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This is a long, boring set of instructions on how to do calculations and complete exhibits from the annual return. It's 88 pages long but simply trying to read these instructions is not the best way to learn the relevant material. Instead, we'll take a look at the types of exam problems that have been asked in the past. Once you have a handle on that, it's much easier to use the source paper as a reference. There are roughly half-a-dozen types of calculation questions in this paper. The excess/deficiency ratio shows up on virtually every exam.

There are no deep concepts here, but it helps if you've taken a university course in accounting. You're then at least familiar with the layout of a financial statement. If you haven't had a university accounting course, it's not a great problem, but it might be a little harder to understand the big picture. Nonetheless, there are only a small number of problem types you need to know.   Forum

Pop Quiz

If you know the formulas you need for this Pop Quiz, that's great. If you don't, that's ok - just study the solution in the POP QUIZ ANSWERS at the bottom of this article. Calculating equity is a basic skill that you'll need both here and for other types of calculation problems on the exam. Equity will be covered in more detail in this article.

If an Insurer has:

  • A(x) = 20m (Assets for year x)
  • L(x) = 14m (Liabilities for year x)

Calculate E(x) (Equity for year x)

If the same Insurer has:

  • NI(x) = 3m (Net Income)
  • OCI(x) = 1m (Other Comprehensive Income)

Calculate: E(x-1) (Equity for year x-1)


Based on past exams, the main things you need to know (in rough order of importance) are:

  • Excess(Deficiency) margin or ratio, both discounted & undiscounted versions (This is the most frequently asked question on Exam 6-Canada.)
  • investment income from insurance operations
  • net commissions exhibit 80.10
  • equity, net income, comprehensive income
  • loss ratio, combined ratio
reference part (a) part (b) part (c) part (d)
E (2018.Spring #15) Excess(Deficiency):
- discounted margin
E (2017.Fall #14) net income:
- calculate 3
see Odo.FinReg
E (2017.Fall #16) Excess(Deficiency):
- margin / ratio
E (2017.Spring #11) Excess(Deficiency):
- margin / ratio 1
E (2016.Fall #14) investment income:
- from operations
- margin / ratio
E (2016.Spring #14) Excess(Deficiency):
- undiscounted margin
investment income:
- calculate
- discounted margin
E (2016.Spring #18) net commissions:
- calculate
- non-deferrable comm.
E (2015.Fall #13) Excess(Deficiency):
- margin
- ratio
see CIA.Discnt
E (2015.Fall #22) equity:
- calculate
see MSA.Ratios see MSA.Ratios
E (2015.Spring #21) Excess(Deficiency):
- margin / ratio
E (2015.Spring #22) investment income:
- from operations
E (2015.Spring #24) net loss ratio:
- calculate
net combined ratio:
- calculate
net income:
- calculate
E (2014.Fall #16) net commissions:
- calculate
E (2014.Fall #19) Excess(Deficiency):
- undiscounted margin
- discounted margin
E (2014.Fall #20) investment income:
- from operations
E (2014.Spring #18) Excess(Deficiency):
- margin / ratio
E (2013.Fall #21) Excess(Deficiency):
- margin / ratio
see CIA.Discnt
E (2012.Fall #23) Excess(Deficiency):
- margin / ratio
1 Part of a larger problem on completing Exhibit 60.30
3 Part of a larger problem involving APV(claim liabilities) and bond values. (See also CIA.MfAD, CIA.Accting)

In Plain English!

Types of Problems

Source Readings: BattleActs covers all material from past exams. It also covers significant material that has not appeared on past exams but that I've judged to be important. Still, it's a good idea to spend at least little time reviewing the source readings. You may have a different opinion on what's important and what you can skip. You cannot read all 2,500 pages in depth, but BattleActs give you the necessary background knowledge so that the time you do spend on the source readings will be much more efficient.

