Agencies’ Ratings vs Internal Ratings
 Comparing agencies’ credit ratings to internal credit rating systems involves examining the methodologies, outcomes, and purposes of each system to understand their similarities and differences.
 Credit rating agencies (CRAs) such as Moody’s, Standard & Poor’s (S&P), and Fitch provide external credit assessments of entities and financial instruments. These ratings are used by investors, regulators, and the market at large to gauge creditworthiness.
 On the other hand, internal credit rating systems are developed and used by financial institutions like banks to assess the credit risk of their borrowers, including individuals, corporations, and other entities.
 Purpose, scope and application: CRAs provide a standardized, public rating useful for a wide range of market participants, facilitating capital market activities. Internal ratings are confidential, tailored to the institution’s specific credit policies, and used for loan approval processes, pricing, and managing credit risk.
 Methodology: CRAs’ methodologies are standardized and designed to be applicable across different entities and sectors. In contrast, internal systems can be more flexible, allowing institutions to incorporate their own experiences, portfolio considerations, and risk appetite.
 Regulatory Oversight: CRAs are subject to regulatory scrutiny and must adhere to international standards such as those set by the IOSCO (International Organization of Securities Commissions). Internal rating systems are regulated concerning their use in capital adequacy calculations under frameworks like Basel III but have more leeway in their development and application.
 Impact: CRA ratings can significantly impact an entity’s access to capital and terms of borrowing. In contrast, internal ratings primarily affect the terms and availability of credit within the issuing institution.
 Rating Agencies – Stability and Market Responsiveness: Credit rating agencies such as Moody’s, S&P, and Fitch focus on the longterm creditworthiness of entities, leading to relatively infrequent changes in ratings. This approach aims to ensure stability in the ratings, which is crucial for bond traders and institutional investors who rely on these ratings to adhere to investment mandates or regulatory requirements. Such stakeholders prefer stability to avoid the high transaction costs that would come with frequent rating adjustments due to short term market fluctuations.The principle of “rating through the cycle” means that agencies look beyond shortterm economic downturns or market volatility, focusing instead on the longerterm outlook of a company’s ability to meet its debt obligations. This approach avoids reactionary rating changes, which could force bondholders to rebalance portfolios unnecessarily, incurring additional costs.
 Contrasting with MarketSensitive Models: Contrary to the stabilityfocused approach of rating agencies, firms like Moody’s KMV and Kamakura offer models that estimate a company’s probability of default based on more volatile factors, such as equity prices and other marketdriven variables. These models tend to react more quickly to market information, providing a more immediate reflection of changing credit conditions but without the objective of maintaining rating stability over time.
 Internal Credit Ratings, Coverage and Approach: Internal credit rating systems address a gap left by rating agencies: the assessment of creditworthiness for companies that do not have publicly traded debt and therefore lack external ratings. These systems are essential for banks and financial institutions to evaluate the credit risk of a wide range of borrowers, from small and mediumsized enterprises (SMEs) to individual retail clients. Banks employ various financial metrics to assess creditworthiness, including profitability ratios (like return on assets) and balance sheet ratios (such as the current ratio and debttoequity). Recognizing the importance of cash flow over profit figures in loan repayment, banks often convert a company’s financial information into a cash flow statement. This process allows them to better gauge a company’s capacity to service its debt, providing a direct assessment of credit risk tailored to the bank’s lending criteria and risk appetite.
Linear Discriminant Analysis And Altman’s Z Score
Example
 Consider a company with the following parameters
 Working Capital: $250,000
 Total Assets: $800,000
 Sales: $3,000,000
 Market Value of Equity: $500,000
 Total Liabilities: $300,000
 Retained Earnings: $350,000
 Earnings Before Interest and Taxes (EBIT): $90,000
 Since, the calculated Zscore of 6.109 is much higher than 3, this company is also not in danger of defaulting in the near future.
