Valuing Floaters at Par on Reset Dates

Introduction

In our last lesson, we introduced the Floating Rate Note (FRN), modeled its cash flow structure, and derived an essential insight: a "standard" floater (with no spread) is valued at its par value on any reset date. This result is a fundamental building block in fixed income, and today we will formalize it.

This lesson is dedicated to proving this principle mathematically from two distinct perspectives. Your background in rigorous mathematical thinking will be valuable here, as we will explore both a direct summation proof and a more conceptual no-arbitrage argument. Understanding why this rule holds underpins the valuation of nearly all floating-rate instruments.

The Par Valuation Principle

Let's formally state the principle we aim to prove:

On any coupon reset date, the value of a floating rate note whose coupon is set equal to the market reference rate for that period is exactly equal to its par value, provided the cash flows are discounted using a curve derived from that same reference rate.

We will now prove this using two different methods.

Proof 1: Direct Valuation via Telescoping Sum

This first proof follows directly from the cash flow decomposition we performed in the previous lesson. It's a straightforward approach that relies on the mathematical definition of forward rates.

Recap of the Derivation

Recall the structure:

The value of the floater is the sum of the present values of all its future cash flows (coupons + principal).
We use the forward rate curve to get the market's expectation for future floating coupon rates.
The present value of a single forecasted floating coupon paid at time $T_{i}$ was shown to be:
$P V (C F_{i}) = N \times (P (0, T_{i - 1}) - P (0, T_{i}))$
where $N$ is the notional and $P (0, T)$ is the discount factor (price of a zero-coupon bond) for maturity $T$ .
Summing the present values of all coupons from $i = 1$ to $M$ creates a telescoping series that simplifies to:
$P V (All Coupons) = N \times (P (0, T_{0}) - P (0, T_{M}))$
Since we are valuing the bond at time $t = T_{0}$ , the discount factor $P (0, T_{0})$ is 1. The value of the coupons is therefore $N (1 - P (0, T_{M}))$ .
Adding the present value of the principal repayment, $N \times P (0, T_{M})$ , gives the total price:
$Price = PV of Coupons N (1 - P (0, T_{M})) + PV of Principal N \times P (0, T_{M}) = N$

This completes the direct proof. A standard floater is worth par on its reset date.

For a clear visual and verbal walkthrough of this exact mathematical process, the following video is an excellent resource.

How to find the Value (and Duration) of a Floating Rate Note?

This video from finRGB provides a step-by-step derivation of a floating rate note's value.

Watch the segment from the valuation proof. The presenter sets up the summation of discounted cash flows and uses the definition of forward rates to simplify the expression into a telescoping sum, arriving at the same conclusion we just did: the value of the floater is its face value.

Proof 2: Valuation by Replication (No-Arbitrage)

A more elegant and intuitive way to prove the par valuation principle is through a no-arbitrage argument. If we can construct a trading strategy that perfectly replicates the cash flows of the floater, the cost of initiating that strategy must be equal to the price of the floater. Any difference would present an arbitrage opportunity.

The replicating strategy for a floater is a "roll-over" strategy using short-term bonds that are issued at par.

Consider a semi-annual floater with maturity $T$ . The following resource from NYU Stern outlines the replication argument.

Floating Rate Notes

These slides explain how to replicate a floater's cash flows using a dynamic trading strategy.

Start by reading the section titled "Replicating a T-Year Floater with 0.5-Year Par Bonds". It describes a strategy of buying a 6-month bond, and upon its maturity, using the proceeds to buy another 6-month bond, and so on. Then, continue to the next page, "A Semi-Annual-Coupon Floater is Equivalent to a 0.5-Year Par Bond". This section connects the dots and concludes why this equivalence means the floater must be worth par on its reset dates.

