Mock CAS Exam 9 Fall Solutions

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Silas Underwood
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1 Mock CAS Exam 9 Fall Solutions

2 1. a. i. Not differentiate unfairly between risks ii. Based upon relevant data iii. Respect personal privacy iv. Risks identify naturally with classification b. Can increase confidence that information used is useful in predicting the future and will result in stable results over time. In addition, it can make the results more acceptable to the public. In practice, it is used in a general sense, implying a plausible relationship between risk characteristics and hazards insured against. c. Desirable - close association with an effort to reduce hazards / general acceptability by the public Undesirable potential close association to manipulation, impracticality and irrelevance to predictability of future costs

3 2. Just be consistent in notation there are many correct answers for part (a) a. Vector of Observations: Y = Male Urban = 1,250 Male Rural = 700 Female Urban = 500 Female Rural = 350 Design Matrix: X = Vector of Parameters: β = β 1 β 2 β 3 β 1 corresponds to males, β 2 corresponds to female, and β 3 is the urban effect. b. The link function specifies the relationship between the random component, Y, and the systematic component, X β. It is differentiable and monotonic. E[Y] = g -1 (X β). The error function is used to calculate the likelihood function, which is then maximized to produce the parameter estimates. c. i. Standard errors Use multivariate version of Cramer-Rao lower bound to define standard errors for parameter estimates, which will give intervals around parameter estimates of a GLM. ii. Deviance A measure of how much fitted values differ from observations. Can assess statistical significance of particular factors in a GLM.

4 3. a. Credibility should vary in proportion to average claim frequency, if variation of individual insureds chances for accidents were the same within each class. b. Credibility is greater when the variability of hazards within classification is higher. This is a natural conclusion from experience rating. c. Should vary in proportion to the number of years, as long as individual insureds chances for an accident are constant from year to year, and if no risks are entering or leaving the class (see Hazam true for small credibilities).

5 4. a. i. Ruin Theory says P(L + E < E[L] + E + R + V) < e Standardizing L yields z = (R + V) / S V = zs R V V = z(s S ) r Where L = loss, E = expense, R = return, V = surplus, e = probability of ruin, z = standard normal value corresponding to e, the prime denotes after the addition of account x to the portfolio Marginal Surplus depends on S S, which depends on the correlation between the portfolio and x. Then they are fully correlated, MS is a function of the standard deviation of x, whereas if they are independent, MS is a function of the variance of x. ii. CAPM says required surplus is zs, but Ruin Theory says required surplus is zs R. So modify ruin theory to say that P(L + E + R < E[L] + E + R + V) < e, such that breaking even is not enough anymore. Then the required surplus matches that of CAPM. (You can also talk about how (S-S )/p * P/S can be seen as a beta and go that route, where p = x s premium, P = industry premium) iii. CAPM assumes investors have utility functions of a specific form, that is, risk averse functions with known first and second moments. There is an ability to use more complex utility functions where more than just the first two moments are known; use of CAPM will then evolve into use of APM. b. 3 options here (others are industry vs. company leverage and how to compute covariance) What is the industry leverage ratio? It is not known, but must be selected, based on risk appetite of senior management. The value of z is also not known.

6 5. a. Any company can offer S 0 to attract low risk consumers away from S 1 because S 0 lies on a higher utility curve for low risks, consistent with the average rate. Then as more low risks purchase S 0, so S 1 becomes unprofitable and must be withdrawn. b. High risks then purchase S 0 along with low risks. Suppose a firm offers policy S 2 (located in the area below EH 0 L, above EH 0 H and above P L. This policy will only be profitable when low risks purchase it (and they prefer it, while high risks do not). As more low risks do (and they will since it is on a higher utility curve) then S 0 becomes unprofitable and must be withdrawn. Then high risks will purchase S 2 and thus it will become unprofitable and must be withdrawn. c. It will be possible to achieve equilibrium if firms offer two contracts, one at full coverage at a high unit cost, and another at partial coverage at a low unit cost. Insureds will then self-select, and this will give rise to the Nash separating equilibrium. Know the graph on page 39 of Cummins (NSE equilibrium) d. Neither. NSE is not Pareto optimal because under the Wilson subsidizing equilibrium (WSE), high risks' position does not change, low risks are better off, and aggregate profits are not reduced. NSE is not socially optimal either because the expected utility of low risks is higher than that of high risks.