Rating Migration Matrix
 To assess the performance of different bond ratings, the probability of default of issuers with specific ratings can be examined over time. This analysis is based on cumulative average default rates reported by Standard & Poor’s (S&P) for the period from 1981 to 2020. Table CR91 here provides a small subset of the full data. It reveals the probability that an issuer rated within a certain category will default within a given timeframe, ranging from one year to five years.
 The probability of a bond defaulting in a future year, known as the unconditional default probability, is determined by the difference in default probabilities between successive years. For example, the unconditional default probability of a BBrated bond in its fifth year is found by subtracting the fouryear default rate from the fiveyear default rate, which is Similarly, the probability of default of all bonds defaulting during any year can be calculated. The results of these calculations are shown in table CR92 here. For reference, the previously calculated figure of 1.44% is circled.
 This approach allows us to calculate the annual probability of default for bonds across all ratings. These probabilities are termed “unconditional” because they do not consider whether the bond has survived without defaulting up to a certain point.
 In contrast, the conditional default probability considers the bond’s survival up to a certain year and assesses the likelihood of default in the following year. For example, to calculate the conditional default probability for a BBrated bond in its fifth year, the probability of the bond surviving the fourth year is determined first, which can be calculated from table CR91 as
Given that the unconditional probability of default in the fifth year is , the conditional probability of default, assuming no prior default, is calculated to be . These conditional probabilities offer a nuanced view of default risk by accounting for a bond’s survival to a specific point in time before evaluating its risk of defaulting in the subsequent year. Table CR93 shows the results of these conditional default probability calculations for all rating categories and all future years. For reference, the previously calculated figure of 1.52% is circled.
Note: The detailed calculations for tables CR91, CR92, and CR93 are available in the Excel sheet added to the LMS, against the same chapter, accessible to all MF candidates.

1 
2 
3 
4 
5 
AAA 
0.00 
0.03 
0.10 
0.11 
0.10 
AA 
0.02 
0.04 
0.05 
0.10 
0.09 
A 
0.05 
0.08 
0.09 
0.11 
0.13 
BBB 
0.16 
0.27 
0.32 
0.39 
0.40 
BB 
0.63 
1.31 
1.56 
1.58 
1.52 
B 
3.34 
4.61 
4.28 
3.56 
2.89 
CCC/C 
28.30 
13.99 
8.25 
5.20 
4.14 
Table CR93 : Conditional Percentage Default Probabilities Calculated from Table CR91 and Table CR92
Borrower Ratings And Probability Of Default
 Borrower ratings are utilized to predict default probabilities, under the assumption of a uniform group of borrowers and the application of the law of large numbers, which suggests that observed frequencies are reliable estimators of true probabilities.
 It has been noted, however, that expectations are surpassed by lower initial ratings, while higher ratings often fail to meet them. Migrations between rating categories over time exhibit dependent correlations rather than randomness
 Typically, the lower the borrower’s rating, the higher the likelihood of default; conversely, the higher the rating, the lower the chance of default. Additionally, for investmentgrade bonds and certain noninvestmentgrade bonds, the likelihood of default tends to rise over the initial years. This suggests that issuers start off stable but face growing risks of financial decline over time. For example, as seen from the previous table 2, a bond with an initial BBB rating has unconditional default probabilities of in the first year, increasing to in the second year, in the third year, and reaching in the fourth year.
 Conversely, with some noninvestmentgrade bonds, the risk of default may actually diminish over time. The critical period for these issuers is the early years. If they survive this phase, it is often assumed their financial condition is more robust than first thought. For example, as seen from the previous table 2, a bond rated B initially has default probabilities of in the first year, increasing to in the second, but then decreasing to in the third year and further to in the fourth year. This shows a peak in the second year, followed by a downward trend in the subsequent years.
 Default rates, when tracked over a period, allow for the calculation of average frequencies of defaults per rating class. These calculated frequencies can then help in the derivation of default probabilities, which may be employed to predict future default rates.
Hazard Rates
Hazard Rates – Example 1
 As an example, suppose that the hazard rate is constant at 1% per year.
