Modeling Credit Risk and the Credit Valuation Adjustment

When you think of a government bond, you probably note that it's pretty much as close as you can get to a risk-free rate. But that's not the case with all bonds. In regards to the risk-free rate, what's the big difference between a corporate bond and a government bond?

Exactly. If the government bond is the risk-free rate, then the corporate bond will have some additional risk. And that risk is captured in the corporate bond's yield, which is usually higher than the government bond's yield. That yield difference is the credit spread (or G-spread), and it captures the compensation that investors receive for taking on _default risk_, which is the possibility that the issuer won't pay on the scheduled date. This is different than credit risk.

Not quite. Corporate bonds and government bonds both have liquid markets.

No. Both corporate bonds and government bonds have various maturity dates.

Credit risk captures both the default probability and the potential amount lost if default occurs. Yet, it's possible that default risk can be high and that credit risk can be low. What type of security would have a high default risk but low credit risk?

Not quite. With a corporate junk bond, both default risk and credit risk would be high.

That's not it. If a zero-coupon bond defaults, investors would lose both interest and principal, so default risk and credit risk are high.

Right! With a highly collateralized corporate junk bond, default risk is high, as evidenced by the rating, but since there's collateral, credit risk is low because the investor will probably be paid back due to the collateral. That's why it pays to focus on credit risk and model it accordingly. Although bonds have other features that influence yields (taxation and liquidity), you're going to focus on credit risk and credit risk modeling here. And the first factor in modeling credit risk is the expected exposure, which is the amount of money that the investor could lose in the event that a default occurs.

These events are more than just bankruptcy or missing a payment. They can also include failing to meet a different obligation or violating a financial covenant. In simple terms, the expected exposure for a one-year annual-pay 5% bond is 105, which is calculated as the principal plus interest. This application can be built upon to develop a true credit model. For example, suppose you're modeling a four-year zero-coupon corporate bond. If you assume that default only occurs at year-end and that a flat government bond yield is 2.5%, then you can calculate the expected exposure by discounting the 100 par value by the government bond yield. So which year-end will have the lowest expected exposure?

Not quite. At the end of year two, the zero-coupon bond is discounted 1.025 to the second power.

No, actually. If default is only assumed at year-end, then the investor would be paid back at the end of year four.

Excellent! At the end of year one, the expected exposure will be the lowest, since the credit risk is the highest. Since there are still three years remaining on the bond, the calculation at the end of year one is >$$\displaystyle \frac{100}{(1 + 0.025)^3} = 92.86$$.

This calculation is then repeated for the other years of the bond, with year four having no discount. This leads to the second factor, calculating the recovery rate, which is the percentage of the loss recovered from a bond in default. This rate can depend on multiple factors like securitization, collateralization, and leverage in the capital structure, along with its place in the capital structure. What's another way to think about the bond's place in the capital structure?

That's not it. The bond's coupon rate doesn't impact the legal obligation to pay proceeds before other securities.

Not quite. Some bonds can have a higher priority even if the maturity is longer than other securities.

Nice work! The bond's seniority in relation to other liabilities on the balance sheet is another influence on the recovery rate because the bond may have higher or lower priority to be paid in bankruptcy. So there are lots of factors that go into the recovery rate. To calculate the recovery rate, most analysts assume 40% and then times that percentage by each year's expected exposure. For year one, that's >$$92.86 \times 0.40 = 37.14$$.

From this value, you can then calculate the loss given default, which is the amount of loss if a default occurs. It's the difference between the expected exposure and the recovery rate: >$$92.86 - 37.14 = 55.716$$. This can also be put into percentage terms, which is called the loss severity. If the recovery rate is 40%, what do you think the loss severity is?

No. The investor will recover 40% of the investment in the bond.

That's not it. The recovery rate is 40%, meaning that 40% is the percentage of what's recovered, not what's lost.

Yes! If 40% is recovered, then 60% is lost, which makes 60% the loss severity. Essentially, it's the difference between 100% and the recovery rate. Note, however, that the recovery rate and loss severity rate aren't actually the bond's probability of default (POD) or the probability that a bond issuer will not meet the contractual obligations. Here, it's important to recognize that the risk-neutral probability, which uses the risk-free rate, is different than the actual (historical) default probability. One reason for the difference between the risk-free probability and the actual default probability is that actual default probabilities don't capture the uncertainty over the timing of possible default losses. Another reason is due to the liquidity and tax considerations on a bond with risk versus a risk-free bond.

In building out the credit model, analysts assume a conditional probability of default, which means that the year-by-year POD assumes no prior default. To calculate the POD, you start with an assumed rate, say 1.5%, and build each year's POD off of the base rate. Think about how a zero-coupon bond's probability of default will change over time. As the bond gets closer to maturity, the issuer knows that the bond is coming due and probably has a plan to pay it back. Plus, there's also the time factor associated with the potential default. So what would you expect to happen to the POD over time?

