CIA.Duration

Most of the information in this reading is contained in other readings. For example, the premium liability reading covers the duration calculation for premium liabilities. Also, the reading on discounting covers the cash flow matching model. Nonetheless it's a good summary of concepts related to duration. In particular, it clarifies the calculation of the Macaulay, modified, and effective durations.   Forum

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

In Plain English!

My First Thoughts

• The first thing I check when beginning a new paper is the # of pages. This one is 20, about average. 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 8 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 spreadhsheets, to focus on duration calculations.

• Appendix A:

• duration of bonds, claim liabilities, premium liabilities
• interest rate risk margin

• 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.
• 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 added a syllabus reading specifically on duration for 2017.Fall, 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...

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 = APV
duration calculation on combined basis:	use effective duration
when interest rate is small:	modified duration and effective duration are approximately the same

• Note that these 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.)
• For premium liabilities, there are a couple of extra considerations:

accident date adjustment #1:	adjust duration calculation for future accident date
accident date adjustment #2:	adjust future accident date for policy terms of other than 12 months

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

A.1: Bond Duration

• First, note there is an error in this exhibit. The exhibit gives the formula for columns (11) and (12) as: (11) = (3) x (8), and (12) = (3) x (9). It should be: (11) = (3) x (9), and (12) = (3) x (10)
• It's typical for bonds to pay semi-annual coupons, as is the case for the example in the exhibit.
• Other than that, the calculation for the modified duration is pretty easy. Just practice writing out the 5 columns. (It's probably ok to ignore the calculation for effective duration.) Make sure to go through the calculations for each of the 3 bonds, and also the part at the end where the overall duration is calculated as a weighted average. weights = fair or market value.
  • Recall that the Macaulay duration is calculated first, then just divide by (1 + discount rate) to get the modified duration.
  • 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.)

A.2-3: Claim Liability Duration

• The example in the Excel spreadsheet is complicated because it shows how to do the calculation for two LOBs and for all AYs. Actually, it isn't that hard to understand if you spend 10 or 15 minutes going through it, but here is a super-simple example of how to calculate the claim liability duration:

• Suppose that for AY 2018, you're given:

net unpaid @ 12 months:	69,000
discount rate:	11%

net PAID @ 12 months:	30%
net PAID @ 24 months:	55%
net PAID @ 36 months:	100%

• Then we calculate the discounted cash flows as follows:

period	lag	% PAID in period	PV(cash flows)
12-24	0.5	0.35741	23,390
24-36	1.5	0.64286	37,930

• Then Macaulay duration = weighted average of lag using PV as weights = 1.1186
  • Note 1: % PAID in period is calculated as:
    • 0.357 = (55% - 30%) / (100% - 30%)
    • 0.643 = (100% - 55%) / (100% - 30%)
  • Note 2: PV(cash flows) is calculated as:
    • 23,390 = 0.35741 x 69,000 / 1.11^0.5
    • 37,930 = 0.64286 x 69,000 / 1.11^1.5
• Finally, modified duration = 1.1186 / 1.11 = 1.0077

A.4: Premium Liability Duration

• The pattern for calculating the duration of premium liabilities is in the Excel Exhibits for CIA's Duration paper as per the syllabus.
• This calculation is illustrated for an old exam problem in the BattleWiki article on premium liabilities CIA.PrLiabs. See the section called Duration of Premium Liabilities.

A.5: Interest Rate Risk Margin

• This is discussed in the pop quiz, and at the beginning of this wiki article. The Excel spreadsheet is just a more elaborate example.

Appendix B

• This is a review of the cash flow matching model from the CIA reading on discounting: CIA.Discnt.

Appendix C

• Appendix C is a long, drawn-out example of calculating DPAE. See the article CIA.PrLiabs for details on how to do this calculation.

BattleCodes

• Memorize:
  • why is the concept of duration important
  • what does duration measure
  • 4 considerations in calculating duration for claim & premium liabilities
  • 2 extra considerations for the premium liability duration (relates to FAD or future accident date)

• 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 defns of these terms.)

• Calculational:
  • Macaulay duration
  • modified duration
  • (I could be wrong, but I don't think you'd be asked to calculate effective duration on an exam - just know that it accounts for changes in cash flows if interest rates change, whereas the modified duration does not.)

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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