Repricing a Swap After Spot Rates Change

Why a Swap Must Be Repriced When the Curve Changes

An interest rate swap fixes contractual terms at the time the parties enter the agreement, but the market’s valuation of those terms changes as interest rates move. In particular, the fixed rate written into an outstanding swap does not reset when the spot curve changes. The floating payments that have not yet been set must instead be projected using forward rates implied by the revised curve, and both legs must be discounted using that curve.

This distinction is central to repricing: the contract stays the same, but the estimated present values of its future cash flows change. The methods below use an annual-pay swap valued immediately after a reset date, so each future floating coupon is determined by a forward rate. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Model and assumptions: The swap has notional principal $N$, annual payments at times $t_1,\ldots,t_n$, and no exchange of notional. The fixed rate is $K$ per year. Valuation occurs just after a reset date, and the same internally consistent curve is used to project floating rates and discount payments. There is no default risk, collateral adjustment, tax, or transaction cost. For annual periods, the accrual factor is $\alpha_j=t_j-t_{j-1}=1$, with $t_0=0$. The formulas do not directly apply unchanged if a floating coupon has already been set for a period that has not yet ended.

Projecting Payments From the Revised Curve

Let $D_j$ denote the revised discount factor from the valuation date to $t_j$, with $D_0=1$. A discount factor is the present value of one unit paid at its stated future date. For a period from $t_{j-1}$ to $t_j$, define the implied simple forward rate $F_j$ by

1+\alpha_j F_j=\frac{D_{j-1}}{D_j}, \qquad F_j=\frac{D_{j-1}/D_j-1}{\alpha_j}.

This rate is the rate over that particular future accrual period implied by the two relevant discount factors. It is not generally equal to the spot rate for either payment date. The projected floating coupon is $N\alpha_jF_j$, while the fixed coupon is $N K\alpha_j$. At valuation, each coupon is multiplied by the discount factor for its payment date.

Valuation result: For the model above, the value to the party that receives floating and pays fixed is $N\left[1-D_n-K\sum_{j=1}^{n}\alpha_jD_j\right]$. The value to the party that receives fixed and pays floating is the negative of this amount. The contractual $K$ is unchanged by a market-rate movement.

Why the Floating-Leg Formula Works

The floating leg’s present value can be calculated coupon by coupon. The forward-rate definition gives $\alpha_j F_jD_j=D_{j-1}-D_j$. Therefore, the present value of the $j$th projected floating coupon is $N(D_{j-1}-D_j)$. Adding these values causes the intermediate discount factors to cancel.

\begin{aligned} \operatorname{PV}(\text{floating}) &=N\sum_{j=1}^{n}\alpha_jF_jD_j\\ &=N\sum_{j=1}^{n}(D_{j-1}-D_j)\\ &=N(1-D_n). \end{aligned}

This telescoping identity explains why a fully reset floating leg has a compact value even when its individual projected coupons vary. The fixed leg is a level stream of known contractual payments, so its present value is $NK\sum_{j=1}^{n}\alpha_jD_j$. Subtracting the fixed-leg value from the floating-leg value gives the receive-floating, pay-fixed value. Reversing the two positions reverses the sign.

Hold the contract terms fixed.
Record the existing notional, payment dates, accrual factors, and contractual fixed rate. Do not replace that fixed rate with today’s market par rate.

Use the revised curve.
Obtain the new discount factors at every future reset and payment date. Derive a forward rate for each floating accrual period.

Value each leg consistently.
Project the floating coupons, discount them, and separately discount the contractual fixed coupons. Subtract according to the position’s sign.

Worked Repricing Examples

Worked Example: A Two-Year Swap After a Curve Shift

A party receives floating and pays fixed on a two-year annual-pay swap with notional $N=\$1{,}000{,}000$ and contractual fixed rate $K=4\%$. At initiation, the discount factors were $0.98$ and $0.94$. The revised discount factors are $D_1=0.96$ and $D_2=0.90$. Find the swap value before and after the change, and explain how repricing uses the new curve.

