Collateral and Discounting

If you value or manage the risk of bonds and interest rate swaps, you choose the discount rate for each trade according to the terms of its collateral agreement.

Collateral terms and the discount rate

The rate used to discount a collateralized derivative is set by the collateral terms. Under a collateral agreement, the party for which the trade has a positive market value receives collateral from the other party and pays that party interest on the collateral it receives. A 2009 paper by Fujii and others describes the usual practice. Collateral is commonly cash in the currencies of developed countries, such as the US dollar, the euro, and the yen. The interest on cash collateral is usually the overnight rate of that currency¹.

Collateral on OTC derivatives is generally exchanged under a credit support annex (CSA), the collateral clause attached to the master agreement for derivatives². The main terms set in a CSA, and their effect on discounting, are as follows.

Figure description

The first question asks whether the full market value is exchanged as cash collateral (almost) continuously. Daily exchange comes close to this. If the answer is no, the trade is outside the assumptions of discounting at the collateral rate, and the part up to the threshold and the changes in market value between transfers have to be treated separately. If yes, go to the next question. The second asks whether there is a single collateral currency, the same as the currency of the cash flows. If the answer is no, a difference due to the currencies is added to the collateral rate, and where the posting side can switch currencies at any time, the currency with the highest discount rate is chosen. If yes, discount at the interest rate on the collateral. Treatment outside the assumptions is covered in "Discounting trades without collateral", and the currency difference in "Trades with collateral in another currency".

Why the collateral rate is used for discounting

For trades in which cash collateral covering the full market value is exchanged continuously, future cash flows are discounted at the interest rate paid on the collateral⁴. Suppose Bank L holds a trade under which it will receive cash in the future, and the trade has a positive market value. Bank L immediately receives cash collateral equal to the market value, pays interest on it at the collateral rate, and returns it in full at the end. This is the same as Bank L borrowing at the collateral rate to fund its position in the trade. Conversely, if the market value is negative, the collateral Bank L posts is equivalent to a loan to the other party at the same rate⁵. In both cases the position is funded at the collateral rate, so the discount rate is also the collateral rate.

As a formula, the value $V(t)$ at time $t$ of a trade that pays $V(T)$ at maturity $T$ is as follows⁴.

$$V(t) = E^{Q}_t\left[ e^{-\int_t^T c(s)\,ds}\, V(T) \right]$$

$c(s)$ is the collateral rate at time $s$ (annual rate), and $E^{Q}_t$ is the expectation in Risk-Neutral Valuation, given the information at time $t$. For a collateralized trade, the collateral rate, not the risk-free rate, enters the discounting.

For a trade with a fixed amount to receive, the expectation of $e^{-\int_t^T c(s)\,ds}$ is itself the discount factor. If the collateral rate is the overnight rate, this discount factor can be found from the fixed rates in the market for OIS, which exchanges a fixed rate for the compounded overnight rate⁶. Building the curve of discount factors is covered in The Multi-Curve Framework.

Deep diveDeriving the discounting formula from the collateral account

Let $X(s)$ be the balance of the collateral account, $V(s)$ the value of the trade, and $r(s)$ the risk-free rate, and let $y(s) = r(s) - c(s)$. Collateral received can be invested at the risk-free rate, and the collateral rate is paid to the other party, so the account balance grows at the difference $y(s)$. Holding $a(s)$ units of the trade, the account balance changes as follows⁷.

$$dX(s) = y(s)\,X(s)\,ds + a(s)\,dV(s)$$

If the starting balance equals the value of the trade, $X(t) = V(t)$, and the number of units held is $a(s) = e^{\int_t^s y(u)\,du}$, the balance at maturity is $X(T) = e^{\int_t^T y(s)\,ds}\, V(T)$. Finding the present value of this account as the expectation discounted at the risk-free rate, and using $r(s) - y(s) = c(s)$, gives the formula in the main text.

$$V(t) = E^{Q}_t\left[ e^{-\int_t^T r(s)\,ds}\, X(T) \right] = E^{Q}_t\left[ e^{-\int_t^T c(s)\,ds}\, V(T) \right]$$

Discounting trades without collateral

Trades without collateral are valued in line with the rate that matches how the funds are raised (the funding rate), not the collateral rate. The idea is that, just as collateralized trades use OIS rates that match the collateral, using a discount rate that matches the source of funding is consistent in theory. A 2018 paper by the Bank of Japan's Institute for Monetary and Economic Studies said that valuing at LIBOR appeared to be market practice at that time⁸. Panel bank publication of yen LIBOR later ended at the end of December 2021⁹.

