**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 6, authored by Mr. Stolyarov. This is Section 44 of the Study Guide. See an index of all sections by following the link in this paragraph.**

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.

Some of the problems in this section were designed to be similar to problems from past versions of Exam 6, offered by the Casualty Actuarial Society. They use original exam questions as their inspiration – and the specific inspiration is cited to give students an opportunity to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

**Formulas for Brosius’s Least-Squares Method**

y = a + b*x

b = ((xy)- – x-*y-)/((x2)- – (x-)2)

a = y– b* x-

This is least-squares linear regression applied to loss development. The data points x are the independent variables (earlier time periods’ data), and the data points y are the dependent variables (later time periods’ data). The value a is the y-intercept, and b is the slope. The – superscripts denote sample means and are equivalent to bars over the entire symbol, which are not expressed here due to notational difficulties.

**Sources:**

Brosius, E., “Loss Development Using Credibility,” CAS Study Note, March 1993.

Friedland, Jacqueline F. *Estimating Unpaid Claims Using Basic Techniques**.* Casualty Actuarial Society. July 2009.

Past Casualty Actuarial Society exams: 2008 Exam 6 and 2009 Exam 6.

Teng, M.T.S.; and Perkins, M.E., “Estimating the Premium Asset on Retrospectively Rated Policies,” *PCAS* LXXXIII, 1996, pp. 611-647.

**Original Problems and Solutions from The Actuary’s Free Study Guide**

**Problem S6-44-1. Similar to Question 2 from the 2009 CAS Exam 6.** You know the following information about cumulative paid losses for Insurer Ψ by accident year (AY):

**Cumulative Paid Loss,** expressed in the format(Amount at development year 0, Amount at development year 1, Amount at development year 2). **AY 2022:** (343, 444, 500)** AY 2023:** (360, 500, 555)

**(320, 466)**

AY 2024:

AY 2024:

**AY 2025:**(350)

An accident-year model of the following nature is fitted to the data above:

y(j) = α + ε0 for j = 0;

y(j) = α + k=1jΣ(γk) + εj for j = 1, 2.

**Definitions:**

y = ln(Incremental paid loss)

j = year of development – either 0, 1, or 2

α, γj (for j = 1, 2) = constants

εj (for each j) = error terms with means of zero

Find the median value of the estimated incremental paid loss during development year 1 for accident year 2025.

**Solution S6-44-1.** First we find the incremental paid losses during development year 1 for each of the other three accident years:

**AY 2022:** 444-343 = 101** AY 2023:** 500-360 = 140

**466-320 = 146**

AY 2024:

AY 2024:

For year 1, our formula becomes y(1) = α + γ1 + ε1.

The average (arithmetic mean) of the natural logs of the known incremental paid losses should be a good approximation of α + γ1, since ε1 has a mean of zero.

Thus, we estimate y(1) = (ln(101) + ln(140) + ln(146))/3 = 4.846789854, and the desired estimated incremental paid loss would therefore be ey(1) = e4.846789854 = **127.330982**.

**Problem S6-44-2. Similar to Question 3 from the 2009 CAS Exam 6.** You are again analyzing the following information about cumulative paid losses for Insurer Ψ by accident year (AY):

**Cumulative Paid Loss,** expressed in the format (Amount at development year 0, Amount at development year 1, Amount at development year 2). **AY 2022:** (343, 444, 500)** AY 2023:** (360, 500, 555)

**(320, 466)**

AY 2024:

AY 2024:

**AY 2025:**(350)

**(a)** Use Brosius’s least-squares method to find the expected losses at development year 1 for AY 2025.

**(b)** Is the least-squares method proper to use in a situation such as the one in part (a)? Explain why or why not.

**Solution S6-44-2.**

**(a)** First we find the various averages necessary for the least-squares method.

Let x be experience at development year 0, and let y be experience at development year 1.

x- = (343 + 360 + 320)/3 = 341.

y- = (444 + 500 + 466)/3 = 470

(xy)- = (343*444 + 360*500 + 320*466)/3 = 160470.666667

(x2)- = (3432 + 3602 + 3202)/3 = 116549.666667

(x-)2 = 3412 = 116281

Now we find b = ((xy)- – x-*y-)/((x2)- – (x-)2) =

(160470.666667 – 341*470)/(116549.666667 – 116281) = b = 0.7468982642.

Now we find a = y– b* x- = 470-0.7468982642*341 = 215.3076919.

For AY 2025, x = 350, so y = a+bx = 215.3076919 + 0.7468982642*350 = **476.77220844**.

**(b)** It **is proper** to use the least-squares method in this situation, because b > 0.

**Problem S6-44-3. Similar to Question 7 from the 2008 CAS Exam 6.**

You are again analyzing the following information about cumulative paid losses for Insurer Ψ by accident year (AY):

**Cumulative Paid Loss,** expressed in the format (Amount at development year 0, Amount at development year 1, Amount at development year 2). **AY 2022:** (343, 444, 500)** AY 2023:** (360, 500, 555)

**(320, 466)**

AY 2024:

AY 2024:

**AY 2025:**(350)

An accident-year model of the following nature is fitted to the data above:

y(j) = α + ε0 for j = 0;

y(j) = α + k=1jΣ(γk) + εj for j = 1, 2.

