Section A: Estimating Parameters Flashcards

Question

Mack (1994): Mack Chain Ladder Assumption 2

Answer 1

**_Mack Chain Ladder Assumption 2_** Losses are _independent_ between accident years {C_i,1,...,C_i,I} and {C_j,1,...,C_j,I} between different accident years i≠j are independent. * a good estimator (f^hat_k)is unbiased and is as long as we can assume that accident year are independent E[f^hat_k]=f_k * cannot make this assumption for triangles impacted by calendar year effects such as changes to claim handling practices or case reserving which affect several accident years similarly

Answer 2

**_Mack Chain Ladder Assumption 3_** Variance of losses in the *next* development period is _proportional_ to losses-to-date with proportionality constant, **⍺²_k**, that varies by age. **Var[C_i,k+1 | C_i,1,...C_i,k] = C_i,k \* ⍺²_k** * stems from the fact that using a volume weighted average has a smaller variance than using a simple average for the LDFs

Answer 3

1. E[C_i,k+1| C_i,1,...,C_i,k] = C_i,k\* LDF 2. Losses are independent between accident years 3. Var[C_i,k+1 | C_i,1,...C_i,k] = C_i,k \* ⍺²_k

Answer 4

If Assumption 1 holds, subsequent development factors are **uncorrelated** because the expected value of f_k (the LDF) is not dependent on prior loss development. **_Impact:_** If the book of business typically shows a smaller-than-average increase, Loss_k+1 / Loss_k \< LDF_k, after a larger-than-average increase, Loss_k / Loss_k-1, then the chain ladder method would not be appropriate. * you would need to make adjustments to the triangle before analysis can be done

Answer 5

* remember to take the square root to get the standard error * **s.e.(R^hat_i) = s.e.(C^hat_iI)**

Answer 6

* σ is a parameter for lognormal and is not the sd of the reserve - don't mix these up (same for µ)

Answer 7

* The estimators R^hat_i are all influenced by same age-to-age factors, f^hat_k, resulting in positive correlation between accident year estimates.

Answer 8

* **f_k0** → assumes that the variance of C_j,k+1 is **proportional to 1** * C²_ik weighted average of the individual development factors * violates the third assumption of chain ladder as this assumes variance is proportional to C²_ik * **f_k1** → assumes the variance of C_j,k+1 is **proportional to C_jk** * C_ik weighted average of individual development factors * the usual chain-ladder age-to-age factor f_k * **f_k2** → assumes that the variance of C_j,k+1 is **proportional to C²_jk** * unweighted average or simple average of individual development factors * proportional to 1 and violates the third assumption of the chain ladder method

Answer 9

1. Calculate the CI for T 2. Reject the null hypothesis that development factors are uncorrelated if T lies outside the CI

Answer 10

1. First calculate the LDFs 2. Convert age to age factors in each column to ranks 3. Convert table ranks to S's, L's and \*'s 4. Count the number of S's and L's for each diagonal starting top left except ignore the first one as there is only 1 element 5. Using formulas, calculate the CI 6. Reject the null hypothesis that there are not calendar year effects if Z lies outside the CI

Answer 11

Use lognormal when the distribution is **skewed** or the confidence interval can't be **negative**. * Normal distribution will _allow_ the CI to have negative lower limits even if a negative reserve is not possible

Answer 12

R^hat_i = exp[µ_i + σ²_i/2] s.e.(R^hat_i)² = (exp[2µ_i + σ²_i]) \* (exp[σ²_i] - 1) \*Solve for µ & σ which are needed for the C.I.

Answer 13

1. The square of the standard error of R-hat is not the sum of each of the (s.e. (R^hat_i))² since each estimator of R-hat is influenced by the same age-to-age factors. * **not independent** - you would need a covariance term 2. To get the CI by accident year, **allocate** the upper and lower limit of the CI to each accident year in such a way that each accident year has the same confidence level * e.g. might be that 65% CI for each AY results in 80% CI of overall reserve

Answer 14

* Lower empirical limit comes from applying the minimum age-to-age factors for each development period to incurred losses * Upper empirical limit results from applying the maximum age-to-age factors for each development period to the incurred losses

Answer 15

The purpose of looking at global T is that it allows us to test the entire triangle for correlation. There are _two main reasons_ driving this: 1. Some correlation will occur in subsequent LDF purely due to **random chance**; and 2. It is important to know whether correlations prevail globally rather than finding a small part of the triangle with correlations.