 The probability of a default by the end of the third year is
 The unconditional probability of a default occurring during the fourth year is
 The conditional probability of defaulting in the fourth year, given that it has survived until the end of the third year, is
Recovery Rates
 When a company declares bankruptcy, its creditors submit claims to seek repayment. During this process, a reorganization may occur, leading creditors to settle for partial repayment. Alternatively, a company’s assets may be liquidated, and the generated funds are distributed to settle claims to the extent possible, with certain claims receiving priority and thus a higher degree of fulfillment.
 The bond recovery rate is typically measured by its market value approximately one month following default, expressed as a percentage of its face value. Historical data on the average recovery rates for various bond categories in the US shows a range from 54.6% for first lien bonds to 22.3% for those positioned below other creditors in the repayment hierarchy. First lien bonds hold the premier right to collateral in the event of a default, whereas second lien bonds have a secondary claim. In the absence of collateral, senior unsecured bonds are prioritized over senior subordinated bonds, followed by subordinated bonds, and finally, junior subordinated bonds, each descending in the order of claim priority.
Class 
Average Recovery Rate (%) 
1^{st} lien bond 
54.6 
2^{nd} lien bond 
44.3 
Senior unsecured bond 
38.0 
Senior subordinate bond 
31.0 
Subordinate bond 
32.1 
Junior subordinate bond 
22.3 
Figure CR91: Recovery Rates on Corporate Bonds as a Percent of Face Value, 1983 to 2021, Issuer Weighted Source – Moody’s
Recovery Rates And Default Rates
 A key insight from the Global Financial Crisis of 20072008 is the inverse relationship between the average recovery rate on mortgages and the mortgage default rate. As more mortgages default, leading to increased foreclosures, there is a surplus of houses on the market, which causes house prices to drop. This drop in prices, in turn, leads to lower recovery rates.
 This phenomenon is similarly observed in the corporate bond market. In periods characterized by a minimal number of bond defaults, which often coincide with robust economic conditions, the average recovery rate for defaulting bonds can become as high as 60%. In contrast, during periods marked by higher default rates, indicative of adverse economic conditions, the average recovery rate on these bonds can decline to approximately 30%.
 In essence, the negative correlation between default rates and recovery rates poses a significant challenge for lenders during economic downturns, as they are characterized by both high default rates and diminished recovery rates.
Credit Default Swaps
 A Credit Default Swap (CDS) is a financial derivative used to exchange credit risk among market participants. This market saw significant growth until 2007, after which it began to decline. At its core, a CDS serves as insurance against the possibility that a specific company, referred to as the reference entity, will go through a credit event. Credit events may include non payments, late payments, debt restructuring, bankruptcy, etc. Restructuring might not always be included, especially in North American highyield debt scenarios. The CDS buyer gains the right to sell the company’s bonds at their full value to the CDS seller if such a credit event occurs. The face value of bonds that can be sold under this agreement is called the notional principal of the swap. To maintain this protection, the CDS buyer pays the seller periodic fees until the swap’s expiration or a credit event takes place, typically on a quarterly basis and in arrears.
 As an example, assume that two parties agree on a threeyear CDS contract on January 1, 2025. The notional principal of the contract is $50 million, and the buyer agrees to pay the seller 100 basis points (1% or 0.01 in decimals) annually (paid quarterly in arrears) for protection against default by the reference entity.
 In this scenario, if there are no credit events, then the buyer does not receive any payout. The buyer’s payments to the seller would be on April 1, July 1, October 1, and January 1 of each of the years 2025, 2026, and 2027.
 However, if a credit event occurs, say, on August 15, 2026, the buyer may be entitled to a substantial payout. If the contract requires physical settlement, the buyer could sell the reference entity’s bonds, with a face value of $50 million, to the seller for the same amount.
 If the contract is settled in cash, as is commonly the case, the value of the payout is determined through an auction process after the credit event. Assuming the auction finds the cheapest bond worth $40 for every $100 of face value, the cash settlement amount would be $30 million ($50 million face value minus $20 million market value).
