Epic immunization question

CFA Level III Fixed Income
#1

Ok guys, I have a question about immunization.

I’ve read through a lot of posts from the past but there has never been a specific answer.

So let’s start with something basic.

Let’s assume that your PV of liabilities are $100 million and the average liability is due in 10 years, i.e the Macaulay Duration of the liability is 10 years. Let’s also assume that the Modified Duration of the liability is 8, i.e the value of the liability will go down by 8% or up by 8% if the interest rate moved up by 1% or down by 1% (since bond price and interest rates are inversely related).

Now you are this awesome bond portfolio manager and you’re asked to immunize this liability and you get to work.

You start assembling bonds. According to Chapter 21, PV of assets = PV of liabilities. Chapter 22 says PV of assets >= PV of liabilities. It makes a little more sense to me that we have PV of assets >= PV of liabilities.

Question 1 - For the purpose of the exam, if you are asked to write down steps on how you immunize the liability, would you go with PV of assets = PV of liabilities or PV of assets >= PV of liabilities?

Let’s move a little bit further into our process. Now you need to match the characteristics. I understand that in an ideal world you would love to have a 10 year zero coupon Treasury Bond that pays $100 million in 10 years. But you understand the constraints of the real world, and get a whole bunch of coupon paying Investment Grade corporate bonds and off-the run Treasury Bonds (to get higher yield) with varying maturities. This brings me to my next questions.

Question 2 - Do I assemble my asset portfolio so that the Macaulay Duration of the Assets is 10 years, which happens to be the Macaulay duration of the liabilities?

Question 3 - Do I assemble my asset portfolio so that the Modified Duration of the Assets is 8, which happens to be the Modified duration of the liabilities? That is, we want the value of both our assets and liabilities to either move up together by 8%, or move down together by 8%, if interest rates goes down or up by 1%.

Question 4 - Do I assemble my portfolio so that both my Macaulay Duration and Modified duration of assets are 10 years and 8, respectively?

I would appreciate if someone can answer my questions. I don’t think I’ve seen this question asked in such a manner before. Hopefully we get a really awesome answer and that we can make this a sticky because I feel like a lot of people would have the same question that I have.

Hey you can even treat this as an AM essay practice.

If you want this to be an item set, here you go:

A. Match Macaulay Duration only while making sure PV of assets >= PV of liabilities

B. Match Modified Duration only while making sure PV of assets >= PV of liabilities

C. Match both Modified and Macaulay Duration while making sure PV of assets >= PV of liabilities

#2

A

#3

I have changed it to multiple liabilities.

Are you saying A is the answer because you thought it was a single liability?

Sorry for the unclear question.

#4

for single liabilities we want to match Macaulay Duration. For multiple we’d want to match on dollar duration/BPV. Also for multiple liabilities, the PV A should be greater than or equal to the PV L, and the assets should have a higher convexity than the liabilities.

#5

Awesome, thanks dude!

I was holding off for s2000 to give a slam dunk answer, but your answer has cleared up so many doubts. It’s crystal clear now.

I guess I would also say that if it’s a single liability then you want convexity as little as possible, but in case of multiple liabilities you want slightly higher convexity (for reasons unknown ). Is this correct?

#6

I would probably enjoy reading an answer from S2K also, just in case I am wrong! I’d like to know before the 23rd!

You want higher convexity for multiple liabilities so that when interest rates rise, your assets fall in value less than the liabilities will, and when interest rates fall your assets will increase in value quicker than the liabilities.

#7

Im curious too, because I think the question isn’t whether you want greater assets or not (because you always should want more), but on if you do, because if you do then you can take an AO (asset only) approach on the surplus amount and use the ALM approach on the rest of the portfolio to immunize the liabilities. For immunization purposes I believe that Macualay duration is the most appropriate because it is a measure that is directly related to timing of cash flows rather than just price sensitivity of the bonds. I would also like to hear for S2000. No offense StageRight!

#8

Lol…thanks for the responses

DPBass88 - I didn’t even think about the contingent immunization possibility if PV of Assets is >>> PV of liabilities.

I think Macaulay duration should be matched if you’re trying to immunize a single liability. For this you also want as low a convexity as possible - since you know it’s one duration, you want your cash flow to be concentrated around it (ideally a zero coupon bond that matures on that day i.e convexity of 0).

I think Modified Duration, or as StageRight said, PVBP/BPV/Dollar Duration/Money Duration (all same things) should be matched for multiple liabilities. For this you want high convexity, but again not too high (for reasons unknown).

All of this is just my understanding so far.

I hope it is right.

I wish that a senior board member chimes in to either correct or confirm my conclusions above in bold.

#9

This is the way I think about it:

You want convexity higher than that of your liability because in a rising rate environment a bond with more convexity is going to perform better with a bond with less convexity – if the asset has less convexity then the liability and rates go up, the price of the liability will decrease at a lower rate than that of the asset supposedly “immunizing” it, which would cause underfunding.

You want to minimize convexity because convexity isn’t free and has a cost (i.e. lower yield).

Hope this helps.

#10

Which would be better for immunizing multiple liabilities?

  1. Convexity of assets = convexity of liabilities

  2. Convexity of assets > convexity of liabilities

#11
#12
  1. for multiple liabilities

If it’s a single liability, you want to minimize convexity.

Again, would like a board member to approve of our “conclusions”

#13

Yes, I agree with Fin. You want minimize convexity because the goal is to immunize the liabilities and you can’t do that if the sensitivities are different due to a difference in convexity.

#14

Minimize convexity to limit structural risk, but you still want your convexity of assets to be higher than the convexity of the liabilities for any parallel shifts in the yield curve. If your convexity is slightly higher, than the asset gain more for a decrease in yields and lose less for an increase in yields.

The same goes for multiple liabilities. You want the convexity to be slightly higher than the liabilities but to have it minimized for structural risk.

Structural risk comes into play when the shift in the yield curve is not parallel. In this case, the dispersion of cash flows around the liability (directly linked to convexity) can create problems. If you have a barbell portfolio and the curve steepens (LT yields increase, ST yields decrease) than the PVA will be less than the PVL. If the increase in portfolio IRR due to the increase in yields is not high enough, the FV of your assets might not match the future value of your liabilities. Likewise, if you have a barbell portfolio, if the curve flattens (LT rates decrease, ST rates increase) the PVA will be greater than the PVL. While this is favorable it does not mean it will be successful. If the IRR of the portfolio falls enough, the FVA might not be enough to meet the FVL, hence, structural risk.

For single liability you are going to focus on MacDur, for multiple you will focus on PVBP/ModDur/DollarDur. You can still match MacDur but it’s harder.

#15

Thank you!