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Exam FM · Swaps and Derivative Hedging · Tutorial 876 of 998

Worked Examples in Plain-Vanilla Swap Pricing

Use payment accrual factors and discount factors to price both legs, solve for a par swap rate, and determine each party’s value.

Advanced 11 min read

What You'll Learn

  • Calculate fixed-leg present values using payment accrual factors and discount factors.
  • Derive the floating-leg value from a sequence of implied forward rates.
  • Solve for the par fixed rate when a swap begins at zero value.
  • Value an existing pay-fixed or receive-fixed swap from either party’s perspective.
  • Handle semiannual and irregular payment periods without confusing accrual fractions and discounting.
  • Check rate calculations, dollar values, and position signs for consistency.

Pricing a Plain-Vanilla Swap

A plain-vanilla interest rate swap exchanges fixed-rate interest payments for floating-rate interest payments on an agreed notional principal. In the standard structure considered here, the notional is not exchanged: it is used only to calculate the coupons. Pricing requires two present values, one for each leg. The difference between them is the value to one party, with the opposite value applying to the other party.

The calculations below use supplied discount factors and accrual fractions. The key is to keep the payment schedule, coupon amounts, discount factors, and perspective of the party consistent. This approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let \(N\) be the notional principal, \(K\) the annual fixed swap rate, \(\alpha_j\) the year fraction for payment period \(j\), and \(P_j\) the discount factor from the valuation date to payment date \(t_j\). The present value of the fixed leg is \(NK\sum_{j=1}^{n}\alpha_jP_j\). For a standard floating leg valued immediately after a reset date, under the single-curve assumptions described below, its value is \(N(1-P_n)\). The value to the fixed-rate receiver is fixed-leg value minus floating-leg value; the value to the fixed-rate payer is the negative of that amount.

Model and Pricing Assumptions

Use the following model before applying these formulas:

  • The swap has a constant notional \(N\), with no exchange of principal at the start or end.
  • At valuation, a floating-rate reset has just occurred. Each future floating coupon is set at the beginning of its accrual period and paid at its end.
  • The floating index and discounting use the same curve. In particular, the forward rates used to estimate floating coupons are consistent with the supplied discount factors.
  • The discount factors \(P_j\) are positive and apply to the corresponding payment dates. Each \(\alpha_j\) is the year fraction for the coupon’s own accrual period.
  • There are no default, funding, collateral, tax, or transaction-cost adjustments.

These assumptions matter. For example, if the valuation date is between reset dates, the next floating coupon may already be partly determined; the simple floating-leg formula \(N(1-P_n)\) should not be used without adjusting for that known coupon. Likewise, if the floating index and discount curve differ, the single-curve derivation below does not apply as written.

Why the Floating-Leg Formula Holds

Let \(P_0=1\), and let \(L_j\) be the forward rate for period \(j\), quoted as a simple annual rate over the accrual fraction \(\alpha_j\). Consistency with the discount factors gives \(\alpha_jL_j=P_{j-1}/P_j-1\). The expected floating coupon for that period is therefore \(N\alpha_jL_j\), paid at \(t_j\). Its present value is:

$$ N\alpha_jL_jP_j =N\left(\frac{P_{j-1}}{P_j}-1\right)P_j =N(P_{j-1}-P_j). $$

Summing over all periods makes the intermediate terms cancel:

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

This telescoping sum explains why, just after a reset, the floating leg’s value depends on the final discount factor rather than on a separate sum of all its projected coupons. No principal is paid in the swap; the formula is a present value of the floating coupons under the stated assumptions.

The Par Fixed Rate and Party Values

A newly initiated swap is at par when its value is zero. Set the fixed-leg value equal to the floating-leg value and cancel the common notional:

$$ NK\sum_{j=1}^{n}\alpha_jP_j=N(1-P_n) \qquad\Longrightarrow\qquad K_{\text{par}}= \frac{1-P_n}{\sum_{j=1}^{n}\alpha_jP_j}. $$

The denominator is the discounted value of one unit of annual fixed rate applied to the scheduled accrual periods. If an existing swap has fixed rate \(K\), define the fixed receiver’s value by \(V_{\text{rec}}\). Then:

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

A positive value to the fixed receiver means that the present value of the fixed coupons received exceeds the present value of floating coupons paid. The same swap has an equal and opposite value to the fixed payer.

