# AN ACTUAL FLOATER

On February 3, 2004, Citigroup Global Markets issued at par value a 17- year floating-rate note that pays 6-month LIBOR + 0.20%. Figure 7.2 displays its Bloomberg Description page. This is a Euro medium-term note (MTN) but is denominated in U.S. dollars. This floater resets the coupon rate semiannually and uses a 30/360 day-count. Most FRNs in the U.S. domestic issuance market reset quarterly, are tied to 3-month LIBOR, and accrue interest on an actual/360 basis.

Figure 7.3 presents the Bloomberg Yield and Spread Analysis (YAS) page for this floater for settlement on March 28, 2014. The (flat) price is

**FIGURE 7.1** Macaulay Duration of a Floating-Rate Note

93.08 per 100 of par value. The FRN is trading at a discount because the quoted margin of 20 basis points is less than the discount margin, shown to be 127.372125 basis points. The settlement date is 55 days into the 180-day semiannual period.

The coupon rate for the current period is 0.537%. This rate was set at the beginning of the period on February 3, 2014 (actually it is set two

**FIGURE 7.2** Bloomberg Description Page, Citigroup Global Markets Floating-Rate

Note

Used with permission of Bloomberg.com © 2014. All rights reserved.

business days before). Presumably, 6-month LIBOR was 0.337% – that plus the quoted margin of 0.20% determines the interest rate for the period. The interest payment due at the end of the semiannual period is 0.2685 per 100 of par value: 0.00537 * 180/360 * 100 = 0.2685. The accrued interest is 0.082042 (= 55/180 * 0.2685). That plus the flat price of 93.08 gives a full price of 93.162042 per 100 of par value. The Bloomberg YAS page shows those amounts in the Invoice section for $1 million in face (or par) value.

Before trying to interpret some of the other numbers reported in Figure 7.3, let's use the somewhat more complex valuation model on this actual floater. The inputs to equation 7.8 to get the estimated discount margin are: * QM =* 0.0020,

*93.162042,*

**MV=***55/180,*

**t/T=***0.2685,*

**INT=***100,*

**FV=***= 2, Z = 7. The remaining parameter is*

**PER***the interest rate per period for discounting the annuity based on the difference between QM and*

**y,***Fortunately, the Bloomberg page provides a quote for the Fixed Equivalent Yield at the bottom of the page corresponding to the maturity of the floater. That yield is determined by assuming the coupon rate on the floater is swapped to a synthetic fixed rate and then solving for the internal rate of return. It's*

**DM.****FIGURE 7.3 **Bloomberg Yield and Spread Analysis Page (YAS), Citigroup Global

Markets Floating-Rate Note, Settlement on March 28, 2014

Used with permission of Bloomberg.com © 2014. All rights reserved.

shown to be 3.745%. I'll presume it's on a semiannual bond basis so that * y* = 0.018725 (=3.745%/2).

The model gives a discount margin of 123.081 basis points.

The same inputs can be used in equation 7.9 to estimate the Macaulay duration for the floater.

This is the duration in terms of semiannual periods; annualized it is 0.136569 (= 0.273137/2). The annualized modified duration is 0.134058 (= 0.136569/1.018725). This is the * rate duration* statistic – it indicates the sensitivity of the market value to changes in benchmark interest rates, in particular, to the yield used to discount future cash flows. As expected, it is close to zero.

This FRN is trading at a discount, so its duration is less than the time until the next reset date. In Figure 7.1 in the lower panel, the duration is sliding down the 45-degree line and might even become negative. As an experiment, suppose that the flat price and the Fixed Equivalent Yield remain the same as the next coupon date on August 4, 2014, nears. For instance, on July 4 there will be just one month to go so that 1 – * t/T =* 1 – 150/180. The accrued interest goes up to 0.22375 (= 150/180 * 0.2685) per 100 of par value. The full price, that is, the market value, would be 93.30375.

Annualized, the modified duration would be -0.154175. Negative duration!

Another approach to getting the risk statistics is to use the approximation formulas, as in equations 7.2 and 7.3. But first it is useful to show that the estimated discount margin of 123.081 basis points is consistent with the current market value. To do that, the inputs are substituted into equation 7.7.

This is no surprise at all because equations 7.7 and 7.8 are the same, just rearranged algebraically. This is * MV(initial)* in the approximation formulas.