The cost of funding can be put into the discount rate, or it can be treated separately as a valuation adjustment (FVA). Either way, the funding cost of an uncollateralized trade is reflected in its market value. After the financial crisis, adjustments based on a firm's own funding rate became common for uncollateralized trades, including trades with only partial collateral¹⁰. The part not covered by collateral also reflects adjustments for the credit risk of the counterparty and of the firm itself (CVA and DVA)¹¹.

In Japan around 2018, collateral was not always required in trades between nonfinancial companies, or in client clearing, where a financial institution passes a nonfinancial company's trades on to a central counterparty for clearing. For this reason, a certain share of uncollateralized trades was expected to remain, mainly with nonfinancial companies, until market practice and regulation spread¹².

Trades with collateral in another currency

When the currency of the cash flows differs from the currency of the collateral, the difference between the two currencies is added to the discount rate. Let $i$ be the currency of the cash flows and $j$ the collateral currency, and write the difference between the risk-free rate and the collateral rate of currency $k$ as $y^{(k)} = r^{(k)} - c^{(k)}$. With the difference between the two currencies $y^{(i,j)} = y^{(i)} - y^{(j)}$, the value of the trade is as follows¹³.

$$V^{(i)}(t) = E^{Q_i}_t\left[ e^{-\int_t^T \left(c^{(i)}(s) + y^{(i,j)}(s)\right) ds}\, V^{(i)}(T) \right]$$

$Q_i$ is the risk-neutral probability when discounting at currency $i$ rates, and $V^{(i)}$ is the value in currency $i$. If the collateral is in the same currency as the cash flows, $y^{(i,i)} = 0$, and the formula reduces to the collateral-rate discounting formula. If $y^{(i,j)}$ is negative, the discount rate falls, and the present value of cash flows covered by currency $j$ collateral is larger than with currency $i$ collateral.

The risk-free rate itself cannot be observed in the market, so $y^{(i,j)}$ is found from values in the cross-currency swap market. A 2010 paper extracted the differences in $y$ between the euro and the dollar, and between the yen and the dollar, from spreads in the cross-currency swap market (the cross-currency basis). Over the paper's period, a large part of the movement in cross-currency swap spreads came from movements in this difference¹⁴.

In the market around 2009, the US dollar was often used as collateral for trades with cash flows in several currencies, and dollar-yen cross-currency swaps usually had dollar cash collateral¹⁵. In a paper using 2010 market data, the present value of a yen cash flow 10 years ahead was about 3% larger when covered by dollar collateral instead of yen collateral. In other words, dollar collateral tends to be expensive for the side that posts it¹⁶.

Contracts with a choice of collateral currency

If a CSA allows several currencies as collateral and the posting side can switch currencies at any time, the posting side chooses the currency that favors it¹⁷. The side that posts collateral has a negative market value and wants to make the size of that value smaller, so it chooses the currency with the highest discount rate. With $C$ as the set of eligible currencies, the discount factor for currency $i$ is the discount factor at the collateral rate multiplied by the following value¹⁷.

$$E^{Q_i}_t\left[ e^{-\int_t^T \max_{j \in C} y^{(i,j)}(s)\,ds} \right]$$

The 2010 paper calls this right to choose a "cheapest-to-deliver" (CTD) option. In an example of a contract that allows either the euro or the dollar (market data of March 16, 2010), choosing the currency optimally raised the effective discount rate by about 50bp a year, even with an annual volatility of only 50bp for the difference in $y$¹⁸. The adjustment that values this right is called CTDVA. Valuation adjustments with complex option features like CTDVA can make hedging harder. Around 2017, some firms limited the assets eligible as collateral to avoid these option features¹⁹. Around 2010, there was also a tendency to move toward CSAs with a single collateral currency to reduce operational work, while contracts that allowed both the home currency and the dollar as collateral seem to have been not unusual¹⁸.

Worked example: 10 billion yen received in 5 years

Bank L has a trade under which it will receive 10 billion yen from Company E in 5 years. It values the trade with only the collateral terms changed. Consider a hypothetical example in which rates are constant over the period and expressed with continuous compounding. The discount rate with the yen overnight rate as the collateral rate is 1.00%, the yen-dollar difference $y^{(\text{JPY},\text{USD})}$ is −0.30%, and Bank L's funding rate used for an uncollateralized trade is 1.50%. The discount factor is then $e^{-\text{discount rate} \times 5}$.

The discount rate with dollar collateral is 1.00% − 0.30%, or 0.70%. The present value is 144 million yen larger than with yen collateral and 379 million yen larger than with no collateral²¹. With no collateral, it is 235 million yen smaller than with yen collateral. If Bank L were the side paying 10 billion yen, the present value of the liability would change by the same amounts under the same terms. Compared with an uncollateralized trade, in a collateralized trade the side with a positive market value gains by funding at the low collateral rate, and the side with a negative market value loses because the collateral it posts earns only the collateral rate⁵.