Worked Pricing Examples

Worked Example: Find the Par Rate and Value an Existing Annual-Pay Swap

A hypothetical three-year swap has annual payments, notional \(N=\$2{,}000{,}000\), and discount factors \(P_1=0.96\), \(P_2=0.91\), and \(P_3=0.85\). The valuation date is immediately after a reset. Find the par fixed rate. Then find the value to a party receiving fixed at \(K=5.25\%\), and state the other party’s value.

Method. First calculate the discounted accrual sum, using \(\alpha_j=1\) for each annual period. Then use the par-rate formula and the signed value formula. The model assumptions are those stated above; no principal is exchanged.

The fixed-leg factor is:

$$ \sum_{j=1}^{3}\alpha_jP_j =1(0.96)+1(0.91)+1(0.85) =2.72. $$

The floating-leg value is:

$$ N(1-P_3) =\$2{,}000{,}000(1-0.85) =\$2{,}000{,}000(0.15) =\$300{,}000. $$

Thus the par fixed rate is:

$$ K_{\text{par}}=\frac{0.15}{2.72} =0.0551470588\ldots =5.515\%. $$

For the existing receive-fixed position, the fixed-leg value is:

$$ \$2{,}000{,}000(0.0525)(2.72) =\$2{,}000{,}000(0.1428) =\$285{,}600. $$

Therefore:

$$ V_{\text{rec}}=\$285{,}600-\$300{,}000=-\$14{,}400, \qquad V_{\text{payer}}=+\$14{,}400. $$

The fixed receiver’s contract rate is below the current par rate, so the negative value is consistent with the result. The party values sum to zero. Common mistake avoided: treating the par rate as the value of the swap. The par rate is a coupon rate; an existing swap’s value is the dollar difference between the two legs.

Worked Example: Semiannual Payments and Both Party Values

A hypothetical two-year swap has four semiannual payment periods, \(N=\$5{,}000{,}000\), and discount factors \(0.985,\ 0.967,\ 0.946,\ 0.922\). The swap is valued immediately after a reset. Find its par fixed rate. Then value an existing position in which the fixed receiver receives \(K=4.40\%\) per year.

Method. Apply \(\alpha_j=0.5\) to every coupon period, both when calculating the fixed-leg factor and when determining the fixed coupon amounts. The discount factors already correspond to the four payment dates.

First add the discount factors and apply the half-year accrual factor:

$$ \sum_{j=1}^{4}\alpha_jP_j =0.5(0.985+0.967+0.946+0.922) =0.5(3.820) =1.910. $$

The floating-leg value is:

$$ N(1-P_4) =\$5{,}000{,}000(1-0.922) =\$5{,}000{,}000(0.078) =\$390{,}000. $$

The par fixed rate, quoted as an annual rate, is:

$$ K_{\text{par}}=\frac{0.078}{1.910} =0.0408376963\ldots =4.0838\%. $$

For the existing swap, the fixed-leg value is:

$$ NK\sum_{j=1}^{4}\alpha_jP_j =\$5{,}000{,}000(0.044)(1.910) =\$5{,}000{,}000(0.08404) =\$420{,}200. $$

The fixed receiver’s value is \(\$420{,}200-\$390{,}000=+\$30{,}200\), and the fixed payer’s value is \(-\$30{,}200\). The result is positive for the receiver because the contract rate of 4.40% exceeds the current par rate of about 4.0838%. Common mistake avoided: using a full-year accrual for each semiannual coupon. Omitting the \(0.5\) factor would overstate the fixed-leg value and give an incorrect par rate.

Worked Example: Irregular Accrual Periods and a Below-Par Contract

A hypothetical swap has notional \(N=\$3{,}600{,}000\), three future payment dates, discount factors \(P_1=0.990,\ P_2=0.975,\ P_3=0.950\), and accrual fractions \(\alpha_1=0.25,\ \alpha_2=0.25,\ \alpha_3=0.50\). The valuation date is immediately after a reset. Find the par fixed rate and the value to a party receiving fixed at \(K=5.00\%\).