Answer 16

Under the Mack assumptions; the Chain Ladder method gives the _minimum variance_ unbiased linear estimator of future claims emergence.

Answer 17

E[q(w,d+1)|data to w+d] = f(d)\*c(w,d) * expected losses to emerge are proportional to the cumulative losses emerged to date

Answer 18

This assumption would be _violated_ if there are calendar year effects * e.g. latest diagonal shows upward shift due to case strenghtening or perhaps a speed up in settlement

Answer 19

**Var[q(w,d+1)|data to w+d] = a[d,c(w,d)]** * variance of the next incremental loss is a function of age and the cumulative losses to date * 'a' does NOT vary by accident year

Answer 20

1. Significance of development factors, f(d) 2. Superiority of the CL method to alternative emergence patterns 3. Linearity of the model * Review residuals vs. Loss_k 4. Stability of development factors * Review residuals vs. time 5. No correlation among columns of development factors 6. No particularly high/low diagonals (calendar year effects)

Answer 21

A factor is considered significant (so not zero) if the factor, **|f(d)|**, is at least **twice** its standard deviation. **|f(d)| ≥ 2σ** * can be tested if distribution is known; can assume normal or if the distribution is positively skewed then use lognormal * if f(d) predicts cumulative losses rather than incremental then test the significance from 1 instead of zero.

Answer 22

1. Linear with a constant: E[q(w,d+1)|data to w+d] = f(d)\*c(w,d)+g(d) **E[IncLoss_d] = f(d)\*Loss_k + g(d)** * states that the next period's expected emerged loss is a linear function of the previous cumulative losses plus a constant 2. Factor times parameter **E[IncLoss_d] = f(d) \* h(w)** * states that the next period's expected emerged loss is a lag factor times the expected ultimate loss amount for an AY 3. Including calendar year effects **E[IncLoss_d] = f(d)\*h(w)\*g(w+d)** * states that the next period's expected emerged loss is a lag factor times the expected ultimate loss amount for an AY times a CY effect factor

Answer 23

**Adjusted SSE = SSE / (n-p)²** p = # of parameters in the model n = # of incremental loss observations (exlude the first column)! For example, a 4 x 4 triangle would have 6 observations.

Answer 24

**AIC = SSE \* e^2p/n** * penalizes less heavily than the adjusted SSE and BIC goodness of fit tests

Answer 25

**BIC = SSE \* n^p/n**

Answer 26

* Often significant at development age 0 to age 1 especially for highly variable and slowly reporting lines (e.g. excess reinsurance) * If **constant is significant** and the **factor is not**, the additive chain-ladder method may be appropiate * i.e. if g(d) is significant, then this emergence is more strongly supported than the chain ladder method

Answer 27

E[IncLoss_d] = f(d) \* h(w) * number of parameters = 2m-2 where m is the number of AYs * the "-2" is due to removing the first development period and the first accident year as these are known and don't need to be estimated * to see if BF method is better than CL, calculate the SSE test statistic for each method and see which has lower SSE taking number of parameters into account * if BF is better, then loss emergence is **more accurately represented as a proportion to ultimate losses** rather than as a percentage of previously emerged losses

Answer 28

1. Assume several accident years in a row have the same mean level 2. Assume subsequent periods all have the same expected percentage development (as percentage of ultimate) 3. Fit a trend line through the ultimate loss parameters 4. Group AY's using apparent jumps in loss levels and fit a single h parameter to each group 5. Use a Cape Cod method

Answer 29

A special case of the BF method is the _Cape Cod_ method. It sets _h(w) = h_ and requires the same number of parameters as the _chain-ladder_ method.

Answer 30

Cape Cod says that the next period's emergence is a lag factor, f(d), times the expected ultimate loss amount, h. Since **h does not vary by AY** this acts as a _constant_ similar to g(d) in the CL constant model. When we fit the CC we can set g(d) = f(d)h for the additive chain ladder model or we can fit the additive chain ladder model by defining f(d)h = g(d) for the Cape Cod model.

Answer 31

Implications 1 & 2 can be quickly tested by graphing the age d+1 loss against the age d loss 1. A factor only model would show a straight line through the origin with a slope equal to the development factor 2. A constant only model would show a horizontal line at the height of the constant

Answer 32

1. Get the f(d) using the chain ladder - get the cumulative LDF's and take reciprocal to get the lags. Then take the difference between the lags to get the incremental lag or f(d). 2. Now use the formulas for h(w) and f(d) to iterate the process until convergence occurs.

