Pricing a CDS

This calculation comes up when a CDS is traded and the payments between the two sides are set, and when a CDS you hold is valued.

The two legs and the value of a CDS

The value of a credit default swap (CDS) is the difference between the present values of what the protection buyer receives and what the buyer pays. What the buyer receives is called the protection leg, and what the buyer pays is called the premium leg. In the pricing formulas, any credit event on the reference entity is simply called a default. How a CDS works (the protection buyer and seller, the reference entity, and credit events) is covered in Credit Default Swaps (CDS).

The protection leg is the amount the protection seller pays the buyer if the reference entity defaults before maturity. This amount is the notional amount multiplied by (1 − recovery rate)¹. With a notional of 1 billion yen and a recovery rate of 40%, the seller pays 600 million yen on default. Nobody knows when, or even whether, the reference entity will default, so the value of the protection leg is an expected value weighted by how likely default is.

The premium leg is the premium the protection buyer pays the seller every quarter. Premium payments stop when a default occurs. Under most contracts, the protection buyer also pays, on default, the premium accrued from the last payment date to the default date¹. With a notional of 1 billion yen and a premium of 100bp (1% a year), each payment is 1 billion yen × 1% × 0.25 years = 2.5 million yen.

The annual premium rate at which the two legs have equal value is called the par spread. A CDS premium is set so that the expected future cash flows of the protection buyer and the protection seller are equal². If the credit quality of the reference entity changes after the trade, the par spread moves, and a contract struck at a fixed premium shows a mark-to-market gain or loss.

Consider a hypothetical example in which Bank E, the protection buyer, enters a CDS with a notional of 1 billion yen, a premium of 100bp, and payments every three months. After the second payment and before the next payment date, the reference entity defaults, and the recovery rate is 40%.

Figure description

The figure shows the cash flows from the side of Bank E, the protection buyer, on a CDS with a notional of 1 billion yen and a premium of 100bp (1% a year). The recovery rate is 40%, and the amounts in the figure are in millions of yen. Bank E pays a premium of 2.5 million yen on each of the payment dates at 3 months and 6 months (1 billion yen × 1% × 0.25 years). If the reference entity then defaults before the next payment date, Bank E receives a payout of 600 million yen from the protection seller (1 billion yen × (1 − 40% recovery rate)). Premium payments stop after the default. The length of each arrow is a rough guide to the size of the amount, and the spacing between times is not proportional to the actual number of days. The figure does not show the premium accrued from the previous payment date, which is paid on default, or the upfront payment exchanged at the trade date for CDS after the 2009 standardization.

If no default occurs, Bank E keeps paying the premium and receives no protection. The value of the protection leg is the expected value of these two cases, weighted by how likely default is. The premiums paid can end up either larger or smaller than the amount received on default.

Hazard rate, survival probability, and recovery rate

The values of the two legs depend on the hazard rate, which measures how likely default is, and the recovery rate on default. The hazard rate is the probability, expressed as an annual rate, that the reference entity defaults in the short period just after a given time, given that it has not defaulted up to that time³. A Bank of Japan paper calls the same quantity the default intensity².

If the hazard rate is $\lambda$ and stays constant until maturity, the probability that the reference entity has not defaulted by year $t$ (the survival probability) $Q(t)$ is given by the following formula³.

$$Q(t) = e^{-\lambda t}$$

With a hazard rate of 2%, the survival probability at year 5 is $e^{-0.1}$ = 90.48%, and the probability of default within the five years is 9.52%. The survival probability has the same form as a discount factor. The hazard rate plays the role of the interest rate, and the survival probability plays the role of the discount factor³.

The recovery rate is the share of the face value of the defaulted reference entity's bonds that can be recovered. The protection leg of a CDS covers the loss, which is the face value minus the recovered amount, so the higher the recovery rate, the smaller the value of the protection leg. The actual recovery rate on default is set by an auction or a similar process¹. In pricing, the recovery rate is usually treated as a constant, unrelated to interest rates or the hazard rate⁴.

A hazard rate backed out of CDS spreads is a value under the risk-neutral probability used for pricing, and differs from the real-world probability of default³⁵. According to the Bank of Japan paper, sovereign CDS premiums contain a reward for taking on risk (a risk premium) in addition to each country's credit risk, and a default probability derived without allowing for the risk premium is overstated⁶.

