CIA.Duration

Reading: Educational Note: Duration Considerations for P&C Insurers (Candidates are responsible for the Excel illustrations attached to the Educational Note.)

Author: Canadian Institute of Actuaries

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BA Quick-Summary: Duration This reading is a good summary of concepts related to duration, which you would be familiar with from previous preliminary exams. In particular, it clarifies the calculation of: Macaulay duration Modified duration Effective duration. This reading also provides examples of how to calculate the PAA LRC, GMA LRC, ARC as well as LIC.

Pop Quiz

• Alice the Actuary helped me with today's pop quiz. She's already studied this paper and knows what's important. It's a difficult multi-part quiz. (You'll thank her later!)

1. What are the 4 risk categories that are considered in calculating MCT? Hint: IMCO
2. What are the components of market risk in the MCT calculation? Hint: Mr. IFER
3. What is the formula for CapReq(IntRt)? (In other words, what is the formula for calculating the capital required for interest rate risk in the MCT calculation?)
4. Given a change in interest rate of 175 bps and the following info, calculate CapReq(IntRt): (Assume there are no other assets or liabilities)

Asset or Liability	Item	Fair Value	Modified or Effective Duration
Asset	Bonds & Debentures	750,000	5.25
Liability	net UCAE	550,000	1.50

• Recall: UCAE = Unpaid Claims & Adjustment Expenses

Keywords

Macaulay duration, Modified duration, Effective duration, IFRS17

In Plain English!

My First Thoughts

• The first thing I check when beginning a new paper is the # of pages. This one is 11, which is slightly shorter. The next thing I check is whether it's a concept paper or a math paper. This one is a math paper. That usually means there's not too much memorization, but you'll have to learn how to do some specific calculations. Page 10 of the reading has a nice table of contents for the calculations in the appendices. Since the title of this paper is Duration Considerations for P&C Insurers, I'd expect these appendices, which are Excel spreadsheets, to focus on duration calculations. Since this is a revamp to an old paper, expect a lot of IFRS17 concepts to be displayed and to be tested on it. Reading both the old and new paper side by side, this paper is substantially more difficult.

• Appendix A:

• interest rate risk margin calculation in the MCT

• Interesting! What's interest rate risk margin doing here? Well, if you did the pop quiz, you'll recall that...

...you need the duration of the assets & liabilities to calculate the interest rate risk margin.

• Now, if you get a question on the interest rate risk margin ( or CapReq(IntRt) ), I would really like to know whether the exam committee will give you the necessary durations, or require you to calculate them. It's fairly involved, and if you get it wrong, or don't know how to do it, you won't be able to proceed with the margin calculation. Note that...
  • ...on (2016.Spring #13), you had to calculate the claim and premium liability durations
  • ...on (2016.Fall #15a), you had to calculate the claim and premium liability durations
  • ...on (2017.Spring #20a), you were given all necessary durations (asset, claim liability, premium liability)
• The 2 questions where you actually had to calculate the durations were incredibly long. It was possible to do it, but you had go into the exam knowing those calculations backwards and forwards. That means you had to practice them 10 or 20 times over a period of weeks to really get them into your brain. I know from my experience as a math professor that most students simply don't have the time or inclination to do that. I suspect those questions had very low average scores. Maybe that's why they decided to just give you the durations on the 2017.Spring exam.
• The concept of premium liabilities is no longer relevant under IFRS17 and I would skip anything to do with that. They are now replaced with the LRC and ARC where the calculation is conceptually different.
• Also on the 2017.Spring exam, parts (a) & (b) were very easy:
  • (2017.Spring #20a): Define the interest rate risk.
    • Answer: risk of loss FROM market changes in interest rates
  • (2017.Spring #20a): Contrast effective duration and modified duration.
    • Answer: Effective duration accounts for situations where the cash flows may change as a result of changes in interest rates. Modified duration does not.
  • Then part (c) is where they ask you to do the calculation of the interest rate risk margin, but since they provided you with the duration, it was pretty easy. But even if they had not given you the duration, you could still have written down the margin formula, and filled in the information you were given, and received partial credit. Recall the formula from the pop quiz:
    • CapReq(IntRt) = (chg.A - chg.L), where chg(A or L) = (Fair Value) x chg(IntRt) x (modified duration)

Because the exam committee updated the syllabus reading specifically on duration for IFRS17 with great sample calculation exhibits for 2024.spring, I predict that the topic will come up more frequently on future exams. Learn it!!

