Credit Analysis Models
READING 28
CREDIT ANALYSIS MODELS
EXAM FOCUS
This topic review augments credit analysis covered at Level I with newer models of credit analysis including structural models and reduced-form models. Be able to differentiate between the two and know the merits and drawbacks of both. Be able to compute the credit valuation adjustment (CVA) and expected change in a bond's price given credit migration, and credit spread given CVA. Finally, understand how credit analysis of asset-backed securities (ABS) differs from credit analysis of corporate debt.
This topic review provides a broad overview of evaluation of credit risk. Do note that while credit risk and default risk are often used interchangeably, credit risk is a broader term. Credit risk includes the risk of default, as well as the risk of worsening credit quality (even if default does not occur).
MODULE 28.1: CREDIT RISK MEASURES
LOS 28.a: Explain 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 present value (based on the risk-free rate) of the bond's remaining cash flows. A bond's expected exposure changes over time as cash flows are received or discounted for fewer remaining periods.
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 (or exposure × (1 - recovery rate)).
Probability of default (PD) is the likelihood of default occurring in a given year. The hazard rate is the conditional probability of default in a small interval given that default has previously not occurred. Probability of survival (PS) over time t is 1 minus the cumulative probability of default up to t. If we assume a constant hazard rate h, then the probability of survival for time period t (assuming discrete annual periods) is:
PS_t = (1 - hazard rate)^t
From this expression it can be seen that the probability of survival decreases over time. The PD in period t depends on the survival probability at the end of period t - 1:
PD_t = hazard rate × PS_{t-1}
In the first year, the probability of default equals the hazard rate because PS_0 = 1 at inception. In subsequent years PD_t will be less than the hazard rate if PS_{t-1} < 1.
The expected loss for any period is the loss given default (LGD) for that period multiplied by the probability of default (PD) for that period.
Credit valuation adjustment (CVA) is the sum of the present values of 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:
CVA = price of risk-free bond - price of risky bond.
We illustrate the computation of CVA in the following worked examples.
EXAMPLE: Credit Valuation Adjustment (CVA), Part 1
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 expected exposure, probability of survival, probability of default, loss given default, CVA, and the credit spread on the bond.
Answer
Exposure at the end of each year (present value of remaining cash flows, discounted at the benchmark risk-free rates):
Exposure at end of Year 3 = face value = $100.
Exposure at end of Year 2 = present value of $100 discounted 1 period at 3% = $100 / 1.03.
Exposure at end of Year 1 = present value of $100 discounted 2 periods at 3% = $100 / (1.03)^2.
Loss given default (LGD) expressed as a percentage = 1 - recovery rate = 1 - 0.60 = 0.40 = 40% of exposure.
Probabilities of survival (PS) for Years 1, 2 and 3 (with hazard rate h = 0.02):
PS_1 = (1 - 0.02)^1 = 0.98.
PS_2 = (1 - 0.02)^2 = 0.9604.
PS_3 = (1 - 0.02)^3 = 0.941192 ≈ 0.9412.
Probability of default (PD) in each year:
PD_1 = hazard rate = 0.02 = 2.00%.
PD_2 = hazard rate × PS_1 = 0.02 × 0.98 = 0.0196 = 1.96%.
PD_3 = hazard rate × PS_2 = 0.02 × 0.9604 = 0.019208 = 1.9208%.
Compute LGD in dollars for each year (LGD$ = exposure × 40%):
LGD$_1 = (100 / (1.03)^2) × 0.40.
LGD$_2 = (100 / 1.03) × 0.40.
LGD$_3 = 100 × 0.40 = 40.0.
Expected loss in dollars for each year = PD_t × LGD$_t.
Example numeric expected loss for Year 3:
expected loss Year 3 = PD_3 × LGD$_3 = 0.019208 × 40 = 0.76832 ≈ $0.7683.
Discount factors (DF) using the benchmark rate of 3%:
DF_1 = 1 / 1.03 = 0.970873786.
DF_2 = 1 / (1.03)^2 = 0.942595909.
DF_3 = 1 / (1.03)^3 = 0.91514166.
Present value of expected loss for each year = DF_t × expected loss_t.