Now let's raise and lower the Fixed Equivalent Yield by one basis point, up from 3.745% to 3.755% and then down to 3.735%. That entails changing * y,* the yield per semiannual period, to 0.018775 and to 0.018675, holding the discount margin and the other variables constant.

Substitute these results into equation 7.2 for the approximate rate duration.

This is virtually the same number for modified duration that is obtained above starting with the closed-form formula for **MacDurFRN.**

Now we can use the same approach to estimate the credit duration. Raise and lower the discount margin by one basis point, up from 123.081 basis points to 124.081 and down to 122.081, holding all the other variables the same.

Substitute these into equation 7.3 for approximate credit duration.

The somewhat more complex valuation model estimates these risk statistics for the Citigroup Global Markets floater: a low modified duration of 0.134 with respect to changes in market interest rates and a much higher modified duration of 6.064 with respect to changes in credit risk. The estimated discount margin is 123.081 basis points, significantly above the quoted margin of 20 basis points because the FRN is trading at a discount below par value.

Now let's see how those results compare to those on the Bloomberg YAS page. In Figure 7.3 the estimated discount margin is 127.372125 basis points. Two modified duration numbers are shown, 0.325 for the next rate reset date on August 4,2014, and 6.343 for OAS duration. The former looks to be the sensitivity to interest rates (hence is a low number close to zero) and the latter the sensitivity to credit risk (hence a much higher number similar to a comparable maturity fixed rate bond).

Frankly, I'm disappointed with the differences between these reported results and those that I obtain. In the first edition of this book and in an earlier academic journal article where I first developed the FRN valuation model, my numbers and those reported on Bloomberg were very close and sometimes identical. I expect some variance for the discount margin because I use a single yield for discounting future cash flows and I understand that Bloomberg uses the forward curve. I have found that our DM results get closer when the yield curve gets flatter. Currently, the yield curve is rather steep, so the difference between 123 and 127 basis points is not unreasonable.

However, I cannot figure out how Bloomberg gets the reported modified duration statistics. I decided to try another settlement date for the same Citigroup Global Markets floater. Figure 7.4 shows the YAS page for settlement on April 17, 2014. The flat price is up a bit to 93.3 per 100 of par value, and the discount margin is down to 124.676965 basis points. The modified duration to the next reset date on August 4, 2014, is now 0.277. It appears to me that Bloomberg calculates this number as a small adjustment (perhaps in how days are counted) to the fraction of the year remaining in the coupon period. I suspect it will slide down toward zero as August 4 nears and then jump up after the coupon reset. It is as if its formula is the 1 – * t/T* term in equation 7.9 but does not include the terms that capture the effect of the bond being priced at a premium or discount.

The real shocker for me on this page is the OAS modified duration statistic of -0.107. How can it go from 6.343 on March 28 to -0.107 a few

**FIGURE 7.4** Bloomberg Yield and Spread Analysis Page (YAS), Citigroup Global

Markets Floating-Rate Note, Settlement on April 17, 2014

Used with permission of Bloomberg.com © 2014. All rights reserved.

weeks later? We know that duration can be negative and that it changes as time passes and yields change, but not like that! I thought it reflected credit duration and was calculated by bumping the discount margin and using the approximation formula. Part of the mystery can be explained by looking at the YASN Floater Analysis page shown in Figure 7.5. Look at the Stochastic Risk section. This reports some of the Greeks (delta, gamma, vega) usually associated with options. [I've been known to tell classes with a serious face that “vega” is the new Greek letter discovered on a submerged urn recently found by some option traders while scuba diving off an island in the Aegean Sea.] It also reports the OAS modified duration to be 6.3180 and the Market modified duration to be -0.1068. Somehow the definition of OAS duration gets reversed between the YAS and YASN pages as -0.1068 is rounded to -0.107.

The moral of the story is the importance of being able to calculate these risk statistics using a simple, transparent model and not always having to rely on what is a “black box” to the user. If I owned this Citigroup floater, I'd use my own rate and credit duration numbers.

**FIGURE 7.5** Bloomberg Floater Analysis Page (YASN), Citigroup Global Markets Floating-Rate Note, Settlement on April 17, 2014 Used with permission of Bloomberg.com © 2014. All rights reserved.