Floating Rate Instruments

Floating-rate instruments, such as floating-rate notes (FRNs) and most loans, differ from fixed-rate bonds in their periodic payment dynamics. Their interest payments fluctuate based on a reference interest rate, ensuring the borrower’s base rate remains aligned with prevailing market conditions. For lenders or investors, this means minimized price risk amidst interest rate volatilities. For instance, an FRN issued by a company will adjust its interest payments depending on a prevalent market reference rate (MRR), such as LIBOR. This mechanism ensures that even amidst unstable interest rates, the FRN’s price remains steady as its cash flows recalibrate with rate changes.

Market Reference Rate (MRR)

The MRR on an FRN or loan is usually a short-term money market rate. The reference rate is determined at the beginning of the period, and the interest payment is made at the end of the period. This payment structure is called in arrears. For example, if the MRR is based on the 3-month LIBOR, the interest payment for a period will be determined by the LIBOR rate at the start of the period. The most common day-count conventions for calculating accrued interest on floaters are actual/360 and actual/365. To this MRR, a specified spread, called the quoted margin, is either added or subtracted.

Quoted Margin

The quoted margin, a specified spread over or under the reference rate, compensates the investor for the differential in credit risk associated with the issuer relative to the implications of the reference rate. For example, consider Tesla. If it possesses a greater credit risk relative to a government treasury and issues an FRN, the associated quoted margin would surpass that of an FRN put forth by the government. This increased margin acts as a form of compensation, offsetting the increased risk that investors assume when opting for Tesla over a more secure government note.

Required Margin

The required margin (discount margin) is the yield spread over or under the reference rate such that the FRN is priced at par value on a rate reset date. It is determined by the market. Changes in the required margin usually come from changes in the issuer’s credit risk. Changes in liquidity or tax status can also affect the required margin. For example, if the issuer’s credit rating is downgraded, the required margin may increase as investors demand a higher return for the increased risk.

Relationship between the Required Margin and Floater’s Price at Reset Date

When the required margin is greater than the quoted margin, the floater tends to be priced at a discount. This phenomenon can often be traced back to changes in the issuer’s credit risk. Specifically, at the reset date, if the required margin overshadows the quoted margin, it results in the floater making what is termed a “deficient” interest payment. This deficiency, in turn, leads to the floater being priced below its par value, solidifying its position at a discount.

Conversely, when the required margin is exactly equal to the quoted margin, the floater is said to be priced at par. The rationale behind this is that the alignment between the required and quoted margins naturally drives the floater’s flat price towards its par value, especially as the next reset date comes into the horizon.

Lastly, in situations where the required margin is less than the quoted margin, the floater enjoys premium pricing. This is indicative of favorable market conditions for that particular floater.

Valuation of Floating-Rate Note

Valuing a floating-rate note (FRN) necessitates the use of a pricing model. For a fixed-rate bond, the price is determined based on a market discount rate, denoted as r, and a coupon payment per period, denoted as PMT. The formula to calculate this price is as follows:

\[PV = \frac{PMT}{(1 + r)} + \frac{PMT}{(1 + r)^{2}} + \ldots + \frac{PMT + FV}{(1 + r)^{N}}\]

Where:

• \(PV =\) present value, or the price of the bond
• PMT = Coupon payment per period
• \(r =\) the market discount rate
• FV = the future value paid at maturity, or the par value of the bond
• \(N =\) the number of evenly spaced periods to maturity

In the context of an FRN, PMT is derived from the Market Reference Rate (MRR) and the quoted margin. r is influenced by the MRR and the margin. The price of an FRN is calculated using the formula:

For an FRN, the price is calculated using the following formula:

\[PV = \frac{\frac{(MRR + QM) \times FV)\ }{m}}{\left( 1 + \frac{MRR + DM}{m} \right)} + \frac{\frac{(MRR + QM) \times FV)\ }{m}}{\left( 1 + \frac{MRR + DM}{m} \right)^{2}} + \ldots + \frac{\frac{(MRR + QM) \times FV)\ }{m} + FV}{\left( 1 + \frac{MRR + DM}{m} \right)^{N}}\]

Where:

• \(PV =\) present value, or the price of the floating-rate note
• MRR = the market reference rate, stated as an annual percentage rate
• \(QM =\) the quoted margin, stated as an annual percentage rate
• FV = the future value paid at maturity, or the par value of the bond
• \(m =\) the periodicity of the floating-rate note, the number of payment periods per year
• \(DM =\) the discount margin \(=\) required margin stated as an annual percentage rate
• \(N =\) the number of evenly spaced periods to maturity

Example: Calculating the Price of a Floating-Rate Note

Suppose we are pricing a three-year, semi-annual FRN that pays MRR plus \(0.75\%\). Assume MRR is \(1.50\%\) and the yield spread required by investors is 50bps. Given:

• MRR = 0.0150
• QM = 0.0075
• FV = 100
• m = 2
• DM = 0.0050
• N = 6

Calculate the price of the floating rate instrument.

Solution

Determine the coupon payment for each period:

\[PMT = \frac{MRR + QM}{m} \times FV\]

\[PMT = \frac{0.0150 + 0.0075}{2} \times 100\]

\[PMT = \frac{0.0225}{2} \times 100\]

\[PMT = 1.125\]

Calculate the present value of each cash flow: For \(i = 1\) to \(N\) :

\[PV_{i} = \frac{PMT}{\left( 1 + \frac{(MRR + DM)}{m} \right)^{i}}\]

E.g., for i=1

\[PV_{1} = \frac{1.125}{\left( 1 + \frac{0.0150 + 0.0050}{2} \right)^{1}} = \ 1.1139\]

Finally, the price of the FRN is the sum of all present values plus the present value of the face value at maturity:

\[\text{~Price~} = \sum_{i = 1}^{N}\mspace{2mu} PV_{i} + \frac{FV}{\left( 1 + \frac{MRR + DM}{m} \right)^{N}}\]

The price of the FRN has been calculated as: 100.724

Question

A floating-rate note has a quoted margin of +60bps and a required margin of +85bps. On its next reset date, the note is said to be priced at:

1. par
2. A discount
3. A premium

Solution

The correct answer is B.

When the required margin is greater than the quoted margin, the floater tends to be priced at a discount. This phenomenon can often be traced back to changes in the issuer’s credit risk. Specifically, at the reset date, if the required margin is greater than the quoted margin, it results in the floater making what is termed a “deficient” interest payment. This deficiency, in turn, leads to the floater being priced below its par value, solidifying its position at a discount.

A is incorrect: When the required margin is exactly equal to the quoted margin, the floater is said to be priced at par. The rationale behind this is that the alignment between the required and quoted margins naturally drives the floater’s flat price towards its par value, especially as the next reset date comes into the horizon.

C is incorrect:

When the required margin is less than the quoted margin, the floater is said to be priced at a premium. This is indicative of favorable market conditions for that particular floater.