Reserve Estimation Using the Extended Link Ratio Family and the Probabilistic Trend Family of Models

Some of the questions here ask for short written answers. This is meant to give the student practice in answering questions of the format that will appear on Exam 6. Students are encouraged to type their own answers first and then to compare these answers with the solutions given here. Please note that the solutions provided here are not necessarily the only possible ones.

Sources:
Barnett, G.; and Zehnwirth, B, "Best Estimates for Reserves," PCAS LXXXVII, 2000, pp. 245-303.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S6-67-1.

(a) What is the central objective of the Barnett and Zehnwirth paper? (See Barnett and Zehnwirth, p. 245.)

(b) Why do Barnett and Zehwirth pursue this objective (i.e., what problem are they trying to remedy)? (See Barnett and Zehnwirth, p. 245.)

(c) According to Barnett and Zehwirth, what are the "three critical stages" of arriving at a reserve figure? (See Barnett and Zehnwirth, pp. 245-246.)

Solution S6-67-1.

(a) Barnett and Zehnwirth seek to explain how to use the "extended link ratio family" (ELRF) of regression models to test the assumptions of standard link ratio techniques and to "compare their predictive power with modeling trends in the incremental data" (Barnett and Zehnwirth, p. 245).

(b) Barnett and Zehnwirth state that data in most loss arrays do not fulfill the assumptions of standard link ratio techniques, and ELRF approach is more consistent with empirical data. But the ELRF approach is itself deficient and offers a bridge from standard link ratio techniques to more thoroughly statistical modeling approaches (Barnett and Zehnwirth, p. 245).

(c) According to Barnett and Zehwirth, what are the "three critical stages" of arriving at a reserve figure are as follows:

1. Extracting trend and stability information from the data, including distributions about the trends;
2. Formulating assumptions about the future → forecasting paid loss distributions;
3. Considering correlations between lines and how they affect the desired security level (Barnett and Zehnwirth, pp. 245-246).

Problem S6-67-2.

(a) Fill in the blank: Barnett and Zehnwirth propose a statistical modeling framework based on the analysis of the _________ of the incremental data. (See Barnett and Zehnwirth, p. 247.)

(b) What are the four components of interest in the Barnett/Zehnwirth statistical modeling framework? (See Barnett and Zehnwirth, p. 247.)

(c) What is the name of the family of models based on the four components of interest in part (b)? (See Barnett and Zehnwirth, p. 247.)

(d) Fill in the blanks: The statistical nature of the Barnett/Zehnwirth modeling framework allows for the separation of __________ and _________. (See Barnett and Zehnwirth, p. 247.)

(e) What five beneficial functions does the Barnett/Zehnwirth modeling framework enable, aside from the answer to part (d)? (See Barnett and Zehnwirth, p. 247.)

Solution S6-67-2.

(a) Barnett and Zehnwirth propose a statistical modeling framework based on the analysis of the logarithms of the incremental data.

(b) The four components of interest in the Barnett/Zehnwirth statistical modeling framework are as follows (Barnett and Zehnwirth, p. 247):

1. Trend in the development period
2. Trend in the accident period
3. Trend in the payment/calendar period
4. Distribution of data about the trends

(c) The name of the family of models based on the four components of interest in part (b) is the Probabilistic Trend Family (PTF).

(d) The statistical nature of the Barnett/Zehnwirth modeling framework allows for the separation of parameter uncertainty and process variability.

(e) The Barnett/Zehnwirth modeling framework enables the following five beneficial functions (Barnett and Zehnwirth, p. 247):

1. Checking that the data satisfy all the model's assumptions;
2. Calculating reserve forecast distributions and the total reserve;
3. Calculating distributions of future payment streams and the correlations between such payment streams;
4. Pricing for future underwriting years, including for excess layers and aggregate deductibles;
5. Updating models based on new data and tracking forecasts with relative ease.

Problem S6-67-3. On page 248, Barnett and Zehnwirth describe previous findings that influenced their paper.

(a) Fill in the blank: According to Brosius, using link ratio techniques corresponds to fitting a regression line without _______.

(b) How do Mack's findings relate to the statement in part (a)?

(c) What is heteroscedastic normality?

Solution S6-67-3.

(a) According to Brosius, using link ratio techniques corresponds to fitting a regression line without an intercept term.