Valuing the legs and finding the par spread

The value of the protection leg is the sum, over every possible default time, of the probability of default at that time multiplied by the present value of the amount received, $N(1-R)$. Here $N$ is the notional, $R$ is the recovery rate, $T$ is the maturity in years, and $D(t)$ is the discount factor at year $t$. With a constant hazard rate, the probability of default in a short period $dt$ at year $t$ is $\lambda e^{-\lambda t} dt$. The value of the protection leg, $PV_{\text{prot}}$, is therefore given by the following formula⁴.

$$PV_{\text{prot}} = N (1 - R) \int_0^T \lambda e^{-\lambda t} D(t) \, dt$$

The value of the premium leg is proportional to the annual premium rate $C$. So let $A$ be the value of the premium leg when the premium is paid at an annual rate of 1 (100%). $A$ is the sum of two parts. The first adds up, for each payment date $t_i$, the length of the period $\tau$ (0.25 years for a quarter) multiplied by the discount factor and the survival probability. The second is the accrued premium paid on default⁷.

$$A = \sum_{i=1}^{n} \tau \, D(t_i) \, Q(t_i) + (\text{accrued premium on default})$$

The value of the premium leg is $N C A$. The amount per basis point, $N \times A \times 0.0001$, is called the RPV01 (risky PV01). The RPV01 shows how much the value of the premium leg changes when the premium changes by 1bp⁸.

The par spread $S_p$ is the annual premium rate at which the two legs are equal. You find it by dividing the value of the protection leg by the value of the premium leg per unit of annual rate⁹.

$$S_p = \frac{PV_{\text{prot}}}{N A}$$

The value $V$, from the protection buyer's side, of a CDS struck at an annual premium rate $C$ is the protection leg minus the premium leg. From the formulas above, it can be written as the difference between the par spread and the premium, multiplied by the value of the premium leg per unit of annual rate⁹.

$$V = PV_{\text{prot}} - N C A = N (S_p - C) A$$

If the premium is paid continuously and the hazard rate is constant, the par spread is exactly the hazard rate × (1 − recovery rate). This relationship is called the credit triangle¹⁰.

$$S_p \approx \lambda (1 - R)$$

In actual contracts, where the premium is paid quarterly in arrears, this formula is an approximation. It serves as a rule of thumb for converting between spreads and hazard rates, and as the starting value when solving for a hazard rate¹⁰. With a hazard rate of 2% and a recovery rate of 40%, the spread is about 2% × 0.6 = 1.2% (120bp). Across the simulator's input ranges (hazard rate 0.1% to 10%, recovery rate 0% to 90%, interest rate −1% to 5%, maturity 1 to 10 years), the gap between the approximation and the par spread is less than 1% of the approximation.

Deep diveClosed-form formulas when the hazard rate and interest rate are constant

Take the interest rate as a constant continuously compounded rate $r$, so that $D(t) = e^{-rt}$. Setting $k = r + \lambda$, the integral in the protection leg can be solved as follows.

$$PV_{\text{prot}} = N (1 - R) \frac{\lambda}{k} \left( 1 - e^{-kT} \right)$$

The premium payments part is the sum of $\tau e^{-k t_i}$ at the end of each period, $t_i = i\tau$. The accrued premium on default is the integral of the time from the start of the $i$-th period, $t_{i-1}$, to the default time, multiplied by the probability of default at that time and the discount factor. It is given by the following formula.

$$\int_{t_{i-1}}^{t_i} (t - t_{i-1}) \lambda e^{-kt} \, dt = \lambda e^{-k t_{i-1}} \frac{1 - e^{-k\tau}(1 + k\tau)}{k^2}$$

If the premium is paid continuously, the premium leg per unit of annual rate becomes $\int_0^T e^{-kt} dt = (1 - e^{-kT})/k$. Dividing the protection leg formula by this gives a par spread of exactly $\lambda(1 - R)$. With quarterly payments in arrears, each payment comes later. So when interest rates are positive, the premium leg per unit of annual rate is slightly smaller, and the par spread is slightly larger than $\lambda(1 - R)$.

