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The most complex computational problem on the syllabus is calculating APV(premium liabilities) and DPAE/PDR.
It is NOT the same as calculating APV(claims liabilities).   Forum

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Given a discount rate of 5%, calculate APV(claims liabilities) for $10,000 of payments made during the 12-mth development period. (When calculating APV for claims liabilities, we assume payments are made in the middle of the year, so the discounting period for 12-mths would be 0.5 yrs)


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

  • DPAE/PDR: calculation (DPAE = Deferred Policy Acquisition Expense, PDR = Premium Deficiency Reserve)
  • durations of premium liabilities (usually part of a larger MCT problem)
  • definition of the DPAE asset
  • components of the policy liabilities in connection with the UPR (Unearned Premium Reserve)

Important: Questions from 2015.Spring and prior are from the old premium liabilities reading.

reference part (a) part (b) part (c) part (d)
E (2018.Spring #14) DPAE/PDR:
- calculate 4
E (2017.Fall #15) DPAE/PDR:
- calculate 1
E (2017.Spring #15) DPAE/PDR:
- calculate
- does it exist
E (2016.Fall #15) calculate:
- durations 2,3
E (2016.Fall #27) see OSFI.AA see OSFI.AA (c) see OSFI.AA
(d) see OSFI.AA
(e) UPR liabilities
(f) see OSFI.AA
E (2016.Spring #13) calculate:
- durations 2,3
E (2015.Fall #12) DPAE/PDR:
- calculate 2
E (2015.Spring #28) DPAE/PDR:
- outdated
E (2014.Fall #25) DPAE/PDR:
- outdated
- DPAE (+ example)
E (2014.Spring #19) DPAE/PDR:
- outdated
E (2013.Fall #17) DPAE/PDR:
- outdated
E (2013.Fall #20) DPAE/PDR:
- outdated
- order of operations
E (2012.Fall #26) DPAE/PDR:
- outdated
1 This is the first time the examiner's report used the duration method from the 2016 version of the source reading.
2 Uses the pre-2016 version of the duration calculation. Do not use this method. This is explained in detail in the wiki article.
3 Part of a larger MCT problem (see OSFI.MCT)
4 There is not enough information provided to solve this problem. They do not provide the interest rate shock. The first sample answer in the examiner's report uses 1.25% but you could not have known this (and they did not mention the omission in their solution.) The second sample answer assumes the interest rate shock equals MfAD(inv). This is not a valid assumption, but it's the only way you could solve the problem.

In Plain English!


Focus on the appendices! The appendices are printouts of Excel spreadsheets that show you how to calculate the APV of the premium liabilities. Other than that, there are just a few simple definitions & concepts. (These are covered in the BattleCards.)

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.


  • components of UPR (Unearned Premium Reserve)
  • definition of DPAE asset
  • purpose of DPAE asset

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The Calculation

It is not that easy to learn this calculation from the appendices. There is a calculational problem template in BattleActs that explains it more concisely.

Don't do it now, but at some point, go over Appendix B, Sheet 5 from the Excel exhibits.

  • This explains how to calculate the discount factor for premium liabilities.
  • Note that this calculation was greatly simplified in the 2016 update of this paper. You will see the old version of the discount factor calculation in examiners reports for 2016.Fall and prior. (Sometimes they're nice and give you the discount factor, as in 2017.Spring #15a)

Overview of the 5 Sections in the Paper

For the sake of completeness, here is a brief summary of each section of the paper:

Section 1: General definitions. Just memorize the BattleCards.
Section 2: A discussion of familiar pricing concepts related to the DPAE/PDR calc. Skip it. The vital info is in the appendices and the BattleActs calculational problems.
Section 3: Basic formulas for 'max DPAE' and 'PDR'. Again, the vital info is in the appendices and the BattleActs calculational problems.
Section 4: Discusses 'Other Net Liabilities', namely commission & premium adjustments. You might want to know the definitions of these, but my guess it that it's too detailed to be asked on the exam. If you've covered everything else and need something to do, you can read this section!
Section 5: Subsequent events. This is a brief review of the ice-storm example from the CIA.Subseq. Skip it. It's all covered in CIA.Subseq.