There are 3 types of exam problems that have appeared on multiple exams:

  • excess/deficiency ratio (appears on every exam)
  • investment income from operations
  • net commissions

I don't want you to do any BattleCards right now, but click here just to see the BattleCards for the old exam problems. Notice that they threw in a few non-calculation questions, but that's not the thrust of the paper. Aside from the 3 types of questions listed above, you also need to have the skills to calculate the following:

  • equity
  • net income
  • comprehensive income
  • net loss ratio
  • net combined ratio

Calculating Equity

Sometimes you need the equity but are not provided with that information directly. The pop quiz above covered the 2 basic methods for calculating equity from other quantities. The first method was the Fundamental Accounting Equation, which states that:

Assets = Liabilities + Equity

The second method relies on knowing E (Equity) for the prior year, and CI (Comprehensive Income) for the current year x, then:

E(x) = E(x-1) + CI(x)
It's also useful to know the formula relating CI to NI (Net Income) and OCI (Other Comprehensive Income) for a specific year:
But, there is a potential complication with this method: (Shout-out to cgc2018 for noticing this!)
It assumes the company did not make certain kinds of capital transactions, like dividends for example. If dividends were paid, they must be subtracted from NI to arrive at the correct value for Equity.
It would be wise to spend a moment looking at the following references:
- this ROE (Return On Equity) exam problem, part (a.i), is discussed later in this article
- it's a problem where you have to calculate equity using this second method describe above
- but you have to subtract the dividends from NI to arrive at the correct value
- this is because if you think of dividends as being paid out of income, this would also reduce the equity (dividends leave the company to go to shareholders)

A third method for calculating equity is when you have the required balance sheet items. Just sum the following quantities to get shareholder's equity:

  • shares issued & paid (common & preferred)
  • contributed surplus
  • retained earnings
  • reserves
  • AOCI (Accumulated Other Comprehensive Income, includes unrealized gains/losses on OCI)

The first mini BattleQuiz has a problem where you have to calculate E(2014), the equity for year 2014, (2015.Fall #22a). You also have to calculate NI, but we haven't covered that yet, so I'm going to give you a hint and tell you that:

  • NI(2014) = 1,460

Just do part (a) of this problem. It's easy if you already know NI(2014). We'll learn how to calculate NI from balance sheet items later. (The quiz also has a couple of non-calculation questions, but if this is your first time through this paper, you can skip those and come back to them later.)

mini BattleQuiz 1 You must be logged in or this will not work.

Excess/Deficiency Ratio - Part 1

Easy Version

I mentioned above that this question appears on every exam. They vary the problem by providing the necessary information in different ways. Let's first cover the most basic version of the problem. In this version, based on (2015.Spring #21), you're given:

  • investment yields for the relevant CYs
  • paid loss triangle (incremental)
  • unpaid loss triangle (discounted & including PfADs OR equivalently, APV)

You need to get those 3 items into your head. One of the variations on this is to provide the cumulative paid loss triangle instead of the incremental triangle. It's amazing how many people don't notice that in the heat of the exam! More than any other paper on the syllabus, this one is very much skills-based. That means you've got to practice. Start with easy versions of the problems and work your way up over the course of your studying.

If you're following the BattlePlan in the recommended order, you'll already have done this problem, but now is a good time to review it:

mini BattleQuiz 2 You must be logged in or this will not work.

Actual Exam Problems

All of the old exam problems on the excess/deficiency ratio are organized into the tables below. The mini BattleQuiz provides links to the actual problems and solutions. You should do these problems in the order they are listed since I've organized them from easy to hard.

problem difficulty PAID loss triangle format UNPAID loss triangle format must calculate... tricks
1 (2015.Spring #21) easy incremental discounted with PfADs ratio for AY 2012 as of CY 2014 none
2 (2014.Spring #18) easy incremental discounted with PfADs ratio for AY 2010 as of CY 2012 none
3 (2015.Fall #13) medium incremental discounted with PfADs ratio for AY 2012 as of CY 2014 - they leave out one of the paid loss values
- you have to solve a linear equation to get it

You should now scroll down to the next mini BattleQuiz and do problems #1,2,3. (You can then return to this point in the wiki article and continue with problems #4,5,6,7.)