CVA = sum of the three PVs of expected loss = $2.15 (rounded as given in the example).
Value of an identical benchmark (risk-free) zero-coupon bond (price today):
price_risk_free = 100 / (1.03)^3 = $91.51 (approximately).
Value of the credit-risky bond = price_risk_free - 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 (solve for I/Y with N = 3, PV = -89.36, PMT = 0, FV = 100):
Computed YTM (risky) = 3.82% (as given).
Step 3: Credit spread = YTM (risky) - YTM (risk-free) = 3.82% - 3.00% = 0.82% (82 basis points).
EXAMPLE: Credit Valuation Adjustment (CVA), Part 2
Continuing the previous example: now suppose that the bond pays a 4% annual coupon and everything else is the same. Calculate the expected exposure, probability of survival, probability of default, loss given default, and CVA.
Answer
The probability of default and discount factors remain the same as before because hazard rate and benchmark rates are unchanged. The expected exposure changes due to the coupon payments.
Exposure at the end of Year 3 = face value + final coupon = $100 + $4 = $104.
Exposure at the end of Year 2 = present value of Year 3 cash flows discounted 1 period at 3% plus the Year 2 coupon:
Exposure_2 = (104 / 1.03) + 4 = 100.971 (rounded) + 4.00 = $104.971.
Exposure at the end of Year 1 = present value of Year 3 cash flows discounted 2 periods + present value of Year 2 coupon discounted 1 period + Year 1 coupon:
Exposure_1 = (104 / (1.03)^2) + (4 / 1.03) + 4 = $105.913 (rounded as given).
Other calculations (LGD$, PDs, PV of expected loss, and CVA) follow the same process as in the zero-coupon example.
Risk Neutral Probability of Default
In the previous example, we used a probability of default derived from expected likelihoods. In practice, we can compute the risk-neutral probability of default, which is the probability implied by current market prices.
Example: 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 year end there are two possible outcomes: no default (payment of $100) or default (recovery payment of $60).
Assume risk-neutral default probability = p. Then expected year-end cash flow = 60p + 100(1 - p) = 100 - 40p.
Present value (discount using risk-free rate 3%) = (100 - 40p) / 1.03.
Set present value equal to market price $95 and solve for p:
(100 - 40p) / 1.03 = 95
100 - 40p = 95 × 1.03 = 97.85
40p = 100 - 97.85 = 2.15
p = 2.15 / 40 = 0.05375 ≈ 5.38%
Note: while estimating risk-neutral PDs we assumed a recovery rate of 60%. For a given market price, the risk-neutral PD and implied recovery rate are positively correlated: if we assume a higher PD, the implied recovery rate for the same market price would be higher, and vice versa.
PROFESSOR'S NOTE
This is an important statement. There are several inputs in the bond valuation model-some are known (such as the bond's coupon rate, maturity, and the benchmark term structure) while two of the inputs are estimates (probability of default and recovery rate). We have to assume one to calculate the other implied in the current market price.
ESG Considerations
Analysts should also consider environmental, social, and governance (ESG) factors when evaluating a company's default risk. For example, polluters may violate environmental regulations, resulting in fines or business curtailments. Companies with poor labour 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 jurisdictions provide tax incentives for investing in green bonds (bonds that fund environmentally beneficial projects).
An example of a specialised instrument is the pandemic bond issued by an international body in 2017 that offered a high interest rate but, in the event of a qualifying pandemic, the principal would be used as aid. Due to the COVID-19 pandemic, by July 2020, the entire principal of such bonds was wiped out-illustrating event-based structural risk.
MODULE QUIZ 28.1
1.
Manny Zeld is evaluating a 5%, annual pay, $100 par, 5-year Barry Corp. bond and has calculated the CVA as $12.67. Benchmark rates are flat at 2%. Zeld would most appropriately value Barry Corp. bonds at:
A. $101.47.
B. $110.22.
C. $114.76.
2.
If the annual hazard rate for a bond is 1.25%, the probability of default (PD) in Year 1 is:
A. less than 1.25%.
B. equal to 1.25%.
C. greater than 1.25%.
3.