Method. Use each supplied accrual fraction with its corresponding discount factor. Although the periods have different lengths, the floating-leg formula still uses the final discount factor under the stated single-curve, post-reset assumptions.

The discounted accrual sum is:

$$ \sum_{j=1}^{3}\alpha_jP_j =(0.25)(0.990)+(0.25)(0.975)+(0.50)(0.950) =0.2475+0.24375+0.475 =0.96625. $$

The floating-leg value is:

$$ N(1-P_3) =\$3{,}600{,}000(1-0.950) =\$3{,}600{,}000(0.05) =\$180{,}000. $$

The par rate is:

$$ K_{\text{par}}=\frac{0.05}{0.96625} =0.051746\ldots =5.1746\%. $$

The fixed-leg value at the contract rate is:

$$ NK\sum_{j=1}^{3}\alpha_jP_j =\$3{,}600{,}000(0.05)(0.96625) =\$3{,}600{,}000(0.0483125) =\$173{,}925. $$

Thus the fixed receiver’s value is \(\$173{,}925-\$180{,}000=-\$6{,}075\). The fixed payer’s value is \(+\$6{,}075\). As a check, the contract rate is below the par rate, so a fixed receiver should have negative value. Common mistake avoided: replacing the irregular accrual fractions with equal periods. Each fixed coupon uses its own accrual fraction; the supplied final discount factor determines the floating-leg value here.

Checks and Common Exam Traps

A reliable calculation keeps the rate, accrual period, and discount factor attached to the same payment date. Before finalizing a result, check that the fixed-leg factor is \(\sum \alpha_jP_j\), not simply \(\sum P_j\), unless every accrual fraction is one. For an existing contract, also check the perspective: receiving fixed means fixed-leg value minus floating-leg value; paying fixed reverses the sign.

For instance, in the semiannual example, an incorrect calculation might set the fixed-leg factor to \(3.820\) rather than \(1.910\). That error implicitly treats each half-year coupon as if it accrued for a full year. The correction is to multiply each discount factor by \(0.5\), giving \(1.910\), and then use that same factor in both the par-rate and fixed-leg-value calculations.

A second trap is to report \(N(1-P_n)\) as the value to the fixed receiver. That is only the floating-leg value under the model, not the net swap value. Subtract it from the fixed-leg value to get the receiver’s value, and negate that result to obtain the payer’s value. In a par swap, the two leg values agree and both party values are zero.

Key takeaway: With a reset just completed and a consistent single curve, calculate \(A=\sum \alpha_jP_j\) and \(F=1-P_n\). Then \(K_{\text{par}}=F/A\), \(V_{\text{rec}}=N(KA-F)\), and \(V_{\text{payer}}=-V_{\text{rec}}\). Check period accruals and party signs before rounding.

Check Your Understanding

For each problem, assume a standard single-curve swap valued immediately after a floating-rate reset, with no principal exchange or additional adjustments.

  1. A two-payment annual swap has discount factors \(0.97\) and \(0.93\). Find its par fixed rate. (Answer: 3.6842%)
  2. A three-payment semiannual swap has discount factors \(0.990,\ 0.975,\ 0.955\). Find the annual par fixed rate. (Answer: 3.0822%)
  3. An annual-pay swap has \(N=\$4{,}000{,}000\), discount factors \(0.98,\ 0.95,\ 0.91\), and a fixed rate of 4.00%. Find the value to the fixed receiver. (Answer: +\$94,400)
  4. A swap has discount factors \(0.99,\ 0.97,\ 0.94\), with accrual fractions \(0.25,\ 0.50,\ 0.25\), respectively. Find the par fixed rate. (Answer: 6.2016%)
  5. A three-payment semiannual swap has \(N=\$10{,}000{,}000\), discount factors \(0.99,\ 0.96,\ 0.92\), and an existing fixed rate of 3.50%. Find the value to the fixed payer. (Answer: +\$297,750)

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