 After the credit event, the buyer stops making regular payments to the seller. However, a final payment for the quarter in which the default occurs may still be required. In this example, with a default on August 15, 2026, the buyer would need to pay the prorated (or accrued) amount for the period from July 1, 2026, to August 15, 2026, which would be approximately $41,666.67. No further payments would be needed post this settlement.
 The CDS spread is the annual fee, expressed as a percentage of the notional principal, paid to secure protection against default. For example, a spread of 90 basis points (0.9%) might be typical. Major banks often act as intermediaries in the CDS market, offering to buy CDS protection at one rate (e.g., 250 basis points or 2.5%) and sell it at a slightly higher rate (e.g., 260 basis points or 2.6%).
 CDS contracts cover a wide range of companies and countries as reference entities. Although fiveyear contracts are the most common, contracts of 1, 2, 3, 7, and 10 years are also available. Contracts tend to expire on set dates (March 20, June 20, September 20, December 20), meaning the actual duration might slightly differ from the nominal term. For example, initiating a five year CDS in November 2024 could extend the contract to December 2029, with the first payment due in December 2024 covering the initial period until the next quarter. Payments are then made quarterly.
 The CheapesttoDeliver Bond – A CDS often allows for various bonds of the same seniority to be delivered if a default occurs. These bonds might not have the same market value post default, granting the CDS holder the “cheapesttodeliver” option. Typically, an auction process determines the value of this bond and the corresponding payout to the protection buyer.
 Credit Indices – In the credit market, participants use indices to monitor credit default swap (CDS) spreads. By 2004, the industry saw some standardization with key portfolios like:
 CDX NA IG: This includes 125 investmentgrade companies in North America.
 iTraxx Europe: This comprises 125 investmentgrade companies in Europe.
These portfolios are refreshed semiannually, in March and September, swapping out companies that fall below investment grade and adding new ones that meet the criteria.
 Suppose the fiveyear CDX NA IG index is quoted at 140 basis points (bid) and 142 basis points (ask) by a market maker. This indicates that to acquire CDS protection on the full index of 125 companies, an investor would need to pay 142 basis points (ask) for each company. If an investor seeks $1,000,000 in protection for each company, the annual expense is calculated as
 On the flip side, selling protection at 140 basis points (bid) could yield an annual premium of
If a default occurs by one of the companies within the index, the investor is entitled to a standard CDS compensation, and the yearly cost is reduced by an amount equal to
This is same as 142 basis points on $1,000,000
 The trading of CDS index protection remains active across 3, 5, 7, and 10year maturities, with the contracts set to expire on June 20 and December 20. This scheduling means a “fiveyear” contract might actually span between 4.75 and 5.25 years. In essence, the index provides a mean CDS spread of the companies listed in the portfolio.
 The Use of Fixed Coupons – When trading Credit Default Swaps (CDS) and CDS indices, the process mimics how bonds are traded, with a few additional steps to accommodate the unique nature of CDS contracts:
 Fixed Coupon and Recovery Rate: Each trade sets a fixed interest rate, known as the coupon, and a predetermined recovery rate in case of default.
 Protection Payments: The payments made by the buyer of protection (essentially the insurance premium) are equal to the fixed coupon rate.
 CDS Spread Quoting: The market quotes a CDS spread, reflecting the cost of buying protection as a percentage of the notional amount. This spread varies based on market conditions and perceived risk.
 Net Present Value Calculation: When a trade occurs, the difference between the payments based on the quoted spread and those based on the coupon rate for the remaining duration of the CDS contract is calculated. This involves estimating the present value (the current worth of a future sum of money or stream of cash flows given a specified return rate) of this difference.
 Settlement at Trade Time:
 If the quoted spread is lower than the coupon, it means the market perceives lower risk than when the contract was issued. In this case, the seller of protection (the one offering insurance) pays the present value difference to the buyer as a sort of adjustment for the overpriced protection.
