The Multi-Curve Framework

In interest rate swap valuation, whether the discount and projection curves are separate changes the value. The same trade is worth a different amount under one curve than under separate curves.

Separate curves for discounting and projection

The multi-curve framework splits the curves used to value an interest rate swap by their role. One is the discount curve, which converts future payments into present value. The other is used to project the amounts of the floating rate payments, and gives the forward rate for each future period. The sequence of forward rates backed out from market swap fixed rates is also called the projected curve¹. Using one curve for both discounting and projection is the single-curve approach.

The discount curve depends on the collateral terms. When cash is posted as collateral, the interest paid on it (the collateral rate) is usually the overnight rate of the collateral currency. For trades where collateral is fully exchanged, future payments need to be discounted at that collateral rate². An OIS is an interest rate swap that exchanges a fixed rate for a floating rate calculated by compounding the overnight rate over a set period. Because yen cash collateral earns the yen overnight rate, the curve built from yen OIS fixed rates is the discount curve.

Projection curves are separated by reference rate and by tenor. Unsecured interbank rates such as Japanese Yen TIBOR (Tokyo Interbank Offered Rate) include the credit risk of banks, which makes them different in nature from the overnight rate³. Even for the same interbank rate, the value differs by tenor. A tenor swap that exchanges two LIBOR (London Interbank Offered Rate) tenors does not balance unless a nonzero spread is added to one side⁴. The margin added to one side when exchanging two floating rates that differ in reference rate or tenor is called the basis spread.

How one curve became several

Before the financial crisis, derivatives were usually discounted at LIBOR. While bank credit risk was not prominent, treating LIBOR as the risk-free rate caused no particular problem⁵. The traditional method finds discount factors from LIBOR swap fixed rates and takes LIBOR forward rates from the same curve. Before 2007, LIBOR swap fixed rates were seen as a workable stand-in for the risk-free curve⁶.

In the US market, the gap between 3-month LIBOR and the OIS fixed rate for the same period was usually 5–10bp until August 2007. It first jumped on August 9, 2007, and reached about 350bp after the Lehman Brothers bankruptcy was announced on September 15, 2008⁷. As bank credit risk rose and the gap widened, OIS rates came to be seen as a better risk-free rate for discounting than LIBOR⁸.

After the crisis, exchanging collateral became standard practice in derivatives trading. A 2018 paper from the Bank of Japan's Institute for Monetary and Economic Studies says the trend is to value trades where cash collateral is received by discounting at OIS rates⁸. Projection curves split as well. In 2009, the spread on tenor swaps exchanging 3-month and 6-month yen LIBOR often exceeded 10bp⁹. The 3-month and 6-month projection curves can be built separately from the discount curve. Where tenor swap markets for other tenors exist, the same method extends to 1-month and 12-month LIBOR. Where a market for swaps exchanging LIBOR and TIBOR exists, it extends to TIBOR as well⁹.

Building the discount curve and the projection curve

The curves in the multi-curve framework are built in this order¹⁰.

1. Build a curve of discount factors $P(t)$ from OIS fixed rates by bootstrapping.
2. Collect the market fixed rates, by maturity, of swaps whose floating rate is the reference rate to be projected (TIBOR, for example).
3. Solve for the forward rates of that reference rate, starting with the shortest maturity, so that the swaps in step 2 have zero value when discounted with the discount factors from step 1.
4. Project the floating rate of the swap being valued with the forward rates from step 3, and discount with the discount factors from step 1.

Figure description

The figure shows how steps 1 to 4 connect. The discount factors come from the OIS fixed rates, and the TIBOR forward rates are then solved from those discount factors and the fixed rates of the TIBOR swaps. The swap being valued has its floating payments projected with those forward rates and is discounted with the same OIS discount factors to give its value. There are two differences from a single curve. The discount factors are determined by a separate market, OIS, and the TIBOR forward rates are found from the condition that the swaps have zero value rather than from a ratio of discount factors.

