Valuing an Existing Swap from Either Party's Perspective

Why an Existing Swap Has Value

A plain-vanilla interest rate swap has no initial exchange of notional principal, but its value can change after the contract begins. If market interest rates move, the fixed payments already specified in the contract may become more or less attractive than the floating payments implied by current market rates. The swap’s value is the present value of the cash flows a party expects to receive minus the present value of the cash flows that party must pay.

This tutorial values a swap immediately after a payment and reset date, so there is no accrued payment for a period already in progress. Assume a standard single-curve framework: the same discount factors are used to discount cash flows and infer the floating forward rates, the floating leg has no spread, and there is no credit, collateral, or funding adjustment. The notional is not exchanged. These assumptions matter: the formulas below do not directly value an off-reset-date swap or a swap with different projection and discount curves.

Definition: At a valuation date immediately after a reset, the value to the fixed-rate payer is the present value of the floating payments it receives minus the present value of the fixed payments it pays. The fixed-rate receiver has the opposite value. A positive value is an asset to the party being valued; a negative value is a liability.

Valuing the Two Legs

Let $N$ be the notional, $K$ the annual fixed rate, and $P_j$ the discount factor from the valuation date to the $j$th remaining annual payment date. Set $P_0=1$. For $n$ remaining annual periods, the fixed payment is $NK$ at each date. Its present value is

PV_{\text{fixed}}=NK\sum_{j=1}^{n}P_j.

The floating leg is valued differently. At the reset date, under the stated assumptions, it is equivalent to a floating-rate note that is worth par immediately after reset and matures with a final principal repayment. The swap does not exchange that principal; the note is a valuation device used to calculate the floating coupons’ present value. The result is

PV_{\text{floating}}=N(1-P_n).

Thus, for the fixed-rate payer and receiver, respectively,

V_{\text{payer}}=N(1-P_n)-NK\sum_{j=1}^{n}P_j, \qquad V_{\text{receiver}}=NK\sum_{j=1}^{n}P_j-N(1-P_n) =-V_{\text{payer}}.

These expressions value the remaining swap cash flows from the specified valuation date. They do not include an exchange of notional, and they are not the amount of the next net settlement.

Why the Floating-Leg Formula Holds

The floating-leg identity follows from the relationship between discount factors and annual forward rates. In the single-curve model, the forward rate $f_j$ for the period from date $j-1$ to date $j$ satisfies

1+f_j=\frac{P_{j-1}}{P_j}, \qquad f_jP_j=P_{j-1}-P_j.

The floating coupon at date $j$ is $Nf_j$, so its present value is $Nf_jP_j=N(P_{j-1}-P_j)$. Summing the coupon values gives a telescoping sum:

PV_{\text{floating}} =N\sum_{j=1}^{n}(P_{j-1}-P_j) =N(P_0-P_n) =N(1-P_n).

The intermediate discount factors cancel, leaving only the initial value $P_0=1$ and the final discount factor. This derivation requires a reset-date valuation, annual accrual periods, and forward rates consistent with the same curve used for discounting. It should not be applied unchanged to a valuation between reset dates, when the next floating coupon is already partly determined.

Worked Example: A Fixed-Rate Payer With a Positive Value

A company pays a fixed rate of 4.5% and receives floating on a three-year swap with notional $N=\$1{,}000{,}000$. The valuation date is immediately after a reset and payment. Annual discount factors for the remaining payment dates are $P_1=0.96$, $P_2=0.91$, and $P_3=0.85$. Find the swap’s value to each party.

Method. Calculate each leg’s present value, then subtract the fixed-leg value from the floating-leg value for the payer. The common trap is to report only the next net payment rather than the value of all remaining payments.

Floating leg. The final discount factor is $P_3=0.85$, so

PV_{\text{floating}} =\$1{,}000{,}000(1-0.85) =\$150{,}000.

Fixed leg. The discount factors sum to $0.96+0.91+0.85=2.72$. Each fixed payment is $\$1{,}000{,}000(0.045)=\$45{,}000$, giving

PV_{\text{fixed}} =\$45{,}000(2.72) =\$122{,}400.

Value to the fixed-rate payer.

V_{\text{payer}} =\$150{,}000-\$122{,}400 =\$27{,}600.

The payer has a positive value of $\$27{,}600$; the receiver’s value is $-\$27{,}600$. The check is that the two parties’ values sum to zero, as their contractual cash flows are exact opposites.

Use the Sign Convention Carefully

A fixed-rate payer receives floating and pays fixed. Its value is therefore $PV_{\text{floating}}-PV_{\text{fixed}}$. A fixed-rate receiver receives fixed and pays floating, so its value is $PV_{\text{fixed}}-PV_{\text{floating}}$. If the fixed rate in an existing contract is above the current market par swap rate, the fixed payments are relatively costly to the payer, which tends to make the payer’s value negative. If the contract rate is below the market par rate, the payer’s value tends to be positive.

The market par swap rate is the fixed rate that would make the two leg values equal at the valuation date. Under the assumptions used here,

K_{\text{par}}=\frac{1-P_n}{\sum_{j=1}^{n}P_j}.

This provides a useful sign check. A fixed-rate payer’s value can be written as

V_{\text{payer}} =N\left(\sum_{j=1}^{n}P_j\right)(K_{\text{par}}-K).

Since $N$ and the sum of discount factors are positive, the payer’s value is positive when $K<K_{\text{par}}$, zero when $K=K_{\text{par}}$, and negative when $K>K_{\text{par}}$. The receiver has the reverse sign.