If the annual hazard rate for a bond is 1.25%, the probability that the bond does not default over the next three years is closest to:
A. 94.32%.
B. 95.20%.
C. 96.30%.
4.
For a risky bond, the expected loss for a specific year is most appropriately calculated as:
A. exposure multiplied by the recovery rate multiplied by the hazard rate.
B. loss given default multiplied by the probability of default.
C. exposure multiplied by the probability of default.
5.
The CVA for a risky bond is most likely the:
A. price discount for a credit risky bond compared to an otherwise identical risk-free bond.
B. cumulative amount of expected loss.
C. mean cumulative present value of expected loss.
6.
Joel Abramson, CFA, is the fixed income portfolio manager of VZ Bank. Based on his analysis of bonds issued by Tinta Corp, including an evaluation of Tinta's balance sheet, Joel estimates the hazard rate for the bond. Abramson would most accurately consider the bond to be an attractive purchase if the:
A. estimated hazard rate is less than the bond's risk-neutral probability of default.
B. estimated hazard rate is less than 2%.
C. estimated hazard rate is greater than the risk-neutral probability of default.
7.
Given a risky bond's market price, its risk-neutral probability of default is most likely to be:
A. positively correlated with the assumed recovery rate.
B. negatively correlated with the assumed recovery rate.
C. independent of the assumed recovery rate.
MODULE 28.2: ANALYSIS OF CREDIT RISK
Consider the earlier example of a 3-year, $100 par zero-coupon corporate bond with 60% recovery and benchmark rates flat at 3%. We calculated the exposures at the ends of Years 1-3 as $94.26, $97.09 and $100 respectively. CVA was $2.15 and the bond price today was $89.36. Assume that default only occurs at year-end.
The cash flows on this bond and the corresponding internal rates of return (IRR) in the event of default at various dates, and the IRR if the bond does not default, can be computed as shown below.
If the bond defaults at the end of Year 1, recovery = 60% of exposure at Year 1 = 0.6 × 94.26 = $56.556 ≈ $56.56.
If default occurs at the end of Year 2, recovery = 0.6 × 97.09 = $58.254 ≈ $58.25.
If default occurs at the end of Year 3, recovery = 0.6 × 100 = $60.00.
IRR calculations (using PV = -89.36 as the purchase price):
In case of default in Year 1: PV = -89.36, N = 1, FV = 56.56 → CPT I/Y = -36.71% (negative return).
In case of default in Year 2: PV = -89.36, N = 2, FV = 58.25 → CPT I/Y = -19.26%.
In case of default in Year 3: PV = -89.36, N = 3, FV = 60.0 → CPT I/Y = -12.43%.
If the bond does not default over its life: PV = -89.36, N = 3, FV = 100.0 → CPT I/Y = 3.82% (this matches the YTM computed earlier for the risky bond).
Relative Credit Risk Analysis
When comparing the credit risk of several bonds, the metric that combines both probability of default and loss severity is the expected loss. All else equal, for a given period, the higher the expected loss the higher the credit risk.
EXAMPLE: Relative risk evaluation
Elsa Jaitley is comparing three corporate bonds for inclusion in her fixed-income portfolio. For the next year, Jaitley has collected exposure and recovery information for three bonds (data given per $100 par). On a relative basis, which bond has the highest risk? Based on this information, what would be an appropriate trading strategy?
Answer
Credit risk is evaluated based on expected loss.
Given that exposure and recovery amounts are per $100 par:
Loss given default per $100 par = exposure - recovery.
For example, for bond Y: LGD = 88 - 45 = $43.
Expected loss = LGD × PD.
Based on expected loss calculations (using the PDs provided), bond Z is the most risky while bond Y has the least credit risk.
Because we are not provided with market prices for the three bonds, no trading strategy can be recommended based solely on expected losses.
MODULE QUIZ 28.2
1.
A 3-year, zero-coupon corporate bond trades at a price of $89.49. The benchmark yield curve is flat at 3%. Given a recovery rate of 45%, what is the IRR on the bond if it defaults in Year 2?
A. -24.66%.
B. -27.43%.
C. -30.13%.
2.
Given the following information, which bond has the least amount of credit risk?
