ALM: Present value of your liabilities=Value of your bond portfolio Duration is sensitivity of your bond to parallel shift in rates, i.e., amount the bond value changes for a unit change in rate. If your liability payments look exactly like a 10 year bond and if you own a 10 year bond, irrespective of how rates change you have a perfect match. If rates go down, your liabilities will go UP, as you discount the future cash flows with a lower rate now. And on the other hand your bond value goes UP by definition. As time passes your liability terms goes to 9 years and your bond term goes to 9 years. Assets (long on bond) and Liabilities (short on bond) move together. Fair and Square. In reality, your liabilities are not as straight forward(they may go up/down and payments not spread evenly), you do not own a single perfect bond but own a bond portfolio(various maturities and rates), rates do not move as plainly as parallel shift, the ALM equation is disturbed. To keep up with the fluctuations you match(not an exact science) duration(sensitivity to rates). The assumption is that, as a portfolio manager the liabilities are not under your control, however, the bond portfolio is. You keep chasing the liabilities with your bond portfolio to the end. 0=0.
Thanks a lot for the inputs! It was really helpful.
Duration here means Macaulay Duration or Modified/Effective Duation ? Since only the Macaulay Duration of zero coupon bond (asset) equals its maturity and the maturity of liability is its duration (in case of only one liability), therefore, in ALM, I think the duration of the asset shall mean its Macaulay Duration (rather than its Modified or Effective Duation). Am I right ?
Effective duration takes under account the optionality embedded in bonds (i.e. callable, putable, MBS etc) in order to calculate their sensitivity to changes in interest rates. Since a zero-coupon Treasury Note/Bond is option free, Effective and Modified Duration will be similar. You are correct. The Macaulay duration of a zero is its maturiy in years. To find the Modified Duration, you divide the Macaulay Duration by (1+Y/n). I found that most of the time that “duration” is mentioned by Fabozzi and Tuckerman, they are referring to modified duration. The CFA textbooks do not make a distinction, which makes it simple.
I’m still confused about example 7 on page 31. It’s now one year later. The duration of the liability has lessened. Why are we dialing the dollar duration back up to the initial value? Initially, the asset and the liability each had a duration approximating 5. Now the liability has a duration of 4, but it appears we’re increasing the asset to bring its duration back to 5. What am I missing here?
Example 7: “…Our requirement is to maintain the portfolio dollar duration at initial level…”. The initial level is $111,945 as per Example 6. Forget about ALM, think about this as the IPS dictating you to maintain that duration on your portfolio. Why is IPS saying so? May be the liability itself has a constant duration. May be the client himself is doing the ALM and assigned you a piece that requires maintaining bond portfolio with constant duration. The example just demonstrates how you increase your portfolio duration if you need to.
duration of your liability has lessened, but value HAS NOT. you are not increasing assets to bring DURATION back to 5 - but changing the amount of assets to meet the value of the liabilities. Your assets have fallen in value (in terms of dollar duration) so you need to dial it back up - so you can meet the liabilities when they become due.
janardhanc, I agree that resetting the DD to it’s initial level is a given. But the text at the bottom of p. 30 suggests that we’re looking at an ALM situation. So we are looking at liability matching.
(capitals is just for stress). again as I said - your dollar duration of your ASSET portfolio is REDUCING. since dollar duration = market value * duration * 0.01 so now you need more assets - to meet the liabilities. originally with a 111,965$ dollar duration on your portfolio - you met the liabilities. even though it is 1 year later - the value of Dollar duration being smaller means you have fewer assets to meet your liabilities. If interest rates rose - the liabilities would be falling further in value, wouldn’t it… so your gap would increase. and your assets being bonds as well, would face the same shortfall.
I get the math, but not the reasoning behind it. Yields didn’t change all that much. FMV of assets decreased from $3,078k in example 6 to $3,030K in example 7. Presumably, yields changed slightly over the year. But now, the cash required at the end of example 7 suggests we increase those assets by $1,079k to $4,109k. At the beginning, we thought that roughly $3 million would settle the debt, but at the end it’s $4 million? What happens next year and the year after that? The economic rationale is not making sense to me.
rates are rising. so liabs are increasing too … that is the economic rationale. if rates keep rising - the liabs would continue to increase …
you mean, decreases. Presumably if ALM is duration-matched, a change in interest rate should affect both sides the same.