Method. First value both legs using each curve’s discount factors. For the revised curve, calculate the period forwards from adjacent discount factors. Keep the contract rate at $4\%$ throughout.

Before the curve change. The floating-leg present value is $N(1-0.94)=\$60{,}000$. The fixed-leg present value is $\$40{,}000(0.98+0.94)=\$40{,}000(1.92)=\$76{,}800$. The receive-floating value is therefore $\$60{,}000-\$76{,}800=-\$16{,}800$.

New forward rates. For year 1, $F_1=1/0.96-1=0.0416667$, or about $4.1667\%$. For year 2, $F_2=0.96/0.90-1=0.0666667$, or about $6.6667\%$. The projected floating coupons are $\$1{,}000{,}000(0.0416667)=\$41{,}666.67$ and $\$1{,}000{,}000(0.0666667)=\$66{,}666.67$, to the nearest cent.

New present values. Discount the projected floating coupons: $\$41{,}666.67(0.96)=\$40{,}000.00$ and $\$66{,}666.67(0.90)=\$60{,}000.00$. Thus the floating-leg value is $\$100{,}000.00$. The fixed coupons remain $\$40{,}000$ each, so their present value is $\$40{,}000(0.96+0.90)=\$74{,}400.00$. The revised receive-floating value is $\$100{,}000-\$74{,}400=\$25{,}600$.

Check and mistake avoided. The compact formula gives $\$1{,}000{,}000[1-0.90-0.04(1.86)]=\$25{,}600$, agreeing with the coupon calculation. The position has moved from a negative to a positive value. The common mistake avoided is resetting the contractual fixed rate to a new market rate; the $4\%$ rate belongs to the existing contract.

Worked Example: Repricing Along a Three-Year Curve

An investor receives floating and pays fixed on a three-year annual-pay swap. The notional is $\$2{,}000{,}000$, the original fixed rate is $3.5\%$, and the revised discount factors are $D_1=0.97$, $D_2=0.92$, and $D_3=0.86$. Calculate the new value and show how the implied forwards relate to the floating-leg present value.

Method. Derive each annual forward from successive discount factors, calculate its projected coupon, and discount each coupon at its payment-date factor. Then compare that leg with the existing fixed leg.

Forward rates and coupons. The first forward is $F_1=1/0.97-1=0.03092784$. Its projected coupon is $\$2{,}000{,}000(0.03092784)=\$61{,}855.67$. The second forward is $F_2=0.97/0.92-1=0.05434783$, giving a coupon of $\$108{,}695.65$. The third is $F_3=0.92/0.86-1=0.06976744$, giving a coupon of $\$139{,}534.88$. Coupon amounts are rounded to cents here; the valuation uses the unrounded forward rates.

Floating-leg value. Discounting the unrounded coupons gives $\$2{,}000{,}000(0.97F_1)=\$60{,}000$, $\$2{,}000{,}000(0.92F_2)=\$100{,}000$, and $\$2{,}000{,}000(0.86F_3)=\$120{,}000$. Their sum is $\$280{,}000$, which also equals $\$2{,}000{,}000(1-0.86)=\$280{,}000$.

Fixed-leg value and answer. Each fixed coupon is $\$2{,}000{,}000(0.035)=\$70{,}000$. Its present value is $\$70{,}000(0.97+0.92+0.86)=\$70{,}000(2.75)=\$192{,}500$. The receive-floating swap is worth $\$280{,}000-\$192{,}500=\$87{,}500$. The check using the compact formula is $\$2{,}000{,}000[1-0.86-0.035(2.75)]=\$87{,}500$.

Common mistake avoided. Using one spot rate as the floating rate for every period would ignore the curve’s period-by-period forward rates. The forward-rate construction is what makes each projected coupon consistent with the revised discount factors.

Worked Example: Value From the Receive-Fixed Party’s Perspective

A party receives fixed and pays floating on a three-year annual-pay swap with notional $\$500{,}000$ and contractual rate $K=5\%$. The old discount factors were $0.97$, $0.93$, and $0.89$; the revised factors are $0.94$, $0.88$, and $0.81$. Find the value before and after the curve change.

