Floating-Rate Note (FRN) Discount Margin Calculation

Suppose you had a five-year, USD 10,000 floating-rate bond priced at 98.82. The bond pays a semi-annual coupon of a 3-month market reference rate (MRR) plus 90 basis points. Assume the MRR is expected to be a stable 1.20% going forward, with simplified 180/360 calendar periods.

The full form of a simplified discount model across N periods for a floating rate note (FRN) looks a little overwhelming at first glance: $$\displaystyle PV = \frac{\frac{R+QM}{m} \times FV}{\left(1 + \frac{R+DM}{m}\right)} + \frac{\frac{R+QM}{m} \times FV}{\left(1 + \frac{R+DM}{m}\right)^2} + ... + \frac{\frac{R+QM}{m} \times FV+FV}{\left(1 + \frac{R+DM}{m}\right)^N}$$ where _PV_ is present value, _R_ is the reference rate, _QM_ is the quoted margin, _m_ is the periodicity of the bond, _FV_ is the face value, and _DM_ is the discount margin. As complicated as it looks, it's just estimated floating coupons and the face value discounted at an estimated floating rate. The discount margin is the piece of this discount rate which is provided by market forces, and therefore provides an important signal of issuer risk.

No, the coupon payment here is the correct annual amount, but this is a semiannual coupon bond, so the coupon cash flows are cut in half.

No, there must be a principal payment in the cash flow series as well.

That's right! The bond is a series of cash flows which are certainly not guaranteed to be level payments of USD 105, but an assumption of the constant 3-month MRR will lead you to the coupon payment of: $$\displaystyle Coupon = \frac{R + QM}{m} \times FV $$ where _R_ is the reference rate, _QM_ is the quoted margin, _m_ is the periodicity, and _FV_ is the face value of the bond.

Now with those assumed cash flows, a standard bond price is calculated based on a consistent discount rate: $$\displaystyle 9,882 = \frac{105}{(1+r)} + \frac{105}{(1+r)^2} + \frac{105}{(1+r)^3} + ... + \frac{10,105}{(1+r)^{10}} $$

How would you interpret the variable _r_ in this formulation?

No, this isn't a bad answer, but a yield to maturity measure is expressed as an annual rate, and this is not an annual rate.

Yes! And can you calculate the periodic discount rate for this bond?

If you wanted to calculate the discount margin used for this bond, you would probably want to start with setting up the expected cash flows for the bond. What are they?

No, it's possible to obtain this answer by finding the periodic discount rate for a series of just five coupons instead of 10. Make sure you have n=10 if using a financial calculator.

No, this is an annualized rate, but the periodic discount rate for this bond is lower.

Correct! Calculating the discount rate is straightforward once the other values are input; this is a periodic rate. [[calc: 10 n 9882 sign pv 105 pmt 10000 fv cpt iy , 10 n 9882 chs pv 105 pmt 10000 fv i]]

Now there's just one final step. Knowing the discount rate is 0.01176, you can go back to the general form of the FRN model and look at the form of the denominator: $$\displaystyle PV = \frac{\frac{R+QM}{m} \times FV}{\left(1 + \frac{R+DM}{m}\right)} + \frac{\frac{R+QM}{m} \times FV}{\left(1 + \frac{R+DM}{m}\right)^2} + ... + \frac{\frac{R+QM}{m} \times FV+FV}{\left(1 + \frac{R+DM}{m}\right)^N}$$ The discount rate _r_ is represented by: $$\displaystyle 1 + r = 1 + \frac{R+DM}{m} $$

What does the discount margin have to be?

No, this can't be correct. The bond is trading at a discount, and so the discount margin must be greater than the quoted margin of 90 basis points. It's possible to obtain this answer with a slight algebra error in simplifying.

No, it's possible this represents a small algebra error. Make sure you solve DM = 2r - R, using the notation from the example.

No, the bond is a floating-rate note, and so there isn't a set coupon rate. Even if there were, the coupon belongs in the numerator of these terms.

To summarize this discussion: [[summary]]

Yes! Since $$\displaystyle 1 + 0.01176 = 1 + \frac{0.012+DM}{2} $$ It must be the case that: $$\displaystyle DM = 2(0.01176) - 0.012 = 0.0115 $$ or 1.15%. This is qualitatively in the right area, since the bond is trading at a discount. You know the discount margin has to be greater than the quoted margin of 90 basis points. So based on this information, investors have decided that, for whatever reason, this bond owes them a 1.15% spread over the 3-month MRR instead of the quoted margin of just 0.90%.

The coupon rate of the bond

The six-month periodic bond discount rate

The bond's yield to maturity

1.176%

1.295%

2.352%

Ten payments of USD 105

Nine payments of USD 210, followed by USD 10,000

Nine payments of USD 105, followed by USD 10,105

0.56%

1.12%

1.15%

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