A. Bond P.
B. Bond Q.
C. Bond R.
3.
A bond trader observes a 5% annual-pay, 3-year, corporate bond trading at $103. Benchmark rates are flat at 2.50%. The trader has collected the following information on the bond:
Based on the trader's analysis the bond is most likely:
A. correctly priced.
B. overvalued.
C. undervalued.
MODULE 28.3: CREDIT SCORES AND CREDIT RATINGS
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LOS 28.b: Explain credit scores and credit ratings.
Credit scoring is predominantly used for individuals and small businesses. Higher credit scores indicate better credit quality. One well-known credit scoring model in the United States is FICO. FICO scores are higher for those with (a) longer credit histories (age of oldest account), (b) absence of delinquencies, (c) lower utilisation (outstanding balance divided by available credit line), (d) fewer credit inquiries, and (e) a wider variety of credit types used.
Credit ratings are issued for corporate debt, securitised debt (asset-backed securities), and government and quasi-government debt. Like credit scores, ratings are ordinal (higher = better). Three major global rating agencies are Moody's Investor Service, Standard & Poor's, and Fitch Ratings.
The issuer rating for a company typically refers to its senior unsecured debt. Ratings on other classes of the issuer's debt are often notched-that is, lowered by one or more rating levels to reflect higher loss severity (LGD) for subordinated debt. Notching accounts for LGD differences between debt classes of different seniority.
In addition to a letter grade, rating agencies provide an outlook (positive, negative, stable). Higher-rated bonds generally trade at lower spreads relative to their benchmarks.
LOS 28.c: Calculate the expected return on a bond given transition in its credit rating.
Bond portfolio managers often evaluate how a bond's return would change if its credit rating migrates (i.e., a downgrade or upgrade). A change in rating typically reflects a change in credit risk. The resulting change in bond price depends primarily on the bond's modified duration and the change in credit spread implied by the rating migration.
Δ%P = - (modified duration of the bond) × (Δ spread)
EXAMPLE: Credit migration
Suppose a bond with modified duration 6.32 is downgraded from AAA to AA. Typical AAA spread = 60 bps; typical AA spread = 87 bps. Calculate the percentage change in price assuming the bond was priced at typical spreads.
Change in spread = 87 bps - 60 bps = 27 bps = 0.0027.
Δ%P = -6.32 × 0.0027 = -0.017064 = -1.71% (approximately).
MODULE QUIZ 28.3
1.
Credit ratings incorporate:
A. default probabilities only.
B. loss given default only.
C. both the default probability and the loss given default.
2.
Credit scores and credit ratings are:
A. ordinal.
B. cardinal.
C. optimized.
3.
In credit rating, the practice of notching accounts for differences in:
A. probability of default.
B. business cycle impact.
C. loss given default.
4.
Bjørn Johansen, portfolio manager for Agnes Advisors, is concerned about one of the holdings in Agnes's fixed income portfolio. Beta, Inc., bonds are currently rated A, but with a negative outlook. Beta bonds have a modified duration of 9.20. Johansen collects the following information about average spreads by ratings class.
Assuming that the spread on Beta bonds is equal to the average spread in its rating class, what is the expected change in price of Beta, Inc., bonds if Beta gets downgraded to BBB?
A. -1.01%.
B. -0.82%.
C. -0.92%.
MODULE 28.4: STRUCTURAL AND REDUCED FORM MODELS
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LOS 28.d: Explain structural and reduced-form models of corporate credit risk, including assumptions, strengths, and weaknesses.
STRUCTURAL MODELS
Structural models of corporate credit risk are grounded in the firm's balance sheet and exploit an analogy to option pricing theory.
Option Analogy
Consider a company financed by equity and a single issue of zero-coupon debt. The value of the firm's assets at any time is the sum of the value of equity and the value of debt.
Equity has limited liability. Thus, shareholders have the payoff of a European call option on the firm's assets with strike equal to the face value of debt K at debt maturity T. At time T:
value of equity_T = max(0, A_T - K)
Value of debt at time T equals firm assets less equity:
value of debt_T = A_T - value of equity_T = min(A_T, K)
where A_T is the value of the company's assets at time T and K is face value of debt.