Step 1 is covered in Bootstrapping a Yield Curve. The formula for step 3 is as follows. Let $S_n$ be the market fixed rate of the $n$-year TIBOR swap, $t_i$ the $i$-th payment date, $\tau_i$ the length of that period in years, and $F_i$ the TIBOR forward rate for that period. If the fixed and floating payment dates are the same, the condition for zero value at the market fixed rate is the following, per 1 of notional amount¹¹.

$$S_n \sum_{i=1}^{n} \tau_i P(t_i) = \sum_{i=1}^{n} \tau_i F_i P(t_i)$$

The left-hand side is the present value of the fixed payments, and the right-hand side is the present value of the floating payments. Once the forward rates up to period $n - 1$ are known, $F_n$ is given by the following formula.

$$F_n = \frac{S_n \sum_{i=1}^{n} \tau_i P(t_i) - \sum_{i=1}^{n-1} \tau_i F_i P(t_i)}{\tau_n P(t_n)}$$

With a single curve, the forward rate comes from the ratio of discount factors, $F_i = (P(t_{i-1})/P(t_i) - 1)/\tau_i$. In the multi-curve framework, TIBOR forward rates are not taken from the ratio of OIS discount factors. As long as there is a nonzero basis spread between the discounting rate and the projected rate, this ratio relationship does not hold for the projected rate¹².

Suppose the swap being valued has the same payment dates as the market swap and pays a fixed rate $K$. Its value $V$ is given by the formula below. $N$ is the notional amount, and $A$ is the sum of the OIS discount factors each multiplied by the length of its period.

$$V = N \sum_{i=1}^{n} \tau_i (F_i - K) P(t_i) = N (S_n - K) A$$

The second equality follows from the zero-value condition above. With a single curve, $A$ is calculated from the discount factors of the same curve used for projection. In the multi-curve framework, $S_n$ stays the market fixed rate of the TIBOR swap, and only $A$ changes to the value calculated from the OIS discount factors.

Deep diveHow the present value of floating payments relates to the basis spread

Consider a swap that starts now ($P(t_0) = 1$). Let $F^{\mathrm{OIS}}_i = (P(t_{i-1})/P(t_i) - 1)/\tau_i$ be the OIS forward rate found from the ratio of OIS discount factors, and $S^{\mathrm{OIS}}_n$ the $n$-year OIS fixed rate. Because the OIS curve is a single curve, the present value of the OIS floating payments per 1 of notional is as follows.

$$\sum_{i=1}^{n} \tau_i F^{\mathrm{OIS}}_i P(t_i) = 1 - P(t_n) = S^{\mathrm{OIS}}_n A$$

The present value of the TIBOR floating payments, on the other hand, is $S_n A$ from the zero-value condition. This gives the following relationship.

$$\sum_{i=1}^{n} \tau_i F_i P(t_i) = 1 - P(t_n) + (S_n - S^{\mathrm{OIS}}_n) A$$

The present value of the TIBOR floating payments exceeds the OIS value, one minus $P(t_n)$, by $(S_n - S^{\mathrm{OIS}}_n) A$. This is because the TIBOR swap fixed rate is higher than the OIS fixed rate. In other words, consider a TIBOR floating-rate note that repays a principal of 1 at maturity. With OIS discounting, its value (the present value of the floating payments plus $P(t_n)$) is 1 plus $(S_n - S^{\mathrm{OIS}}_n) A$. That is above the principal when the TIBOR swap fixed rate is above the OIS rate. The same relationship holds for a floating-rate note paying LIBOR¹³.

Now consider a basis swap that exchanges the OIS floating rate plus $b$ for the TIBOR floating rate. If its payment dates and period lengths are the same as the TIBOR swap's, the $b$ that gives zero value is set by the following condition.

$$\sum_{i=1}^{n} \tau_i (F^{\mathrm{OIS}}_i + b) P(t_i) = \sum_{i=1}^{n} \tau_i F_i P(t_i)$$

The left-hand side is $(S^{\mathrm{OIS}}_n + b) A$ and the right-hand side is $S_n A$, so $b = S_n - S^{\mathrm{OIS}}_n$. The difference between the TIBOR swap and OIS fixed rates of the same maturity corresponds to the basis swap spread. For trades with different payment dates or period lengths, this relationship does not hold as it stands.

