Why a Floating-Rate Leg Is Worth Par at Reset

Why the Reset Date Matters

A floating-rate note resets its coupon periodically to a reference interest rate. Immediately after a reset, the coupon for the next accrual period has been set to the rate currently required for that period. Under standard no-arbitrage assumptions, this makes the note’s value equal to its notional, even though its future coupons are not all known yet.

The result is useful in swap valuation, but it must be applied carefully. A floating-rate note pays its notional back at maturity. A plain-vanilla swap generally exchanges interest payments only: its notional is used to calculate those payments but is not itself exchanged. Thus a note can be worth par at reset while the present value of a swap’s future floating coupons is less than par.

Definition: A floating-rate note with notional $N$ is a debt instrument whose interest payment for each accrual period is based on a specified floating reference rate. Immediately after a reset means just after the coupon for the preceding period has been paid and the rate for the next period has been set.

Assumptions Behind the Par Result

Use payment dates $t_0,t_1,\ldots,t_n$, where $t_0$ is the reset date and $t_n$ is maturity. Let $P(t_0,t_j)$ be the discount factor from $t_j$ to $t_0$, with $P(t_0,t_0)=1$. Let $\alpha_j$ be the year fraction for the accrual period from $t_{j-1}$ to $t_j$, and let $L_j$ be the floating rate applicable to that period.

The derivation below uses a single-curve, deterministic-rate model: the rates used to set the floating coupons are consistent with the discount factors used to value them. The note has no default risk, no payment delay, and no contractual spread above the reference rate. The coupon is paid at the end of its accrual period, and the note repays $N$ at maturity. Under these assumptions, the rate implied by the discount factors is $L_j=\big(P(t_0,t_{j-1})/P(t_0,t_j)-1\big)/\alpha_j$.

In more general market settings, the same par result requires that the floating index and discounting framework be consistent and that credit, liquidity, or other contractual spreads do not alter the value. A mismatch between the rate used to set coupons and the rate used for discounting can invalidate the simple result.

Deriving the Floating-Rate Note Value

For period $j$, the note pays a coupon $N\alpha_jL_j$ at $t_j$. Substitute the rate implied by the discount factors and discount that coupon back to $t_0$:

P(t_0,t_j)N\alpha_jL_j =N\left(P(t_0,t_{j-1})-P(t_0,t_j)\right).

Adding the present values of the coupons makes the intermediate discount factors cancel. The note also repays its notional at maturity, whose present value is $NP(t_0,t_n)$. Therefore:

\begin{aligned} \text{Note value at }t_0 &=N\sum_{j=1}^{n}\left(P(t_0,t_{j-1})-P(t_0,t_j)\right) +NP(t_0,t_n)\\ &=N\left(P(t_0,t_0)-P(t_0,t_n)\right)+NP(t_0,t_n)\\ &=N. \end{aligned}

The last equality uses $P(t_0,t_0)=1$. The principal repayment is essential: the floating coupons alone have value $N(1-P(t_0,t_n))$, not $N$.

Set the valuation time.
Use the instant immediately after a coupon payment and reset. The next period’s coupon rate has just been set.

Value each reference-rate coupon.
Under the stated single-curve assumptions, its present value is $N(P(t_0,t_{j-1})-P(t_0,t_j))$.

Add the maturity principal.
The principal has present value $NP(t_0,t_n)$, which completes the telescoping sum to $N$.

What the Result Means for a Swap Floating Leg

A swap’s floating leg consists of floating interest payments, usually calculated using a notional principal. It does not normally include a repayment of that notional. One way to identify its reset-date value is to compare it with the floating-rate note just derived: remove the note’s maturity principal payment. That principal has present value $NP(t_0,t_n)$, so the present value of the swap’s future floating coupons is

V_{\text{floating coupons at }t_0} =N-NP(t_0,t_n) =N\left(1-P(t_0,t_n)\right).

This formula assumes the same reset-date, payment, and curve conditions as the note derivation. It values the future floating coupons only; it does not add the swap notional as a cash flow. In a receive-floating position, this is the value of the floating side before accounting for the other side of the swap. The value of a complete swap also depends on the fixed-side cash flows and the position’s sign.

The formula also gives a useful reasonableness check. If maturity is farther away and the final discount factor is less than one, then $N(1-P(t_0,t_n))$ is less than $N$. Treating the swap’s floating leg as worth $N$ would wrongly include principal that the swap does not exchange.

Worked Example: A Floating-Rate Note at Reset

A hypothetical note has notional $N=\$1{,}000{,}000$, annual payments, and three years remaining. Immediately after a reset, the discount factors for the next three payment dates are $0.96$, $0.92$, and $0.88$. Assume the single-curve, deterministic-rate model described above, no credit risk, and no contractual spread. Find the note’s value by calculating its coupons and principal separately.

Method. Infer each annual reference rate from consecutive discount factors, discount each coupon, and include the maturity principal. This explicitly checks the difference between coupon value and note value.

For the first year, the implied rate is $1/0.96-1=0.0416667$, or $4.16667\%$. The coupon is $\$1{,}000{,}000(0.0416667)=\$41{,}666.67$, and its present value is $\$41{,}666.67(0.96)=\$40{,}000.00$.

For the second year, the implied rate is $0.96/0.92-1=0.0434783$, or $4.34783\%$. The coupon is $\$1{,}000{,}000(0.0434783)=\$43{,}478.26$, and its present value is $\$43{,}478.26(0.92)=\$40{,}000.00$.

For the third year, the implied rate is $0.92/0.88-1=0.0454545$, or $4.54545\%$. The coupon is $\$1{,}000{,}000(0.0454545)=\$45{,}454.55$, and its present value is $\$45{,}454.55(0.88)=\$40{,}000.00$.

