Fixed Income - Multiple Liability Immunization

SS9: I don’t understand one of the requirements to immunize mutliple liabilitites. In particular, why does the range of durations of assets have to be greater than the range of duration of liabilities? According to CFAI and Schweser it has something to do with needing principals to cover cash outflows. What is the link between range of duration and principals? If anything I would think a greater range of duration would imply longer maturities and less frequent principal repayments. Any help is appreciated. Thanks. Mike.

Your first liability is $1 million due in 1 year. The shortest bond in your portfolio matures in 2 years and has a par value of $1 million. You will have to sell the bond in 1 year to cover the liability, which may or may not be more than par depending on how interest rates moves. There is still risk.

Thanks, but the wording in Schweser still doesnt make much sense to me. If anything the above would imply that the range of the distribution of duration of liabilities exceeds the distribution of duration of assets. I’ll attribute my conufusion to sloppy wording in Schweser…

In the Schweser video it mentions that you want to minimize maturity variance, which explains the point that bpdulog is making. However, I don’t understand what is meant by the range of durations of assets having to be greater than the range of duration of liabilities?

if your liabs range in duration from say 7 to 10 years, then your assets must bracket that range - say 6-12 years… e.g. (6-12 is just an example) so any untoward change in factors such as interest rates that cause the liability to be due earlier - can be covered…

Thanks. After understand this I’m confused by an aspect comparing multiple liability immunization and cash flow matching. In Schweser it says “an immunized portfolio is only required t have sufficient value on the date of each liability because funding is achieved through portfolio rebalancing.” How is rebalancing involved? Isn’t the point of having the range of the duration of assets exceed liabilities so that you cover the liabilities? I’m really confused as to how this aspect of multiple liability immunization and cash flow matching differ

Here is a summary of the difference “Multiple liability immunization method” is technically superior than “cash flow matching method” since CF needs more capital. However, CF matching easier to understand. CF matching: Cash balances may be substantial since the cash (received from repayment of coupons and principals) will not be fully reinvested in new bonds --> most of the cash will be deposited in short term cash (e.g., bank deposits or CD) --> return is quite low. The reason for this cash is not reinvested but sits in the bank is that one will use this cash to pay liabilities. Because of this method, one uses conservative reinvestment assumption. On the other hand, Immunized portfolio method only requires to have sufficient VALUE (not cash): funding is achieved through portfolio rebalancing (i.e, sell some bonds to get the needed cash). Therefore, it assumes that all cash (whenever it is received from coupons and principals payments) will be fully reinvested (at higher rate than short term cash flow in CD or bank deposits as in CF matching method).,985010,985050#msg-985050 “maturity variance i think is related to the dispersion of the asset maturities around the liability maturity. the manager will attempt to keep this to the minimum so that the immunized portfolio works better. it works better because the closer the distributions of asset maturities to the liability maturity the lower the reinvestment risk. key point: the lower the maturity variance the lower the immunization risk.” The point I’m struggling with is that from watching the video and reading more about this it seems like the point about the range of durations of assets having to be greater than the range of duration of liabilities is to minimize the maturity variance, so that you don’t have to see off a bond when the liability comes due in order to minimize reinvestment risk. This seems to be the opposite of just needing sufficient value of the portfolio on hand and seems exactly like cash flow matching. Isn’t it saying you should have a bond mature just before every liability so you won’t have to liquidate it to pay the liability. You should also have one mature at the same time or after the last liability so there isn’t reinvestment risk associated with the last asset maturing early? With cash flow matching, aren’t you also trying to get the coupon plus the maturity to occur just before the liability and reinvesting at those lower liquid rates? I don’t understand how on the one hand with multiple liability you want to minimize the maturity variance and not want to liquidate securities to pay off liabilities, but then on the other hand also say that it only requires having sufficient value (not cash) and that you achieve funding through rebalancing, allowing you to reinvest at higher rates.

You are reading too much into it, and way beyond what is necessary. I don’t understand much of your question, so answer with general explanations in hope that it answers your concerns: Let’s step back. 1. To ensure that there is necessary CF (without too much selling), you need a wider distribution of assets than distribution of liabilities. 2. But you don’t want TOO MUCH distribution since it will increase your immunization risk, so you want minimum maturity variance, subject to the above condition. So what you’ll do is to run a linear program (LP) with the following Min (maturity variance,M2) with constraints: o Weighted duration = average duration of liabilities. o Necessary duration dispersion (i.e., > dispersion of the liabilities). There is always reinvestment risk (unless you have a bullet portfolio meeting each CF precisely, but it is not often feasible), so you want to minimize this risk. Besides, as you see at the constraints, you don’t aim to meet the precise date of each CF, but only aim to meet the AVERAGE duration of all liabilities, relying on partly on selling off when you need as well as coupon and principals repayments to meet the liabilities. The portfolio is therefore always fully invested. CF matching on the other hand, aims to meet the individual CF as precisely as possible. When not possible, it will aim to have cash available BEFORE each deadline. The cash available then just put into bank (or CD) to be readily available. This is all you need to know for the exam.

I’s struggling with this as well. I can conceptualise why you would want the distributions wider on the near end in order to have cash available to meet the liabilities, but why do the distributions need to be further out at the far end? Doesn’t that expose us to price risk? Or is it done to balance the near end widening? Thanks.

Elcfa has answered the questions raised succinctly. @ Mutton When you make the duration of the assets to go to the far end, you minimize the impact of lowered interest rates. If rates drop dramatically and you have asset durations that extend from the near end and stop short of the longest liability duration, this will expose the risk of the immunization not matching the liabilities. The liability will rise faster more than the asset and will cause an erosion of value but with a longer asset duration, you will have at least one security rising in price faster than the longest liability duration.