7 6. a. i. Equity holders contribute 10,000 / 2 = 5,000 as supporting suplus only (since 10,000 10,000 = 0, there is no U/W loss). Total assets = 10, ,000 = 15,000 will be invested in the four year coupon, growing to 15,000 * 1.10 ^ 4 = 21, The loss is paid, so the equity holders get 11, back (in four years), implying an IRR of (11, / 5,000) ^ = 24.4%, so accept. ii. Now, they contribute (25,000 10,000) + 25,000 / 2 = 27,500 at policy inception. Total assets grow from 37,500 to 54, The loss is paid and equity holders get 29,903.75, yielding an IRR of 2.1%, so reject. b. This simply means that the IRR is less than the investment yield, so if we let x be the expected loss, we want {[(x 10,000) + x / ,000] * 1.10 ^ 4} x <= 1.10 ^ 4 * [(x 10,000) + x / 2)] 3x/2 * x <= * (3x/2 10,000) 1.196x <= 2.196x 14,640 x >= 14,640 [This problem is probably worth 1 point, not half!] c. This highlights that even though a contract may be very unprofitable, the IRR could still be positive, which, when the actuary makes a rate filing, might not tell the whole story and could cause issues with regulatory acceptance. A positive number looks better than a negative number, so the actuary should publish the NPV as well, since that will be negative and will complete the story.

8 7. a. T/S = I/A * (1 + R/S) + U/P * P/S 12% = 7% * ( ) + U/P * (1.75/50%) U/P = -2.1% CR = 1- U/P = 102.1% b. i. Increase: Because I/A = 7% > -2.1% ii. Not Increase: Subjects surplus to more risk (P/S ratio > 3) c. i. Cost of debt capital is fixed; cost of insurance leverage has an expected value and a variance ii. Increase in debt capital increases interest rate; expanding insurance portfolio has an unpredictable effect (increase in volume reduces volatility of underwriting results and may offset added risk)

9 8. a. An empirically determined table of charges for the aggregate of deductible losses, where D is the deductible amount, reflecting the distribution of limited losses b. Like Table M it only includes the insurance charge for the aggregate limit, not for the charge in excess of the deductible Like Table L it is indexed by three variables: expected (limited) losses for the policy, the deductible and the entry ratio c. The ICRLL can be used to map the three indices to the two used by the unlimited Table M because the entry ratio and size category are modified to account for the deductible

10 9. a. i. It would imply that it would be good to give more weight to the insured's experience, because that would mean greater safety incentives. However, it would be unfair to charge back losses that were random and unpredictable. ii. Charging insureds for past loss experience to the extent that it is predictable of the future produces the best allocation of resources, and makes all insureds (credit and debit insureds) equally desirable. b. Necessary condition credit risks and debit risks equally produce the permissible loss ratio (in addition, each subgroup, with sufficient volume, should reproduce the permissible) Sufficient condition risks within a group cannot be subdivided on any basis in order to produce different loss ratios The word "produce" reflects the experience that would emerge in the prospective period

11 10. a. i. No-split plan: Losses are not split into primary and excess portions for the purposes of experience rating ii. Single-split plan: Losses are split into primary and excess portions based on one split for the purposes of experience rating iii. Multi-split plan: Losses are split into primary and excess portions, where the primary portion is dependent on the size of loss. b. i. ii. a. E / (E + K) >= 0 because E >=0 and E / (E + K) <= 1 because K > 0 b. dz/de = [(E + K) E] / (E + K)^2 >= 0 because K > 0 c. d(z/e)/de = d/de (1 / (E + K)) = -1 / (E + K)^2 < 0. a. No split: M = [AZ + E(1 Z)] / E, where A = actual losses, E = expected losses, M = experience modification M = [A(E / (E + K)) + E(1 - E / (E + K))] / E => M = (A*E + K*E) / (E*(E + K)) => M = (A + K) / (E + K) b. Single split/multi split Single split & multi split have the same experience mod formula, except the definition of the primary loss (A p ) is different. M = [Z p * A p + (1 - Z p )* E p + Z e * A e + (1 Z e )* E e ] / E e denotes excess, use K and J in the formula for Z p and Z e, respectively After substitution and simplification, you should get: M = 1 + (A p - E p ) / (E + K) + (A e E e ) / (E + J)

12 11. a. E = 50% Entry Ratio # Risks # Above Losses Above Insurance Charge b. T = (0.2 + E*(1 +.02)) / (0.2 + E) * 1 / (1 3.5%) = e = (1-D)/T - E = 0.885/ = r G r H = (G H) / cet = ( ) / (1.25*0.5*1.051) = 1.40 χ H χ G = (e + E H/T) / ce = 0.58 The entry ratios and insurance charges that satisfy these relations are: r G = 1.80, r H = Therefore, ci = c*e*(0.040 ( ) = Thus, b = 100,000*( *(1.25 1) ) = 22,950 c. i. The variation in loss ratios is the same as that of the Table M ii. Aggregate loss ratio of the risks equal the expected loss ratio Of course, this problem is simplistic in nature, as we made sure the total retrospective premium equals the guaranteed cost premium, such that (i) and (ii) were satisfied.