 If the quoted spread is higher than the coupon, indicating increased risk, the buyer pays this present value difference to the seller. This compensates the seller for providing protection at a nowcheaper rate than the current market conditions would dictate.
 Continued Payments: After this initial adjustment, the buyer continues to make regular payments at the coupon rate on the principal amount for those entities within the index that have not defaulted. This is akin to paying insurance premiums based on the original terms, adjusted for the initial difference in market perception versus the contract terms.
 This mechanism ensures that both buyers and sellers of protection can adjust for changes in market conditions, making the CDS market more flexible and responsive to evolving risk assessments.
CDS Spreads
CDS Bond Basis
 The CDSbond basis represents the difference between the CDS spread and the bond yield spread of a company, defined as:
CDS Bond Basis = CDS Spread – Bond Yield Spread …Equation CR94
Theoretically, this basis is expected to be close to zero based on arbitrage arguments, given earlier. However, in practice, various factors contribute to deviations from this expected value. These factors include:

 Bond Price Deviations from Par: Bonds trading above or below their par value can lead to negative or positive basis values, respectively.
 Counterparty Default Risk in CDS: The risk that the counterparty to a CDS contract may default impacts the basis negatively.
 CheapesttoDeliver Option in CDS: This option allows the protection seller to choose the cheapest bond to deliver in the event of a credit event. This flexibility can increase the CDS spread, pushing the basis up.
 Exclusion of Accrued Interest in CDS Payoffs: When a CDS contract does not compensate for accrued interest on a defaulted bond, it negatively affects the basis.
 Restructuring Clause in CDS Contracts: This clause can trigger a payout even in the absence of a default event, leading to an increase in the CDS spread and a positive basis.
 Differences in Reference Rates: Discrepancies between the “riskfree” rate used to calculate bondyield spreads and the one actually utilized by the market can influence the basis.
 Historical trends have shown fluctuations in the CDSbond basis, with periods of positive basis observed before the 2007 financial market turmoil, notably an average of 16 basis points in 2004 and 2005 as estimated by De Wit. However, during the Global Financial Crisis, the basis became significantly negative, a situation attributed to liquidity shortages among other factors, making arbitrage between bonds and CDSs challenging. Postcrisis, the CDSbond basis has generally been small and negative, indicating a shift in market dynamics and the influence of regulatory and economic changes postcrisis.
Estimating Default Probabilities From Spreads
 The hazard rate, often referred to in the context of credit risk, is defined as the probability of default of a borrower within a given time frame, assuming that the borrower has survived (i.e., not defaulted) up to the start of that time frame. Mathematically, the hazard rate () is expressed as
…Equation CR95
where is the average hazard rate between time zero and time T. s(T) is the credit spread (which should in theory be expressed with continuous compounding) for a maturity of T, and RR is the recovery rate,
 A scenario is considered where a company’s fiveyear credit spread has been set at 300 basis points (bps), equivalent to an annual rate of 3%. In the event of a default, it is expected that a recovery rate of 50% would be achieved. Hence, an annual loss of 3% due to defaults is anticipated by investors holding the company’s corporate bond. This credit spread acts as an indicator of the average loss rate.Given a 50% recovery rate, the hazard formula gives an annual default probability of :
This calculation serves to estimate the average annual probability of default over a fiveyear period, assuming no default has occurred prior to this.
 When credit spreads are available for several different maturities, the term structure of hazard rates can be approximately constructed, as is illustrated by the example that follows.
 Example – Consider the CDS spreads for 3, 5, and 10year instruments to be 80, 100, and 150 basis points respectively, with an expected recovery rate of 40%.