The next group of excess/deficiency problems has three complications:

  • you have to understand the difference between the discounted and undiscounted excess (deficiency) ratio. The difference is that the undiscounted ratio doesn't have a term for investment income.
  • the paid and unpaid data is provided in a less straightforward way
  • you have to read the question very carefully for which AY and CY they want you to do the calculation for (it's very easy to get tripped up on that)

Note: The solution in the examiner's report for problem #4 below, (2016.Spring #14a), is not very clear. For that reason, I've created a separate wiki page with an Explanation for 2016.Spring 14a.

problem difficulty PAID loss triangle format UNPAID loss triangle format must calculate... tricks
4 (2016.Spring #14a) medium incremental undiscounted undiscounted ratio for AY 2012 as of CY 2015 - there are 4 AY/CY years instead of 3
5 (2016.Spring #14bc) 1 medium incremental discounted with PfADs (APV) discounted ratio for AY 2013 as of CY 2015 - there are 4 AY/CY years instead of 3
- they ask for a different AY from part (a)
6 (2014.Fall #19a) harder cumulative case reserves, undiscounted IBNR undiscounted ratio for AY 2012 as of CY 2013 - paid losses are cumulative (not incremental)
- must calculate unpaid as case + IBNR
7 (2014.Fall #19b) harder cumulative case reserves, undiscounted IBNR, effect of discounting discounted ratio for AY 2011 as of CY 2013 - paid losses are cumulative (not incremental)
- must calculate unpaid as case + IBNR + effect of discounting
- they ask for a different AY from part (a)
1 User chrisboersma (Thx!) noticed an error in the CAS solultion for part (b). Their calculation labeled the investment income as being for 2013 & 2014, but it should have been 2014 & 2015. (The numbers are correct - it's just the labels that are wrong.)

You should now scroll down to the next mini BattleQuiz and do problems #4,5,6,7. (You can then return to this point in the wiki article and continue with problems #8,9.)

These last 2 problems are recent. Since the excess (deficiency) ratio problem has been asked so many times, they have to come up with ways to make it suck even harder. :-)

problem difficulty PAID loss triangle format UNPAID loss triangle format must calculate... tricks
8 (2017.Fall #16) hard cumulative discounted with PfADs (APV) margin for CY 2016 - paid losses are cumulative (not incremental)
- they ask for the margin not the ratio
- must sum across all AYs
- must know that margin for AY 2016 is zero
9 (2016.Fall #14) crazy hard!! not given as a triangle discounted but not given as a triangle excess amount for CY 2015 - info not given in triangle format
- they ask for the margin not the ratio
- different method for calculating investment income
- must sum across all AYs
- full explanation in next section

Note that you really can't do problem #9 (2016.Fall #14) until you know how to calculate investment income from operations given balance sheet amounts. This is what we'll cover in the next section before returning to this crazy hard excess (deficiency) ratio problem!

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Investment Income from Insurance Operations

Explanation of (2016.Fall #14a)

The given information covers what you need to calculate Investment Income from Insurance Operations and the excess (deficiency) ratio for calendar year (CY) 2015, which is actually part (b)

AY (Accident Year) net APV @12/31/14 net APV @12/31/15 Paid in CY 2015
2012 2,500 2,000 600
2013 4,500 3,800 650
2014 5,000 4,000 1,200
2015 5,300
12/31/14 12/31/15
Unearned Premium Reserve (UPR) 4,000 4,500
Premium Deficiency Provision (PDR) 500 100
Unearned Commissions (UEComm) 1,000 1,200
Policy Holder Receivables (PH Recvs) 0 200

The formula or concept behind this type of question is that the investment income attributable to insurance operations equals:

  • (yield rate) x (the 'money' the company has lying around)

This money includes UCAE (unpaid claims & adjustment expenses) and other items. The idea is to invest this money in the interim, before it is actually paid to the claimant. Great idea!! Both the reading and the examiner's report provide a formula for this money. Basically, you have to calculate 2-yr averages of all the relevant quantities then add/subtract them as appropriate. Liabilities are added; assets are subtracted.

