Exam 7 Cookbook
Stephen Roll, FCAS
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Copyright © 2016 by Rising Fellow LLC
All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form
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Published in United States
Table of Contents
Preface .................................................................................................................................................1
2016 Exam 7 Reading List ....................................................................................................................7
Brosius
Least Squares Method ..........................................................................................................................9
Bayesian Credibility in a Changing System ......................................................................................... 12
Mack (2000)
Benktender Method ........................................................................................................................... 14
Hürlimann
Credible Loss Ratio Claims Reserves .................................................................................................. 16
Clark
Variance of Reserves (LDF Method) .................................................................................................. 19
Variance of Reserves (Cape Cod Method) ........................................................................................... 22
Normalized Residuals ......................................................................................................................... 25
Variance of Prospective Losses ........................................................................................................... 28
Variance of Calendar Year Development ............................................................................................ 30
Mack (94)
Residual Test ..................................................................................................................................... 32
Calendar Year Test ............................................................................................................................ 35
Reserve Confidence Interval ............................................................................................................... 38
MSE Calculation ............................................................................................................................... 40
Venter Factors
Correlation of Adjacent LDFs ............................................................................................................ 43
Correlation of Development Factors ................................................................................................... 45
Parameterized BF: f(d)h(w) – Constant Variance ................................................................................ 48
Parameterized BF: f(d)h(w) – Var ∝ f(d)h(w) ..................................................................................... 50
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Testing for Significantly High/Low Diagonals.................................................................................... 53
Siewert
Loss Ratio Method............................................................................................................................. 55
Implied Development Method ........................................................................................................... 57
Direct Development Method ............................................................................................................. 59
Credibility Weighted Method ............................................................................................................ 61
Unlimited/Limited/Excess LDF Relationships ................................................................................... 63
Aggregate Excess of Loss Ultimate ..................................................................................................... 66
Service Revenue Asset ........................................................................................................................ 68
Sahasrabuddhe
Determine LDFs at Basic Limits ........................................................................................................ 70
Calculate Layer LDFs from Basic Limit LDFs ................................................................................... 73
Shapland and Leong
Over-Dispersed Poisson GLM Setup ................................................................................................. 76
Expected Incremental Losses from GLM ........................................................................................... 79
Standardized Pearson Residuals.......................................................................................................... 81
Bootstrap Model ................................................................................................................................ 84
Negative Incremental Values .............................................................................................................. 88
Heteroscedasticity Adjustment ........................................................................................................... 91
Multiple Models: Same Random Variables ......................................................................................... 94
Multiple Models: Independent Variables ............................................................................................ 96
Verrall
Incorporating Expert Opinion in the Chain Ladder Method ............................................................... 98
Bayesian Model for the BF Method .................................................................................................. 101
Fully Stochastic BF Model ............................................................................................................... 105
Estimated Reserve from Fully Stochastic BF ..................................................................................... 108
Meyers
p-p Plots & Histogram ..................................................................................................................... 110
Interpret p-p Plots ............................................................................................................................ 113
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Kolmogorov – Smirnov (K-S) Test ................................................................................................... 115
Correlated Chain-Ladder (CCL) Model........................................................................................... 117
Changing Settlement Rate (CSR) Model .......................................................................................... 119
Marshall
Risk Margin Calculation .................................................................................................................. 121
Internal Risk Balanced Scorecard...................................................................................................... 124
Teng and Perkins
PDLD Ratios – Retro Formula ........................................................................................................ 126
PDLD Ratios - Empirical ................................................................................................................ 128
Premium Asset ................................................................................................................................. 130
Patrik
Stanard-Bühlmann (Cape Cod) Method .......................................................................................... 133
IBNR Monitoring ............................................................................................................................ 135
Goldfarb
Dividend Discount Model ................................................................................................................ 137
FCFE Model ................................................................................................................................... 140
Abnormal Earnings .......................................................................................................................... 143
Valuation with Price-Earnings Ratio ................................................................................................ 145
Valuation with Price-Book Value Ratio ............................................................................................ 147
Valuation with Peer Multiples .......................................................................................................... 149
Brehm
Economic Value Added (EVA)......................................................................................................... 151
Efficient Frontier ............................................................................................................................. 152
Marginal Cost-Benefit Analysis ....................................................................................................... 154
Tail-Based Risk Measures ................................................................................................................ 157
Scenario Planning ............................................................................................................................ 159
About the Author ............................................................................................................................. 160
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Preface
When I took Exam 7 in 2013, I built out the original version of the Exam 7 Cookbook to help me
systematically solve calculation problems on exam day. I found this approach to be so useful to help me
pass Exam 7 that I created similar “Cookbooks” for myself for Exams 8 and 9, which I also passed on the
first sittings to get my FCAS. I updated and expanded the original Exam 7 Cookbook with the goal of
helping you see a similar level of success when you take Exam 7 in a few short months.