An equivalent interpretation is that equity holders are long the firm's net assets (A_T - K) and long a put option allowing them to sell the assets at K. Default corresponds to exercising that put. Under the put-call parity analogy:
value of the put option = max(0, K - A_T)
Investors in risky debt can be viewed as holding a risk-free bond and being short a put on the firm's assets:
value of risky debt = value of risk-free debt - value of put option
Recall from CVA discussion:
value of risky debt = value of risk-free debt - CVA
Therefore the value of the put option = CVA.
Figure 28.2 in the original reading shows the distribution of asset values at time T. If asset value A_T falls below the default barrier K, default occurs. The left tail area below K corresponds to probability of default.
Advantages of structural models:
- They provide an economic rationale for default (default occurs when A_T < K).
- They utilise option pricing models to value risky debt using firm asset dynamics.
Disadvantages of structural models:
- They assume a simple balance sheet; complex balance sheets or significant off-balance-sheet liabilities make the default barrier K inaccurate and model outputs unreliable.
- They often assume the firm's assets are traded in liquid markets - a restrictive and often impractical assumption.
REDUCED FORM MODELS
Reduced-form (RF) models do not depend on the firm's balance sheet structure and do not require that the firm's assets trade. RF models do not explain why default occurs; rather, they statistically model when default occurs. Default is treated as an exogenous, randomly occurring event. RF models therefore impose assumptions on structural model outputs (asset values, recovery rates, etc.).
A key input to RF models is the default intensity (or hazard intensity), which is the instantaneous or short-period probability of default. Default intensity can be estimated using regression models with independent variables such as leverage, beta, interest coverage, and macroeconomic indicators.
Advantages of reduced-form models:
- They do not require that firm assets trade.
- Default intensity can vary over time as firm fundamentals or macro conditions change.
Disadvantages of reduced-form models:
- They do not explain why default occurs; they only model the timing probabilistically.
- They treat default as a surprise; in practice defaults are often preceded by observable distress and rating downgrades.
MODULE QUIZ 28.4
1.
The probability of default under the structural model is most likely:
A. endogenous.
B. exogenous.
C. estimated using a regression model.
2.
Under the structural model, risky debt can be thought of as equivalent to a portfolio comprising a long position in a risk-free bond and:
A. a short put option on assets of the company, with a strike price equal to the face value of debt.
B. a long put option on the assets of the company, with a strike price equal to the face value of debt.
C. short put option on assets of the company, with a strike price equal to the present value of debt.
3.
Which one of the following is least likely a disadvantage of structural models?
A. Structural models are not appropriate in the presence of off-balance sheet financing.
B. Structural models assume that company assets trade.
C. Structural models do not explain why default occurs.
4.
Default intensity is least likely to be:
A. estimated using company specific and macroeconomic variables.
B. the probability of default over the next time period.
C. constant over the life of the risky bond.
MODULE 28.5: CREDIT SPREAD ANALYSIS
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LOS 28.e: Calculate the value of a bond and its credit spread, given assumptions about the credit risk parameters.
Credit spread on a risky bond is defined as:
credit spread = YTM of risky bond - YTM of benchmark
The value of a risky bond assuming it does not default is its value given no default (VND). VND is calculated using the risk-free rate (benchmark) to discount the risky bond's cash flows.
EXAMPLE: Credit spread
Jack Gordon, a fixed income analyst, is evaluating an AA corporate bond. The bond is $100 par, 3.50% annual pay, 5-year and is currently priced with a credit spread of 135 bps over the benchmark par rate of 2%. Calculate the bond's CVA implied by its market price.
Answer
Compute VND for the bond (use benchmark YTM = 2%):
N = 5, PMT = 3.50, I/Y = 2.00, FV = 100 → CPT PV = 107.07 (VND).
Value of the risky bond using market YTM = benchmark 2% + credit spread 1.35% = 3.35%:
N = 5, PMT = 3.50, I/Y = 3.35, FV = 100 → CPT PV = 100.68 (market value).
CVA = VND - value of risky debt = 107.07 - 100.68 = $6.39.
We can also introduce interest-rate volatility using a binomial interest-rate tree for the benchmark rates and apply backward induction to value the VND and the expected exposures.