Standard coupons and the upfront payment

Since the 2009 standardization, a CDS pays its premium at one of a few set coupons rather than at the par spread, and the difference is exchanged at the trade date as an upfront payment. CDS originally traded, like interest rate swaps, at the premium that made the value zero at the trade date (the par spread)¹¹. CDS on reference entities with low credit quality had very high par spreads, so even before then they traded with a low coupon combined with an upfront payment¹¹.

The 2009 standardization aligned the coupon levels and payment dates¹². According to a 2013 OpenGamma paper, the standard coupons are 100bp and 500bp in North America, and 25bp, 100bp, 500bp, and 1,000bp in Europe¹¹. CDS with the same reference entity, maturity, and coupon have the same payment terms whenever they are traded, which makes prices easier to compare across trades¹¹.

The upfront payment is the value at the trade date of a CDS struck at coupon $C$, found from $V = N (S_p - C) A$ above. In terms of RPV01, it is (par spread − coupon) in basis points multiplied by the RPV01. If the par spread is above the coupon, the protection buyer pays the seller. If it is below, the seller pays the buyer¹³. The upfront payment expressed as a percentage of the notional is called points upfront (PUF). As of 2013, quoting in terms of a standard coupon and PUF was the market standard¹³.

Under the standard contract the protection buyer pays a full quarter of premium on the first payment date, whenever the trade is made. In return, the protection seller pays the buyer the premium accrued from the previous payment date to the trade date¹⁴.

Worked example

Consider a hypothetical example in which a CDS has a notional of 1 billion yen, a 5-year maturity, a 100bp coupon, and quarterly premium payments in arrears. The hazard rate is 2%, the recovery rate is 40%, and the interest rate is 1% continuously compounded, all constant until maturity. Every period is treated as 0.25 years long.

Intermediate values are shown to enough digits that recalculating from them gives the same results. The survival probability at year 5 is $e^{-0.02 \times 5}$ = 0.904837. Setting the sum of the interest rate and the hazard rate to $k$ = 0.01 + 0.02 = 0.03, the value of the protection leg follows from the closed-form formula in the deep dive above: 1 billion yen × 0.6 × (0.02 ÷ 0.03) × (1 − $e^{-0.15}$). The product 0.6 × (0.02 ÷ 0.03) is exactly 0.4, and $e^{-0.15}$ = 0.86070797643, so the value is 1 billion yen × 0.4 × 0.13929202357 = 55,716,809.43 yen. Rounded to the nearest yen, this is 55,716,809 yen.

The premium leg per unit of annual rate, $A$, is 4.625677714 from the premium payments plus 0.011593159 from the accrued premium on default, for a total of 4.637270873¹⁵. The RPV01 is 1 billion yen × 4.637270873 × 0.0001 = 463,727 yen.

The par spread is 55,716,809.43 yen ÷ (1 billion yen × 4.637270873) = 1.2015%, or 120.15bp. The 0.15bp gap from the credit triangle's 120bp comes from paying the premium quarterly in arrears. The value of the premium leg at a 100bp coupon is 1 billion yen × 1% × 4.637270873 = 46,372,708.73 yen, or 46,372,709 yen rounded. The par spread exceeds the coupon by about 20.15bp, so the protection buyer pays the seller an upfront payment. The amount is the difference between the values of the two legs. Calculated to fractions of a yen, it is 55,716,809.43 yen − 46,372,708.73 yen = 9,344,100.70 yen, or 9,344,101 yen rounded¹⁶. That is 0.93% of the notional.

What moves the values

The par spread moves almost in proportion to the hazard rate and to (1 − recovery rate). With a constant hazard rate, the interest rate and the maturity barely change the par spread, but they change the upfront payment through the RPV01. Changing one input of the worked example at a time gives the following results (100bp coupon, from the protection buyer's side).

Doubling the hazard rate roughly doubles the par spread. Raising the recovery rate from 40% to 60% lowers (1 − recovery rate) from 0.6 to 0.4, so the spread falls to two-thirds. The recovery rate does not change the RPV01, because the recovery rate does not enter the value of the premium leg. With a 10-year maturity, the par spread stays the same but the RPV01 rises by a factor of about 1.9, so the same spread difference produces a larger upfront payment. At a 0% interest rate there is no discounting. The quarterly payments plus the accrued premium on default then add up to the same amount as continuous payment, and the par spread is 120.00bp, the same as the credit triangle¹⁵.