  • But I find it easier to keep track of what I'm doing if I always organize the given information into a table as shown below:
  • For example, the UPR for CY2014 and CY2015 is given as 4,000 and 4,500, so the 2-yr average is 4,250. Since UPR is a liability, we put it in the + items column. Easy!
+ items avg(CY-1, CY) - items avg(CY-1, CY)
UCAE: 13,550 DPAE: 0
UPR: 4,250 receivables: installment premiums 0
UEComm: 1,100 receivables: policy holder + agents & brokers 100
PDR: 300
TOTAL 19,200 100 = 19,100
  • The values for UEComm, PDR, and receivables for policy holders + agents & brokers) are all easy - just take the average of the columns in the 2nd table of the question.
  • On the last line of the table, take the sum of the + items and subtract the sum of the -items to get to the total money to be invested, 19,100
The final step is to multiply this by the given yield rate of 5% to get investment income from insurance operations = 955.
  • The only hard part (of course, there has to be a HARD part!) is the UCAE of 13,550.
  • You have to realize that the net APV is the same thing as the UCAE.
  • If we make the assumption that this company started operations in 2012, then we calculate UCAE for CY2014 and CY2015 as follows:
  • CY2014: SUM(APV at Dec14) over all accident years = 2,500 + 4,500 + 5,000 = 12,000
  • CY2015: SUM(APV at Dec15) over all accident years = 2,000 + 3,800 + 4,000 + 5,300 = 15,100
  • Then the 2-yr average is simply 13,550.
  • THE END!

The next mini BattleQuiz covers the above problem and others on calculating investment income from balance sheet amounts. They all have the same format. (Difficulty level is medium.)

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Excess/Deficiency Ratio - Part 2

Explanation of 2016.Fall #14b

There are 2 things about this problem that are different from most previous exam questions on excess/deficiency:

  1. They ask for the amount here not the ratio.
  2. The time period is a calendar year, not an accident year.

Recall the formula for the excess (deficiency) ratio:

  • ExRatio = [ U(beg) - U(end) - Pd(All) + ii ] / U(beg)

But, for point #1 above, if we want only the amount then we only need the numerator:

ExAmt = [ U(beg) - U(end) - Pd(All) + ii ]

Also, this formula applies to 1 accident year across all CYs. Point #2 above is asking for the opposite: 1 CY across all accident years. We need to do the calculation for each accident year individually (for the given CY only) then sum. For this type of question, the data is normally given in a triangle, so I find it helpful to rewrite the first table as 2 separate triangles as in the BattleCard question referred to above:

Paid Loss during CY

Accident Year (AY) 2012 2013 2014 2015
2012 -- -- -- 600
2013 -- -- 650
2014 -- 1,200
2015 5,000

Discounted UCAE (including PfAD) at END of CY

Accident Year (AY) 2012 2013 2014 2015
2012 -- -- 2,500 2,000
2013 -- 4,500 3,800
2014 5,000 4,000
2015 5,300

Let's now do the calculation for AY2012 and CY2015

  • ExAmt
= U(beg) - U(end) - Pd(All) + ii
= 2,500 - 2,000 - 600 + ii
= -100 + ii
  • The investment income for this period is:
  • ii
= 5% x avg(2500, 2000)
= 112.5
  • So the result is: ExAmt (AY2012, CY2015) = 12.5


  • ExAmt (AY2013, CY2015) = 257.5
  • ExAmt (AY2014, CY2015) = 25

For AY2015, there is no actual calculation. We need at least 2 data points to calculate ExAmt as above and at year-end 2015, we only have 1 data point for AY2015. Our assumption has to be that the reserve of 5300 is accurate and the redundancy/deficiency is 0. (if we didn't think 5300 was accurate, we would have set the reserve to something else. Also, some people tried to use a reserve value of 0 for AY2015 at year-end 2014, but that doesn't makes sense. That reserve isn't 0 at that point - it simply doesn't exist and can't be used in a calculation.)