EXAMPLE: Calculation of VND using an interest-rate tree
For a 3-year, annual-pay, 4% coupon, $100 par bond using a given three-year benchmark interest-rate binomial tree, calculate the VND and the expected exposure for each year.
Answer
The procedure (backward induction) is:
At terminal nodes in Year 3, cash flow = 104 (coupon + par).
Compute present values at Year 2 nodes using the node-specific benchmark rate; then at Year 1 nodes compute the expected value of the two subsequent nodes (average) add coupon and discount one period at the node rate; continue to time 0.
Given the completed tree in the reading, the bottom node value at end of Year 2 = 104 / 1.04825 = $99.21.
Value of bottom node in Year 1 is the present value of the average of the two Year 2 values plus coupon:
Value_node = ( (99.21 + 97.02) / 2 + 4 ) / 1.0388 = $98.30.
Using the same procedure back to time 0 yields VND = $97.24 (as given).
Expected exposure for Year t is the probability-weighted sum of node values at time t plus the coupon for that year:
expected exposure Year 1 = 0.5×98.30 + 0.5×94.02 + 4 = $100.16 (rounded).
expected exposure Year 2 = 0.25×93.92 + 0.5×97.02 + 0.25×99.21 + 4 = $100.79 (rounded).
expected exposure Year 3 = $104 (terminal cash flow).
Once expected exposures are obtained, compute LGD, expected loss, discount, and sum PVs to obtain CVA given unconditional PDs and recovery rate.
MODULE QUIZ 28.5
1.
An analyst has completed the following interest rate tree:
The VND for a 3-year, annual-pay, 3.50% corporate bond is closest to:
A. $99.78.
B. $100.70.
C. $101.54.
2.
Jill Smith, CFA, is evaluating an annual pay 4%, 3-year corporate bond and has compiled the following information. Smith has assumed a recovery rate of 60% and a flat benchmark rate of 2.50%. Unfortunately, some of the information is missing. Complete the missing information to answer the question.
The CVA on the bond is closest to:
A. $1.20.
B. $2.58.
C. $2.82.
3.
Jill Smith, CFA, is evaluating a 4%, annual-pay, 3-year corporate bond with a CVA of $1.88. The bond is trading at $103.53. The benchmark par curve is given here.
Smith is most likely to conclude that the corporate bond is:
A. overvalued.
B. undervalued.
C. correctly valued.
MODULE 28.6: CREDIT SPREAD
LOS 28.f: Interpret changes in a credit spread.
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A benchmark yield should equate to the real risk-free rate plus expected inflation and a premium for uncertainty in future inflation. Credit spreads include compensation for default risk, liquidity risk, and taxation differences relative to the benchmark. Adjustment for all these risk factors together is sometimes referred to collectively as XVA in practice; in this reading the focus is on the default risk component (CVA), which is the most important and commonly used component in practice.
Credit spreads change as investor perceptions about future default probabilities and recovery rates change. Expectations about the economy affect these perceptions: expected recession → higher defaults and lower recoveries → wider spreads; expected expansion → tighter spreads.
EXAMPLE: Changes in probability of default and recovery rates
Joan De Silva is evaluating a 3-year, annual pay, 3% XYZ corporate bond priced at $102. Benchmark curve is flat at 1.75%. She prepares an analysis assuming hazard rate 1.25% and recovery rate 70% and calculates expected exposures. Example results:
expected exposure Year 1 = (104.23 / 1.0175) + 3 = $105.44.
expected exposure Year 2 = (103 / 1.0175) + 3 = $104.23.
expected exposure Year 3 = $103.
The CIO, Susan Collins, believes a slowdown implies hazard rate 1.50% and recovery rate 60%. Using these revised assumptions De Silva prepares updated CVA and bond valuation.
Quiz based on the example
1.
Using De Silva's estimates of hazard rate and recovery rate, XYZ bond is currently most likely:
A. undervalued.
B. overvalued.
C. fairly priced.
2.
Using the market price of the bond, the credit spread on XYZ bond is closest to:
A. 0.44%.
B. 0.49%.
C. 0.55%.
3.