The three coupons therefore have total present value $\$40{,}000+\$40{,}000+\$40{,}000=\$120{,}000$. The maturity principal has present value $\$1{,}000{,}000(0.88)=\$880{,}000$. The total note value is $\$120{,}000+\$880{,}000=\boxed{\$1{,}000{,}000.00}$, rounded to the nearest cent.

Check. The result equals the note’s $\$1{,}000{,}000$ notional. The coupons alone are worth $\$120{,}000$, so they are not themselves worth par.

Worked Example: The Floating Coupons on a Swap

A hypothetical receive-floating swap has notional $N=\$2{,}000{,}000$ and three annual floating payment dates remaining. Just after a reset and payment, the discount factor to the final date is $0.88$. Assume the floating index matches the discount curve, there is no spread, and no notional is exchanged. Find the present value of the remaining floating payments.

Method. Use the floating-rate note identity, but subtract the present value of the note’s maturity principal because the swap does not pay that principal.

The hypothetical floating-rate note with the same notional and dates would be worth $\$2{,}000{,}000$ at reset. Its maturity principal would have present value $\$2{,}000{,}000(0.88)=\$1{,}760{,}000$. Hence the floating coupons alone are worth $\$2{,}000{,}000-\$1{,}760{,}000=\boxed{\$240{,}000.00}$.

Equivalently, $N(1-P(t_0,t_n))=\$2{,}000{,}000(1-0.88)=\$240{,}000$. The answer is the value of the receive-floating cash flows at reset, not a value for the entire swap, which also has a fixed side.

Check. Since the final discount factor is below one, the coupon-only value is below the $\$2{,}000{,}000$ notional, as expected.

Worked Example: A Floating Note with a Contractual Spread

A hypothetical two-year note has notional $\$500{,}000$, annual coupons equal to the reference rate plus a contractual spread of $0.60\%$ per year, and discount factors $0.97$ and $0.93$ immediately after a reset. Assume annual accrual factors, no default risk, and a single curve for the reference rate and discounting. Find the note’s value.

Method. Separate each coupon into its reference-rate part and its spread part. The reference-rate coupons plus principal are worth par under the model; the spread payments add their own present value.

The reference rate for year one is $1/0.97-1=0.0309278$. Its coupon is $\$500{,}000(0.0309278)=\$15{,}463.92$, with present value $\$15{,}463.92(0.97)=\$15{,}000.00$.

The reference rate for year two is $0.97/0.93-1=0.0430108$. Its coupon is $\$500{,}000(0.0430108)=\$21{,}505.38$, with present value $\$21{,}505.38(0.93)=\$20{,}000.00$. Thus, reference-rate coupons have present value $\$35{,}000$, and principal has present value $\$500{,}000(0.93)=\$465{,}000$. Together, these amount to $\$500{,}000$.

The spread payment each year is $\$500{,}000(0.006)=\$3{,}000$. Its present value is $\$3{,}000(0.97)+\$3{,}000(0.93)=\$2{,}910+\$2{,}790=\$5{,}700$. The note is worth $\$500{,}000+\$5{,}700=\boxed{\$505{,}700.00}$, to the nearest cent.

Check. The note is above par because it pays a positive spread in addition to the reference rate. The spread’s present value is added once for each payment period.

Common Exam Trap: Adding Notional to a Swap Leg

A frequent error is to transfer the floating-rate note’s par value directly to a swap’s floating leg. For the swap in the second worked example, the incorrect calculation would be: “The floating side is worth par, so its value is $\$2{,}000{,}000$.” That calculation includes a notional repayment that the swap does not make.

The correction is to remove the note’s maturity principal. Here its present value is $\$1{,}760{,}000$, so the swap floating coupons are worth $\$2{,}000{,}000-\$1{,}760{,}000=\$240{,}000$, not $\$2{,}000{,}000$. Always check the contract’s actual cash flows before using the par result.

Key takeaway: Immediately after a reset, a standard floating-rate note is worth its notional because its floating coupons and maturity principal together telescope to par. A swap floating leg has no principal repayment, so its coupon-only value at reset is $N(1-P(t_0,t_n))$ under the same assumptions.

Check Your Understanding

For each problem, use the stated reset-date assumptions and distinguish note principal from swap cash flows.

A floating-rate note has notional $\$800{,}000$, two annual periods remaining, no spread, and discount factors $0.98$ and $0.95$. What is its reset-date value, and what is the value of the corresponding swap’s floating coupons? (Answer: Note value $\$800{,}000.00$; floating coupons $\$40{,}000.00$)
A hypothetical receive-floating swap has notional $\$1{,}200{,}000$, three annual periods remaining, and final-date discount factor $0.90$. Assume the index and discount curve match, and no principal is exchanged. Find the reset-date value of its floating coupons. (Answer: $\$120{,}000.00$)
A two-year floating-rate note has notional $\$400{,}000$, annual payments, a $0.25\%$ annual spread, and discount factors $0.98$ and $0.95$. Under the single-curve, no-credit assumptions, find its value immediately after reset. (Answer: $\$401{,}930.00$)
A no-spread receive-floating swap has notional $\$2{,}500{,}000$ and final-date discount factor $0.96$. It has several floating payment dates remaining, but no principal exchange. Find the value of its floating coupons immediately after reset. (Answer: $\$100{,}000.00$)
A hypothetical three-year note has notional $\$900{,}000$, annual coupons equal to the reference rate plus $0.40\%$, and discount factors $0.99$, $0.965$, and $0.94$ after reset. Assume no credit risk and consistent single-curve discounting. Find the note’s value, rounded to the nearest cent. (Answer: $\$910{,}422.00$)

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