Worked Example: A Fixed-Rate Payer With a Negative Value

An investor pays fixed at 6.2% and receives floating on a four-year swap with notional $N=\$2{,}000{,}000$. Immediately after a reset, the annual discount factors are $0.98$, $0.94$, $0.89$, and $0.83$. Calculate the value to the payer and receiver, and use the market par rate as a check.

Method. Value the floating leg and the four fixed payments separately. Do not compare the 6.2% contract rate with a single spot rate: the par rate depends on all remaining payment dates.

Floating leg.

PV_{\text{floating}} =\$2{,}000{,}000(1-0.83) =\$340{,}000.

Fixed leg. The fixed payment each year is $\$2{,}000{,}000(0.062)=\$124{,}000$. The discount factors sum to $0.98+0.94+0.89+0.83=3.64$, so

PV_{\text{fixed}} =\$124{,}000(3.64) =\$451{,}360.

Payer’s and receiver’s values.

V_{\text{payer}} =\$340{,}000-\$451{,}360 =-\$111{,}360, \qquad V_{\text{receiver}}=\$111{,}360.

Check. The market par rate is

K_{\text{par}} =\frac{1-0.83}{3.64} =\frac{0.17}{3.64} \approx 0.0467033 =4.6703\%.

The contract rate of 6.2% exceeds the par rate, consistent with a negative value to the fixed-rate payer. The parties’ values are equal in magnitude and opposite in sign.

Worked Example: Interpreting a Receiver’s Value

A pension fund receives fixed at 5.5% and pays floating on a three-year swap with notional $N=\$750{,}000$. At a reset date, the discount factors are $P_1=0.97$, $P_2=0.92$, and $P_3=0.86$. Find the value to the fund and explain what the sign means.

Method. First compute the payer’s value using floating minus fixed, then reverse the sign for the receiver. This avoids accidentally assigning the payer’s sign to the party receiving fixed.

The floating-leg value is

PV_{\text{floating}} =\$750{,}000(1-0.86) =\$105{,}000.

The fixed payment each year is $\$750{,}000(0.055)=\$41{,}250$. The discount factors sum to $0.97+0.92+0.86=2.75$, so

PV_{\text{fixed}} =\$41{,}250(2.75) =\$113{,}437.50.

The fixed-rate payer’s value would be

V_{\text{payer}} =\$105{,}000-\$113{,}437.50 =-\$8{,}437.50.

Therefore the pension fund, which receives fixed, has value

V_{\text{receiver}}=\$8{,}437.50.

The fund’s positive value means that, under this model and at this valuation date, the remaining contractual cash flows are worth $\$8{,}437.50$ more to it than the opposite leg it must pay. The par rate is $0.14/2.75\approx 5.0909\%$, below the contract rate of 5.5%, which is consistent with the receiver having a positive value.

Common Mistakes and Practical Interpretation

Reversing the legs. For the fixed-rate payer, subtract the fixed-leg present value from the floating-leg present value. The receiver’s value is the negative of that result.
Using only the final discount factor for both legs. The floating-leg identity uses $P_n$, but the fixed leg requires the sum of every remaining discount factor.
Exchanging notional in the valuation. Notional is used to calculate contractual interest payments; it is not exchanged in a standard interest rate swap.
Calling the swap value the next net payment. The value is the present value of all remaining cash flows. The next settlement is only one cash flow and may not indicate whether the whole contract is an asset or liability.
Applying the reset-date formula between resets. Between reset dates, the next floating payment may already be partly determined. The simple $N(1-P_n)$ expression in this tutorial assumes an immediate post-reset valuation under the stated model.
Rounding inconsistently. If a displayed discount factor is rounded, arithmetic based on that displayed factor must match the displayed result. Keep factors exact where possible, or show that a result uses unrounded factors.

For a party considering termination or replacement, the sign is an economic guide: a positive value is an asset to that party, while a negative value is a liability. Actual close-out amounts may differ if the contract specifies other valuation conventions or if credit and collateral adjustments apply. The basic two-leg comparison remains the central calculation, aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume a single-curve model and valuation immediately after a reset and payment. All periods are annual and there are no spreads or credit adjustments.

A fixed-rate payer has notional $\$500{,}000$, contract rate 4%, and two remaining annual payments. The discount factors are $0.95$ and $0.90$. Find the value to the payer and receiver. (Answer: payer $+\$13{,}000$; receiver $-\$13{,}000$.)
A company pays fixed at 5.5% on notional $\$1{,}200{,}000$, with three annual payments remaining and discount factors $0.97$, $0.93$, and $0.88$. Calculate both parties’ values and check the payer’s sign against the par rate. (Answer: payer $-\$39{,}480$; receiver $+\$39{,}480$; par rate $0.12/2.78\approx4.3165\%$.)
A fund receives fixed at 4% on notional $\$800{,}000$. Three payments remain, with discount factors $0.96$, $0.91$, and $0.84$. Calculate the floating-leg and fixed-leg values, then find the fund’s value. (Answer: floating leg $\$128{,}000$; fixed leg $\$86{,}720$; receiver value $-\$41{,}280$.)
A proposed swap has three annual payments remaining and discount factors $0.98$, $0.95$, and $0.90$. Determine the fair fixed rate that gives the swap zero value at the reset date, and state the value to a fixed-rate payer if the contract rate is exactly that rate. (Answer: $K_{\text{par}}=0.10/2.83\approx3.5336\%$; payer value $0$.)
A fixed-rate payer has notional $\$2{,}000{,}000$, contract rate 4.8%, and three remaining annual payments. The discount factors are $0.99$, $0.94$, and $0.87$. Find both party values and use the par rate to check the direction of the sign. (Answer: payer $-\$8{,}800$; receiver $+\$8{,}800$; par rate $0.13/2.80\approx4.6429\%$.)

What You'll Learn