Worked example: a 3-year TIBOR swap

Consider a hypothetical example in which the OIS fixed rates are 0.50% for 1 year, 0.75% for 2 years, and 1.00% for 3 years. The fixed rates of swaps with TIBOR as the floating rate are 0.80% for 1 year, 1.05% for 2 years, and 1.30% for 3 years.

Japanese Yen TIBOR has five tenors, from 1 week to 12 months¹⁴. The TIBOR swaps cleared by the central counterparty Japan Securities Clearing Corporation (JSCC), on the other hand, have floating rate tenors of 1, 3, and 6 months¹⁵. To keep the calculation simple, this example uses 12-month TIBOR as the floating rate, with both fixed and floating payments made once a year on the same date. Every period is treated as one year ($\tau_i = 1$), and the day count convention and business day convention are ignored.

The swap being valued is an existing pay-fixed swap with a notional amount of 10 billion yen, 3 years remaining, and a fixed rate of 1.00%. It pays the fixed rate and receives the TIBOR floating rate.

First, find the discount factors from each set of market rates by bootstrapping¹⁶.

The right-hand column shows the single-curve discount factors, which also use the TIBOR swaps for discounting. Because the OIS fixed rates are lower, the OIS discount factors are larger at every maturity.

Next, find the TIBOR forward rates in three ways. The single curve uses the ratio of the discount factors in the right-hand column, and the multi-curve framework uses the $F_n$ formula above with the OIS discount factors. For comparison, the OIS forward rates found from the ratio of OIS discount factors are also shown.

The multi-curve rate for 1–2 years is (1.05% × (0.995025 + 0.985149) − 0.80% × 0.995025) ÷ 0.985149, or about 1.3025%. For 0–1 year, both methods give 0.8000%, the same as the 1-year fixed rate. After the first year, the multi-curve forward rates are slightly lower than the single-curve ones. The example in Smith's teaching note, which uses LIBOR swaps, shows the same direction: when the LIBOR–OIS spread is positive and the curve slopes upward, OIS discounting lowers LIBOR forward rates¹⁷.

Finally, discount the payment on each date as seen from the pay-fixed side (projected floating rate minus fixed rate) and add them up. Receipts are positive and payments are negative.

The multi-curve value is $N (S_n - K) A$ = 10 billion yen × (1.30% − 1.00%) × 2.950667, which is 88,520,007 yen ($A$ calculated unrounded). With a single curve, $A$ becomes 2.933232, the sum of the discount factors from the TIBOR swaps, and the value is 87,996,965 yen. The difference between the unrounded values is 523,043 yen, and it comes from the difference in discount factors. From the receive-fixed side, both values have the opposite sign.

If the OIS curve alone is used for both projection and discounting, the value is 0 yen. This is because the 3-year OIS fixed rate is 1.00%, the same as this trade's fixed rate. When payment dates and periods line up as in this example, the 0.30% difference from the TIBOR swap fixed rate corresponds to the spread on a basis swap that exchanges OIS for TIBOR (derived in the deep dive above). Projecting with the OIS curve alone misses the value of this basis spread: 10 billion yen × 0.30% × 2.950667, or 88,520,007 yen.

How the value changes when the basis spread moves

In the multi-curve framework, the value changes when OIS rates move, even if the TIBOR swap fixed rates do not. In the example above, suppose the TIBOR swap fixed rates stay the same and only the OIS fixed rates fall by 0.20 percentage points at every maturity. OIS rates become 0.30% for 1 year, 0.55% for 2 years, and 0.80% for 3 years, and the basis spread widens from 0.30% to 0.50%.

The OIS discount factors become 0.997009, 0.989077, and 0.976301, and their sum $A$ rises to 2.962386. The pay-fixed value becomes 10 billion yen × 0.30% × 2.962386, or 88,871,592 yen ($A$ calculated unrounded), an increase of 351,585 yen. On the receive-fixed side, the value goes from −88,520,007 yen to −88,871,592 yen, a loss of the same amount. The TIBOR forward rates also fall slightly, to 1.3020% and 1.8086%. However, the value is set by $N (S_n - K) A$, so it changes only through $A$. The single curve uses only the TIBOR swap fixed rates, so its value stays at 87,996,965 yen.

