Overview Overview

This article will model the protection leg, further discuss the modeling methods of the term structure of interest rates and expected recovery rate, and finally use the integrated formula to calculate the value of CDS.

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Protection Leg Valuation Method

The value of the Protection leg is the face value of the insurance paid after the credit event (100% - R). R is the expected recovery rate—precisely, it is the expected price at which CTD debt is insured in the event of a credit event. There may be a delay of up to 72 days between the notification of the credit event and the settlement of the Protection leg portion of the payment, but we generally assume that such payment is immediate.

• When pricing the insurance tranche, it is important to consider the timing of the credit event, as this can have a significant impact on the present value of the insurance tranche – especially for default swaps with longer maturities. In the hazard rate approach, we can solve this timing problem by adjusting each small time interval [s, s+ds] between time tV and time tN over which credit events may occur. The steps are described as follows:

• Calculate the probability of surviving to a certain point in the future s equal to Q(tV,s)

Calculate the probability ds of a credit event occurring in the next small time increment, which is given by λ(s).ds.

• 3. At this point, the amount (100% - R) has been paid and we discount back to today's risk-free rate Z(tV,s).

We then consider the probability of this happening at any time from s = tV to the maturity date tN. Strictly speaking, the duration of a credit event should not be less than one day. However, we simplify the exposition by assuming that a credit event can occur within the same day with little impact on valuation.

We can now derive the discounted value of the expected recovery value, which is:

In the formula, R is the expected recovery price of CTD assets when a credit event occurs. This integral makes this expression cumbersome to compute. It is possible to show that, without any loss of accuracy, we can simply assume that credit events can only occur at a finite number of M discrete points per year. For tN-year default swaps, we have M × tN discrete times, which we denote as M = 1. , M×tN. Then, we have

The smaller the value of M, the less calculations need to be done. However, this also means reduced precision. In terms of spread variation, for a flat risk rate structure, the percentage difference between the calculated spreads in the continuous and discrete cases is r/2M, where r is the continuous compounded default-free rate. The quality of this approximation is shown in Fig. 7 for different values ​​of M and r. For example, assuming r = 3%, M = 12 (corresponding to monthly intervals) we have a percentage error spread of 0.125%, that is, an absolute error of 1 bp with a spread of 800 bp while the continuous case. This accuracy is well within typical bid-ask spreads.

Calibration Expected Recovery

A necessary input that we have not yet discussed is the rate of return R, which is not a market observable input, unlike spreads or the term structure of interest rates. The expected recovery R is not the expected value of the asset during training after default. Instead, it is the price of the CTD asset expressed as a percentage of face value. This is similar to how rating agencies such as Moody's define recovery rate statistics.

There are a few caveats to the rating agencies' recall statistics, though: (i) rating agencies don't consider restructurings to be defaults, while standard default swaps (CDSs) do; (ii) they're heavily skewed toward U.S. companies , because U.S. companies are the largest source of default data, and thus may not be applicable to companies in other countries; (iii) they are historical, not future-oriented, and thus do not take into account market expectations for the future; (iv) they do not specify name or department. Still, for high-quality investment-grade credit, most dealers use recovery data from rating agencies as a starting point.

These bonds typically show average recoveries by seniority and credit instrument type, usually focusing on US corporate bonds. Adjustments may be made for non-US company names and certain industrial sectors.

Using valuation models to extract information about recovery values ​​from bond prices may be one way to overcome this calibration problem. However, for high-quality bonds, this is difficult because the low probability of default means that the recovery rate is only a small fraction of the bond price, and the same order of magnitude as the bid-ask spread. However, at low spread levels, the mark-to-market of default swaps is very insensitive to recovery assumptions. Bonds of much lower credit quality are much more sensitive to recovery rates and, we hope, lower bond prices start to reveal more about market expectations of future recovery rates.

Calculating the CPD Swap Spread

We now propose a model that values ​​the insurance and premium of CDs. The next step is to calculate the survival probability based on the default swap spread quoted in the market. This is the breakeven distribution, i.e.

PV of Premium Leg = PV of Protection Leg.

For a new contract, we have tV = t0, so, substituting and rearranging the formula, we get

For the break-even point, RPV01 is defined as formula (5).

We now have a direct relationship between default swap spreads in the market and their implied survival probabilities. However, this is still not enough to enable us to extract all required survival probabilities. To see this, consider the example of a 1-year CDS with a quote spread of 85 bps. Assuming quarterly payments in the premium phase, monthly discretization frequency (M=12) and premium accumulation, the formula can be rewritten as

In this equation, we know all the accrual factors, we can make an assumption about the recovery rate R, and we can calculate all the Libor discount factors through the Libor discount curve. What we need to know is the maximum survival probability of 12+4=16. Obviously, this equation cannot give all survival probabilities. Therefore, we need to make a simplifying assumption about the term structure of survival probabilities.

First, the full market value of the long-term insurance default swap position is

in

If the premium is part of the contract, 1PA=1, otherwise 0. The value of the current market spread to maturity S(tV,tN) is given by solving

Conclusion

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