In the contract with a choice of yen or dollars, Company E, whose market value is negative, posts the collateral. If $y$ is −0.30%, Company E chooses yen, which has the higher discount rate (1.00%), and the present value is 9.512 billion yen, the same as with yen only. If $y$ moves to +0.20%, the discount rate with dollars becomes 1.00% + 0.20%, or 1.20%, and Company E switches to dollars. The present value seen from Bank L falls to 9.418 billion yen, a loss of 94 million yen. Seen from Company E, the liability becomes smaller by the same amount. This calculation holds rates constant and does not include the option value that comes from movements in $y$.

Interest on variation margin at a central counterparty

For interest rate swaps cleared by Japan Securities Clearing Corporation (JSCC), a central counterparty, variation margin is also exchanged with interest. JSCC's interest rate swap clearing has two kinds of cleared contracts. On Cleared Contracts (Collateral Type), variation margin is exchanged. On Cleared Contracts (Settlement Type), a settlement amount for the daily change in market value (the Mark-to-Market Difference) is exchanged every business day instead of variation margin²².

Under the Collateral Type, JSCC and clearing participants exchange interest on variation margin every business day in the currency of the cleared contract²³. The base rate used as the interest rate is the weighted average of the uncollateralized overnight call rate published by the Bank of Japan (TONA) for yen cleared contracts, and the Federal Funds Rate for US dollar cleared contracts. Yen interest is the variation margin deposited on the previous business day × the base rate × the number of days from the previous business day to the calculation date ÷ 365. Under the Settlement Type, an adjustment amount on the Mark-to-Market Difference is found with the same formula²⁴. Because the interest on yen variation margin is the overnight rate, applying the discounting idea above means that yen cash flows covered by this collateral are discounted at the yen overnight rate.

Assumptions and limitations

The formula that discounts at the collateral rate is derived under simplifications: collateral is exchanged continuously, it covers the full market value in cash, and the threshold is 0. Under these conditions, losses from the counterparty's default can be ignored. Because daily exchange of collateral was spreading as the best market practice, the 2009 paper says this approximation is not far from reality⁷. In Japan around 2017, however, collateral calls (margin calls) were mostly weekly, although contracts with daily calls were increasing²⁷. When the interval between transfers widens, changes in market value in the meantime are not covered by collateral.

Around 2017, a threshold of 0 had become common in trades between banks. Margin rules also require variation margin matching the market value to be exchanged with a threshold of 0³.

The formulas on this page assume cash collateral. Bond prices move, and selling bonds can take time, so a CSA sets a discount (haircut) for each bond. Because prices of even highly rated bonds fell sharply in the financial crisis, cash collateral, which has no price changes, was becoming common around 2017, mainly in Europe and the United States²⁸. The funding cost of initial margin is handled by a valuation adjustment (MVA), separately from the interest on variation margin¹⁰.

Notes

1. Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi, "A Note on Construction of Multiple Swap Curves with and without Collateral" (first version June 2009, revised January 2010), Section 3.1, page 9. ↩

2. Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (XVA taking into account the combined effects of financial regulations), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), "CSA 契約" (CSA agreements) in Section 3 (1), pages 117–118 (in Japanese). The paper lists four main CSA clauses: a. the threshold, b. the minimum transfer amount, c. the valuation frequency, and d. eligible collateral. They correspond to the four terms in the table other than the interest rate on collateral. ↩

3. Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (XVA taking into account the combined effects of financial regulations), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 2 (page 113) and "イ. 信用極度額" (a. Threshold) in Section 3 (1) (page 117) (in Japanese). ↩ ↩²

4. Fujii, Shimada, and Takahashi (first version June 2009, revised January 2010), Section 3.1, equation (3.5) (page 9) and the explanation that follows (page 10). The paper's $h(t)$ corresponds to $V(t)$ on this page. ↩ ↩²

5. The same document, Section 3.1, pages 10–11. The paper says that, compared with an uncollateralized trade (a trade funded at LIBOR), the side gains if the market value is positive and loses if it is negative. ↩ ↩²

6. Fujii, Shimada, and Takahashi (first version June 2009, revised January 2010), Section 3.2, equations (3.14)–(3.16) (page 11). The paper assumes the OIS itself is collateralized with the overnight rate as the collateral rate, and finds the collateral-rate discount factor $D_{t,T}$ from the OIS fixed rate. ↩

7. The same document, Section 3.1, page 9. The derivation is in equations (3.1)–(3.5). The paper writes the collateral account as $V(t)$ and the value of the trade as $h(t)$, which correspond to $X(t)$ and $V(t)$ on this page. ↩ ↩²