  • ExAmt (AY2015, CY2015) = 0
Sum these quantities to get the final answer: 295 = 12.5 + 257.5 + 25 + 0

Please note: This problem was different from previous exam problems on this topic BECAUSE you were asked for:

  1. an AMOUNT, not a ratio
  2. a CALENDAR YEAR value not an accident year value



Pop Quiz A: Suppose this question HAD asked for the excess (deficiency) ratio for AY2012 as of yr-end 2015. Could you calculate it from the given information? Why or why not?

  • Answer: No, there are missing entries in the triangle. On the row for AY2012, you need paid loss for CY2013 & CY2014. Same for the unpaid triangle. (If you don't see this, use the mini BattleQuiz below to review the BattleCard from CCIR.ARINSTR: Excess Ratio (page 60.41) EASY VERSION)

Pop Quiz B: Given ii(CY2014) = 145, calculate UCAE(AY2012 @ 12/31/2013)

  • Answer: 3300 (Solve for U: 5% x (2500 + U)/2 = 145).
You can now go back to mini BattleQuiz 3 and do problem #9 from the Excess (Deficiency) Ratios - Part 1.

Net Commissions

Note that the solution in the examiner's report for the net commissions problem (2014.Fall #16) is all messed up. The numbers they provide in the table are not internally consistent, and you get a different answer depending on how you approach it.

A net commissions problem appeared again on (2016.Spring #18a) but this time the CAS cleaned up their act and got it right. It isn't a very hard problem, provided you memorized the layout of Exhibit 80.10 from the quarterly return. (Note that exhibit 80.10 does not appear in the sample annual return referenced in OSFI.AR2. It only appears in the sample quarterly return OSFI.AR1.)

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Other Basic Accounting Formulas

There are a few more basic accounting formulas that you will often need to solve problems from this paper. Note that these metrics normally pertain to a particular AY (not a single point in time.):

metrics formula
NLR (Net Loss Ratio) = APV(incurred claims) / EP
NCOR (Net Combined Operating Ratio) = [ APV(incurred claims) + operating expenses ] / EP     (operating expenses exclude investment expenses.)
NI (Net Income) = (1 – tax rate) x [ (U/W Income) + (Investment Income) + (Other Income) ]

Note that these formulas were covered on the Exam 5 syllabus, but without the "APV" part. Also, some of the quantities in the above formulas can be broken down further. (Recall that in general, income equals revenue – expenses, whether it's U/W or investment income)

supporting metrics formula
APV(incurred claims) for a particular period = (paid loss in period) + change [ APV(unpaid claims) ]
EP for a particular period = WP – change(UEP)
U/W Income for a particular period = EP – APV(incurred claims) – operating expenses     (operating expenses exclude investment expenses.)
Investment Income for a particular period = (investment revenue) – (investment expenses)

The "particular period" is usually 1 year. Note also that the change in the formulas above generally means (current year-end) – (prior year-end).

Here is a PDF with practice problems using the above formulas similar to (2015.Spring #24). In this problem, you're given the investment revenue. But contrast this with problems like (2017.Fall #14a) where they give you asset/bond values and make you calculate the investment income. Ugh! Alice the Actuary hates when they complicate the problem like that! That's a topic that will be covered in CIA.Accting so don't worry about it for now. Just get the basics down first.

Basic Accounting Practice

Once you're good at calculating NLR, NCOR, and NI, have a crack at the mini BattleQuiz below! (Either that or take a break with with Alice and lcparga at the actuarial cafe. Breaks are just as important as study time! Shout-out to lcparga for inspiring this new section! He deserves a nice hot double-double! Update: lcparga likes his coffee black.)

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Alphabet City

When I was in university, Alice dragged me to a double-feature at the Revue Cinema on Roncesvalles Ave. We saw 2 New York City mafia films. The first one was Mean Streets with Robert DeNiro & Harvey Keitel. It was A-W-E-S-O-M-E. The second one was Alphabet City. It S-U-C-K-E-D, much like the exam problems listed below: (These problems all ask you to calculate unknown quantities A,B,C,D...)