13 12. a. % eliminated = k = [L r + L R (R / (R r)) * (L R r* N R )] / L L r = 90,000 L R = 1,080,000 N R = 2,400 R = 750 r = 250 L = 2,620,000 Therefore, k = b. Total rate for full coverage = 3 = (E N + E M + 3*10%) / (1 4% - 12%) E N + E M = 2.22 but E N = E M, thus E N = E M = 1.11 Rate for coverage = 1.75 = [ f*(k* E N * E M ) + 3*10%] / 84% => f = 92.6%

14 13. a. Many more answers possible here i. Both use actual losses in determining the experience modification, but self-rating (100% credibility) is only possible in ISO and not NCCI. ii. NCCI splits losses into primary and excess portions in order to maximize predictive accuracy. The loss distribution skewness decreases. In ISO, losses are limited to improve predictive accuracy. The ARULL calculation also represents stabilizing the ultimate value of losses. b. i. Employers' financial incentive to not report medical-only claims is reduced, thereby creating a safer work environment. ii. Did not really improve predictive accuracy of plan, since medical only losses are such a small % of total losses.

15 14. a. Safety constraint: A >= (s E[L]) / (1 + E[y]) Variance constraint: A >= SQRT(Var[L] / Var[y]) E[L] = 35, Var[L] = 4,187.50; s = 250 E[y] = 6%, Var[y] =.3075% Therefore, A(safety) >= A(variance) >= 1, Since A(variance) > A(safety), the Variance constraint is dominant b. The risk loaded premium uses R from the dominant constraint, and P = R + E[L] / (1 + rf), where R = (E[y] rf) / (1 + rf) * A So, P = (6% - 3%) / 1.03 * 1, / 1.03 = $67.97

16 15. The formula in Lee for "c" is wrong (there is a thread on this) a. Understand the graph on page 65 if Lee and the three shaded areas b. Numerator = expected excess loss per ground-up claim with indexing Denominator = expected excess loss per ground-up claim without index Discount = 1 (k*$275,000) / [k*$275,000 + k*(1.2 1)*100,000 + c*5,000] Where k = G(R) = and c = G(R/a) - G(R) = =.0667 Thus, Discount = 7.1%

17 16. a. Could define using words or integrals i. Expected amount of unlimited loss in excess of r times expected loss (divided by expected loss) ii. Expected amount by which actual unlimited loss falls short of r times expected loss (divided by expected loss) iii. Expected amount of limited loss in excess of r times expected loss, plus expected difference between actual unlimited loss and actual limited loss (divided by expected loss) iv. Expected amount by which actual limited loss falls short of r times expected loss (divided by expected loss) b. E{L * } / E = 1 + ψ*(r 1 ) - φ*(r 2 ) Know the graph on page 73 of Lee for graphical proof c. φ*(r) = 1 r (if r < 1 k) and φ*(r) = k (if r >= 1 k) Know figure 19c on page 74 of Lee for graphical proof

18 17. a. i. Number of years used in credibility-weighted estimate ii. Covariance between a pair of years with lag k (within covariance) iii. Lag in years (estimated year latest year available) iv. Lagrange multiplier (used to solve for credibilities) b. Using the equation given, you have: (For i = 1) Z 1 *C(0) + Z 2 *C(1) = C(0) + λ/2 (For i = 2) Z 1 *C(1) + Z 2 *C(0) = C(1) + λ/2 Subtract the equations to get rid of λ: 0.10* Z * Z 2 = But we also have that Z 1 + Z 2 = 1 (because this credibilities must add to 1 since the grand mean is not used in this case see Mahler 1 page 267) Thus, solving the two equations yield: Z 1 = 0.25, Z 2 = 0.75 Thus the estimated losing % for 2009 = 0.25*60% *54% = 55.5% c. i. Small chance of large errors deals with the probability that the observed result will be more than a certain percentage different than the predicted result. (lower prob => better solution) ii. Meyers/Dorweiler minimize patterns in errors; calculate correlation between actual/predicted losing percentage and predicted/overall average losing percentage; closer the correlation is to zero, the better the solution.