I worked through the syllabus papers to identify all of the different “problem-types” that I believe are
testable with any degree of reasonability based on the syllabus. You can find a list of these unique,
calculation problem-types by paper in the Table of Contents. By exam day, you should know how to solve
each problem-type.
The Structure
Inside, you’ll find a separate section for each testable problem-type identified in the Table of Contents.
Each section has the following structure with the goal of preparing you to be able to confidently answer a
problem on exam day without wasting time trying to “think through” a problem-solving approach before
writing the solution.
Problem
For each section, there is an original, exam-style problem that demonstrates the problem-type. I wrote
these based off of the syllabus papers, focusing on creating high-quality practice problems that are
commensurate with the difficulty-level and style that you can expect to see on exam day. In the steps of
the solution recipe, the practice problem is solved from start to finish so that you understand how to apply
the different methods.
Solution Recipe
The solution recipe shows the step-by-step approach you should take to answer a problem from the
problem-type. For each step, you’ll see:
•
•
•
•
The description for what to do in the step
The Formula(s) necessary for the step
The Formula(s) translated from symbolic notation to plain-English
Worked step from the example problem
Discussion
Each section includes discussion to add clarity so that you have a more confident understanding of how to
solve the problem-type. I also try to point out subtleties and details that you need to remember on exam
day.
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For many problem-types, I point out potential “twists” to the problem-type that could show up on the
exam that would make the exam problem more difficult. Since you’ve taken actuarial exams up to this
point, you know that straightforward exam problems are more the exception than the rule!
Source
For each problem-type, I include the exact pages in the syllabus reading that you can cross-reference for
more information and details. This is particularly useful to be prepared for “tack-on” essay questions, the
extra 0.5 point problem-part you typically see at the end of a calculation problem. I also reference any
appendices in the papers demonstrating the problem-type.
More Practice
Here, you’ll see references to past CAS problems of a given problem-type. You’ll find this helpful
especially closer to the exam if there are particular types of problems that you are struggling with. This
section includes references to problems from the 2011-2015 exams, the past exams in the current syllabus
structure of Exam 7.
Notation and Formulas
One of the big challenges with Exam 7 is that almost every author uses their own unique set of notation
for losses, LDFs, premiums, etc. The changing notation between papers makes preparing for the exam
more challenging especially for the following reasons:
•
It’s harder to see the big picture and draw connections between the main themes in Exam 7
across multiple papers, which is important to be prepared for higher-level Blooms Taxonomy
questions.
•
It’s more difficult to have an intuitive understanding of the different methods and how they work,
which is critical to be able to solve problems on exam day without wasting time trying to think
through an approach.
To help you avoid getting lost in the notation I also show a plain-English version of the formulas in the
solution recipe steps. If you’re like me and get lost in the symbolic notation in the syllabus papers, you’ll
find that this feature will save you a lot of frustration.
Below are two examples of what the formulas look like in the solution recipe steps.
Mack (2000) – Benktender Method:
U GB = C k + q kU BF
U GB = Loss + (1 − %Paid ) ×Ult BF
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Verrall – Incorporating Expert Opinion in the Chain Ladder Method
(
)
E ⎡⎣Ci , j ⎤⎦ = λi , j − 1 ⋅ Di , j −1
(
)
E ⎡⎣ IncLoss AY ,k ⎤⎦ = LDFAY ,k − 1 ⋅ Loss AY ,k−1
( )
(
)
Var ( IncLoss ) = dispersion ⋅ LDF
Var Ci , j = ϕ ⋅ λi , j ⋅ λi , j − 1 ⋅ Di , j −1
AY ,k
AY ,k
(
)
⋅ LDFAY ,k − 1 ⋅ Loss AY ,k−1
Note that Mack (2000) uses C to indicate cumulative losses and Verrall uses C to indicate incremental
losses. In the plain-English version of the formula I make the distinction between incremental and
cumulative losses as well as spell out what the symbols in the formulas really mean.
I firmly believe you should learn and memorize the formulas in the way that you’ll best be able to
remember and apply to an exam-day problem. I see no evidence that you would be marked off for writing
a formula on the exam with LDFAY ,k instead of λi , j .
In fact, if you look at problem 8 from the 2012 exam, you’ll see just that in the sample solutions. Sample
solution 1 uses the clearer LDFAY ,k notation while sample solution 2 uses the Verrall notation, λi , j .
The exam graders want to see that you understand how to apply the different methods, not whether you
memorized the specific, symbolic notation from a paper written 20 years ago. * Gets off soap box *
Practice Problems
All of the original problems are also compiled in a separate Practice Problems set that you can use to work
the practice problems on your own. For each problem, you’ll see the problem on the front of the page and
the solution on the back.
