Deriving the Fair Fixed Rate of a Plain-Vanilla Swap

Why a Swap Has a Fair Fixed Rate

A plain-vanilla interest rate swap exchanges a stream of floating-rate payments for a stream of fixed-rate payments on a stated notional principal. The notional is used to calculate the payments; it is not ordinarily exchanged. At the start of a new swap, the fair fixed rate, also called the par swap rate, is the rate that makes the two legs equal in present value. Thus, neither party pays an upfront amount to enter the swap under the model used here.

The key calculation is to discount each leg’s cash flows to the valuation date. The fixed leg is a level stream whose payment amounts depend on the fixed rate. The floating leg’s present value can be expressed directly using the discount factors at the payment dates. Equating these values gives the fair fixed rate.

Model and assumptions: Value a new swap at time 0. The notional principal $N$ is constant, there are no upfront payments, and the fixed and floating legs have the same payment dates and accrual periods. Assume a single, arbitrage-free discount curve is also used to determine the floating rates, no credit or transaction costs, and no payment delays. The floating rate for each period is set at that period’s start and paid at its end. These assumptions make the floating-leg formula below apply; with different forecasting and discounting curves, or a spread or other adjustment in the floating leg, the formula must be modified.

Discount Factors, Accrual Fractions, and the Two Legs

Let the payment dates be $t_1,t_2,\ldots,t_n$, with $t_0=0$. Write $D_j$ for the time-0 discount factor to date $t_j$, so $D_0=1$. Let $\alpha_j$ be the year fraction from $t_{j-1}$ to $t_j$, determined by the contract’s day-count convention. If the fixed annual rate is $K$, its payment at $t_j$ is $N K\alpha_j$. The present value of the fixed leg is therefore

PV_{\mathrm{fixed}}=NK\sum_{j=1}^{n}\alpha_jD_j.

The annual rate $K$ is multiplied by the accrual fraction: for a half-year period, for example, the fixed coupon is $NK/2$, not $NK$. The discount factor $D_j$ applies to the payment at $t_j$, not to the start of that period.

To express the floating leg, let $L_j$ be the annualized floating rate set at $t_{j-1}$ for the period ending at $t_j$. Under the single-curve assumptions, the simple-compounding forward rate implied by the discount factors satisfies

1+\alpha_jL_j=\frac{D_{j-1}}{D_j}.

The floating coupon paid at $t_j$ is $N\alpha_jL_j$. Its time-0 present value is $N\alpha_jL_jD_j$. Substituting the discount-factor relation gives a useful expression for the whole leg.

Theorem: Under the stated assumptions, the time-0 present value of the floating coupons is $N(D_0-D_n)=N(1-D_n)$. Therefore the fair fixed rate is

K^*=\frac{D_0-D_n}{\displaystyle\sum_{j=1}^{n}\alpha_jD_j} =\frac{1-D_n}{\displaystyle\sum_{j=1}^{n}\alpha_jD_j}.

Why the Floating-Leg Formula Holds

For each period, rearrange the relation between the forward rate and discount factors:

\alpha_jL_jD_j =\left(\frac{D_{j-1}}{D_j}-1\right)D_j =D_{j-1}-D_j.

Multiplying by $N$ gives the present value of that period’s floating coupon. Summing across all periods makes the intermediate discount factors cancel:

PV_{\mathrm{floating}} =N\sum_{j=1}^{n}(D_{j-1}-D_j) =N[(D_0-D_1)+(D_1-D_2)+\cdots+(D_{n-1}-D_n)] =N(D_0-D_n).

At the par rate, the fixed and floating legs have equal present values. Set $NK\sum_{j=1}^{n}\alpha_jD_j=N(D_0-D_n)$ and cancel the positive notional $N$. Solving for $K$ produces the stated formula. The notional affects each leg’s dollar value but not the fair rate.

Worked Example: Annual Payments and a Three-Year Swap

A new three-year swap has notional $N=\$2{,}000{,}000$ and annual payments, so each accrual fraction is $\alpha_j=1$. The discount factors at years 1, 2, and 3 are $D_1=0.97$, $D_2=0.94$, and $D_3=0.90$. Find the fair fixed rate and verify the equality of the legs.

Method. Use the payment-date discount factors in the par-rate formula. The accrual-weighted fixed-leg factor is

\sum_{j=1}^{3}\alpha_jD_j =(1)(0.97)+(1)(0.94)+(1)(0.90) =2.81.

The floating-leg present value is

PV_{\mathrm{floating}} =\$2{,}000{,}000(1-0.90) =\$2{,}000{,}000(0.10) =\$200{,}000.

Thus the fair annual fixed rate is

K^*=\frac{1-0.90}{2.81} =\frac{0.10}{2.81} =0.035587\ldots

Rounded to four decimal places as a rate, $K^*=3.5587\%$. At the unrounded rate, the fixed-leg present value is $\$2{,}000{,}000(0.035587\ldots)(2.81)=\$200{,}000$, equal to the floating-leg present value. Common mistake avoided: the final discount factor $D_3$ belongs in the floating-leg numerator, while all three payment-date discount factors belong in the fixed-leg denominator.

Applying the Formula at Different Payment Frequencies

The formula does not require equal payment intervals. It requires the correct accrual fraction for every interval and the discount factor for every payment date. For semiannual payments with a half-year accrual convention, use $\alpha_j=0.5$; for irregular periods, use the contract’s stated year fractions. The rate $K^*$ remains an annualized fixed rate, while each coupon is scaled by its own accrual fraction.