Let's break down the logic described in the presentation:

The Strategy: At time $t = 0$ (a reset date), you buy a 6-month bond with a face value of $100 for a price of $100 (i.e., a par bond). The coupon of this bond is set to the current 6-month market rate, let's call it $L_{0}$ .
First Cash Flow: At $t = 0.5$ years, this bond matures. You receive the principal ($100) plus the coupon payment ( $100 \times L_{0} \times 0.5$ ). This coupon is exactly the same as the first coupon you would have received from the floater.
The "Roll-over": You immediately use the $100 principal you just received to buy a new 6-month par bond. Its coupon will be set to the new 6-month market rate, $L_{0.5}$ . The cost is $100, and the cash you received was $100 + coupon$ . The net cash flow to you at this point is just the coupon, matching the floater's cash flow.
Continuing the Strategy: You repeat this process every six months. At each step, you receive a principal and coupon, use the principal to buy the next bond, and pocket the coupon. The coupon you pocket at each step $T_{i}$ is determined by the rate $L_{T_{i - 1}}$ , perfectly matching the cash flow stream of the floating rate note.
Final Step: At the floater's maturity date $T_{M}$ , the last 6-month bond matures, and you receive its principal ($100) and the final coupon. You do not roll over again. This final receipt of principal + coupon matches the floater's final cash flow.

The Conclusion (The No-Arbitrage Argument):
This dynamic strategy of rolling over short-term par bonds perfectly replicates every single cash flow of the floater. The initial cost to set up this strategy at time $t = 0$ was the price of the first 6-month par bond, which is $100. By the principle of no-arbitrage, the price of the floater today must be equal to the cost of its replicating portfolio. Therefore, the floater must be worth par.

Crucial Assumptions and Real-World Caveats

The par valuation principle is elegant but relies on a crucial assumption: the floating coupons are forecasted and discounted using the same, single yield curve.

In the pre-2008 financial world, it was common practice to use a single curve (e.g., a LIBOR-derived curve) for both forecasting LIBOR-based payments and for discounting them. However, the financial crisis revealed the embedded credit risk in rates like LIBOR, leading to the evolution of "multi-curve" valuation.

In a modern, collateralized derivatives market, the standard is to discount cash flows using a risk-free rate curve, such as one derived from Overnight Indexed Swaps (OIS) or SOFR. If you have a floater whose coupons are linked to a different index (which may contain a credit component), you would be forecasting with one curve and discounting with another.

In this multi-curve world, the par-value rule breaks down.

A Teaching Note on Pricing and Valuing Interest Rate Swaps

This document on swap valuation contains an excellent explanation of why the par-value assumption for floaters is challenged in modern valuation.

First, read the paragraph in the introduction on the first page that begins "An alternative approach is to interpret the interest rate swap...". Note the sentence: "In the traditional methodology for swap valuation, the implicit floater maintains its par value on rate-reset dates... With OIS discounting, the result that the implicit floating-rate bond paying LIBOR is priced at par value no longer holds." Then, skip to the conclusion on the second-to-last page and read the paragraph beginning "One important caveat arises...". This section explicitly states that when using OIS discounting, you can no longer assume the floater trades at par and must value it by explicitly projecting the cash flows and discounting them.

This is a key insight for anyone working in quantitative finance today. While the par valuation rule is the theoretical foundation, understanding its limitations is what separates academic exercises from practical, real-world finance. Your interest in technical deep-dives makes this distinction particularly relevant. For our course, we will largely operate in a "single-curve" world for simplicity, but it is vital to know this assumption is being made.

Conclusion

Today, we have rigorously established one of the most important principles in basic fixed-income analysis: a standard floating rate note is valued at par on its reset dates.

Key Takeaways:

This principle can be proven both by direct summation, where the present values of the forecasted coupons form a telescoping series, and by a no-arbitrage replication argument, where the floater's cash flows are duplicated by a strategy of rolling over short-term par bonds.
The direct proof is an exercise in mathematical substitution, leveraging the definition of forward rates.
The replication proof is a more conceptual argument rooted in the idea that two assets (or strategies) with identical cash flows must have the same price.
This principle holds under the critical assumption of a single-curve framework, where the same family of rates is used for both forecasting and discounting. In modern multi-curve frameworks (e.g., OIS discounting), this rule does not hold.

Having proven that a floater is worth par on a reset date, the natural next question is: what is it worth between reset dates? We will tackle this in our next lesson, where we will derive the pricing formula for a floater at any point in time.

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