Valuing an Interest Rate Swap

This value comes up when a company or financial institution that holds an interest rate swap revalues the trade at today's market rates. This page explains how to calculate it, starting with how the curve is built.

What the value represents

The value of an interest rate swap is the amount you get by projecting the remaining payments at today's market rates, converting them to present values, and netting them¹. The fixed rate is set so that the present values of both parties' payments are equal at the time of the trade, so a swap traded at market rates is worth zero right after it is traded². When market rates move after the trade, the value can become positive or negative. The value to one party is the value to the other party with the sign reversed. How the trade works is covered in Interest Rate Swaps.

For example, suppose you hold a pay-fixed swap with a notional amount of 10 billion yen, a term of 5 years, and a fixed rate of 1.00%. In today's market, the same 5-year swap can be traded at 1.10%. A new pay-fixed swap would cost you 1.10% a year, but this trade costs only 1.00%. The difference of 0.10%, or 10 million yen a year, discounted over 5 years, is the value seen from the pay-fixed side. The worked example below gives 48,571,481 yen, and from the receive-fixed side the value is −48,571,481 yen.

How to calculate the value

The value is calculated in the following order.

1. Build a curve of discount factors for each maturity from rates traded in the market (such as the fixed rates of overnight index swaps, or OIS).
2. Project the floating rate payments with forward rates derived from the curve.
3. Discount the fixed and floating payments with the discount factor for each payment date and add them up.
4. Subtract the present value of what you pay from the present value of what you receive.

The fixed payments and the floating payments are each called a leg. The curve in step 1 is built by deriving theoretical spot rates (equivalent to the yields on zero-coupon bonds) from the fixed rates of market swaps³. Discount rates are found in order from the shortest maturity (1 year, 2 years, 3 years, and so on), so that a swap valued at the market fixed rate is worth zero⁴.

The forward rate used in step 2 is the rate for a future period, derived from today's curve⁵. Let the $i$-th payment date be $t_i$ years from now, the start of the trade be $t_0$, and the length of the period (in years) be $\tau_i = t_i - t_{i-1}$. The forward rate $F_i$ from year $t_{i-1}$ to year $t_i$ is found from the ratio of the discount factors $DF$ as follows.

$$F_i = \frac{1}{\tau_i}\left(\frac{DF(t_{i-1})}{DF(t_i)} - 1\right)$$

For the side that pays fixed and receives floating (pay-fixed), the value $V$ is given by the formula below, where $N$ is the notional amount, $K$ is the fixed rate, and $n$ is the number of payments⁶.

$$V = N \sum_{i=1}^{n} \tau_i \,(F_i - K)\, DF(t_i)$$

The value to the receive-fixed side is $-V$. The fixed rate $S$ at which the value is zero is the par rate. If $A$ is the sum of each payment date's discount factor multiplied by the length of its period, $S$ is given by the following formula⁷.

$$S = \frac{1 - DF(t_n)}{A}, \quad A = \sum_{i=1}^{n} \tau_i \, DF(t_i)$$

Using the par rate, the value of a pay-fixed swap can also be written as $V = N (S - K) A$. It is the difference between today's par rate and the fixed rate of the trade, multiplied by the notional amount and by $A$. This formula is the calculation behind the example in "What the value represents," where the difference of 0.10% was discounted over 5 years.

Deep diveWhy the present value of the floating payments is 1 − DF

Rearranging the forward rate formula gives $\tau_i F_i \, DF(t_i) = DF(t_{i-1}) - DF(t_i)$. Adding these from the first payment to the $n$-th, adjacent terms cancel out, leaving the following.

$$\sum_{i=1}^{n} \tau_i F_i \, DF(t_i) = DF(t_0) - DF(t_n)$$

For a swap that starts today, $DF(t_0) = 1$, so the present value of the floating payments per unit of notional is $1 - DF(t_n)$. The present value of the fixed payments is $K A$, so the $K$ at which the two are equal gives the par rate formula. This relationship holds when the same curve is used for both projecting and discounting. It does not hold when different curves are used for projecting and discounting; in that case, the payments projected with forward rates are discounted one by one and added up.

Worked example: a 5-year pay-fixed swap

Consider a hypothetical example in which the overnight rate is 0.50% and the OIS fixed rates are 0.60% at 1 year, 0.75% at 2 years, 1.10% at 5 years, and 1.50% at 10 years. An OIS is an interest rate swap that exchanges a fixed rate for the compounded overnight rate⁸. The curve is built from these rates, assuming annual payments and treating every period as exactly one year. The overnight rate only sets the shortest point on the curve (one day) and has no effect on the value of this 5-year swap. The trade being valued is a pay-fixed swap with a notional amount of 10 billion yen, a term of 5 years, annual payments, and a fixed rate of 1.00%.

The payments on each payment date, seen from the pay-fixed side, are shown in the table below. Receipts are positive and payments are negative⁹.

In the first 2 years, the projected floating rate is below the fixed rate of 1.00%, so the pay-fixed side makes a net payment. From year 3 to year 5, the projected floating rate is 1.3407%, above the fixed rate, so the net flow is a receipt. The present value of the receipts is larger, and the value is positive, at 48,571,481 yen.