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Introduction and Scope

• Why is the concept of duration important?
  • calculating interest rate risk margin
  • calculating investment return rate risk margin
  • matching assets & liabilities
  • modeling market risk

Duration Defined

• What does duration measure?
  • average maturity of fixed future cash flows
  • sensitivity of PV cash flows to interest rate changes

Macaulay Duration

• Don't let the formula at the top of p2 confuse you. The Macaulay duration of a bond is easy to understand if you think about the case where there is only 1 payment per year. (k=1)

Macaulay duration = a weighted average of time where the weights are the cash flows.

• Suppose you have the following cash flow schedule:

time	discounted cash flows
1	15
2	15
3	15
4	1,000

• The Macaulay duration

= the weighted average of (1, 2, 3, 4), using cash flows as the weights

= [ (1 x 15) + (2 x 15) + (3 x 15) + (4 x 1000) ] / 1,045

= 3.9139.

• (If there are k payments per year, you do the calculation exactly the same way, except at the end you divide by k.)

Modified Duration

• The modified duration is even easier than the Macaulay duration. (It measures the sensitivity of the cash flows to the interest rate.)

modified duration = (Macaulay duration) / (1 + yield rate)

• Suppose the yield rate is 1.5% for the previous example. Then the modified duration = 3.9139 / 1.015 = 3.8561

Effective Duration

• The effective duration is harder to calculate, but it gives a similar answer to the modified duration when interest rate changes DO NOT AFFECT future cash flows. (This is true for "normal" bonds, but not true for callable bonds, or interest rate derivatives.)
• Note that either the modified or effective duration is acceptable in calculating the duration of assets and liabilities for the interest rate risk margin in MCT, as long as the one chosen is used consistently. And because both are acceptable, you can probably ignore the calculations for effective duration in Appendices.

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Duration of Interest-Rate-Sensitive Liabilities

• Here's a type a question I can't stand, but it often appears on exams. The examiners simply select a bullet-point list from the reading and expect you to memorize it. This section begins with such a list: considerations for calculating the duration of claim and premium liabilities... Now premium liabilities are no longer relevant, but the following points are still all relevant with some additional IFRS17 considerations

consistency of assumptions:	assumptions for the duration calculation should be consistent with the discounting calculation from the valuation
duration calculation by LOB:	use the same payout patterns as used for discounting THEN total duration is a weighted average where weights = PV of FCF + RA. Could also evaluate duration on a combined basis for all LOBs, but must use effective duration
duration calculation on combined basis:	use effective duration
when interest rate is small:	modified duration and effective duration are approximately the same
when discounting is incorporated in the Risk Adjustment:	If the RA technique requires discounted amounts, then the RA is interest rate sensitive and not otherwise
GMA Considerations:	Future Cash Flows are interest rate sensitive while CSM is not
PAA Considerations:	Loss Component is considered highly leveraged and extremely sensitive to interest rate changes LRC excl LC is not affected by interest rate changes
Special Considerations, when duration is 0:	When claims are expected to be paid out within a year and the LIC and AIC is not discounted When there is a significant financing component

• Note that the non-IFRS17 considerations apply to any calculation of duration, not just the assets and liabilities for MCT. (Recall that for MCT, both modified duration and effective duration are acceptable as long as the choice is consistent.)

