# FRN's & negative duration

April 16, 2009 in

Floating rate notes (FRN's) can exhibit a curious property called negative duration.

An FRN is a bond whose coupon is not fixed, but rather changes each year (or each reset period, to be specific).  For example, the coupon might be quoted as L+100, meaning 100 bps above than Libor.   A familiar if infamous example is an adjustable rate mortgage.

Fixed-coupon bonds bear interest rate risk, because higher interest rates make their coupons and principal worth less due to heavier discounting.  Conversely, lower rates make the bond appreciate.  Since FRN coupons reset to higher interest rates, FRN's have no interest rate risk (actually, they have a little bit of risk, since coupons are set based on Libor at the beginning of the coupon period, but not paid until the end of the period).  In the absence of credit risk, they should always trade at or above par.

For a fixed coupon bond, the following relationship exists (this should be a familiar graph to anyone whose taken a basic finance course):

This, of course, is an illustration of bond duration.  As the discount rate increases, the bond's price falls.  In an extreme case, if rates went to infinity the bond would go to zero, as any value would be instantly discounted away.  If the bond lies on the blue line at the point where the red line is tangent, then the bond's duration, however you choose to define it (and there are many ways), is essentially the slope of the red line in the graph, or the slope of the price/yield curve at the point of tangency.  In other words, it represents the first derivative of the bond's price with respect to the discount rate, or the amount the bond's price drops as rates increase.  Strictly speaking, duration is a positive number, so it is actually -1 times that slope, and indeed almost all bonds exhibit this property with a negatively-sloped line, indicating positive duration.

FRN's are a little different, however.  Since they bear no interest rate risk, a plot like the one above for an FRN would display a horizontal blue line across from 100.

But what would it look like if the FRN were not trading at par?  This situation arises when there are concerns about the issuing company's credit quality - if the issuer were to go bankrupt tomorrow, then it doesn't matter that the bond has a nice coupon feature; it will be worthless.  Supposing we have such an FRN, trading at a discount to par.  If rates went to infinity, it would actually gain value, since the next coupon would be worth an immense amount.  Thus, as illustrated in this extreme case, discount FRN's exhibit negative duration - they increase in value when rates go up.

Negative duration is a very weird thing.  Most corporate finance classes teach duration as time (in years) when 50% of a bond's cashflows have been received (it just so happens that with some math you can show that this happens to be it's rate sensitivity as well).  To say that duration is negative is to imply, by that definition, that 50% of the cashflows were received in the past! I'm not going to worry about that paradox right now, however, except to leave it as evidence for how strange the concept of negative duration is.

But let's plow ahead and investigate it nonetheless.  First, some definitions.  For a given FRN, Let L be the Libor rate, C be the coupon spread (i.e. if the coupon is L+100 then C = 100) and N be the number of payment periods (i.e. N = 5 for an annual-pay 5 year bond).  Additionally, define S as the additional spread which, when added to Libor, is the rate at which the bond's cashflows are discounted (i.e. if S = 100 the cashflows are discounted at L+100) .  When S < C, the bond will trade at a premium (it's coupons more than make up for the discounting); when S > C the FRN will trade at a discount (the discounting overwhelms the coupon payments); and when S = C the bond will trade at par (the coupons offset the loss of value to discounting).

For those following along at home, what follows is based on a simple DCF model and is quite easy to implement in Excel.  Let N = 5, L = 1%, and C = 300 bps. If S = 400 bps, then the FRN (as expected) will trade at a discounted price of 95.67.   Raise L to 1.5% and sure enough, the bond will rise to 95.73: negative duration!  But why?  Start back with a par FRN. Like the discounted bond, the par FRN also has N = 5, L = 1%, and C = 300, but by necessity has S_p = 300. Now the only difference between these two bonds is that in each period, the cashflows of the discount FRN are discounted by an extra 100 bps. I can represent that difference as an N-period annuity with a 100 bp "coupon" discounted at the rate L+S . This is because both bonds have the same nominal cashflows; the only difference is that one bond's cashflows are being discounted more heavily.  The annuity represents that discounted difference, in this case 100 bps.  Now - and this is the important part - note that if I create a portfolio in which I purchase the par FRN and sell the annuity, I have recreated the cashflows of the discount FRN.

The par FRN, as demonstrated, has a duration close to zero.  The annuity has positive duration, like most bonds, since raising L reduces the value of the coupon stream. But because I've shorted the annuity, my portfolio winds up with a negative duration.  Et voila: all it takes is a little security deconstruction to show why discount FRN's exhibit negative duration.