19 18. a. i. Develop, trend and on-level losses ii. Group by accident, then by hazard grade iii. Truncate and shift data at a selected truncation point iv. Normalize data by hazard group and combine for curve fitting b. Fit excess ratios of the truncated and shifted data using a mixed distribution (Pareto and exponential), with most of the weight going to the exponential. c. For retentions below the truncation point, excess ratio comes from the data. For retentions higher than the truncation point, calculate the excess ratio at the truncation point and multiply it by the estimated (curve fit) excess ratio of the retention minus the truncation point (since data was truncated and shifted before curve-fitted).

20 19. Someone can hide their answer in a spoiler in the thread (or I could later)

21 20. a. Know the top graph on page 120 in McClenahan. ROS = 20% / leverage b. Know the graph on the bottom of page 120 in McClenahan. ROS is a constant = 20% / 4 = 5%. c. It forces the regulator to forgo rate equity for rate of return equity. May indicate one insurer earning too high of a return (and therefore rates disapproved), and another insurer having an adequate return (and therefore rates approved), even if they file the same rates. d. i. Natural way to view profit (markup on price) ii. Independent of relationship between premium and equity and results in true rate regulation

22 21. Someone can hide their answer in a spoiler in the thread (or I could later)

23 22. Highlighted stuff will change when using the new manual, so adjust as necessary to make the problem work. I like this problem because it involves no balance equations, but still uses the manual. First, calculate LUGS: LUGS = 2,000,000*75%*1.10*[( *.25/.75) / (1 -.25/.75)] = 3,135,000 => ELG = 27 Look for the entry ratio at which the savings is (for ELG = 27) = 0.22 So, r H = The charge at the minimum, χ H, equals = Now we calculate the insurance charge at the maximum entry ratio: χ H χ G = (e + E H/T) / [c(e ELF)] = => χ G = (which corresponds to an entry ratio of r G = 2.31) Thus, b = e E(c 1) + c(e ELF)*( ) = Thus R = (2,000,000* *(0.25*2M + 500K + 1.8M))*1.045 = 3,608,301.40, but subject to the maximum and minimum. You also have to check G before calculating R (to see if R > G)

24 23. For the CY II Offset, Offset = 6%*(30,000/1,000, %*1/1.20) = 3.43% Thus, U OFFSET = 5% % = 1.57% For the CY ROE Method, Income = (1 0.33)*U ROE *1,000, %*(500,000 + PHSF) Where PHSF = 30, ,000,000*65%*1/1.2 = 571, Equity = 650,000 Thus, ROE = 12% = (670,000* U ROE + 64,300) / 650,000 => U ROE = 2.045% So, U OFFSET - U ROE = 0.475%

25 24. Required rate of return = / % + 1% - 2% = 13.96% This implies that Actual rate of return = 14.66% Actual rate of return = (250K + 2M + 1M 550K) / 15M PHDIV + 1% - 2% = 17% - PHDIV Thus, PHDIV = 17% % = 2.34% The dollars = 2.34% * 15M = $351,000

26 25. a. Do this the short way, not the long way E = 58.5%, k = 1 45% / 58.5% = φ(1.2) = [3/10*(80% %) + 1/10*(100% %) + 1/10*(115% %)] / = φ*(1.2) = [1/10*(85% %) + 1/10*(95% %)] / = Thus, Δ φ(1.2) = = b. This problem demonstrates the interplay between Table L, Table M, and Table M D i. Use the Table L that applies for a deductible level of 100,000. This is because Table L contains both charges and thus can price the deductible and aggregate simultaneously without overlap ii. Can use ICRLL approach or Table M D (d = 100,000) because Table M D calculates the insurance charge for the aggregate, but not for the deductible. You will need an ELF to price the deductible portion. ICRLL allows you to use Table M, adjusting the entry ratio and size indicator to account for the deductible.