Answer 33

* on exam could give you picture and you should be able to know what method to use - weighted least-square or basic

Answer 34

* Same as the BF method except solving for a single h which is summed over all accident years

Answer 35

1. Use a loss ratio triangle 2. Adjust loss ratios for trend and rate level

Answer 36

**_Chain Ladder Assumption_** Assumes future emergence is **proportional to losses** emerged to date for a given accident year. **_BF Assumption_** Assumes expected emergence in each period is a **percentage of ultimate loss**. * Regards losses emerged to-date as a random component that doesn't influence future development. * If this is the case, using the chain ladder will apply factors to the random component and increase error.

Answer 37

Years with low (or high) losses to-date will have the same expected future **_dollar_** development as other accident years.

Answer 38

* Plot the residuals of _incremental_ losses against the prior _cumulative_ loss. * Residuals should be random around zero * If residuals show non-linearity (e.g. positive-negative-positive pattern), the test fails. * If there is non-linearity this suggests emergence is a non-linear function of losses to-date.

Answer 39

* No, since there are **strings of positive and negative** residuals, we can conclude that the age 1 incremental losses are NOT a linear function of the age 0 cumulative losses. * Mack uses **weighted residuals** which is different from this test above as these are unweighted. If weighted/normalized, then should see random scattering around zero.

Answer 40

Residuals over Time * Plot the _incremental_ residuals against time (accident year) * If there are strings of positive and negative residuals in a row, then the development factors may not be stable

Answer 41

**If stable:** All AY's should be used to calculate the development factors to reduce the effects of random fluctuations and minimize variance. **If unstable (factors are changing over time):** Use weighted average of factors with more weight to the recent years. Could also adjust the triangle for instability such as the Berquist-Sherman method which adjusts the triangle to the latest pattern.

Answer 42

Moving Average * Examine the moving average of a specific age-to-age factor * Does the moving average hover around a fixed level * If the moving average shows clear shifts over time, then instability exists * could use weighted average of factors

Answer 43

State-Space Model * the state-space model compares the degree of instability of the observations around the current mean to the degree of instability in the mean itself over time * useful for telling us what years to include such as whether we should include all data or a weighted average that favors recent years

Answer 44

**_Correlation of Development Factors_** Calculates the sample correlation coefficients for all pairs of columns in the development triangle, and then counts how many of these are significant to see if correlation exists.

Answer 45

m = _n-3C₂ = the number of testable pairs \*If there are 3 columns that we tested for the whole triangle then m=3 (e.g. 1_2/24 & 24/36_, _24/36 & 36/48_, and _12/24 & 36/48_) p = 0.1 **µ + 2σ = m\*p + 2 (mpq)** This is Binomial with (m, p).

Answer 46

**_Multiplicative Diagonal Effect_** A diagonal of a loss triangle is w+d and is estimated to be k% higher. Then the adjusted BF estimate of a cell would be: **q(w,d) = (1+k%)\*f(d)\*h(w) for w+d; and f(d)\*h(w) otherwise** **_Additive Diagonal Effect_** Run regression on a loss triangle with sorted into columns with dummy variables. * Chain Ladder method would have the last few columns as the dummy columns * Additive Chain Ladder would replace the non dummy columns with dummy columns

Answer 47

* Ignore the first column in the triangle as we assume this is given to us and is prior cumulative loss * the columns then show the prior cumulative loss * dummy variables ignore the first diagonal as there is only one point (really this is diagonal 2 you are ignoring)

Answer 48

E[q(w,d)|data to w+d-1] = f(d)\*g(w+d) * model assumes we have converted the diagonal effects, g(w+d) from additive to multiplicative using non-linear regression

Answer 49

**E[q(w,d)|data to w+d-1] = h(1+i)^d \* (1+j)^w+d \* (1+k)^w** where, CY effect with cumulative inflation → (1+j)^w+d = g(w+d) AY Effect → h(1+k)^w = h(w) Age Parameter → (1+i)^d = f(d)

Answer 50

h(1 + i)^d(1 + j)^w+d(1 + k)^w can be expanded to: **h(1 + i + j + ij)^d(1 + k + j + jk)^w** The calendar year trend, j, can then be removed since i becomes i+j+ij and k becomes k+j+jk

Section A: Estimating Parameters Flashcards

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