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Remember from Chapter 3 that the reason why a fixed-rate bond trades at a premium or discount is that the coupon rate (what you are promised to receive from the issuer) is more or less than the yield to maturity (what you would need to pay par value). The same idea applies to a floating- rate note – the amount of the premium or discount is the present value of the difference between the fixed margin (the “quoted margin”) over the reference rate and the required margin (which, following market terminology, is called the discount margin) in order for the floater to trade at par value. The quoted margin is what you get; the discount margin is what you need.

A simple model to value a floating-rate note is expressed in equation 7.1.


Here MV = market value, Index = reference rate, QM = quoted margin, FV = future value, PER = periodicity, DM = discount margin, and N = number of periods to maturity. Index, QM, and DM are annual percentage rates. This is a simple model because: (1) MV is for a coupon reset date so that N is an integer (and there is no accrued interest), (2) it implicitly assumes a 30/360 day-count so that PER also is an integer; and (3) the same reference rate for Index is used for all future payments, implying that the yield curve is flat as is the forward curve. Despite these simplifications, the model can illustrate some interesting properties of FRNs.

Suppose that a 10-year floater that pays 3-month LIBOR + 0.50% quarterly is priced at 92 (percent of par value). Clearly, something has happened if the note was originally issued at par value. Probably that something is a credit rating downgrade – investors now require a higher spread over LIBOR than 50 basis points for this issuer. Assume that current 3-month LIBOR is 1.50%. We now can estimate the DM by substituting MV = 92, FV = 100, Index = 0.0150, QM = 0.0050, and N = 40 into equation 7.1.

Combining terms in the numerators, this becomes:

This is now a basic time-value-of-money problem where we can solve for the internal rate of return, y, as in equation 3.4 in Chapter 3.

The solution is that y is 0.7314%, or 0.007314. Therefore, DM = 0.014256.

Because of the credit downgrade, investors now require an estimated spread of 142.56 basis points over LIBOR on the floater that is only paying 50 basis points.

Floating-rate notes require that we think differently about duration as a measure of interest rate sensitivity. On a traditional fixed-income bond, the yield duration statistic estimates the price change following a change in the yield to maturity. The cause of that yield change doesn't matter – it could be a credit rating downgrade or an increase in expected inflation. Floaters change that thinking. Now we need to assess separately the impact of a change in market interest rates, as captured by the reference rate (i.e., Index), and a change in the required spread (i.e., DM). The former is the rate duration and the latter the credit duration. These are also called the index duration and the spread duration, respectively.

The simple model can be used to estimate the rate and credit durations for the 10-year floater paying 3-month LIBOR + 0.50%. To get them, we'll use the approximate modified duration formula from Chapter 6. We could also get the rate and credit convexities, but I'll focus here only on the first- order effects (feel free to have fun with the bond math on your own!). To get the rate duration for the floater, bump the level for Index up and down by five basis points, from 1.50% up to 1.55% and down to 1.45%. The result for MV(up) is 92.019557; MV(down) turns out to be 91.980781. MV (initial) is 92.

Notice the QM of 0.50% and the DM of 1.4256% are held constant as the market reference rate is raised. Also, using the same rate for Index for all future periods presumes a parallel shift to a flat yield curve. A key assumption implicit in the calculation of MV(up) and MV(down) is that the rate changes impact the next cash flow. Therefore, the shift occurs before the rate is set for the current time period.

We can now calculate the approximate rate duration for the floater, using a version of equation 6.18 for approximate annual modified duration.


Substituting into equation 7.2 obtains the perhaps surprising result that the rate duration is negative at -0.4215.

First note that this is a small number, close to zero. That is to be expected for a floating-rate note. By design and intent, the investor should be protected from changes in benchmark interest rates. Intuitively, this is because both the numerator and denominator go up and down together as the flat yield curve shifts up and down. But the negative number implies that this FRN has a value that is positively correlated to market rates. This floater appreciates in value, albeit by a small amount, in rising-rate bear markets when virtually all other debt securities are depreciating. An interesting phenomenon!

Negative duration arises because this floater is trading at a discount. The amount of the discount is the present value of an annuity – the difference between the quoted margin and the required (or discount) margin. If credit quality deteriorates further, the annuity becomes larger and the price of the floater falls, just like a fixed-rate bond. If there is no further change in credit quality or liquidity, the size of the annuity remains the same. Then a higher benchmark rate (i.e., Index) reduces the present value of the annuity and the amount of the discount goes down. So, the floater's price goes up when the benchmark rate goes up – hence, negative duration. A lower benchmark yield raises the present value of the annuity, increasing the size of the discount. The price falls when the yield falls – again, negative duration. It's just the opposite when the FRN is priced at a premium. The rate duration is positive but still close to zero.

Now let's hold the benchmark rate constant and bump the discount margin to measure the impact of a change in credit quality. Index remains at 1.50% while DM is raised and lowered by five basis points, up from 1.4256% to 1.4756% and down to 1.3756%.

The approximate credit duration for the floater follows from equation 7.2.


Substitute these results into equation 7.3 to get a credit duration of 8.9729.

This number should not be a surprise. From the perspective of credit risk, a 10-year floater represents the same risk as a 10-year fixed-rate bond. In the pricing equations, the numerators remain constant while the denominators go up and down. If the discount margin goes up by 1%, this floater will fall in value by about 9%, just like a fixed-rate bond having a modified duration of about 9.

The results of this section are all estimates based on a simple valuation model with simplifying assumptions. That is, the discount margin of 1.4256%, the rate duration of -0.4215, and the credit duration of 8.9729 are statistics conditional on the model on which they are based. Hence, there is model risk. If you change the assumptions to the model, you get different results. Next we'll relax some of those simplifying assumptions. This will allow for the derivation of closed-form equations for the discount margin and the rate duration.

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