If the hazard rate changes after the trade, one side (the protection buyer or the seller) gains and the other loses. Suppose the buyer pays 9,344,101 yen to enter the worked example trade and the hazard rate rises to 4% right afterward. The contract's value to the buyer becomes 61,990,965 yen, a gain of 52,646,864 yen over the upfront payment made. If the hazard rate falls to 1% instead, the value becomes −18,973,089 yen, a loss of 28,317,190 yen. The protection seller's gain or loss is the reverse in each case¹⁵.

Calculation in practice and the ISDA Standard Model

According to a November 2021 IHS Markit paper, CDS with a standard coupon are mainly quoted as a quoted spread or as a price¹⁷. The price (the clean price) is defined as 100% − PUF, expressed as a percentage of the notional¹³. If the price is below 100%, PUF is positive and the protection buyer pays the upfront payment. If it is above 100%, the buyer receives it¹³. For CDS indices quoted as a price, the upfront payment is likewise notional × (100 − price) ÷ 100, the same relationship¹⁸.

The quoted spread is another way of quoting the same trade as PUF¹⁹. The ISDA CDS Standard Model (the ISDA Standard Model) is used to convert between the quoted spread and PUF¹⁷. The Standard Model is also used to build a hazard rate curve from quotes²⁰²¹. The Standard Model covered in the OpenGamma paper is a calculation program written in C. It builds the interest rate curve and the hazard rate curve with forward rates that are constant between nodes²⁰. Any implementation that uses the same curve assumptions gives the same price²⁰.

To find PUF from a quoted spread, you find the constant hazard rate at which a CDS with the quoted spread as its coupon has a value of zero, and then price the CDS with its actual coupon at that hazard rate. To find the quoted spread from PUF, you reverse these steps. If the quoted spread equals the coupon, PUF is zero¹⁹.

When a reference entity has CDS quotes at several maturities, the hazard rate curve that reproduces all the quotes is solved for in order, from the shortest maturity up (bootstrapping)²². A quoted spread is calculated from a constant hazard rate for each maturity and is not a par spread, so the quotes are converted to PUF before the curve is built²². A curve built from quotes on actively traded maturities is used to price CDS at less traded maturities, and CDS traded some time ago²³. Maturities are standardized, and 5 years is the most liquid maturity¹⁷. A study of trades in 2010 found that 43% of single-name CDS trades had a 5-year maturity²⁴. Quotes for indices, which bundle many names, are covered in CDS Indices (iTraxx, CDX).

Assumptions and limitations

The formulas and the simulator on this page use a simple model in which the hazard rate, interest rate, and recovery rate are all constant until maturity and unrelated to each other. Actual pricing uses a hazard rate curve that differs by maturity and a discount curve built from market interest rates²⁰. Spreads and recovery rates show a negative correlation, but pricing usually ignores it⁴.

The simulator treats every period as 0.25 years long, and does not take into account day counts, the conventions for maturity and payment dates, the number of days from trade to settlement, or the accrued premium returned at the trade date. So its upfront payments differ in the details from those calculated with the Standard Model. How the recovery rate is set also varies with the type of reference entity and the purpose of the analysis. For example, the Bank of Japan paper assumes 25% for sovereign CDS²⁶.

Notes

1. OpenGamma Quantitative Research, "The Pricing and Risk Management of Credit Default Swaps, with a Focus on the ISDA Model" (October 2013; the "OpenGamma paper"), section 2, "The Standard CDS Contract," page 1. ↩ ↩² ↩³

2. Kei Ikeda, Kazuhiro Hiraki, and Takeshi Yamada, "ソブリンCDS プレミアムの要因分解" (Decomposition of sovereign CDS premiums) (Bank of Japan Working Paper Series No.12-J-9, September 2012, in Japanese), section 3 (1), "誘導型アプローチを用いたCDS プレミアムの評価モデル" (Pricing model for CDS premiums using a reduced-form approach), page 6. ↩ ↩²