Before looking at the old exam problem in the next section, review this next mini BattleQuiz a few times. It's an easy version of the DPAE calculation that was covered in Boot Camp.

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Explanation of 2015.Fall #12

I don't like the way the examiner's report presented the solution to this problem. Below is a more organized way of understanding how to solve it.


Calculate the max allowable DPAE and the premium deficiency, if any. Link to the official question and answer from the examiner's report:  E (2015.Fall #12)

Given Information

year 2016
net UPR 69,580
FutRe (Future Reinsurance) 4,260
ELR (undiscounted) 98%
ULAE 3,160
DiscntRt 3.0%
MfAD(inv) 50 bps
MfAD(clms) 10.0%
MfAD(re) 1.0%
gross PV(L+LAE) 68,260
maintenance 4,175
UEComm (Unearned Commission) 1,045

Accident year payment pattern: (Assume all payments are made in the middle of the year)

age CumPd
12 25%
24 50%
36 75%
48 100%

Whew! That's a lot to take in! But if you're organized, it all fits neatly into the formulas.

Basic Formulas

We're asked to calculate DPAE, so let's start with the basic formula: (All quantities below are net unless it specifically says gross or ceded.)

DPAE = UPR - PolLiabs(UPR) + UEComm

Notice that we already know 2 of the 3 items in this formula:

UPR = 69,580
UEComm = 1,045.

Great start! Now, since the missing piece in the above formula is PolLiabs(UPR), let's write down the formula for that: (PolLiabs is short for policy liabilities.)

PolLiabs(UPR) = APV + FutRe + maint

Again, we know 2 of the 3 items in this formula:

FutRe = 4,260
maint = 4,175

We're really cooking! In fact, it looks like we're almost done, but unfortunately, calculating APV is the hardest part.

Calculating APV

You know from the MfAD paper that to calculate APV, we first need the plain old PV at two different discount rates:

  • the given discount rate of 3.0%
  • the modified discount rate that takes into account MfAD(inv) or 3.0% - 50bps = 2.5%

The PV formula is:

PV = [ ( UPR - FutRe ) x ELR + ULAE ] x PVfctr

If we fill the known (green) items, we get: (Note that PVfctr = present value factor.)

PV = 67,174 x PVfctr

But how do we calculate the PVfctr? This is a long calculation that we'll leave for later. For now, I'll just tell you what the PVfctrs are at the two discount rates:

PVfctr(@3.0%) = 0.9533    ← obtained using old method (with interpolation) to be consistent with examiner's report
PVfctr(@2.5%) = 0.9607    ← obtained using old method (with interpolation) to be consistent with examiner's report

Then just multiply 67,174 by each factor to get:

PV(@3.0%) = 64,034
PV(@2.5%) = 64,536     (Recall, a lower discount rate produces a higher PV)

Finally, we have almost all the pieces we need to calculate the APV:

APV = [PV(@2.5%)] + MfAD(clms) x [PV(@3.0%)] + MfAD(re) x [ceded PV(@3.0%)]

You just need to use the given value of gross PV(@3.0%) to get:

ceded PV(@3.0%)
= gross PV(@3.0%) – net PV(@3.0%)
= 68,260 - 64,034
= 4,226

Substituting all values into the APV formula above gives:

= 64,536 + 10.0% x 64,034 + 1.0% x 4,226
= 70,981

Putting it all together

Now it's just a matter of working your way back up. Using APV = 70,981 we calculate:

= 70,981 + 4,260 + 4,175
= 79,416


= 69,580 - 79,416 + 1,045
= -8,791
But since the result is negative, there is no DPAE. Rather we have a premium deficiency reserve of 8,791.

