Valuing A Swap With Discount Factors

Why Discount Factors Determine Swap Value

A fixed-for-floating interest rate swap exchanges interest payments on a notional amount: one party pays a fixed rate and receives a floating rate, while the other party does the reverse. The notional is used to calculate the payments; in a standard interest rate swap it is not itself exchanged. To value the swap, discount each leg’s expected payments to the valuation date and subtract the value of the payments owed from the value of those receivable.

The payment schedule and discount factors provide a direct route to the fixed-leg value. Under a specified single-curve model, they also give a compact expression for the floating-leg value. The sign of the final result depends on which party’s perspective is requested, so identify that perspective before subtracting.

Definition: Let $N$ be the swap notional, $t_0=0<t_1<\cdots<t_n$ the remaining payment dates, and $\alpha_i$ the year fraction for the accrual period ending at $t_i$. Let $D_i$ be the discount factor from $t_i$ to the valuation date, with $D_0=1$. If $K$ is the fixed annual rate, the fixed coupon at $t_i$ is $NK\alpha_i$. A party’s swap value is the present value of the leg it receives minus the present value of the leg it pays.

The model used for the floating-leg shortcut matters. Assume valuation is immediately after a floating-rate reset, the floating payments are based on simple forward rates implied by the same discount factors used for valuation, and there are no credit, funding, or other spread adjustments. The notional is not exchanged. With these assumptions, the floating rate for period $i$ is the simple forward rate $L_i=(D_{i-1}/D_i-1)/\alpha_i$.

If the floating index is projected using a different curve from the discount curve, the shortcut derived below generally does not apply as written. Nor should it be used without adjustment for a valuation between reset dates, when accrued interest or a currently fixed coupon may need separate treatment. Use the curves and timing conventions specified by the problem.

Present Values of the Two Legs

The fixed leg has a known payment of $NK\alpha_i$ at each $t_i$. Discounting and adding those payments gives

PV_{\text{fixed}}=NK\sum_{i=1}^{n}\alpha_iD_i.

This is the value of the fixed payments, regardless of whether the party in question receives or pays them. For example, a receive-fixed party receives this leg; a pay-fixed party owes it.

For the floating leg, the payment at $t_i$ is $N\alpha_iL_i$. Under the stated single-curve assumptions, its present value is

PV_{\text{float}} =N\sum_{i=1}^{n}\alpha_iL_iD_i =N\sum_{i=1}^{n}(D_{i-1}-D_i) =N(1-D_n).

The middle equality follows by substituting the forward rate expression: $\alpha_iL_iD_i=(D_{i-1}/D_i-1)D_i=D_{i-1}-D_i$. The resulting sum telescopes: $D_0-D_1+D_1-D_2+\cdots+D_{n-1}-D_n=1-D_n$. This explains why the floating-leg value depends only on the final discount factor in this model.

Theorem: Immediately after a reset, under the single-curve assumptions above, the swap values are $$ V_{\text{receive fixed}}=NK\sum_{i=1}^{n}\alpha_iD_i-N(1-D_n), $$ and $$ V_{\text{pay fixed}}=N(1-D_n)-NK\sum_{i=1}^{n}\alpha_iD_i. $$ The two parties’ values are negatives of each other. A positive value is an asset to the party whose perspective is being evaluated.

The par swap rate is the fixed rate $K_{\text{par}}$ that makes a new swap’s value zero at inception. Equating the leg values and solving gives

NK_{\text{par}}\sum_{i=1}^{n}\alpha_iD_i=N(1-D_n), \qquad K_{\text{par}}=\frac{1-D_n}{\sum_{i=1}^{n}\alpha_iD_i}.

The notional cancels, so the par rate does not depend on the size of the swap. It does depend on the payment schedule, accrual factors, and discount factors.

Worked Valuations

Worked Example: Value a Receive-Fixed Swap

A party has a receive-fixed, pay-floating swap with notional $\$1{,}000{,}000$. It is valued just after reset, with three remaining semiannual periods, each having $\alpha_i=0.5$. The fixed rate is $K=5\%$, and the discount factors are $D_1=0.99$, $D_2=0.97$, and $D_3=0.94$. Find the party’s value.

Method. Value each leg from the party’s perspective: fixed leg received minus floating leg paid. This avoids reversing the sign by starting with the opposite party’s position.

First add the discount factors: $0.99+0.97+0.94=2.90$. The fixed-leg value is

PV_{\text{fixed}} =1{,}000{,}000(0.05)(0.5)(2.90) =1{,}000{,}000(0.025)(2.90) =\$72{,}500.

The floating-leg value is

PV_{\text{float}} =1{,}000{,}000(1-0.94) =1{,}000{,}000(0.06) =\$60{,}000.

Therefore, the receive-fixed party’s value is

V_{\text{receive fixed}} =\$72{,}500-\$60{,}000 =\boxed{\$12{,}500}.

Check. The fixed leg is worth more than the floating leg, so a party receiving fixed and paying floating must have a positive value. The common mistake is to report $-\$12{,}500$, which is the value to the counterparty, not to this party.