**Definitions:**

y = ln(Incremental paid loss)

j = year of development – either 0, 1, or 2

α, γj (for j = 1, 2) = constants

εj (for each j) = error terms with means of zero

**(a)** Find the values of α, γ1, and γ2.

**(b)** Find the median value of the estimated incremental paid loss during development year 2 for accident year 2025.

**Solution S6-44-3.**

**(a)** Since the mean error terms are zero, the value of α is the average of the natural logarithms of the development year 0 data: α = (ln(343) + ln(360) + ln(320) + ln(350))/4 = **α = 5.837522157**.

We already know from Solution S6-44-1 that α + γ1 = 4.846789854, so γ1 = 4.846789854 – 5.837522157 = **γ1 = -0.9907323032**.

Now we find the known incremental losses paid for development year 2: **AY 2022:** 500-444 = 56** AY 2023:** 555-500 = 55

α + γ1 + γ2 = (ln(56) + ln(55))/2 = α + γ1 + γ2 = 4.016342438. Since α + γ1 = 4.846789854, it follows that γ2 = 4.016342438 – 4.846789854 = **γ2 = -0.830447416**.

**(b)** Since, by our model, y(2) = α + γ1 + γ2 + ε2, and ε2 has a mean of zero, the median incremental loss in development year 2 will be exp(α + γ1 + γ2) = e4.016342438 = **55.4977477**.

**Problem S6-44-4. Similar to Question 9 from the 2008 CAS Exam 6.**

You are again analyzing the following information about cumulative paid losses for Insurer Ψ by accident year (AY):

**Cumulative Paid Loss,** expressed in the format (Amount at development year 0, Amount at development year 1, Amount at development year 2). **AY 2022:** (343, 444, 500)** AY 2023:** (360, 500, 555)

**(320, 466)**

AY 2024:

AY 2024:

**AY 2025:**(350)

Find the cumulative paid loss amount for AY 2024 at development year 2 using the following methods: **(a)** The development method; **(b)** The budgeted loss method; **(c)** The least-squares method.

**Solution S6-44-4.**

**(a)** We use the development method with a weighted-average loss development factor from year 1 to year 2: (500 + 555)/(444 + 500) = 1.117584746. Our answer is thus 466*1.117584746 = **520.7944915**.

**(b)** The budgeted loss method simply takes the expected value of the known losses at development year 2 and sets that as the loss for AY 2024: (500 + 555)/2 = **527.5**.

**(c)** First we find the various averages necessary for the least-squares method.

Let x be experience at development year 1, and let y be experience at development year 2.

x- = (444 + 500)/2 = 472.

y- = (500 + 555)/2 = 527.5.

(xy)- = (444*500 + 500*555)/2 = 249750

(x2)- = (4442 + 5002)/2 = 223568

(x-)2 = 4722 = 222784

Now we find b = ((xy)- – x-*y-)/((x2)- – (x-)2) = (249750 – 472*527.5)/(223568 – 222784) = b = 0.9821428571.

Now we find a = y– b* x- = 527.5 – 0.9821428571*472 = 63.92857143.

For AY 2024, x = 466, so y = a+bx = 63.92857143 + 0.9821428571*466 = **521.6071429**.

**Problem S6-44-5. Similar to Question 14 from the 2008 CAS Exam 6.**

You have the following information as of December 31, 2050, for a book of retrospectively rated policies for which business first began to be written in 2047.

**Expected Future Loss Emergence** **For Policy Year 2047:** 8000** For Policy Year 2048:** 56000

**For Policy Year 2049:**123000

**For Policy Year 2050:**352000

**Cumulative Premium Development to Loss Development (CPDLD) Ratio** **For Policy Year 2047:** 0.25** For Policy Year 2048:** 0.66

**For Policy Year 2049:**1.04

**For Policy Year 2050:**1.44

**Premium Booked from Prior Adjustments** **For Policy Year 2047:** 222000** For Policy Year 2048:** 180000

**For Policy Year 2049:**150000

**For Policy Year 2050:**0

**Premium Booked** **For Policy Year 2047:** 224000** For Policy Year 2048:** 201000

**For Policy Year 2049:**180000

**For Policy Year 2050:**190000

Calculate the premium asset as of December 31, 2050, using the methodology of Teng and Perkins.

**Solution S6-44-5.** The CPDLD ratio – a ratio of *premium* development to *loss* development – can be used to estimate future premium emergence from expected future loss ratio.

**Expected Future Premium Emergence** **For Policy Year 2047:** 8000*0.25 = 2000** For Policy Year 2048:** 56000*0.66 = 36960

**For Policy Year 2049:**123000*1.04 = 127920

**For Policy Year 2050:**352000*1.44 = 506880

The expected ultimate premium is the sum of the prior booked premium and the expected future premium emergence.

**Expected Ultimate Premium** **For Policy Year 2047:** 222000 + 2000 = 224000 **For Policy Year 2048:** 180000 + 36960 = 216960 **For Policy Year 2049:** 150000 + 127920 = 277920 **For Policy Year 2050:** 0 + 506880 = 506880

The premium asset is the expected ultimate premium minus the premium booked.

**Premium Asset** **For Policy Year 2047:** 224000 – 224000 = 0 **For Policy Year 2048:** 216960 – 201000 = 15960 **For Policy Year 2049:** 277920 – 180000 = 97920 **For Policy Year 2050:** 506880 **–** 190000 = 316880

Our total premium asset is thus 0 + 15960 + 97920 + 316880 = **430760**.

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