No. The issuer makes cash flow plans to pay back the bond, so the POD reflects these plans.

Not quite. As the bond gets closer to maturity, the issuer makes plans to pay it back, and the POD reflects this.

Right! The POD decreases over time as the zero-coupon bond nears maturity. That's because, all else equal, the issuer knows the bond is coming due, and the time that the bond could potentially default is reduced, so the POD gets smaller. Then you can calculate the probability of survival (POS) based on the POD.

For example, assuming a year one POD of 1.5%, the POS is >$$1.000 - 0.015 = 0.985$$. The year two POD rate is based on the year one POS as >$$0.985 \times 0.015 = 0.01478$$. Then, the POS for year two is >$$0.985 - 0.01478 = 0.9702$$. And so on. Thus, after four years, you can sum up the annual POD rates and calculate an overall probability of default. This rate plus the last probability of survival rate will reflect the entirety of possible outcomes. What should the sum of the POD rates and last POS rate equal?

That's not it. The probabilities don't offset each other.

Not quite. There are only two potential outcomes, and they're both reflected in the total sum of probabilities.

That's it! Since the summed POD rates plus the last POS rate capture all the possible outcomes, it stands that the total sum would equal 100%. That's something to remember as you build out a credit model. From there, you can calculate the expected loss for each data as the loss given default times the POD. For year one, that's >$$55.716 \times 0.015 = 0.8357$$. Next, you'll discount these expected loss values by the flat government bond yield rate of 2.5%. Each discount factor is simply calculated as >$$\displaystyle \frac{1}{(1 + 0.025)^n}$$. So year one is >$$\displaystyle \frac{1}{(1 + 0.025)^1} = 0.9756$$.

This gives you the present value of the expected loss, which is the discount factor times the expected loss. For year one, it's >$$0.8357 \times 0.9756 = 0.8154$$ Then you can sum these present values of each year together to calculate the credit valuation adjustment. Check out the table below for the all the values: | Time period | Exposure | Recovery | Loss Given Default | Probability of Default | Probability of Survival | Expected Loss | Discount Factor | PV of Expected Loss | |-------------|----------|----------|--------------------|------------------------|------------------------|---------------|-----------------|---------------------| | 1 | 92.8599 | 37.1440 | 55.7160 | 1.50% | 98.50% | 0.83574 | 0.9756 | 0.8154 | | 2 | 95.1814 | 38.0726 | 57.1089 | 1.48% | 97.02% | 0.84378 | 0.9518 | 0.8031 | | 3 | 97.5610 | 39.0244 | 58.5366 | 1.46% | 95.57% | 0.85191 | 0.9286 | 0.7911 | | 4 | 100 | 40 | 60 | 1.43% | 94.13% | 0.86010 | 0.90595 | 0.7792 | | Totals: | | | | 5.87% | | | | 3.189 | Think about how this total credit valuation adjustment impacts the zero-coupon bond calculation. In year one, the price of the bond without risk of default reflects the zero-coupon characteristic. So what's the year one value?

No, actually. This is the value at the end of year three, since the bond continues to mature.

Nice try, but no. This value is the par value, but this bond pays no coupons.

Yes. At issuance, the default-free zero-coupon bond value is simply the discounted present value of the face value. To incorporate the credit valuation adjustment, you take the discounted present value of the bond and subtract the credit valuation adjustment: >$$90.595 - 3.189 = 87.406$$.

From here, you can use the YTM to find the yield as >$$\displaystyle \frac{100}{(1 + \text{yield})^4} = 87.406$$. This leads to a YTM of 3.422%. This allows you to then calculate the credit spread, which is the additional compensation the investor receives above the 2.5% flat government bond yield. What's the credit spread in this example?

No. That's the bond's yield, which includes both the credit spread and risk-free rate.

Not quite. The credit spread incorporates the government bond yield.

Right. The credit spread is 0.922% and is found by taking the bond's YTM less the flat government bond yield of 2.5%. That's the additional compensation the investor receives. But note that this example assumes defaults only at the end of each year. That's not actually how it happens in real life, which makes the timing of default so critical. So as you build out more sophisticated credit models, it's important to remember these models can be made to be more complex and realistic. In fact, as this example is built upon in future lessons, the government yield curve, POD rates, recovery rates, and discount factors can all vary.

To sum it up: [[summary]]

The risk of default

The liquidity of the bond

The maturity of the bond

Corporate junk bond

Zero-coupon corporate bond

Highly collateralized corporate junk bond

Year one

Year two

Year four

The bond's seniority

The bond's coupon rate

The bond's maturity date

40%

60%

100%

It will increase

It will decrease

It will remain the same

0%

50%

100%

90.60

97.56

100.00

0.265%

0.922%

3.407%

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