Credit Risk Measures
What you'll learn
- LOS 26.aExplain expected exposure, the loss given default, the probability of default, and the credit valuation adjustment.
Expected exposure is the amount of money a bond investor in a credit risky bond stands to lose at a point in time before any recovery is factored in. The expected exposure is equal to the bond’s remaining cash flows, while exposure is the present value (at the risk free rate) of the expected exposure. A bond’s expected exposure decreases over time as coupons are received.
Recovery rate is the percentage recovered in the event of a default. Recovery rate is the opposite of loss severity; given a loss severity of 40%, the recovery rate would be 60%. Loss given default (LGD) is equal to loss severity multiplied by exposure.
Probability of default is the likelihood of default occurring in a given year. The hazard rate is the conditional probability of default given that default has previously not occurred. Probability of survival is 1 − the cumulative probability of default.
If we assume a constant hazard rate, then the probability of survival for time period t is given as:
From this expression, it can be seen that the probability of survival decreases over time. PD, the probability of default, depends on the probability of survival from the prior period.
In the first year, the probability of default = hazard rate (because PS = 1 at inception). In subsequent years, PD < hazard rate. The expected loss for any period is the loss given default (LGD) for that period multiplied by the PD for that period.
Credit valuation adjustment (CVA) is the sum of the present value of the expected loss for each period. CVA is the monetary value of the credit risk in present value terms; it is the difference in value between a risk free bond and an otherwise identical risky bond:
We illustrate the computation of CVA in the following example.
A 3 year, $100 par, zero coupon corporate bond has a hazard rate of 2% per year. Its recovery rate is 60% and the benchmark rate curve is flat at 3%. Calculate the exposure, probability of survival, probability of default, loss given default, CVA, and the credit spread on the bond.
Solution
Exposure at the end of three years is the face value of the bond ($100). Exposure at the end of Year 2 is the present value of $100 discounted for 1 period at 3% (= $100 / 1.03). Exposure at the end of Year 1 is the present value discounted for 2 periods (= $100 / 1.).
Loss given default (LGD) is (1 − recovery rate) or 40% of the exposure.
Probabilities of survival (PS) for Years 1, 2, and 3 are calculated as , , and .
Probability of default is 2% in the first year, and decreases with the PS. PD for Year 2 = 2% of PS for Year 1 = 0.02 × 0.98 = 0.0196 or 1.96%. Similarly, the PD for Year 3 = 0.02 × 0.9604 = 0.019208 or 1.9208%.
DF is the present value of $1. Year 2 DF = .
An identical benchmark bond should trade at $100 / 1. = $91.51. Therefore, the value of credit risky bond = 91.50 − CVA = 91.51 − 2.15 = $89.36.
Credit spread calculation:
Step 1: YTM on risk free bond (given): 3%
Step 2: YTM on risky bond:
Step 3: Credit spread = YTM risky − YTM risk free = 3.82% − 3% = 0.82%
Continuing the previous example: now suppose that the bond pays a 4% annual coupon and everything else is the same.
Calculate the exposure, probability of survival, probability of default, loss given default, and CVA.
Solution
The probability of default and discount factors will remain the same as before, but the expected exposure changes due to the coupon.
Exposure at the end of three years is the face value of the bond plus the final coupon ($104). Exposure at the end of Year 2 is the present value of $104 discounted at 3% for 1 period ($104 / 1.03 = $100.971), plus the $4 Year 2 coupon ($100.971 + $4.00 = $104.971). Exposure at the end of Year 1 is the present value of the Year 3 cash flows discounted for 2 periods, plus the present value of the Year 2 coupon, plus the Year 1 coupon: ($104/1. + 4/1.03 + 4 = $105.913).
Other calculations follow the same process.
Risk Neutral Probability of Default
In the previous example, we used a probability of default that was based on the expected likelihood of default in any given year. However, in practice, we use the risk neutral probability of default, which is the probability of default implied in the current market price.
While the computation of risk neutral probability of default is not needed for the exam, consider a 1 year, zero coupon, $100 par bond trading at $95. The benchmark 1 year rate is 3% and the recovery rate is assumed to be 60%. At the end of the year, there are two possible outcomes: either the bond does not default (pays the promised par value of $100) or defaults (the cash flow is recovery rate × 100 = $60).
Assuming a probability of default of p, the expected year end cash flow = 60p + 100 (1 − p) = 100 − 40p. Using the risk free (benchmark) rate, the present value of this expected cash flow = 100 − 40p / (1.03).
Setting this value to be equal to the bond’s market price:
implies p = 5.38%.
While calculating the risk neutral probability of default, we assumed a recovery rate of 60%. Using similar mechanics, we can also calculate the implied recovery rate in the market price given the probability of default. If we had assumed a probability of default greater than 5.38%, then the implied recovery rate would have been higher (keeping the market price constant).
In general, given the market price (and hence the credit spread), the estimated risk neutral probabilities of default and recovery rates are positively correlated.
ESG Considerations
Analysts should also consider environmental, social, and governance factors when evaluating the default risk of the company. For example, polluters may violate environmental regulations, resulting in fines or business curtailments. Companies with poor labor practices may experience loss of reputation, customers, and profitability.
Similarly, companies with poor governance systems may resort to fraudulent accounting to mask debt service problems. On the flip side, some countries provide tax incentives for investing in green bonds (i.e., bonds that fund green projects). The WHO issued pandemic bonds in 2017 that offered a high interest rate, but in the event of a pandemic, the principal would be diverted as aid to poor countries.
Due to the COVID 19 pandemic, by July 2020, the entire principal of these bonds was wiped out.