of Reserve Error, The Journal of Risk and Insurance, 59: Petroni, K., and M. Beasley, 1996, Errors in Accounting Estimates and Their Relation to Audit Firm Type, Journal of Accounting Research, 34: Petroni, K. R., 1992, Optimistic reporting in the property- casualty insurance industry, Journal of Accounting and Economics, 15: Smith, B. D., 1980, An Analysis of Auto Liability Loss Reserves and Underwriting Results, The Journal of Risk and Insurance, 47: Weiss, M., 1985, A Multivariate Analysis of Loss Reserving Estimates in Property-Liability Insurers, The Journal of Risk and Insurance, 52:

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25 Evaluation Year Table 1: Descriptive Statistics (Including Fit) Mean First Year Errors Std. AIC AIC Skewness Kurtosis Dev. (mu=0) (mu=mean)

26 Evaluation Year Table 2: Descriptive Statistics (Including Fit) Mean Fifth Year Errors Std. AIC AIC Skewness Kurtosis Dev. (mu=0) (mu=mean)

27 Accident Year Table 3: Descriptive Statistics (Including Fit) Mean Five Year Errors Std. AIC AIC Skewness Kurtosis Dev. (mu=0) (mu=mean)

28 Table 4: One Year Reserve Errors Dependent Variable VARIABLES ɛ 8,9 ɛ 7,9 ɛ 6,9 ɛ 5,9 ɛ 4,9 ɛ 3,9 ɛ 2,9 ɛ 1,9 ɛ 0,9 ɛ row *** *** *** *** *** *** *** [0.003] [0.005] [0.004] [0.004] [0.005] [0.006] [0.005] [0.005] ɛ row *** *** *** *** *** *** [0.002] [0.003] [0.003] [0.004] [0.004] [0.004] [0.005] ɛ row *** *** ** *** * [0.002] [0.002] [0.003] [0.003] [0.003] [0.004] ɛ row ** *** ** *** [0.001] [0.002] [0.002] [0.002] [0.003] ɛ row *** *** *** [0.001] [0.001] [0.001] [0.002] ɛ row ** *** [0.001] [0.001] [0.001] ɛ row [0.001] [0.001] ɛ row [0.001] Includes constant term, year effects, and control for log of assets (coefficients suppressed). Table 5: One Year Reserve Errors Dependent Variable VARIABLES ɛ 8,9 ɛ 7,9 ɛ 6,9 ɛ 5,9 ɛ 4,9 ɛ 3,9 ɛ 2,9 ɛ 1,9 ɛ 0,9 ɛ diagonal *** *** *** *** *** * *** [0.006] [0.006] [0.006] [0.007] [0.005] [0.007] [0.007] [0.005] ɛ diagonal *** *** *** *** ** *** *** [0.006] [0.006] [0.007] [0.006] [0.005] [0.007] [0.006] ɛ diagonal *** *** * *** *** [0.006] [0.005] [0.006] [0.005] [0.004] [0.006] ɛ diagonal *** ** *** *** [0.006] [0.006] [0.006] [0.005] [0.005] ɛ diagonal ** * [0.006] [0.005] [0.005] [0.004] ɛ diagonal *** *** [0.006] [0.004] [0.004] ɛ diagonal [0.006] [0.006] ɛ diagonal ** [0.006] Includes constant term, year effects, and control for log of assets (coefficients suppressed). 29

29 Table 6: One Year Reserve Errors Dependent Variable VARIABLES ɛ 8,9 ɛ 7,9 ɛ 6,9 ɛ 5,9 ɛ 4,9 ɛ 3,9 ɛ 2,9 ɛ 1,9 ɛ 0,9 ɛ column *** [0.010] ɛ column *** *** [0.014] [0.008] ɛ column ** *** *** [0.020] [0.011] [0.008] ɛ column *** *** *** [0.029] [0.017] [0.013] [0.009] ɛ column *** * ** *** [0.030] [0.017] [0.013] [0.010] [0.006] ɛ column *** ** *** *** *** [0.037] [0.021] [0.016] [0.012] [0.008] [0.008] ɛ column *** *** *** *** [0.046] [0.026] [0.020] [0.015] [0.010] [0.010] [0.007] ɛ column *** *** *** *** *** *** *** *** [0.044] [0.025] [0.019] [0.014] [0.009] [0.009] [0.007] [0.005] Includes constant term, year effects, and control for log of assets (coefficients suppressed). 30