For the solutions, I try to show work at a similar level of detail to how I would write a full-credit solution
on exam day. I think you’ll find this beneficial if you struggle with balancing how much work to show to
maximize your points, but not showing so much detail that you can’t finish the exam.
Basic Study Schedule and How to Use the Cookbook
Below is a suggested guide for how you can incorporate the Exam 7 Cookbook in your own study
schedule along with the syllabus material and a typical study manual. This is the general approach that I
used when I took my fellowship exams.
For each of those exams I had a main study manual as well as the Exam Cookbook, which I built out
while I studied for the exam (but you don’t need to waste time doing that part!)
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First pass through the syllabus:
While you’re reading a particular paper in the syllabus and your main study manual to learn the material,
use the Exam 7 Cookbook to clearly identify what problem-types you need to know from the paper.
Study the steps in the Solution Recipe to learn how to solve the problem-types.
Second pass through the syllabus:
Review the steps for the problem-types and make sure you have an intuitive understanding of how to
solve the problem. Start working the original practice problems themselves, compiled in the Practice
Problems packet, as well as the problems in your main study manual.
The first level of understanding is to be able to follow the recipe and nod your head as you read the steps.
The next level of understanding is to be able to apply the steps to solve a problem with a blank answer
sheet in front of you. At this point, your goal is to begin to build that level of understanding.
Review and Practice Problems (around 6 weeks to 2 weeks before the exam)
At this point you should have a good understanding of the syllabus and how to use the recipe steps to
systematically solve the different calculation problems. During this period of studying, you should be
doing lots of problems across the whole syllabus and targeting problem-types that you are finding
particularly challenging.
By the end of this phase, you might not have all the formulas memorized, but you should know all the
steps and how to apply them to solve problems without needing to think too much before beginning to
write the solution.
Final Weeks:
In the final weeks you should focus on taking practice exams to see problems from the entire syllabus.
When taking practice exams, make sure to practice your exam strategy to make sure you’re able to finish
the exam and maximize your points.
Prepare for essay problems in the final weeks by using flashcards to make sure that you know all the
details necessary.
Prepare for calculation problems by reviewing the recipes in the Exam 7 Cookbook in a similar fashion to
how you use flashcards for essay problems. Using this approach on my fellowship exams, I was able to
rapidly review the steps and formulas for how to solve each problem-type that might show up on the
exam. This was a huge benefit and gave me a lot of confidence going into the exam.
Exam Day:
I used the original Exam Cookbooks together with a traditional study manual using the approach above
to take my fellowship exams. On exam day, for almost every calculation problem I was able to start
writing the solution without wasting much time trying to think through how to solve the problem. I had
an intuitive understanding of how to solve each of the problems following the step-by-step recipes.
If you follow this approach, you should be able to develop a similar level of understanding and confidence
going into the exam room.
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How to Get Support
I am always accessible for any questions relating to the Exam 7 Cookbook or the Exam 7 syllabus. I want
to see you pass your exam, so reach out to me for questions. Send me an email with your question at
[email protected]. I put my email address in the footer so that you don’t need to waste time trying
to find it.
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2016 Exam 7 Reading List
Overview
This is a suggested reading order for the Exam 7 papers. I made a few switches from the syllabus to
ensure that the order of the concepts is appropriate and to place the more approachable papers of a section
earlier. In a few cases I placed the calculation-heavy papers earlier because I find it’s better to study the
essay-question papers later to avoid forgetting all the details by the time of the exam.
Difficulty Level
The difficulty level is based on how difficult the paper is to read as well as how difficult the concepts are.
Problem Type Indicator
This is a rough measure of whether calculation or essay questions are more likely to appear on the exam
for a given paper. I based this on the 2011-2015 exam problems as well as the syllabus learning objectives.
Syllabus
Problem Type:
Weight
Pages Difficulty
Calc – Essay
!!!""
#$$$$
Brosius
4%
19
!""""
#$$$$
A.1
Mack (2000)
4%
5
!!!""
#$$$$
Hürlimann
4%
18
!!!!"
#$$$$
Clark
6%
50
!!!!!
$#$$$
Mack (1994)
6%
82
A.2-3
!!!!"
$#$$$
Venter Factors
6%
41
!!!!"
#$$$$
Siewert
3%
28
A.4
!!!!!
#$$$$
Sahasrabuddhe
3%
23
!!!!"
$$#$$
Shapland and Leong
6%
66
!!!!!
$$#$$
Verrall
5%
18
A.5-10
!!!!"
$#$$$
Meyers - new
6%
54
!!"""
$$$#$
Marshall
6%
42
!!!!"