8. 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), Section 2 (1), page 123 (in Japanese). ↩

9. Bank of Japan and Financial Services Agency, "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, May 2022, in Japanese), the summary on page 1. An English version is available: "Review of JPY LIBOR Transition and Future Initiatives" (Bank of Japan Review 2022-E-4, May 2022). The page number refers to the Japanese version. ↩

10. Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (XVA taking into account the combined effects of financial regulations), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 1, page 110 (in Japanese). ↩ ↩²

11. 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), Section 2 (1) and Figure 1, page 124 (in Japanese). ↩

12. The same document, footnote 15 on page 130. ↩

13. Masaaki Fujii, Akihiko Takahashi, "Choice of Collateral Currency" (first version April 2010, revised December 2010), equation (2.4) in Theorem 1 of Section 2 (page 3) and equation (2.6) (page 4). This page rewrites $r^{(i)} - y^{(j)}$ in equation (2.4) as $c^{(i)} + y^{(i,j)}$. The paper's $h^{(i)}$ corresponds to $V^{(i)}$ on this page. ↩

14. The same document, Section 4.2 (page 8) and Section 5 (page 11). The paper says the risk-free rate cannot be observed and is contained in the collateral rate $c$ and in $y$. ↩

15. Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi, "A Note on Construction of Multiple Swap Curves with and without Collateral" (first version June 2009, revised January 2010), Section 3.4, pages 12–13. ↩

16. Masaaki Fujii, Akihiko Takahashi, "Choice of Collateral Currency" (first version April 2010, revised December 2010), Section 5 (pages 9–10) and Figure 6. ↩

17. The same document, Section 1 (page 2) and equation (5.3) in Section 5 (page 10). ↩ ↩²

18. The same document, Section 5 (page 10) and Figure 9. The paper models the movement of the difference in $y$ with the Hull-White model. ↩ ↩²

19. Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (XVA taking into account the combined effects of financial regulations), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 2 (page 113) and footnote 4 (in Japanese). ↩

20. Masaaki Fujii, Akihiko Takahashi, "Choice of Collateral Currency" (first version April 2010, revised December 2010), Section 5 (page 10) and equation (5.3). ↩

21. The amounts multiply 10 billion yen by the discount factor before rounding, then round to the nearest million yen. The differences are calculated from the rounded values shown in the table. ↩

22. Japan Securities Clearing Corporation, "金利スワップ取引清算業務に関する業務方法書" (Business rules for the interest rate swap clearing business, in Japanese), Article 2, items (35)-2, (36)-2, and (36)-3, page 8. The English names Cleared Contract (Collateral Type), Cleared Contract (Settlement Type), and Mark-to-Market Difference are those used in JSCC's English form of the clearing brokerage agreement (Form 3-2 in "金利スワップ取引清算業務に関する業務方法書の取扱い" (Handling of the business rules for the interest rate swap clearing business), pages 122 and 131). Checked on September 26, 2026. ↩

23. The same document, Articles 83 and 83-2, page 68. Checked on September 26, 2026. ↩

24. Japan Securities Clearing Corporation, "金利スワップ取引清算業務に関する業務方法書の取扱い" (Handling of the business rules for the interest rate swap clearing business, in Japanese), Article 42, pages 40–41. Checked on September 26, 2026. ↩ ↩²

25. Japan Securities Clearing Corporation, "金利スワップ取引清算業務に関する業務方法書" (Business rules for the interest rate swap clearing business, in Japanese), Articles 83 and 83-2, page 68. The paying clearing participant pays by 11:00 on the business day after the calculation date, and JSCC pays the receiving participant at or after 13:30 on that day; for cleared contracts in foreign currencies, the times and the way business days are counted differ. Checked on September 26, 2026. ↩

26. Japan Securities Clearing Corporation, "金利スワップ担保・決済" (Interest rate swap collateral and settlement, in Japanese), "当初証拠金、日中証拠金及び清算基金の差入れ及び返戻並びに変動証拠金の決済方法" (Methods for depositing and returning initial margin, intraday margin, and the clearing fund, and for settling variation margin). Interest on variation margin is exchanged together with variation margin, coupons, and upfront fees as "variation margin, etc.," and the yen portion is exchanged through BOJ-NET. Checked on September 26, 2026. ↩

27. Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (XVA taking into account the combined effects of financial regulations), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), "ハ. 評価頻度" (c. Valuation frequency) in Section 3 (1), page 117 (in Japanese). ↩

28. Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (XVA taking into account the combined effects of financial regulations), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), "ニ. 適格担保" (d. Eligible collateral) in Section 3 (1), page 118 (in Japanese). ↩