Method. Value receive-fixed as the fixed-leg present value minus the floating-leg present value. The fixed coupons remain $\$500{,}000(0.05)=\$25{,}000$ per year.

Old curve. The floating-leg value is $\$500{,}000(1-0.89)=\$55{,}000$. The fixed-leg value is $\$25{,}000(0.97+0.93+0.89)=\$25{,}000(2.79)=\$69{,}750$. The receive-fixed value is $\$69{,}750-\$55{,}000=\$14{,}750$.

Revised curve. The floating-leg value is $\$500{,}000(1-0.81)=\$95{,}000$. The fixed-leg value is $\$25{,}000(0.94+0.88+0.81)=\$25{,}000(2.63)=\$65{,}750$. The revised receive-fixed value is $\$65{,}750-\$95{,}000=-\$29{,}250$.

Check and sign interpretation. The revised value is also $\$500{,}000[0.05(2.63)-(1-0.81)]=-\$29{,}250$. The negative sign means the receive-fixed party’s position has negative value under the stated assumptions; the counterparty’s value is $+\$29{,}250$. Using the receive-floating sign for a receive-fixed position would reverse the answer.

Common Spot-Rate Trap

A spot rate applies from the valuation date to a particular maturity. A forward rate applies to a future interval between two dates. Those rates are related through discount factors, but they are not interchangeable. For example, in the two-year revised curve above, the one-year spot rate is $1/0.96-1=4.1667\%$, while the two-year effective annual spot rate is $1/\sqrt{0.90}-1$, approximately $5.4093\%$. The second-year forward rate is instead $0.96/0.90-1=6.6667\%$.

Wrong calculation: using the two-year spot rate as the second-year floating rate gives an approximate second floating coupon of $\$1{,}000{,}000(0.054093)=\$54{,}093$, with present value about $\$48{,}683$. Adding the first-year floating present value of $\$40{,}000$ gives about $\$88{,}683$, not the correct $\$100{,}000$.

Correction: use the second-year forward rate of $6.6667\%$. The second coupon’s present value is then $\$66{,}666.67(0.90)=\$60{,}000$. The floating-leg value is $\$40{,}000+\$60{,}000=\$100{,}000$, consistent with $N(1-D_2)$. The error arose from applying a rate over the wrong time interval.

Key takeaway: Repricing changes the market value, not the contract. Keep the existing fixed rate, derive each future floating rate from the revised discount factors, discount the resulting cash flows, and apply the sign convention for the party being valued.

Check Your Understanding

For each problem, assume annual payments, no exchange of notional, and valuation immediately after a reset date. Use the stated discount factors as the revised curve.

A party receives floating and pays fixed on a two-year swap with notional $\$800{,}000$, contractual rate $3.2\%$, and discount factors $D_1=0.98$, $D_2=0.93$. Find the swap value. (Answer: $\$7{,}104$)
A party receives floating and pays fixed on a three-year swap with notional $\$1{,}500{,}000$, contractual rate $4\%$, and discount factors $0.95$, $0.90$, and $0.84$. Find the value using the revised curve. (Answer: $\$78{,}600$)
A party receives fixed and pays floating on a three-year swap with notional $\$750{,}000$, fixed rate $4.5\%$, and discount factors $0.96$, $0.91$, and $0.85$. Find the value from this party’s perspective. (Answer: $-\$20{,}700$)
A party receives floating and pays fixed on a two-year swap with notional $\$1{,}000{,}000$ and contractual rate $3\%$. The old discount factors were $0.98$ and $0.95$; the new factors are $0.96$ and $0.92$. Find the value before and after the change. (Answer: Old value $-\$7{,}900$; new value $+\$23{,}600$)
A party receives fixed and pays floating on a two-year swap with notional $\$2{,}000{,}000$, contractual rate $3.75\%$, and discount factors $0.97$ and $0.91$. Find the value from this party’s perspective. (Answer: $-\$39{,}000$)

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