Page 30: “In a number of ALM applications, the investor’s goal is to reestablish the dollar duration of a portfolio to a desired level. This rebalancing involves the following steps: …”. Moody, investor wants to maintain a desired level of duration. Why he wants to do that? I do not care. Example 7 just shows how you rebalance. As per economic sense, if yields move in such a direction that increases your portfolio duration, then you would sell off some bonds. So it could go either way. It is also not uncommon to have growing liabilities that demand increase in duration on bond portfolio. Ex: pension liabilities that grow as every additional year passes.
duration matching does not guarantee that the changes remain the same. Duration matching only works if parallel changes in rates happen. in case there is a twist in the yield curve, that goes out of the window. and yes I meant decreases. (sorry). see the example on pg 37-38 - when rates increase vs. when rates fall - the assets and liabs go up/down by differing amounts. Both the asset side and the liability side are composed of differing types of components (bonds) which do not necessarily react the same to change in interest rates. (especially if the other types of changes - twists, and change in spreads between treasury and non-treasury happen).
I’m with you on this except for one thing. What happens if the yield curve is completely flat and stays that way? You have a situation where the market value should not change, but both the duration and the DD decline as maturity draws nearer. The math suggests that you should be throwing buckets of cash into your bonds to restore the DD to it’s original level – for what reason? In a setting where time to maturity is 5 years, under normal circumstances, could duration ever increase over the course of a year?
Answer is YES. they state that in the chapter. Textbooks often illustrate immu- nization by assuming a one-time instantaneous change in the market yield. In actuality, the market yield will fluctuate over the investment horizon. As a result, the duration of the portfolio will change as the market yield changes. The duration will also change simply because of the passage of time. In any interest rate environment that is different from a flat term structure, the duration of a portfolio will change at a different rate from time. (Level III 2012 Volume 4 Fixed Income and Equity Portfolio Management, 5th Edition. Pearson Learning Solutions p. 29). they also state that a flat rate term structure is almost kinda impossible. According to the theory, if there is a change in interest rates that does not correspond to this shape-preserving shift, match- ing the duration to the investment horizon no longer assures immunization.21 Non-shape-preserving shifts are the commonly observed case. (Level III 2012 Volume 4 Fixed Income and Equity Portfolio Management, 5th Edition. Pearson Learning Solutions p. 33).
Well, look, the passage of time has to have a much greater impact on duration than a change in rates – at least when you’re only 5 years out. If rates did cause a significant change in duration, then the asset value would also change in the opposite direction such that the DD would remain relatively constant. But that’s not what happened. With the first bond in the example, both the value AND the duration declined, leading to a significant drop in DD. That drop in duration (from 5.025 to 4.246) was mostly due to the passage of time. Our time horizon has shortened on the liability side. So why do we need to ratchet this back up?
Moody, say I have liabilities to pay $x/year for eternity. So I spend $y to buy a bond/bond-portfolio that pays me $x/year. I receive $x end of every year and I pay off my liabilities. There comes a year boom my bonds mature and I receive $y back. I buy same bond/bonds back again and go on and on and on. The equation matches because you are saying rates are not changing. When rates are not changing duration is 0. No rebalancing is required. Other instance when I have no rebalancing will be when I can match duration of assets and liabilities exactly, because when rates change they just keep moving together. In this example the investor is asking to keep the duration constant. Rate changes could take my portfolio duration in either direction and may produce profit or loss on portfolio. To keep the duration constant I may have to either put cash in or take cash out. The movement is bidirectional, so I do not have to keep poring money in always to keep my duration.
janardhanc, mathematically speaking, how could the dollar duration ever increase from years 5 to 4 or 4 to 3 such you’d be pulling money out of the portfolio?
Duration of $x ==> portfolio value goes up $x for 100 points drop in rates and vice-versa. Duration 3 years ==> cumulative bond portfolio duration is equal to duration of 3 year bond, ie, portfolio value is changing like a simple 3 year bond If you want to buy a 10 year bond, you could either buy just one 10 year bond or buy a bond portfolio that has a duration equivalent to a 10 year bond, meaning a portfolio that matches 10 year sensitivity to rates. Now 5 years after, your bond duration is supposed to be 5 years. But it may not be so. Your 5 year bond with the old rate may be close to sensitivity to rates as 7 year bond. Your bond is expiring in 5 years but the duration/sensitivity to rate changes is equal to a 7 years bond sensitivity, your bond duration is 7 years. On the other hand bond portfolio may be behaving close to sensitivity to rates as a 3 year bond. Expiring in 5 years but its value is changing with rates as a 3 year bond, your bond duration is 3 years.