Calculating the Present Value Factors - Ugh!! (NEW version without interpolation)

This is the PVfctr (Present Value Factor) calculation (discount factor) for the NEW 2016 version of the premium liability reading. It's easier than the old version. The answers here will be different than in the examiner's report, and different from what was stated in the solution above. If you have to calculate the Pvfctr on the exam, you should use the following method:

  • In the solution above, we used the PVfctrs calculated using the old version to be consistent with the examiner's report. But if you get a similar problem on a subsequent exam, you'll have to use the method in the Premium Liabilities Excel Exhibit Appendix B, Sheet 5 (GrossMisc tab)
  • Recall from above: discount rate = 3.0%, discount rate with MfAD = 2.5%
APD CumPd IncrPd setup for PVfctr setup for PVfctr w/ MfAD
(2) (3) (4) (6) = (4) / 1.03(2) (9) = (4) / 1.025(2)
0.5 25% 25% 24.63% 24.69%
1.5 50% 25% 23.92% 24.09%
2.5 75% 25% 23.22% 23.50%
3.5 100% 25% 22.54% 22.93%
(10) Total = 0.9431 (10) Total = 0.9521

It appears from Appendix B, Sheet 5, that row (11) will always be 0.5, and row (12) will always be 1/3 or 0.3333. Then PVfctr = (10) x DiscRt ^ [(11) - (12)]:

PVfctr for 3.0% = 0.9431 x (1.030)(0.5 - 0.3333) = 0.9477
PVfctr for 2.5% = 0.9521 x (1.025)(0.5 - 0.3333) = 0.9560

(Note that the PVfctrs calculated using the new version (without interpolation) are slightly smaller than the result using the old method.)

Duration of Premium Liabilities

Once you've got the above table, the Macaulay duration calculation is easy: (This is not part of the DPAE calculation, but it's related, so I've included it here as well. The duration calculation is discussed in detail in CIA.Duration.)

Step 1: Take the SUMPRODUCT of columns (2) and (6) DIVIDED by the SUM of column (6) to get 1.9616 (This is the same method as for claim liabilities.)
(It's the weighted average of the time from column (2) where the weights are the PV factors from column (6))
Step 2: This is an extra step in the duration calculation for the premium liabilities to adjust for the FAD (Future Accident Date). It's easy, just subtract (0.5 - 0.3333):
The final Macaulay duration is:
1.9616 - (0.5 - .3333) = 1.7949
And the modified duration is:
1.7949 / (1.03) = 1.7426

Now you're ready to tackle the harder version of the DPAE problem.

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2016.Fall #15 Redone

Before leaving this topic, let's take a quick look at the calculation of the PV factors and premium liability duration for E (2016.Fall #15)

This problem asked you to calculate the capital required for interest rate risk. As you now know, this long problem involves calculating the PV factors, and the duration of the premium liabilities, but the examiner's report is now outdated. It uses the old method of interpolation. If you had this problem on a subsequent exam, you would have to use the new method explained above.

Click the link 2016.Fall Q15 Redone to see that portion of the calculation redone with the new method.

The old (2016) version of the premium liability duration calculation is no longer relevant and has been archived to 2015.Fall Q12 Old Version.

Explanation of 2017.Spring #15a

This is very similar to 2015.Fall #12 above. The only extra step is calculating gross PV(L+LAE). (In 2015.Fall, you were given this value directly.) To do this, apply the PV formula, but since we're calculating gross of reinsurance, we set FutRe = 0. (Note that we're not given a value for ULAE so we assume it's 0.)

PV = [ ( UPR - FutRe ) x ELR + ULAE ] x PVfctr

The result is: PV = (3,000 - 0) x 60% x 0.98 = 1,764

The rest of the problem is the same as 2015.Fall. You try this problem on your own. Note that a DPAE problem appears on almost every exam. Unfortunately the methodology was changed a couple of years ago and most of the old exam problems are no longer relevant.

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Maintenance Expenses

A lot of people have asked about the discounting of maintenance expenses in the DPAE calculation. There are 2 forum threads on this topic, but the short answer is that you do not have to discount maintenance expenses. The confusion is that sometimes the examiner's reports do discount them (see 2016.Spring Q13) and sometimes they don't (see sample answer 2 for 2018.Spring #14). The Excel exhibits for the most recent version of the CIA's premium liability paper do not discount maintenance expenses.



  • components of the policy liabilities connected to the UPR
  • defn of DPAE/PDR
  • purpose of DPAE asset


  • How to calculate DPAE/PDR

Expect 2-3 pts from this paper on the exam

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APV(ClmsLiabs) = 10,000 / 1.050.5 = 9,759