Worked Example: Semiannual Payments over Two Years

A new swap has notional $N=\$5{,}000{,}000$, four semiannual payment periods, and discount factors $D_1=0.985$, $D_2=0.968$, $D_3=0.949$, and $D_4=0.927$. Each accrual fraction is $0.5$. Find the fair fixed rate and the amount of each fixed coupon, rounded to the nearest cent.

First compute the floating-leg present value:

PV_{\mathrm{floating}} =\$5{,}000{,}000(1-0.927) =\$5{,}000{,}000(0.073) =\$365{,}000.

Next compute the fixed-leg factor:

\sum_{j=1}^{4}\alpha_jD_j =0.5(0.985+0.968+0.949+0.927) =0.5(3.829) =1.9145.

The fair annual fixed rate is

K^*=\frac{0.073}{1.9145} =0.03813006\ldots =3.8130\%\quad\text{to four decimal places}.

Each fixed coupon is the annual rate times the half-year accrual fraction times the notional:

\$5{,}000{,}000(0.03813006\ldots)(0.5) =\$95{,}325.15\quad\text{to the nearest cent}.

Using the unrounded rate, the fixed-leg present value is

\$5{,}000{,}000K^*(1.9145) =\$5{,}000{,}000(0.073) =\$365{,}000.

This matches the floating-leg present value. Common mistake avoided: using a full-year coupon $NK^*$ for each half-year period would double the fixed payments and would not represent the stated annualized rate.

Worked Example: Irregular Accrual Periods

A new swap has notional $N=\$1{,}200{,}000$, with three payment dates whose accrual fractions are $0.25$, $0.50$, and $0.25$. The discount factors at those dates are $0.99$, $0.96$, and $0.94$, respectively. Find the fair annual fixed rate and the three fixed coupons, to the nearest cent.

The floating-leg present value is

PV_{\mathrm{floating}} =\$1{,}200{,}000(1-0.94) =\$1{,}200{,}000(0.06) =\$72{,}000.

Weight each discount factor by its own accrual fraction:

\sum_{j=1}^{3}\alpha_jD_j =(0.25)(0.99)+(0.50)(0.96)+(0.25)(0.94) =0.2475+0.4800+0.2350 =0.9625.

Therefore,

K^*=\frac{0.06}{0.9625} =0.062337662\ldots =6.2338\%\quad\text{to four decimal places}.

The first and third coupons each use an accrual fraction of $0.25$:

\$1{,}200{,}000(0.062337662\ldots)(0.25) =\$18{,}701.30.

The middle coupon uses an accrual fraction of $0.50$:

\$1{,}200{,}000(0.062337662\ldots)(0.50) =\$37{,}402.60.

As a check, the present value of the fixed leg using the unrounded rate is $\$1{,}200{,}000(0.062337662\ldots)(0.9625)=\$72{,}000$, equal to the floating-leg present value. Common mistake avoided: replacing all three accrual fractions with a single assumed frequency would misstate the fixed-leg factor.

Rate Quotes, Checks, and Common Traps

The par-rate formula is useful because it separates the curve information from the notional. The numerator $1-D_n$ is determined by the final payment-date discount factor, while the denominator is the present-value factor for one unit of annual fixed rate, with each payment discounted at its own date. The rate is therefore not generally the simple average of the implied floating rates.

When using a rate quoted in percent, convert it to decimal form before calculating cash flows. For example, a rate of $4\%$ is $0.04$, and a half-year coupon on notional $N$ is $N(0.04)(0.5)$. Keep full precision in intermediate rate calculations and round only the final displayed rate or payment.

Using the wrong discount factor in the numerator. The numerator is $D_0-D_n=1-D_n$, not $1-D_1$, unless there is only one payment period.
Discounting at the start of a period. A coupon paid at $t_j$ is discounted using $D_j$.
Ignoring accrual fractions. Each fixed coupon is $NK\alpha_j$, so the denominator is $\sum \alpha_jD_j$, not simply $\sum D_j$ unless every $\alpha_j=1$.
Including an exchange of notional. A standard interest rate swap uses the notional to calculate interest payments; the notional itself is not added to either leg’s coupon present value.
Applying the single-curve result outside its assumptions. If the floating rate is forecast from a different curve than the one used to discount payments, derive the floating-leg value using those specified rates and discount factors instead.

For a new swap, the fair fixed rate is the rate that sets the initial value of either side to zero before transaction costs. If one party receives fixed and pays floating, its initial value is the fixed-leg present value minus the floating-leg present value; the other party has the opposite value. At the par rate, both are zero.

Check Your Understanding

For each problem, use the stated single-curve assumptions and retain full precision until the final answer.

A two-year swap pays annually on a notional of $\$4{,}000{,}000$. The payment-date discount factors are $0.98$ and $0.95$. Find the fair fixed rate and verify the floating-leg present value. (Answer: $K^*=2.5907\%$; floating-leg present value $=\$200{,}000$.)
A three-period swap has notional $\$3{,}000{,}000$, accrual fractions $0.5$ for each period, and discount factors $0.990$, $0.975$, and $0.955$. Find the fair annual fixed rate and the fixed coupon per period. (Answer: $K^*=3.0822\%$; each fixed coupon $=\$46{,}232.88$.)
A swap has accrual fractions $0.25$, $0.75$, and $0.50$, with payment-date discount factors $0.992$, $0.971$, and $0.945$. Calculate the par annual fixed rate. (Answer: $K^*=3.7964\%$.)
A two-payment annual swap has discount factors $0.97$ and $0.93$, notional $\$1{,}000{,}000$, and a quoted fixed rate of $4.00\%$. Calculate the par rate and the time-0 value to the party receiving fixed and paying floating at the quoted rate. (Answer: par rate $=3.6842\%$; receive-fixed value $=\$6{,}000$.)

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