$#$$$
Teng and Perkins
4%
78
A.14
!!!""
$$#$$
Patrik
7%
31
A.11-13
!!!""
#$$$$
B.1-3
Goldfarb
10%
49
!""""
$$$$#
Brehm Ch. 1
4%
20
C.1
!""""
$$$$#
Brehm 2.1,2.3-2.4
2%
21
C.2
!!!""
$$$#$
Brehm
2.5
2%
25
C.2
!!!!"
$$#$$
C.3-4
Brehm 2.2
3%
12
!!"""
$$$$#
Brehm 3.1-3.2
2%
14
C.5-6
!!!!!
$$$#$
Brehm 3.3
2%
13
C.5-6
!""""
$$$$#
C.7
Brehm 4.1-4.2
4%
29
!!"""
$$$$#
C.8
Brehm 5.4
3%
15
Notes: Make sure to get the errata for Sahasrabuddhe, Shapland and Leong, and Verrall. Teng and
Perkins includes a discussion paper by Feldblum. You’re not responsible for the proofs in Hürlimann.
Section
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Least Squares Method
Brosius
Problem
Given the following information:
Cumulative Reported Losses ($000)
Accident Year
2009
2010
2011
2012
2013
•
36 Months
260
840
130
2,160
3,610
48 Months
830
3,540
1,860
3,240
Earned
Premium
4,120
5,350
6,540
7,780
8,010
60 Months
1,240
3,960
2,840
The tail factor from 60 months-to-Ultimate is 1.25
Calculate the estimated unpaid losses for accident year 2013 using the Least Squares method.
Solution Recipe
1) Convert losses to a cumulative loss ratio triangle and apply the tail factor.
Loss Ratio =
Losses
Earned Premium
260
= 6.31%
4,120
= 30.1% × 1.25 = 37.6%
Loss Ratio2009, 36mo =
Ult Loss Ratio2009
!
AY
2009
2010
2011
2012
2013
36
6.31%
15.7%
1.99%
27.8%
45.1%
48
20.1%
66.2%
28.4%
41.6%
60
30.1%
74.0%
43.4%
!
Ult L/R
37.6%
92.5%
54.3%
2) Calculate the least squares a and b parameters for each development period iteratively. Start with the
most mature development period. Use undeveloped loss ratios as the ‘x’ values and ultimate loss
ratios as the ‘y’ values.
b=
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xy − x ⋅ y
x2 − x 2
a = y − b ⋅x
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Starting with the 48mo development period (for accident year 2012):
x
y
(48mo L/R )
(Ult L/R)
48mo
x
20.1%
37.6%
.3825
y
66.2%
92.5%
.6148
28.4%
54.3%
2
.1864
x
xy
.2808
.2808 − .3825 × .6148
b=
= 1.138
.1864 − .3825 2
b
1.138
a
a = .6148 − 1.138 × .3825 = 0.1795
0.1795
3) Calculate the ultimate loss ratio estimate from the loss ratio to-date for each accident year. Start with
the oldest accident year without an ultimate loss ratio.
AY 2012:
ŷ = a + bx
= 0.1795 + 1.138 × 0.416
!
Ult
Loss Ratio = a + b × Latest Loss Ratio
= 65.3%
4) Repeat steps 2 and 3 iteratively to calculate the ultimate loss ratios for the remaining accident years.
x
(36mo L/R )
6.31%
15.7%
1.99%
27.8%
y
(Ult L/R)
37.6%
92.5%
54.3%
65.3%
From Step 3
x2
xy
36mo
.1294
.6244
.0265
.0903
b
a
.9709
.4988
x
y
AY 2013:
= 0.4988 + .9709 × 0.451
= 93.6%
AY
2012
2013
Ult Loss Ratio
65.3%
93.6%
5) Calculate unpaid losses using the estimated ultimate loss ratios
Unpaid Loss = Earned Prem ×Ult Loss Ratio − Losses
AY 2013 Unpaid Losses:
= 8,010 × 93.6% − 3,610
= 3,890
Discussion
The least squares method can be used with either actual losses or loss ratios. The reason for using loss
ratios instead of losses is because of the large premium grown. Using a loss ratio puts the accident years on
a more equal basis to run the least squares method.
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The key thing to understand is that steps 2 and 3 are an iterative process. We first use 48mo loss ratios as
x and ultimate loss ratios as y in order to get the estimated ultimate loss ratio for AY 2012. Now, when
we iterate through with 36mo loss ratios as x, we can use our estimated ultimate loss ratio for AY 2012
(65.3%) as an additional data point.
Note that you might have slightly different numbers due to rounding throughout the problem.
Source
Brosius – pg. 16-18
More Practice
CAS 2012 – 4
CAS 2011 – 1
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