Worked Example: Value a Pay-Fixed Swap

A pay-fixed, receive-floating swap has notional $\$3{,}000{,}000$, three remaining quarterly periods with $\alpha_i=0.25$, and fixed rate $K=4\%$. The discount factors are $0.992$, $0.978$, and $0.961$. Find the value to the pay-fixed party immediately after reset.

Method. Calculate the floating leg received and subtract the fixed leg paid. Write the party’s position before doing arithmetic so that the subtraction order is explicit.

The sum of the discount factors is $0.992+0.978+0.961=2.931$. The fixed-leg value is

PV_{\text{fixed}} =3{,}000{,}000(0.04)(0.25)(2.931) =3{,}000{,}000(0.01)(2.931) =3{,}000{,}000(0.02931) =\$87{,}930.

The final discount factor is $D_3=0.961$, so the floating-leg value is

PV_{\text{float}} =3{,}000{,}000(1-0.961) =3{,}000{,}000(0.039) =\$117{,}000.

The pay-fixed party receives floating and pays fixed:

V_{\text{pay fixed}} =\$117{,}000-\$87{,}930 =\boxed{\$29{,}070}.

Check. The received floating leg is larger than the fixed leg owed, so the value to this party is positive. Subtracting floating from fixed would give $-\$29{,}070$, the value to the receive-fixed counterparty.

Worked Example: Find the Par Swap Rate

A new swap has three annual payment periods, each with $\alpha_i=1$. The discount factors are $D_1=0.96$, $D_2=0.92$, and $D_3=0.88$. Find the annual par fixed rate, and verify that the two legs have equal value for a notional of $\$2{,}000{,}000$.

Method. Divide the floating-leg value per unit of notional by the fixed-leg annuity factor per unit of notional. The floating-leg factor is $1-D_3=1-0.88=0.12$. The fixed-leg annuity factor is

\sum_{i=1}^{3}\alpha_iD_i =(1)(0.96)+(1)(0.92)+(1)(0.88) =2.76.

Thus,

K_{\text{par}} =\frac{0.12}{2.76} =0.0434782609\ldots.

The par rate is $4.348\%$ per year, rounded to four significant figures. For $N=\$2{,}000{,}000$, the floating-leg value is

PV_{\text{float}} =2{,}000{,}000(0.12) =\$240{,}000.

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

PV_{\text{fixed}} =2{,}000{,}000(0.0434782609\ldots)(2.76) =\$240{,}000.

Check. The leg values agree, so the swap has zero initial value. Using the rounded rate in the verification may produce a small rounding difference; use the unrounded rate for the equality.

Common Exam Traps and a Reliable Process

Swap questions often require more care with setup than with arithmetic. In particular, do not confuse an annual quoted fixed rate with a coupon amount: multiply the rate by the accrual factor for each period. A semiannual coupon at a 5% annual fixed rate is $N(0.05)(0.5)$, not $N(0.05)$. Similarly, do not apply the final discount factor to every floating coupon; the telescoping result already accounts for each payment date.

A useful sequence is:

Record the valuation setup.
Identify the valuation date, remaining payment dates, accrual factors, notional, and discount factors. Confirm whether the single-curve, just-after-reset assumptions needed for the floating-leg shortcut are in force.

Calculate the fixed leg.
Use $NK\sum_i\alpha_iD_i$. Keep the rate as a decimal and use the accrual factor for every payment period.

Calculate the floating leg.
Under the stated assumptions, use $N(1-D_n)$. Do not substitute an intermediate discount factor for $D_n$.

Apply the requested perspective.
For receive-fixed, subtract floating from fixed. For pay-fixed, subtract fixed from floating. Check whether the sign agrees with which leg is larger.

For a par rate, do not first guess a coupon or include the notional as an exchanged payment. Equate the two leg values, cancel $N$, and divide by the fixed-leg annuity factor. The formula assumes the same payment schedule and discount factors for both legs; if a problem specifies different projection and discount curves, follow that model instead of imposing the single-curve shortcut.

Key takeaway: Under the single-curve, immediately-after-reset model, the fixed-leg value is $NK\sum_i\alpha_iD_i$, while the floating-leg value is $N(1-D_n)$. A swap’s value is the received leg minus the paid leg, and the par rate makes those two values equal.

Check Your Understanding

For each problem, use the stated single-curve, just-after-reset assumptions. Values are from the named party’s perspective.

A receive-fixed party has notional $\$500{,}000$, two annual periods with $\alpha_i=1$, fixed rate $6\%$, and discount factors $0.97$ and $0.93$. Find the swap value. (Answer: $\$22{,}000$)
A pay-fixed party has notional $\$1{,}200{,}000$, two quarterly periods with $\alpha_i=0.25$, fixed rate $4\%$, and discount factors $0.99$ and $0.97$. Find the swap value. (Answer: $\$12{,}480$)
A swap has three annual periods, each with $\alpha_i=1$, and discount factors $0.98$, $0.95$, and $0.90$. Find the par fixed rate, rounded to four significant figures. (Answer: $3.534\%$)
A receive-fixed party has notional $\$2{,}000{,}000$, four semiannual periods with $\alpha_i=0.5$, fixed rate $5.5\%$, and discount factors $0.99$, $0.975$, $0.955$, and $0.93$. Find the value and state whether it is an asset or liability to the party. (Answer: $\$71{,}750$, an asset)

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