The simple model in the previous section valued the FRN on a coupon reset date and used the current observation of the reference rate for all future rates. Now we want to consider valuation on a date between interest payments and to allow for some shape to the yield curve. In particular, future cash flows are discounted at a long-term rate, not at the current money market rate. Relaxing these assumptions makes the model somewhat more complex and realistic, but still some accommodations need to be made to keep it tractable. Where possible, the notation follows what has been used in earlier chapters.

Consider a floating-rate note that resets its interest rate PER times per year. As of the beginning of the current period (date 0), the floater has Z years to maturity, or a total of PER *Z periods. Let R0 denote the reference index rate set for date 0 and QM the quoted margin that is added to the reference rate to get the interest payment. Interest is paid in arrears at the end the period, T days later. The interest payment (INT) is (R0 + QM) * (T/Year) * EV, where Year is the assumed number of days in the year and EV is the face (or par) value on the note.

Assume that the floater is priced for settlement on some date t in the current period. The fraction t/T of the period has gone by, and 1 – t/T remains. Let DM represent the fixed margin (i.e., the discount margin) that is required by investors as of date t for the note to trade at its par value on date 1 at the end of the current period. The total value of the FRN on date 1 is the interest payment that is determined on date 0 plus the price of the note on that date. That price is the par value plus the present value of the annuity representing the difference between DM and QM. Tet y be the appropriate interest rate per period for discounting that annuity. As of date 1, there are PER *Z – 1 periods remaining until maturity.

The present value of the annuity as of date 1 is denoted PVANN and defined to be:

(7.4)

The annual amount of the surplus or deficient payment is (QM – DM) * FV. Dividing by PER to get the amount per period is a simplification because most FRNs use an actual/360 day-count. Therefore, the amount typically would vary slightly from period to period but in this case it is constant.

Equation 7.4 contains the sum of a finite geometric series and can be reduced to equation 7.5:

(7.5)

PVANN is negative when QM < DM, meaning the quoted margin is deficient, most likely due to a credit rating downgrade or a loss in liquidity. Investors require a higher margin over the reference rate in order to pay par value on date 1, so the FRN trades at a discount. Similarly, PVANN is positive when QM > DM, so the floater trades at a premium.

The market value of the floater on date t – that is, the full price (including accrued interest) – is denoted MV. It is the forthcoming interest payment plus the price of the FRN on date 1, both discounted back over the remainder of the period at the yield per period.

(7.6)

Here you can see that this still is a relatively simple model in that the same yield is used for all the discounted cash flows. You could argue that a better

discount rate in equation 7.6 would be a short-term money market rate, but then the model is even more complicated.

Combining equations 7.5 and 7.6 provides a general valuation formula for the floater, given the discount margin.

(7.7)

A formula for DM given the current market value is obtained by rearranging equation 7.7.

(7.8)

A payoff from building this more complex model is to obtain a general formula for the Macaulay duration (MacDurFRN) of the floating-rate note for date t in the coupon period, as shown in equation 7.9. The details to the derivation are in the Technical Appendix, but in brief entail using calculus and algebra on the first derivative of MV in equation 7.7 given a change in the yield per period.

(7.9)

We get to a numerical example of this equation in the next section, but for now let's examine the expression. Substitute equation 7.6 into equation 7.9 and rearrange the middle term.

(7.10)

Although not obvious, the third term in parenthesis is always positive – it's the Macaulay duration of a fixed-payment annuity maturing in PER*Z – 1

periods. Therefore, the sign of the Macaulay duration for the FRN depends on the middle term. If QM = DM, PVANN = 0 and MacDurFRN = 1 – t/T. That is the standard result for a risk-free floater having no change in credit risk, liquidity, or taxation – the Macaulay duration is just the fraction of the period remaining until the next reset date. When QM < DM and PVANN < 0, it's possible for MacDurFRN to be negative, especially for long-term FRNs (so that PER*Z is large) that are near a payment date (so that t/T nears 1). The corollary is that when QM > DM because of a credit rating upgrade, RVann > 0 and MacDurFRN is greater than 1 – t/T.

Figure 7.1 is my effort at illustrating this property of a floating-rate note. In the upper panel, QM = DM so that there has been no change in the credit risk since issuance and the floater resets at par value on the next coupon payment date. Just before LIBOR is fixed for the period, the duration is 0; immediately after the fixing, the duration jumps up to 1. Then as t goes from 0 to T (moving right to left), the duration slides down the 45-degree line. When the fraction of the period gone by is t/T, for instance when there are n + 1 – t/T periods remaining until maturity, the MacDurFRN is 1 – t/T. In the middle panel, QM > DM and the floater will trade at a premium above par value on the next rate reset date. The key observation is the duration is above 1 – t/T. In the lower panel, we see the possibility of negative duration when QM < DM. The FRN trades at a discount to par value. MacDurFRN is less than 1 – t/T and could be below zero.

It's easy to get the modified duration for the floater from MacDurFRN – just divide by one plus the yield per period. The key point is that these are rate duration statistics, not credit duration. They indicate the change in value following a change in the benchmark interest rate, not in the spread over that rate. To see these equations in action, it's time to look at an actual floating-rate note.

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