27 26. a. LDD WC Premium = [350,000*( ( )*0.05) + 500,000*( )] / (1 8% - 3% - 3%) = 158, => Expense ratio = 8% + 3% + [350,000*(0.08+( )*0.05) + 500,000*0.06] / 158, = 49.6% b. Excess WC Premium = [350,000*(0.08)*( ) + 500,000*0.06*(1-0.40)] / (1 (8% - 5%) 3% - 0%) = 51, => Expected Loss Ratio = [350,000*(0.08) / 1.10] / 51, c. = 49.6% i. LDD over XS a. Advantage burden of payment on insurer b. Disadvantage relies on insurer's claim handling ability (expense cost is a large portion of premium) ii. XS over LDD a. Adv can control and service own losses under retention (more control) b. Disadvantage burden of payment on insured

28 27. a. Portion of rate from CAT = 1 1 / = 29.33% Portion of rate from hurricane = 75% * 29.33% = 22% Current average hurricane rate = 22% * $5.55 = $1.22 Current average non-hurricane rate = $ $1.22 = $4.33 Indicated average non-hurricane rate = $3.25 * (1 + 5% + 10%) / (1 12%) = $4.25 => Indicated rate level change for non-hurricane = $4.25 / $ = -1.9% Risk margin for hurricane = 5% * 100 / 10 = 50% Indicated average hurricane rate = [2.14*( ) ] / (1 12%) = $5.07 => Indicated rate level change for hurricane = $5.07 / $ = % b. Approved average total rate = $5.55 * ( ) = $8.36 Approved average non-hurricane rate = $ $4.47 = $3.89 Premium level change for non-hurricane coverage = $3.89 / $ = -29.9% This figure will be used in the future to separate out non-hurricane premium from unbundled premium in ratemaking analyses. It is not necessary if non-hurricane and hurricane premium will be separated in future analyses.

29 28. a. 4 possible answers here.2 listed below: i. Improper rate relativities in other classification dimensions, combined with an uneven distribution of insureds by these other classifications ii. Random fluctuations in loss experience a credibility issue b. Assume equal exposures in each cell since none given, and assume Male Rural is the base class. LR Relativities Urban Rural Male: Female: Loss Cost Relativities Urban Rural Male: Female: Given x 1 = 1.25, x 2 = 0.60, y 1 = (1.528 / / 0.60) / 2 = y 2 = (1.000 / / 0.60) / 2 = then, x 1 = (1.528 / / ) / 2 = x 2 = (0.750 / / ) / 2 = 0.619

30 29. a. The entry ratio at 50,000 for undeveloped losses is: [ 50K 100K (y 50,000) * 1 / 100,000 dy] / E[X] = 12,500 / 50,000 = 25% b. The density function for developed losses = K/y f(y r)*h(r) dr = K/y r/100,000 * 2r/3 dy = 2.22 x 10 9 / y x 10-8 The excess ratio at 50,000 for developed losses is then: [ 50K 100K/0.25 (y 50,000) * [2.22 x 10 9 / y x 10-8 ] dy] / (E[X] * avgdev) Note that the average development = h(r)/r dr = 1 The excess ratio for developed losses is = 29.7%

31 30. a. Science i. Purpose reproduce hurricane (simulation) ii. Two inputs radius of maximum wind, forward speed of hurricane b. Engineering i. Purpose estimates damage sustained by a property exposed to a hurricane ii. Two inputs coverage (building vs. contents), resistance of material c. Insurance i. Purpose uses results of first two modules and looks at loss effect on in-force book of business ii. Two inputs coinsurance, insurance to value

32 31. a. End of qtr Incurred Expense GAAP Equity Discount PV GAAP Qtr Surplus Statutory GAAP DAC End of qtr During qtr Factor Equity Total Ann. PVE: b. PVI / PVE = 20% => PVI = 20% * = PVI = PV(U/W Income) + PV(II) PV(Income Tax) PVI = PV(U/W Inc) = PV(U/W Inc) = 3.89 PV(U/W Inc) = PV(EP) 25*( ) 10* PV(U/W Inc) = PV(EP) = 3.89 PV(EP) = PV(EP) = Premium*0.926 = => Premium = c. Profit = Premium Loss Expense = = => U/W Profit Provision = / = -5.74% d. i. Advantages a. Return is comparable to GAP ROE b. Return is a generalization of definition of interest rate ii. Disadvantages a. Need to select discount rate for Investment Income b. Need to choose target present value return

Formulas. Experience Rating Equity and Predictive Accuracy : Venter

Formulas. Experience Rating Equity and Predictive Accuracy : Venter Formulas B.1 Prospective Rating xperience Rating quity and Predictive Accuracy : Venter The formula that minimizes the expected squared error, subject to the linearly constraint A = Actual Loss = xpected