(b) Mack presented diagnostics suggesting that actual data warrant using an intercept term with a regression line, indicating a possible flaw in link ratio techniques.

(c) Heteroscedastic normality is situation of data following a normal distribution without a constant variance for all the errors. (For some data, the variance may be greater or less than for other data.)

Problem S6-67-4. (a) On page 249, Barnett and Zehnwirth state that the link ratio trend or average method is based on the formula y(i) = b*x(i) + ε(i), where Var(ε(i)) = σ²*x(i)^δ.

Explain the meaning of the variables x(i), y(i), b, ε(i), σ², and δ. (See Barnett and Zehnwirth, pp. 249-250.)

(b) What happens when δ = 1, and what commonly used development technique does this situation reflect? (See Barnett and Zehnwirth, pp. 250-251.)

(c) What happens when δ = 2, and what commonly used development technique does this situation reflect? (See Barnett and Zehnwirth, p. 251.)

(d) What happens when δ = 0, and what commonly used development technique does this situation reflect? (See Barnett and Zehnwirth, p. 251.)

Solution S6-67-4. (a) x(i) for some accident period i is the cumulative developed value at some development period (j-1).

y(i) for the same accident period i is the cumulative developed value at some development period j, i.e., at the development period after the one pertaining to x(i).

b is the slope of the "best" line that includes the origin and the data points (x(i), y(i)). Essentially, the formula y(i) = b*x(i) + ε(i) expresses a value at one development period as a function of the value at the previous development period.

ε(i) is the error term in the regression formula.

σ² is the base variance, or underlying variance, that is the same for the entire development period.

The parameter δ is a "weighting parameter" through which the variance of the error term depends to a certain extent on the value of x(i).

(b) When δ = 1, it is the case that the average value of y(i) is b*(x(i)), so b can be estimated (using least-squares estimation) as Σ(x(i)* y(i)/x(i))/Σ(x(i)) = Σ(y(i))/Σ(x(i)). Thus, the estimate of b is the volume-weighted-average chain-ladder ratio between the value at one development period and the value at the previous development period. That is, when δ = 1, we arrive at the chain-ladder method of reserving.

(c) When δ = 2, it is the case that the weighted least-squares estimator of b is (1/n) Σ(y(i)/x(i)), or the simple arithmetic average of the link ratios between each pair (x(i), y(i)). This is also the chain ladder method, except the averages used are simple arithmetic means instead of volume-weighted averages.

(d) When δ = 0, the least-squares estimator of b is an average weighted by volume-squared, corresponding to the technique of ordinary least-squares regression through the origin.

Problem S6-67-5.

(a) What are two advantages described by Barnett and Zehnwirth for estimating link ratios using regressions? (See Barnett and Zehnwirth, p. 251.)

(b) Refer to the formulas in Problem S6-67-4. What is the formula for the standardized errors? (See Barnett and Zehnwirth, p. 251.)

(c) What is the assumption needed to be made regarding the standardized errors in order to render the weighted least-squares estimator of b "efficient"? (See Barnett and Zehnwirth, p. 251.)

Solution S6-67-5.

(a) Estimating link ratios using regressions has the following advantages (Barnett and Zehnwirth, p. 251):

1. It is possible to obtain both the standard errors of the forecasts and the standard errors of the parameters in the average method selection.
2. It is possible to test the assumptions made by this method.

(b) The formula for the standardized errors is ε(i)/√(Var(ε(i)) = ε(i)/√(σ²*x(i)^δ) = ε(i)/(σ*x(i)^δ/2) for all applicable values of i. Dividing by the square root of the variance (i.e., the standard deviation) of the error term is necessary for the standardized error to reflect a standard deviation of 1. (See the answer to part (c)).

(c) In order to render the weighted least-squares estimator of b "efficient", it is needed to assume that the standardized errors are normally distributed with mean 0 and standard deviation 1 (Barnett and Zehnwirth, p. 251).

Problem S6-67-6. Barnett and Zehnwirth describe a basic assumption of the link ratio method that can be expressed in two equivalent ways:

(a) E(y(i)│x(i)) = bx(i)
(b) E((y(i) - x(i))│x(i)) = (b-1)x(i)

Give a verbal interpretation of each of the above two expressions of this assumption. (See Barnett and Zehnwirth, p. 252.)