Duration of Interest-Rate-Sensitive Assets

• Actuaries can either calculate asset durations themselves or may use duration estimates derived by the company's investment specialists, but these estimates should be reviewed for reasonableness and methodology (i.e. which duration measure was used.)

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Appendices

Appendix A

• This appendix ties everything together in the MCT and is dependent on the calculations from the other appendices.
• Recall that the Macaulay duration is calculated first, then just divide by (1 + discount rate) to get the modified duration.
• Effective duration is used here, but you should be okay to proceed with modified duration in the exam
• To get the interest rate charge, multiply the duration by 0.0125 and with each of the LRC, ARC, LIC and AIC.
  • Both the modified and effective duration are accepted in MCT, so long as you are consistent.
  • Effective duration should be used when cash flows are dependent on changes in discount rates. (The modified duration doesn't account for this.)

Appendix B Sh1 - LIC

• The example in the Excel spreadsheet seems complicated at first glance due to the numerous moving parts. Actually, it isn't that hard to understand if you spend 10 or 15 minutes going through it.
• Generally, when calculating the LIC you need a couple of things:
  • Undiscounted future cash flows
  • Payment patterns
  • Yield curve used for discounting

• Step 1: We will work through AY 2023 to first get the future payment in each period for this AY. Every other AY will follow the same logic

Undiscounted future cash flow @ time 0:

2500

net PAID @ 12 months:	35%
net PAID @ 24 months:	70%
net PAID @ 36 months:	80%
net PAID @ 48 months:	85%
net PAID @ 60 months:	95%
net PAID @ 72 months:	100%

• Then we calculate the cash flows as follows:

period	lag	% PAID in period	Cash flow in period
12-24	0.5	0.5385	1346
24-36	1.5	0.1538	385
36-48	2.5	0.0769	192
48-60	3.5	0.1538	385
60-72	4.5	0.0769	192

• To calculate the amounts paid for 24-36, we must first find the amount of remaining CF paid at time 24-36. This is done by taking 0.1/0.65. We then multiply that with 2500 to get the amount paid in the period. The reason we do this is that at time 12, there are only 65% of cash flow remaining which is represented by 2500.
• An alternative method, is to find the total amount of cash flow expected at time 0 which is 2500/0.65 (Since we already paid 35%). We can then apply 0.1 directly to this number to find the amount paid in the period.

• Step 2:

• Discounting is pretty simple once we have the cash flows. Two examples shown below:
  • AY 2023, paid in 2024: 1346/(1.03)^0.5
  • AY 2023, paid in 2025: 385/(1.032)^1.5

• Risk adjustment is then added to the discounted cash flows. In general, if not stated always apply the RA on discounted cashflows
• Other cash flows are assumed to be non-discounted, although if they were to be discounted, proceed as you would with the regular CFs.

• Effective Duration:

• Getting the effective duration is actually quite a bit simpler than it looks. All you have to do is repeat step 2, and add 0.1% to your interest rate curve when discounting to get V+0.1%. For example, your year 1 discount rate would be 3.1% instead of 3%.
• Do the same thing for V-0.1% but subtract 0.1% instead of adding.
• Effective Duration = (V+0.1% - V-0.1%)/(V0*2*0.001), where V0 is the discounted CFs calculated in step 2.
• I could see them making this calculation more challenging by maybe varying the payment pattern under a +0.1% or -0.1% scenario, but conceptually it would be exactly the same thing; you just would not be able to use the same CFs in every scenario.

• Modified Duration:

• This is quite a bit simpler, and is already covered in the prior section.
• The main difference here is that rather than using a fixed discount rate, you would use a yield curve that varies by year to calculate the Macaulay duration. Everything else would be exactly the same!
• Calculating the Modified duration is a little tricky, since we do not have a single discount rate. What is done in the appendix is the CAS attempting to find a single 'implied' discount rate that will provide us with the same present value as what we have above. Modified duration is then Macaulay duration/(1+ 'implied' rate)

Appendix B Sh2 - AIC

• Exactly the same as the prior section, except on the AIC instead of the LIC

Appendix C Sh1 - LRC(onerous)

• This appendix is very similar to the LIC calculation with a bit more nuance as we have to deal with expenses as well as premiums.