 The average hazard rate over three years is approximately 0.008/1 – 0.4 = 0.01333
 The average hazard rate over five years is approximately 0.01/1 – 0.4 = 0.01667
 The average hazard rate over ten years is approximately 0.015/1 – 0.4 = 0.025
From these calculations, it can be estimated that
 The average hazard rate between year 3 and year 5 is
 The average hazard rate between year 5 and year 10 is
Comparison Of Default Probability Estimates

 Default probability estimates derived from historical data are compared with those obtained from credit spreads. Table CR94 displays:
 The sevenyear average cumulative default probability for companies of various credit ratings, sourced from S&P’s published data.
 The average credit spread for bonds of different credit ratings from December 1996 to June 2007, derived from Merrill Lynch’s bond yield data and an assumption of a reference riskfree rate that is 10 basis points below the sevenyear swap rate. These bonds had an average maturity of about seven years.
Rating 
Cumulative 7Year Default Probabilities (%), 19812020 
7Year Credit Spread (bp) 19962007 
AA 
0.51 
35.74 
AA 
0.49 
43.67 
A 
0.76 
68.68 
BBB 
2.27 
127.53 
BB 
8.89 
280.28 
B 
20.99 
481.04 
CCC/C 
50.75 
1103.70 
Table CR94 : Cumulative 7Year Default Probabilities Compared with Credit Spreads
Data on bond yields is considered only up to the beginning of the 2008 Global Financial Crisis. Inclusion of the crisis period, where credit spreads sharply increased, would significantly amplify the findings that hazard rates calculated from credit spreads exceed those based on historical default probabilities.


 Hazard rates can be derived from the average cumulative default probabilities using the equation CR94 as For t = 7, this gives For example, for a AA rated company, the sevenyear average cumulative default probability is Hence
 Hazard rates can be derived from credit spreads using the equation CR95 as Here RR is assumed to be 40%. So, for example, for a AA rated company, the 7year credit spread is 43.67 bps or 0.004367 Hence
Table CR95 presents a summary of the average 7year hazard rates across the full spectrum of credit ratings, utilizing the methodologies previously discussed.
Rating 
Rate from Default Probabilities (%) 
Rate Implied by Credit Spreads (%) 
Difference 
AA 
0.073 
0.596 
0.523 
AA 
0.070 
0.728 
0.658 
A 
0.109 
1.145 
1.036 
BBB 
0.328 
2.216 
1.798 
BB 
1.330 
4.671 
3.341 
B 
3.366 
8.017 
4.651 
CCC/C 
10.118 
18.395 
8.207 
Table CR95 : Average SevenYear Hazard Rates
 This table illustrates that hazard rates implied by credit spreads before the 2008 Global Financial Crisis exceed those derived from extensive historical data. Here are key points regarding this difference:
 Hazard Rate Differences: The gap between the two hazard rates grows as credit quality decreases, suggesting better compensation for investors taking on higher credit risks, exceeding actuarial cost estimates of defaults.
 Excess Returns Fluctuations: Excess returns, or the additional compensation for taking on credit risk, are not constant over time and tend to rise during periods like the 2008 crisis when credit spreads are high.
 Reasons for Hazard Rate Discrepancies:
 Market Liquidity: The relative illiquidity of corporate bonds requires higher returns to compensate, contributing marginally to excess returns.
 Subjective Default Risk: Bond traders might factor in severe depression scenarios, possibly explaining part of the excess returns.
 Correlated Defaults: Bonds don’t default independently, leading to systematic risk akin to equities. This risk varies annually, with bond traders earning returns for bearing it. This systematic risk is heightened by overall economic conditions and phenomena like “contagion”.
 Systematic v Unsystematic Risks: Similar to equities, bond traders are compensated for bearing systemic risks influenced by economic conditions. Unlike equities, the unique risk profile of bonds, with their limited upside and significant downside potential, makes nonsystematic risks harder to diversify, requiring extensive diversification which is often impractical. So, bond traders not only earn for systemic risks but may also receive additional returns for undiversifiable nonsystematic risks, especially in less diversified portfolios.