(c) What is a way to test this assumption, using a plot of fitted values on the horizontal axis and weighted standardized residuals on the vertical axis? What is typically empirically observed as a result of such tests? (See Barnett and Zehnwirth, pp. 252-253.)

(d) What do the empirical observations discussed in part (c) imply about the differences between actual results and the predictions of the link ratio method?

Solution S6-67-6.

(a) Given that we know the cumulative development at period (j-1), all we need to do to obtain the mean cumulative development at period j is to multiply the cumulative development at period (j-1) by the ratio b.

(b) Given that we know the cumulative development at period (j-1), in order to find the mean incremental development at during j, all we need to do is to multiply the cumulative development at period (j-1) by (b - 1), i.e., the ratio b, minus 1 so that we do not count the development through period (j-1).

(c) To test this assumption, plot fitted values on the horizontal axis and weighted standardized residuals on the vertical axis. If the assumption is correct, there should be a random distribution of weighted standard residuals versus fitted values, with no way to fit an intelligible line or curve through them. However, empirically, it has been observed that there is a downward-sloping line that can be fitted to this plot.

(d) The link ratio method overpredicts large values and underpredicts small values. (E.g., for large values, the fitted values are too high, and, for small values, the fitted values are too low.)

Problem S6-67-7. (a) How did Murphy modify the equation from Problem S6-67-4? Give the modified equation and explain the meaning of any additional term(s). (See Barnett and Zehnwirth, p. 253.)

(b) What does Murphy's modification imply regarding the ability of prior cumulative development x(i) to predict subsequent incremental development y(i)-x(i)? Does this correspond to observations from empirical data in situations where b = 1? (See Barnett and Zehnwirth, p. 255.)

Solution S6-67-7. (a) Murphy modified the equation from Problem S6-67-4 as follows:
y(i) = a + b*x(i) + ε(i), where Var(ε(i)) = σ²*x(i)^δ.

The new term added is a, an intercept that enables the regression line used in estimating the link ratio not to go through the origin.

(b) Murphy's modification implies that x(i) can no longer predict y(i)-x(i). This is because the intercept a is estimated via a weighted average of the incremental development in period j pertaining to the values of y(i). Thus, the value of a is independent of any x(i).

This implication does correspond to empirical data in situations where b = 1, which show virtually zero correlation between prior cumulative development and subsequent incremental development.

Problem S6-67-8. (a) How can one further modify the equation from Solution S6-67-7 to reflect a situation where there is a trend in incremental development down the accident periods? Describe the meaning of any additional term(s) used. (See Barnett and Zehnwirth, p. 256.)

(b) How can the Cape Cod (Stanard-Bühlmann) method be described in terms of the formula from part (a)? (See Barnett and Zehnwirth, p. 257.)

Solution S6-67-8. (a) The modified formula is as follows:
y(i) = a₀ + a₁*i + b*x(i) + ε(i), where Var(ε(i)) = σ²*x(i)^δ.

The old intercept a becomes a₀ and serves the same purpose. The new addition is the parameter a₁, which relates to the accident period i. This is the accident-period trend parameter; it can capture changes in cumulative (and corresponding incremental) development in later accident periods, as compared to earlier ones.

(b) The Cape Cod method can be seen as a special case of the formula in part (a), where it is assumed that b = 1 and a₁ = 0. (That is, there is no accident-period trend, and there is no effect of prior cumulative development on subsequent incremental development.) The Cape Cod formula becomes y(i) = a₀ + x(i) + ε(i), where Var(ε(i)) = σ²*x(i)^δ.

Problem S6-67-9. In general terms, what does the Cape Cod method assume regarding the relationship among incremental values in the same development period? Contrast this to the approach of the chain ladder method and discuss the implications regarding coefficients of variation in incremental values for each method. (See Barnett and Zehnwirth, pp. 261-262.)

Solution S6-67-9. The Cape Cod method assumes that incremental values in the same development period are randomly drawn from the same distribution. The chain ladder method makes no such assumption; instead, the incremental values are related to previous cumulative values via the same link ratios for each accident period. Because the Cape Cod method inherently assumes a relationship among incremental values in the same development period, we can expect to observe lower coefficients of variation for such values, as compared to the coefficients of variation in a chain-ladder scenario.