3. OpenGamma paper, section 3.3, "Credit Curve," page 6. Equations (6) and (7) with a constant hazard rate. ↩ ↩² ↩³ ↩⁴

4. The same paper, section 4.1, "The Protection leg," page 7. The formula here is the paper's equation (14), with the recovery rate, interest rate, and hazard rate independent of each other. The negative correlation between spreads and recovery rates is in note 13. ↩ ↩² ↩³

5. Kei Ikeda, Kazuhiro Hiraki, and Takeshi Yamada, "ソブリンCDS プレミアムの要因分解" (Decomposition of sovereign CDS premiums) (Bank of Japan Working Paper Series No.12-J-9, September 2012, in Japanese), section 3 (1), page 7. ↩

6. The same paper, "要旨" (Summary), page 1. ↩

7. OpenGamma paper, section 4.2, "The Premium Leg," page 8. Equations (15) to (18); the $A$ on this page is equation (18) divided by $NC$. ↩

8. The same paper, page 9, the end of section 4.2, "The Premium Leg," and equation (19). It defines RPV01 per unit of coupon (the $A$ on this page) and notes in note 17 that the usual definition is per basis point. ↩

9. The same paper, section 4.5, "The Par Spread," page 10. The paper's equations (27) and (28). ↩ ↩² ↩³

10. The same paper, section 4.6, "The Credit Triangle," page 11. The paper's equation (32). ↩ ↩²

11. The same paper, section 2.2, "The Coupons," page 3. The standard coupon levels are in note 5. ↩ ↩² ↩³ ↩⁴

12. Chen et al., "An Analysis of CDS Transactions: Implications for Public Reporting" (Federal Reserve Bank of New York Staff Reports No.517, September 2011), page 15 and note 1 on page 2. ↩

13. OpenGamma paper, section 2.2 (page 3), and section 4.3, "CDS Valuation," and section 4.3.1, "Points Upfront" (page 9). ↩ ↩² ↩³ ↩⁴

14. The same paper, page 2 (the main text of section 2), and page 4, section 2.4, "Differences before and after 'Big Bang'." ↩

15. Calculated with the same formulas as the simulator's calculation module (src/lib/simulators/cds.ts). The premium payments part is $0.25 \times \sum_{i=1}^{20} e^{-0.0075 i}$, and the accrued premium on default is the sum of the deep-dive formula over 20 periods. ↩ ↩² ↩³

16. The values of the protection leg and the premium leg in the table are rounded to the nearest yen, so the difference between the two table values is 9,344,100 yen, 1 yen off the upfront payment. The 20.15bp and the RPV01 of 463,727 yen are also rounded, and 20.15 × 463,727 yen is 9,344,099 yen. ↩

17. IHS Markit, "CDS Indices Primer" (November 2021; the "IHS Markit paper"), page 9, "Fixed coupons, upfront amounts and quoted spreads" and "Standard maturities, coupon dates and rolls." The conversion with the ISDA Standard Model is in note 12 on the same page. ↩ ↩² ↩³

18. The same paper, pages 20–21, the trade example (CDX.NA.HY) under "How do I trade?" ↩

19. OpenGamma paper, section 6.1.1, "Conversion between PUF and Quoted Spreads," page 16. ↩ ↩²

20. The same paper, section 5, "The ISDA Model," page 12. ↩ ↩² ↩³ ↩⁴

21. The same paper, the beginning of section 6, "Calibration of the Credit Curve," page 15. ↩

22. The same paper, section 6.2, "Multiple CDSs," and section 6.2.1, "Calibration from Quoted Spreads," pages 16–18. ↩ ↩²

23. The same paper, section 2.3, "Pricing," page 3. ↩

24. Chen et al., "An Analysis of CDS Transactions: Implications for Public Reporting" (September 2011), page 15. The data are trades from May 1 to July 31, 2010 (page 6). ↩

25. OpenGamma paper, section 2.1, "Dates" (page 2), and section 3.1, "Day Count Conventions" (page 4). ↩ ↩² ↩³

26. Kei Ikeda, Kazuhiro Hiraki, and Takeshi Yamada, "ソブリンCDS プレミアムの要因分解" (Decomposition of sovereign CDS premiums) (Bank of Japan Working Paper Series No.12-J-9, September 2012, in Japanese), note 8, page 8. ↩