Drawing the fixed and floating columns of the table as payments before discounting gives Figure 1 below.

Figure description

The figure shows the payments before discounting, seen from the pay-fixed side, for a 5-year pay-fixed swap with a notional amount of 10 billion yen and a fixed rate of 1.00%. The fixed payment is 100 million yen each year. The floating receipts are projected from the curve: 60 million yen in year 1, 90.14 million yen in year 2, and 134.07 million yen in each of years 3 to 5. The floating receipt is below the fixed payment in years 1 and 2 and above it from year 3. Amounts in the figure are in millions of yen (¥m), and the length of each arrow is a rough guide to the size of the amount.

Looking at each leg, the present value of the fixed payments is −485,714,808 yen and the present value of the floating receipts is 534,286,289 yen. The floating leg equals the notional amount multiplied by 1 − DF(5 years) = 1 − 0.946571. The sum of discount factors $A$ is 4.857148, and the 5-year par rate is (1 − 0.946571) ÷ 4.857148 = 1.10%. Because the curve was built from the 5-year OIS fixed rate, the par rate comes back to the 5-year OIS fixed rate. Calculating with $N (S - K) A$ gives 10 billion yen × 0.10% × 4.857148 = 48,571,481 yen, the same as the total in the table.

The forward rates for years 3 to 5 are all 1.3407% because, between the 2-year and 5-year points used to build the curve, the logarithms of the discount factors are joined with a straight line. With this interpolation, the forward rate is constant between those points. A different interpolation method would change the projected floating rates and the value for years 3 to 5.

How the value changes when rates move (DV01)

The change in value when all market rates rise by 1bp (0.01%) is called DV01¹⁰. A Bank of Japan training document calls the same kind of quantity BPV (basis point value) and gives a formula that subtracts the current present value from the present value after raising the rates for all maturities by 1bp. The document describes BPV as the "decrease in present value" (our translation), while its worked example shows BPV as a negative number¹¹. Check the direction of the sign before comparing figures. The DV01 figures below are positive when the value rises as rates rise.

On the curve in the worked example, raising all five market rates (the overnight rate and the OIS rates) by 1bp and rebuilding the curve increases the value of the pay-fixed swap by 4,841,366 yen. Lowering them all by 1bp reduces it by 4,844,229 yen. Taking half the difference between the rise and the fall as DV01 gives 4,842,798 yen. The pay-fixed side gains when rates rise and loses when they fall. For the receive-fixed side, the signs of both the value and DV01 are reversed.

The table below puts the value before and after the rates move side by side. The values after the rise and after the fall are found by adding and subtracting the changes above from the value before the move.

The value of the pay-fixed side rises when rates rise and falls when they fall. For the receive-fixed side, the same rate moves change the value in the opposite direction. Half the difference between the values after the rise and after the fall is 4,842,798 yen, the same as the DV01 above.

DV01 is close to $N \times A \times 0.01\%$ = 10 billion yen × 4.857148 × 0.0001 = 4,857,148 yen. This is because a 1bp rise in rates raises the 5-year par rate by exactly 1bp, increasing $S$ in $V = N (S - K) A$. The actual DV01 is smaller because higher rates make the discount factors smaller, so $A$ also falls. For a trade whose fixed rate equals the par rate of 1.10%, the value is zero, so the effect of the fall in $A$ almost disappears and DV01 is 4,857,148 yen.

DV01 also depends on which rates are moved by 1bp. The figures above come from moving the fixed rates of the OIS traded in the market and rebuilding the curve. Moving the annually compounded spot rates instead by 1bp at each maturity, as in the Bank of Japan document's BPV, gives a DV01 (half the difference between the rise and the fall) of 4,824,279 yen for the same trade¹². When comparing DV01 across internal reports or systems, make sure the figures were calculated by moving the same rates.

Early termination and variation margin at clearing houses

The value serves as a guide to the amount paid when a trade is ended early. A swap can be ended early by one party paying the other the value of the swap¹⁴.

A 2018 document by a study group with the Bank of Japan as its secretariat says that almost all yen OIS trades are cleared through central counterparties⁸. The same document says that where derivatives with collateral (variation margin) are valued on a discount curve built from OIS fixed rates, the risk of that discount curve moving can be hedged with OIS¹⁵. For the interest rate swaps it clears, Japan Securities Clearing Corporation (JSCC) sets variation margin as the amount that covers the change in value (NPV) calculated with the previous business day's yield curve and the current day's yield curve¹⁶.

Assumptions of the worked example and differences from actual trades

The worked example uses the same single OIS curve for both discounting and projecting the floating rate. Term rates such as TIBOR differ in nature from the overnight rate because they include the credit risk of banks¹⁸. Between instruments linked to the overnight rate and those linked to TIBOR, there is a risk that the difference between the two (the spread) moves, and OIS/TIBOR basis swaps exist to hedge it¹⁹. Because this difference moves, projecting the floating rate of a TIBOR-referencing swap only with the forward rates of an OIS curve built from the overnight rate cannot reflect the difference. Using different curves for projecting and discounting is covered in The Multi-Curve Framework.