Real World vs Risk Neutral Default Probabilities
 The choice of estimates for market variables is influenced by their intended application. Here’s how this applies to different types of financial analysis:
 Valuation:
 RiskNeutral Estimates: For valuation purposes, estimates are often made under the assumption of a riskneutral world, where investors do not seek additional returns for taking on risk. This approach aligns with the use of riskneutral valuation techniques.
 Use of Credit Spreads: In the context of valuing bonds, estimates derived from credit spreads are utilized similarly to other riskneutral estimates. These estimates of default probabilities are used alongside the riskfree rate for discounting purposes, aligning with the principles of risk neutral valuation.
 Scenario Analysis:
 RealWorld Estimates: For scenario analysis, which involves assessing potential future scenarios under realworld conditions, it is more appropriate to use estimates based on historical data. This approach accounts for the actual risk perceptions and behaviors of market participants.
Merton Model
 Using tables to estimate a company’s realworld default probability depends on its credit rating, which is infrequently updated. This limitation has prompted some analysts to suggest that equity prices, which are updated more regularly, could provide a timelier method for estimating default probabilities.
 As discussed in previous chapters, the Merton model assumes a leveraged firm with a single debt issue, no dividends, and perfect markets. The debt is couponless and matures at T. At this time, if the firm’s asset value is less than the debt principal F, the firm is bankrupt, equity becomes worthless, and debt holders claim the firm. Conversely, if assets exceed the debt, equity holders receive any excess. This structure resembles the payoff of a call option on the value of the firm’s assets
 The current value of equity can be derived as: …Equation CR96 where N is cumulative normal distribution function, F is the face value of the debt (equal to the market value of equity and net debt), A is the current value of the firm’s assets, r is the riskfree rate of return t is the remaining time to maturity of the debt, is the instantaneous volatility (standard deviation) of the firm’s assets
 Under Merton’s model, default occurs when the option is not exercised, where probability of default can be show to be N(d2). For this calculation, A and are needed but are not directly observable. If the company is publicly traded, E can be observed, Additionally, the volatility of equity can be estimated. Using calculus, it can be shown that
…Equation CR97
 Since there are two equations (CR96 and CR97)and two variables (A and ), the values of these two variables can be obtained quite easily by solving them.
 Merton model explains the discrepancy in default probabilities between the riskneutral and realworld contexts due to differences in asset growth rates:
 RiskNeutral Asset Growth: In the riskneutral world, a company’s assets are expected to grow at the riskfree rate.
 RealWorld Asset Growth: In the real world, the asset growth rate typically exceeds the riskfree rate, reflecting the market demanded risk premium.
 Impact on Default Probability: Because of the more conservative growth rate in the riskneutral world, the likelihood of assets falling below the debt’s face value at a future time is greater compared to the realworld expectation.
Merton Model And Distance To Default
 In credit risk assessments, the Merton model highlights Distance to Default (DD) as a crucial metric. DD measures the number of standard deviations by which a company’s asset value exceeds its debt’s face value. A greater DD indicates a lower probability of default, reflecting stronger financial health. The computation for , equivalent to ଶ in the model, calculates this margin of safety at time , with the understanding that a higher suggests a reduced risk of default and vice versa.
 It is given by
…Equation CR98
 Ideally, the expected return on assets () should replace the riskfree rate (r) in the above formula, but if it is not given, then r can be used.
 Default Probability Ranking: Merton’s model is proficient at ranking default probabilities, applicable to both riskneutral and realworld scenarios.
 Monotonic Transformation: The model’s output can undergo a monotonic transformation to accurately estimate realworld or riskneutral default probabilities.
 Calibration Services: Companies like Moody’s KMV and Kamakura offer services to adjust Merton’s model outputs to reflect realworld default probabilities.
 Credit Spread Estimation: Some entities use the model to calculate credit spreads, which are indicative of riskneutral default probabilities.
 RiskNeutral Assumption: Even though is theoretically riskneutral, Moody’s KMV and Kamakura use it for realworld default probability estimates, operating under the assumption that riskneutral and realworld default probability rankings, as well as those from Merton’s model, align consistently.