• Step 1:

• Similar to step 1 for the LIC, with an addition of premiums received (which should be a negative since this is an inflow), future claims, insurance acquisition cash flows and maintenance costs.
• All of this are summed up to obtain a future cash flow number.

• Step 2:

• Discounting and application of the risk adjustment is also the same as with the LIC, but using each quarter's midpoint instead.
• We sum up all of these discounted cash flows with risk adjustment in each quarter to obtain the fulfillment cash flows (The GMA estimate of the LRC).
• At the bottom, they are also calculating the PAA estimate, which is the LRC excl LC (Recall this is UEP - DAC).
• Given that we do not have any DAC in this example, the LRC excl LC is just the UEP.
• The loss component is defined as MAX(0, GMA - PAA).
• Effective and modified duration is similar to what we described above.

Appendix C Sh2 - LRC (non onerous)

• When a group of contract is non-onerous, the PAA LRC excl LC is not discounted (yay) which means there is nothing to do here :)

Appendix C Sh3 - ARC

• This appendix is one that I would spend some additional time on as I believe it is the only place in the syllabus where an actual example of how the ARC is calculated is provided.
• Step 1:

• The key components of the PAA approach for the ARC is to get the reinsurance premiums paid (1500 as noted in the year-end information) and also the allocation of reinsurance premium paid (basically the earned premium for the reinsurer. Since 50% of the premium has been paid, we get 2000*0.5 = 1000.
• The ARC excl LRECC is just the Reinsurance Premium paid - Earned premium for the reinsurer. This is very similar to our LRC excl LC.
• Since this reinsurance contract is a quota share, we obtain the LRECC by multiplying the underlying loss component calculated in App C sh1 - LRC (onerous) by 25%.
• Summing up the ARC excl LRECC and the LRECC gives you the final ARC.

Appendix D - Alternate

• This section shows how to calculate the duration of only the loss component.
• Step 1:

• Just a copy of App C sh1 - LRC(onerous)

• Step 2:

• The effective duration is also the same as what we calculated in the prior section, except this time we strip out the LRC excl LC to get the discounted Loss Component at +-0.1%.
• Apply the usual effective duration formula to get the effective duration of only the LC.
• You can see in cell G31 how leveraged it is relative to the effective duration of the FCF.
• We then use this calculated duration as the duration of the LRECC in our prior section.

BattleCodes

• Memorize:
  • Why is the concept of duration important
  • What does duration measure
  • Considerations in calculating duration, with more focus on the IFRS17 concepts

• Conceptual:
  • Do actuaries always have to do the duration calculation for assets themselves? (No, they can rely on investment specialists. The actuary would be the enquiring professional, and the investment specialist would be the responding professional. See CIA.CSOP for definitions of these terms.)

• Calculational:
  • Macaulay duration
  • Modified duration
  • Effective duration (Given a choice, I would go with modified duration but I could see the CAS forcing candidates to use effective duration)
  • IFRS17 Cash Flow Calculations (LRC, LIC, ARC)

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POP QUIZ ANSWERS

1. MCT risk categories: InsRsk, MktRsk, CrdRsk, OpnRsk (insurance, market, credit, operational)
2. Market Risk components: Interest rate risk, Foreign exchange risk, Equity risk, Real estate risk
3. CapReq(IntRt) = (chg.A - chg.L), where chg(A or L) = (Fair Value) x chg(IntRt) x (modified duration)]
4. CapReq(IntRt) = 68,906.25 - 14,437.50 = 54,468.75

• chg.A = 750,000 x (175/10,000) x 5.25 = 68,906.25
• chg.L = 550,000 x (175/10,000) x 1.50 = 14,437.50