The worked example assumes annual payments and treats every period as exactly one year. As one example of the basic trading terms of yen OIS, the study group's document gives a formula that compounds the overnight rate for each business day. It converts the number of calendar days to years by dividing by 365. It also gives settlement as 2 business days after the maturity date¹⁷.

The value in the worked example does not include the risk that the counterparty defaults or the cost of funding collateral.

Notes

1. Society of Actuaries, "Interest Rate Swaps" (May 2017), Section 6 "Market Value," page 15. The document defines the market value as the present value of the remaining payments discounted at the spot rates at the valuation time, and says that the market value to one party is the negative of the market value to the other party. ↩

2. The same document, Section 2 "Definitions," page 5. It says the fixed rate (swap rate) is set so that the present values of both parties' payments are equal. That the value is zero at the time of the trade is in Section 6, page 15. ↩

3. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (Understanding and managing market risk, June 2012, in Japanese), "（参考）割引率（スポットレート）の定義" (Reference: definition of the discount rate (spot rate)), page 10. It has a diagram deriving spot rates in theory from swap rates (fixed rates). ↩

4. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "【補足資料】スポットレートとその応用" (Supplementary material: spot rates and their applications, August 2012, in Japanese), "キャッシュフロー分解" (Decomposing cash flows), page 7, and "スポットレートの計算式" (Formulas for spot rates), page 8. The document says the 1-, 2-, and 3-year discount rates are backed out from market rates in that order. ↩

5. Society of Actuaries, "Interest Rate Swaps" (May 2017), Example 1, page 5. It calculates the rate from year 1 to year 2 from today's spot rates and calls it the forward rate. ↩

6. The same document, Section 3 "General Formula," formula (3.7), page 10. The formula is written in effective rates per settlement period, and $\tau_i F_i$ and $\tau_i K$ on this page correspond to the document's forward rate and swap rate. ↩

7. The same document, Section 4 "Special Formula," formulas (4.1) and (4.2), page 12. The document's $P_t$ corresponds to the discount factor $DF(t)$ on this page, and its $R$ corresponds to $\tau_i S$ when the settlement periods are of equal length. ↩

8. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (Yen OIS: overview of transactions and examples of use, April 2018, in Japanese), "２．OIS とは何か" (2. What is OIS), page 3. ↩ ↩²

9. The amounts in the table are calculated from unrounded discount factors and forward rates and then rounded to the nearest yen. The forward rates and discount factors shown are rounded. The total adds the unrounded values and then rounds. ↩

10. Office of Financial Research, "Short-term interest rate futures (DV01)" (Hedge Fund Monitor). It describes DV01 as "dollar value of a one basis point change in yield." Checked on September 24, 2026. ↩

11. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (Understanding and managing market risk, June 2012, in Japanese), "（１）ＢＰＶ（ベーシス・ポイント・バリュー）" ((1) BPV (basis point value)), page 13, and "ＢＰＶ、ＧＰＳの計算例" (Calculation example of BPV and GPS), page 15. In the example on page 15, BPV is −4.84 million yen. ↩

12. The annually compounded spot rates $r_t$ ($DF(t) = 1/(1+r_t)^t$) are derived from the 1- to 5-year discount factors of the bootstrapped curve, and the trade is revalued with discount factors after moving every maturity by ±0.01%. A 1bp rise increases the value by 4,822,857 yen, and a 1bp fall reduces it by 4,825,701 yen. ↩

13. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (Understanding and managing market risk, June 2012, in Japanese), "（２）ＧＰＳ（グリッド・ポイント・センシティビティ）" ((2) GPS (grid point sensitivity)), page 14, and "ＢＰＶ、ＧＰＳの計算例" (Calculation example of BPV and GPS), page 15. ↩

14. Society of Actuaries, "Interest Rate Swaps" (May 2017), Section 6 "Market Value," page 15, footnote 8. ↩

15. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (Yen OIS: overview of transactions and examples of use, April 2018, in Japanese), "（デリバティブのディスカウント・リスクのヘッジ）" (Hedging the discount risk of derivatives), page 10. ↩

16. Japan Securities Clearing Corporation, "金利スワップ取引に係る証拠金の種類" (Types of margin for interest rate swap transactions, in Japanese), "変動証拠金" (Variation margin) and "日中証拠金" (Intraday margin). Checked on September 24, 2026. ↩

17. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (Yen OIS: overview of transactions and examples of use, April 2018, in Japanese), "基本的な取引方式" (Basic trading terms), page 5. The document says actual trading procedures need to be confirmed with the counterparty, and gives the floating rate formula as one example. ↩ ↩²

18. The same document, "（１）IBOR と OIS レートの性質の違い" ((1) Differences in the nature of IBOR and OIS rates), pages 7–8. ↩

19. The same document, "（OIS/IBOR ベーシス・スワップ）" (OIS/IBOR basis swaps), page 10. ↩