E (2018.Fall #16)
E (2015.Spring #25)
E (2014.Spring #17)
E (2013.Fall #21)
E (2012.Fall #22)
E (2012.Fall #23)

Note that not all of the problems used letters for the unknowns. Sometimes the examiners just put questions marks to indicate the unknown quantities in the tables. (Lazy much?) These types of problems used to appear regularly, but then there was a longish gap between 2015.Spring and 2018.Fall. We're going to cover #16 on the 2018.Fall exam in detail. It's long and very difficult. It was one of the 3 worst done questions on the 2018.Fall exam, the other poorly done questions being #17 and #28. The average score on #16 was roughly 1.50 - 1.75 out of a possible 4.75. (See Old Exams for a listing of problems from that exam.)

This 2018 problem was worth 4.75 points, the most of any question on the exam. If you follow the time management tips from On BattleDay, this is the first problem you would do. There is over a page of given information and it takes time to wrap your head around it. Ideally, you would spend the whole 15-minute reading period thinking through it so that when the exam starts, you could immediately start calculating. This is not the type of problem you want to be doing 2-3 hours into the exam when your brain is starting to get fried. So, let's get started. Click the link below and spend a few moments looking at it but don't do any calculations yet.

E (2018.Fall #16)

In that mountain of information, my first foothold was noticing that they provided the following:

  • paid & unpaid triangles
  • a payment pattern
  • MfADs

This is what you need to calculate APV of the claims liabilities. Even if you had no idea how to solve the problem, you could start by calculating the APV of the claims liabilities for AY 2016 and 2017. But wait, there's one more thing - you need the discount rate, which isn't given. They do, however, provide:

  • bond yields

You can use the information about the bond portfolio to calculate the discount rate. This is easy and is explained in CIA.Discnt. Anyway, that's how you get started. I wrote out all the details in this 5-page pdf:

Solution to 2018.Fall #16

And here are some practice problems with different numbers: (Alice the Actuary was feeling particularly evil this morning and she inserted a data error into two of these problems. You won't be able to pinpoint a specific error, but you know there is something wrong because you get a negative answer in some cases. She explained to me that in the real world there are sometimes (often?) errors in your data. That means you can't just apply formulas blindly. You have to think about whether your results make sense. Can you find the errors? Make Alice proud!)

Alphabet City practice (problems 1-4)
Alphabet City practice (problems 5-8)

Regarding the other Alphabet City problems, I don't think they're quite as hard as the 2018.Fall version. See if you can work your way through them.

Also, I've noticed a pattern in how the CAS creates new types of exam questions. Often, a new type of question is introduced and is poorly done by most candidates (because it's very different from anything that's appeared before). Then a similar question reappears on subsequent exams and people get better and better at solving it. I don't know if this will happen with this particular Alphabet City question, but it's a good one to practice on because it covers so much ground. You have to understand a lot of things about financial statement quantities.

Slay the beast!


  • Memorize:
    • The various formulas for E(x)
    • Formula for Excess/Deficiency Ratio (This is asked on EVERY exam, with different combinations of given info)
  • Conceptual:
    • definition of 'non-deferrable commission'
    • how to account for the time value in money in calculating the ExRatio
  • Calculational:
    • E(x), given Balance Sheet info and/or Income Statement info from current or prior years
    • ExRatio (Excess/Deficiency Ratio)
    • Investment Income from Insurance Operations
    • NLR (Net Loss Ratio), NCOR (Net Combined Operating Ratio), NI (Net Income)
    • Net Commissions (Exhibit 80.10 in the annual return)
  • This is covered in the Calculation Problems and in old exam problems. Practice makes perfect! (Not as much memorization here as in other parts of the syllabus.)
  • Expect about 3 pts from this paper on the exam

Full BattleQuiz You must be logged in or this will not work.



  • E(x) = A(x) - L(x) = 20m - 14m = 6m
  • Since E(x) = E(x-1) + NI(x) + OCI(x), simple substitution yields E(x-1) = 2m