The side with a positive value gains when $A$ increases, and the side with a negative value loses. If the contracted fixed rate is above the current TIBOR swap fixed rate, so that the pay-fixed value is negative, the same move causes a loss on the pay-fixed side.

Yen projection curves after LIBOR ceased

All tenors of yen LIBOR ceased publication at the end of December 2021²⁰. In December 2021, JSCC converted the yen LIBOR interest rate swaps it was clearing, all at once, into OIS referencing TONA (the Tokyo Overnight Average Rate, the uncollateralized overnight call rate)²¹. The history and practice of the transition are covered in The Transition from LIBOR.

Among the TIBOR rates, all tenors of Euroyen TIBOR ceased publication at the end of December 2024 (the last publication was on December 30, 2024)²². For Japanese Yen TIBOR, JBA TIBOR Administration said in a June 2024 document that it would continue publication²³, and said in the results of its March 2023 public consultation that it was not considering any permanent cessation²⁴.

Of the yen interest rate swaps that JSCC clears as of September 2026, those exchanging a fixed rate for a floating rate are of two types. One has TONA as the floating rate (OIS), and the other has Japanese Yen TIBOR (1, 3, or 6 months). Swaps exchanging two floating rates include tenor swaps (Japanese Yen TIBOR against Japanese Yen TIBOR, and OIS against OIS) and curve swaps exchanging OIS for Japanese Yen TIBOR²⁵. None of the swaps on this list reference LIBOR.

So when the yen interest rate swaps cleared by JSCC are valued under the multi-curve framework, the projection curves are for TONA and for each tenor of Japanese Yen TIBOR. Trades referencing Japanese Yen TIBOR need TIBOR forward rates for each tenor in addition to the OIS curve. For an OIS referencing TONA, if the collateral rate is also the overnight rate, the same OIS curve can be used for both projection and discounting²⁶.

Assumptions and limitations

The calculations on this page assume trades in which yen cash collateral is fully exchanged every day and earns the overnight rate. The formulas are derived under the simplification that collateral is exchanged continuously, in cash, for the full value of the trade²⁸. For trades where collateral is posted in another currency, that currency's rates enter the discounting, so the discount curve changes²⁹.

For trades without collateral, the value of the exposure that collateral does not cover is adjusted for the credit risk of both the counterparty and the firm itself. Together these are called credit valuation adjustments: the adjustment for the counterparty is CVA (credit valuation adjustment), and the one for the firm itself is DVA (debt valuation adjustment). Such trades were discounted at a funding rate such as LIBOR as of 2018³⁰.

The worked example assumes annual payments with fixed and floating payments on the same dates. In real trades, the two sides can have different payment frequencies³¹, and the length of each period is counted under the day count convention. Forward rates at maturities with no market rate depend on the interpolation method.

Notes

1. Donald J. Smith, "A Teaching Note on Pricing and Valuing Interest Rate Swaps Using LIBOR and OIS Discounting" (June 2012), page 3. ↩

2. Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi, "A Note on Construction of Multiple Swap Curves with and without Collateral" (2009), Section 3.1, pages 9–10. ↩

3. 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), "（１）IBOR と OIS レートの性質の違い" (1. Differences in nature between IBOR and OIS rates), pages 7–8. ↩

4. Fujii et al. paper, Section 1, page 1. ↩

5. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effects of credit valuation adjustments of derivatives on hedging transactions), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2018, in Japanese), Section 2 (1), page 123. ↩

6. Smith note, pages 1, 14, and 15. ↩

7. Smith note, page 17. The note uses the US market as its example. ↩

8. Kawakami paper, Section 2 (1), pages 122–123. ↩ ↩²

9. Fujii et al. paper, Section 2.3, pages 4–5. ↩ ↩²

10. Smith note, pages 17–22. Discount factors are on pages 17–19 and LIBOR forward rates on pages 20–22. ↩

11. Fujii et al. paper, equation (3.17) in Section 3.3, pages 11–12. The paper's $C_M$, $\Delta_m$, $E^c_t[L(T_{m-1}, T_m)]$, and $D_{t,T_m}$ correspond to $S_n$, $\tau_i$, $F_i$, and $P(t_i)$. ↩

