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convexity hedging

Posted: August 10th, 2004, 12:39 pm
by mrblue
Hi, could anybody provide a valid explaination of the concept of convexity hedging? The topic is related to the fixed income world. Sometimes i got across this issue reading fixed income strategy paper and i barely undertood the explaination provided there. It seems that when there is a rally of the yield curve, some big fund dealing with asset backed securities act in order to modify the duration of their portfolio, so doing increasing the effect on yields of the previous rally. This extra effect should be taken in account by who is managing to hedge its portfolio before a change in the yield curve. But i didn't understand well the logic behind this extra effect. If not clear i can provid a deeper etract of the paper. Thank you very much

convexity hedging

Posted: August 10th, 2004, 1:59 pm
by DavidJN
Convexity is a second order effect which indicates how much first order effects (delta/duration) change when rates change. Most fixed income instruments exhibit convexity (futures excepted) but some have more convexity than others - a good example being MBS. If you are long MBS then you are also short options (and short convexity) because the mortgages underlying the MBS pools have embedded prepayment options. Options are, relatively speaking, highly convex instruments (unless they are deep in or out-of-the-money).Suppose you are long MBS (and thus short options) and delta hedge this by shorting Treasuries. Suppose interest rates rise unexpectedly and mortgage borrowers exercise their prepayment options at a lower than expected rate. The duration of your MBS holdings thus rises and if you were delta hedged with vanilla bonds before the change, you are now underhedged and need to short more bonds to readjust your hedge back in line. Selling bonds in large quantities in a rising rate environment would tend to aggravate the situation.

convexity hedging

Posted: August 10th, 2004, 6:27 pm
by Gmike2000
Basically, it is good to own (be long) convexity. When you have MBS, you are short convexity (which is bad). So what you do to offset this is to buy some very positively convex instruments into your portfolio. This used to be done via US Treasuries (because of their reliable correlation to rate movements), but is now mostly done via swaps and swaptions. What happened last summer was that due to falling rates, loads of people prepaid their mortgages and banks had to buy a lot of convexity (e.g. by going long treasuries, swaps, etc), which in turn drove rates even further down, which caused even higher prepays, which caused even more convexity buying, etc. It was like a dog chasing its own tail. When rates came back up, banks laid off their convexity hedges, which drove rates higher, etc. The MBS market has grown so big that its influence on the curve (via convexity buying/selling) is likely going to persist in the future.

convexity hedging

Posted: August 11th, 2004, 6:37 am
by mrblue
ok thank you very much, the main topic framework is clearer now, but still i can't understand an aspect of this issue. If rates fall, many people prepay their mortage, for example: if i had subscribed a 30years mortgage and after 2 years rates fall, i can reschedule my (now 28 years to maturity) mortgage into a new one at a lower rate for the remaining 28 years. So at the end i still have a 28 years mortgage to pay (the same as in the case i don't exercise my option). And the same for many other people, so way to say that duration decreases? In the second case rates rush, why you said: "mortgage borrowers exercise their prepayment options at a lower than expected rate"? i expect that nobody would exercise its option, and also in this case duration remains the same. Could you please give me an insight view of how this mechanism works? thank you very much. Bye

convexity hedging

Posted: August 11th, 2004, 10:15 am
by Gmike2000
In this case, think about duration from the perspective of the bond owner (the fund manager for example), i.e. the one who receives mortgage payments. Think of it (as in all textbooks) as the avg time for you to get your money back. Don't think in terms of the guy who pays the mortgage.If people prepay you (i.e. the mortgage receiver/bond owner), then you get your money back earlier. Duration decreases. Of course, when you reinvest the prepaid money then you get duration back....but we are talking about single bonds here, not the whole portfolio.

convexity hedging

Posted: August 11th, 2004, 12:30 pm
by mrblue
ok, so if i own such kind of a bond if rates fall duration decreases and i'm unhappy, that is clear. Let's speak about what happens in the market overall. Fund with portfolio composed by this bonds will however issue new bonds backed by the new mortgages rescheduled at a lower rate, and that should balance the shortening duration effect without the need of buy new treasuries or swap in order to enhance duration of portfolio...but i know that the macro effect of exacerbating movements of rates is provoked by thise portfolio duration rebalancing. Of course i'm wrong but i can't understand where is my mistakeAnd on the contrary, why to state that if rates rally duration increase?People simply don't prepay the bonds and duration should remain the samethank you

convexity hedging

Posted: August 11th, 2004, 3:18 pm
by Gmike2000
You are confusing the various kinds of players in the market. I was talking about the holder of a mortgage bond who gets cash early when people prepay earlier than anticipated. Earlier cash means shorter duration. In the same manner, if people do not prepay then you get cash later. Later cash means higher duration (in your portfolio). If you manage a fixed income portfolio versus an index, the mortgages will take your duration in the opposite direction as your index when rates move. This will cause tracking error. You are forced to hedge that by buying convexity in the market. What happens on the issuer side is completely irrelevant (not really, because issuers hedge themselves too, but to me it does not really matter).

convexity hedging

Posted: August 12th, 2004, 6:33 am
by mrblue
ok i begin to understand... probably i should study better how index works in compairason to single bond. Still not very clear how duration of a bond could increase if rates rally (i thought it would just remain the same)...do you know some good paper about that? It's an interesting issue. I read somewhere that also non callable bond show negative convexity. This is actually a situation in which convexity does matter; normally i experienced working with govies traders that they usually don't really worry too much about that. thank you very much however

convexity hedging

Posted: August 19th, 2004, 6:13 am
by caroe
QuoteOriginally posted by: mrblue Still not very clear how duration of a bond could increase if rates rally (i thought it would just remain the same)...do you know some good paper about that? If duration remains constant, the price-rate curve would be a straight line (as duration is rate sensitivity with respect to parallel shifts of the rate curve). Obviously, this is not possible for mortgage bonds, since debtors have a prepayment option and bond holders receive the resulting index amortizing swap. Depending on borrower prepayment behaviour, bond prices may well be above par but tend to revert to at some point where (prepayment-adjusted) duration vanishes.

convexity hedging

Posted: August 19th, 2004, 12:27 pm
by mrblue
ok thanks. I know that for NORMAL bond the price - rate relationship is not linear, but i was wandering about MBS. But why should duration increase if rate increase? The opposite process is clear..probably i just didn't understand well how tracking error is generated in this case....

convexity hedging

Posted: August 19th, 2004, 4:17 pm
by riskguru
Think of the situation where you own a portfolio of mortgages. Given today's yield curve there is an 'expected prepayment' pattern embedded in the valuation of the portfolio. If rates fall unexpectedly, prepayments will increase overall, reducing the duration of the portfolio. If rates rise, prepayments decline (relative to what was originally expected), thereby extending out the duration of the portfolio.

convexity hedging

Posted: August 20th, 2004, 11:33 am
by SidK
Hi. On a related theme, suppose we enter into a Constant Maturity Swap for 2Y, which exchanges Libor for 10YCMS. What would the duration of this swap be? Would it be approx 10Y throughout the tenor of the swap?