Problem S6-67-10. (a) In what way can the models described by the equation in Solution S6-67-8 be considered illustrative of the bridge that the ELRF forms to the statistical models in the PTF? (See Barnett and Zehnwirth, p. 264.)

(b) What assumption do the ELRF models make regarding the weighted standardized errors, which is rarely true for real-world reserving data? What is generally observed empirically instead, and what approach does this observation suggest? (See Barnett and Zehnwirth, p. 264.)

Solution S6-67-10. (a) As a bridge to the PTF, the ELRF models are capable of more accurately incorporating real-world data and phenomena than the traditional link ratio techniques. What they can do is identify payment-period trend changes diagnostically, but not identify them directly or forecast on their basis; thus, they still fall short of the statistical models in the PTF (Barnett and Zehnwirth, p. 264).

(b) The ELRF models make the assumptions that the weighted standardized errors are normally distributed, which is rarely true in the real world. Instead, it is generally observed that weighted standardized errors are skewed to the right; that is, more of them are positive than are negative. This observation suggests that a logarithmic scale (analyzing the logarithms of the weighted standard errors) would be better suited to analyzing them (Barnett and Zehnwirth, p. 264).

Problem S6-67-11. Explain how the use of logarithms can introduce linearity into observed dollar trends. (See Barnett and Zehnwirth, p. 266.)

Solution S6-67-11. Observed real-world dollar trends are typically percentage trends (e.g., growth at X% per year). But logarithms of incremental data that follow percentage trends themselves follow linear trends.

Problem S6-67-12.(a) If the development year can be expressed by the variable j and the accident year can be expressed by the variable i, find the expression for the payment year (t). (See Barnett and Zehnwirth, p. 266.)

(b) Suppose that all data being used are adjusted for wage and price inflation, but the payment-year trend is still positive. What phenomenon does this observation indicate? (See Barnett and Zehnwirth, p. 267.)

Solution S6-67-12. (a) The expression for payment year is t = i + j. That is, loss amounts that develop j years after a loss occurs in year i will be paid in year (i + j).

(b) If all data being used are adjusted for wage and price inflation, but the payment-year trend is still positive, this is indicative of social inflation (e.g., higher average legal verdicts and increases in medical costs above the increases in the costs of medical inputs).

Problem S6-67-13. Let y(j) = α + _k=1^jΣ(γ_k) + ε_j, where k = 1, ...., j are the development years, ε_j is the error term (normally distributed with a mean of zero), α is the expected development in the first development year, each γ_k is the mean trend between development years (k - 1) and k.

Fill in the blanks (See Barnett and Zehnwirth, p. 271-272):

(a) y(0) has a mean of _____ and a variance of σ².

(b) Let p(j) = exp(y(j)). Then the median of p(j) is ________.

(c) The mean of p(j) is the median of p(j), multiplied by _______.

(d) The standard deviation of p(j) is the mean of p(j), multiplied by _______.

(e) The distribution of p(j) is ________.

Solution S6-67-13.

(a) y(0) has a mean of α and a variance of σ².

(b) Let p(j) = exp(y(j)). Then the median of p(j) is exp(α + _k=1^jΣ(γ_k)).

(c) The mean of p(j) is the median of p(j), multiplied by exp(0.5σ²).

(d) The standard deviation of p(j) is the mean of p(j), multiplied by √(exp(σ²) - 1).

(e) The distribution of p(j) is lognormal.

Problem S6-67-14. (a) Give the general formula for the Probabilistic Trend Family (PTF) of models and explain the meaning of each term. (See Barnett and Zehnwirth, p. 273.)

(b) Based on this formula, with relation to accident year i and development year j, what is the mean trend between (i, j -1) and (i, j)? (See Barnett and Zehnwirth, p. 273.)

(c) Based on this formula, with relation to accident year i and development year j, what is the mean trend between (i-1, j) and (i, j)? (See Barnett and Zehnwirth, p. 273.)