12. Fujii et al. paper, equation (2.2) in Section 2.2, page 4. In an example that uses US dollar LIBOR as the discounting rate, the relationship does not hold as long as the cross-currency swap basis spread is nonzero. ↩

13. Smith note, pages 3 and 23. ↩

14. JBA TIBOR Administration, "金利指標改革を踏まえた全銀協TIBORの現状および今後の展望" (The current state and outlook of JBA TIBOR in light of benchmark reform, June 2024, in Japanese), "①全銀協TIBORとは" (1. What is JBA TIBOR), page 1. ↩

15. Japan Securities Clearing Corporation, "清算対象取引（金利スワップ）" (Transactions eligible for clearing: interest rate swaps, in Japanese), "5．変動金利の期間" (Floating period). An English version is available: "Clearing Products," "5．Floating Period." Checked on September 24, 2026. ↩

16. Discount factors and forward rates are calculated unrounded and rounded to the displayed digits. Recalculating from the displayed discount factors shifts the values by a few yen. ↩

17. Smith note, page 21. ↩

18. Each figure is the difference in value after moving only the rates in that row by 1bp and rebuilding the curves from the worked example, calculated unrounded. ↩

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

20. Bank of Japan Financial Markets Department and Financial System and Bank Examination Department, and Financial Services Agency Strategy Development and Management Bureau and Supervision Bureau, "円LIBOR移行対応の振り返りと今後の取り組み" (A review of the transition from yen LIBOR and future initiatives) (Bank of Japan Review 2022-J-8) (in Japanese), May 2022, page 1 and Chart 1. An English version is available: "Review of JPY LIBOR Transition and Future Initiatives" (Bank of Japan Review 2022-E-4, May 2022), page 1 and Chart 1. ↩

21. The same document, footnote 23 on page 8 (page 9 of the English version). ↩

22. JBA TIBOR Administration, "全銀協TIBOR改革" (JBA TIBOR reform, in Japanese), the history under "全銀協TIBOR改革Next" (JBA TIBOR Reform Next). Checked on September 24, 2026. ↩

23. JBA TIBOR Administration, "金利指標改革を踏まえた全銀協TIBORの現状および今後の展望" (The current state and outlook of JBA TIBOR in light of benchmark reform, June 2024, in Japanese), page 4. ↩

24. JBA TIBOR Administration, "全銀協TIBOR改革" (JBA TIBOR reform, in Japanese), "【日本円TIBORを参照する契約へのフォールバック条項の導入】" (Introducing fallback clauses into contracts referencing Japanese Yen TIBOR). It appears as the position taken in the results of the "全銀協TIBORのフォールバックに係る論点に関する市中協議" (Public consultation on issues related to JBA TIBOR fallbacks), published in March 2023. Checked on September 24, 2026. ↩

25. Japan Securities Clearing Corporation, "清算対象取引（金利スワップ）" (Transactions eligible for clearing: interest rate swaps, in Japanese), the table and conditions "3．スワップの種類" (Types of swaps), "4．変動金利の種類" (Types of floating rates), and "11．日数計算式" (Day count fraction). An English version is available: "Clearing Products," the table "List of Clearing Products" and "3．Types of Swaps," "4．Types of Floating Rates," and "11．Day Count Fraction." Checked on September 24, 2026. ↩ ↩²

26. Fujii et al. paper, equation (3.14) in Section 3.2, page 11. ↩

27. Fujii et al. paper, Section 2.5, page 8. ↩

28. Fujii et al. paper, Section 3.1, page 9. ↩

29. Fujii et al. paper, equation (3.13) in Section 3.1, page 10. ↩

30. Kawakami paper, Section 2 (1), pages 123–124. ↩

31. Fujii et al. paper, Section 2.3, page 4, and Section 2.4, page 7. The paper treats yen interest rate swaps as having floating payments every 6 months, and US dollar interest rate swaps as having annual fixed payments and quarterly floating payments. ↩