Solution S6-67-14. (a) The general formula for the Probabilistic Trend Family (PTF) of models is y(i,j) = α_i + _k=1^jΣ(γ_k) + _t=1^i+jΣ(ι_t) + ε_i,j, where
i = accident year
j = development year
t = i+j = payment year
α_i = expected development in the first development year (j = 0)
γ_k = mean trend between development year (k-1) and development year k.
ι_t = mean of the inflation between payment year t and t + 1
ε_i,j = error term (normally distributed with mean 0)
y(i,j) = natural logarithm of incremental paid loss for accident year i, development year j.

(b) The mean trend between (i, j -1) and (i, j) is γ_j + ι_i+j.

(c) The mean trend between (i-1, j) and (i, j) is α_i-α_i-1 + ι_i+j.

Problem S6-67-15. What is a possible problem resulting from several parameters in the same model pertaining to payment-year and accident-year trends? Briefly, how might such a problem be overcome? (See Barnett and Zehnwirth, p. 295.)

Solution S6-67-15. Multicollinearity is a possible problem resulting from several parameters in the same model pertaining to payment-year and accident-year trends. This is because payment year is a linear combination of accident year and development year, and so there might be some definitional overlap among the parameters. To overcome this problem, one could use a varying-parameter stochastic model, especially one in which the α parameter varies, instead of adding new parameters (Barnett and Zehnwirth, p. 295).

Problem S6-67-16. If you were analyzing a simulated cumulative loss development array created using traditional ratio methods, how would using a PTF model enable you to distinguish this array from an array based on actual observed data? (See Barnett and Zehnwirth, p. 297.)

Solution S6-67-16. Using a PTF model to analyze the array would show that, after removing the trends in the direction of development year, there are still clear patterns in the direction of accident year. Since the same ratios are applied to data for all accident years, the initial discrepancies among accident years at early maturities are never mitigated; there is much more volatility among accident years than would be observed for real data (Barnett and Zehnwirth, p. 297).

Problem S6-67-17.

(a) In order for a forecast distribution to accurately predict the future, at least three assumptions about trends need to be true. Identify the three assumptions as listed by Barnett and Zehnwirth on p. 298.

(b) What is the difference between a fitted distribution and a predictive distribution? (See Barnett and Zehnwirth, p. 299.)

Problem S6-67-17.

(a) In order for a forecast distribution to accurately predict the future, at least the following assumptions about trends need to be true (Barnett and Zehnwirth, p. 298):

1. Assumptions regarding mean trends
2. Assumptions regarding standard deviations of trends
3. Assumptions regarding distributions about the trends

(b) A predictive distribution incorporates into itself parameter risk (the risk of estimating a parameter erroneously), whereas a fitted distribution does not (Barnett and Zehnwirth, p. 299).

Problem S6-67-18. What three important observations with regard to risk-based capital do Barnett and Zehnwirth offer on p. 301?

Problem S6-67-18. Barnett and Zehnwirth offer the following three observations regarding on page 301:

1. Loss reserve uncertainty should be based on a future-oriented probabilistic model and does not have a necessary relationship to the company's past reserves.
2. The company may have unique experience, not reflective of the industry's experience as a whole.
3. For each line of the company's business, the reserve uncertainty should be modeled by a probabilistic model unique to that line and using company-specific experience, as a model derived from the experience of another line will typically not be accurate when applied to the line in question.

Problem S6-67-19. What are the three steps for booking a reserve described by Barnett and Zehnwirth on p. 302?

Problem S6-67-19. The following are the three steps for booking a reserve (Barnett and Zehnwirth, p. 302):

1. Extract information about trends for the loss development array - their stability, distribution about them - especially for incremental paid losses. This is done by identifying the best model in the PTF.
2. Formulate assumptions about the future: will future trends be stable or not?
3. Select a security margin for the combined lines of business. Determine the percentile at which the reserve will be established, based on the reserve distribution, the security margin, and the company's available risk capital.

Problem S6-67-20. What two additional benefits does the statistical modeling framework offer? (See Barnett and Zehnwirth, pp. 302-303.)

Problem S6-67-20. The following are two additional benefits of the statistical modeling framework:

1. Incorporation of credibility: It is possible to bring a trend parameter estimate that is lacking credibility closer to the level of the complement of credibility (e.g., the industry estimate).
2. Segmentation and layers: The model can be applied specifically to some of the segments of the company's book of business or some of the insured layers.

See other sections of The Actuary's Free Study Guide for Exam 6.