Assuming the market price changes to reflect Collins's expectations of PD and recovery rate, the new credit spread would be closest to:
A. 0.52%.
B. 0.61%.
C. 0.79%.
Answer (for the example)
1. A Using the benchmark rate, XYZ bond's VND is:
N = 3, PMT = 3, I/Y = 1.75, FV = 100 → CPT PV = 103.62 (VND).
Value of risky bond (market) = VND - CVA = 103.62 - 1.12 = 102.50.
Since market price is $102.00, the bond is undervalued (market price < model value).
2. C YTM for 3-year risk-free bond = 1.75% (given).
YTM for XYZ bond (market price 102): PV = -102, N = 3, PMT = 3, FV = 100 → CPT I/Y = 2.30%.
Credit spread = 2.30% - 1.75% = 0.55%.
3. B Based on revised PD and recovery rate, CVA = 1.79; value of risky bond = 103.62 - 1.79 = 101.83.
YTM for XYZ (PV = -101.83, N = 3, PMT = 3, FV = 100) → CPT I/Y = 2.36%.
Credit spread = 2.36% - 1.75% = 0.61%.
LOS 28.g: Explain the determinants of the term structure of credit spreads and interpret a term structure of credit spreads.
The term structure of credit spreads displays the relationship between credit spreads and maturity. Term structures can be built for bonds issued by a single issuer (across maturities) or across a sector (e.g., AAA corporate spread curve). They are useful to price new issues and to assess relative valuation of existing issues.
Figures in the reading (Figure 28.4 and 28.5) show Treasury yields and high-quality corporate yields and the resulting credit spread curve. The credit spread is inversely related to recovery rate and positively related to probability of default. When a maturity-matched liquid benchmark is unavailable, interpolation or reference to swap curves is common.
To construct a meaningful spread curve, include bonds with similar credit characteristics. Differences in seniority, lien status, or embedded options distort spreads and will produce an inaccurate curve if mixed together.
Key determinants of the shape of the credit spread curve include expectations about future recovery rates and default probabilities. If default probabilities are expected to be higher (or recoveries lower) in the future, the credit curve will be positively sloped. Flat credit curves signal stable expectations over time.
Determinants of Term Structure of Credit Spreads
- Credit quality: AAA term structures tend to be flat or slightly upward sloping; lower-rated sectors tend to have steeper curves reflecting higher uncertainty and greater sensitivity to the business cycle.
- Financial conditions: Spreads narrow during expansions and widen during downturns. During booms, benchmark yields may be higher while credit spreads narrow.
- Market supply and demand: Liquidity premia matter. Less liquid maturities show higher spreads. More heavily traded maturities influence the curve most. New issues are usually more liquid and may display narrower spreads.
- Equity market volatility: Structural models use stock volatility as input; increased equity volatility tends to widen spreads and influence curve shape.
MODULE QUIZ 28.6
1.
Mossimo Gulzar is compiling data on spreads for AAA corporate bonds. Mossimo has compiled the following information from the firm's senior economist.
Using a typical 4% coupon AAA bond for each maturity category, Mossimo would most likely conclude that the credit spread curve is:
A. upward sloping.
B. downward sloping.
C. flat.
2.
Credit spreads are most likely:
A. positively related to probability of default and loss severity.
B. positively related to probability of default and recovery rate.
C. negatively related to probability of default and recovery rate.
3.
Credit spread curves of the highest rated bond sectors tend to be:
A. flat or downward sloping.
B. flat or slightly upward sloping.
C. steeply upward sloping.
4.
Flat credit curves are most likely indicative of:
A. expectations of economic expansion.
B. expectations of a recession.
C. stable expectations about future recovery rates and default probabilities.
5.
The shape of the credit curve is least likely to be affected by:
A. demand and supply forces in the market.
B. sector quality.
C. the swap rate curve.
MODULE 28.7: CREDIT ANALYSIS OF SECURITIZED DEBT
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LOS 28.h: Compare the credit analysis required for securitized debt to the credit analysis of corporate debt.
Securitized debt finances specific assets (for example auto loans, credit-card receivables, and mortgages) rather than the issuer's entire balance sheet. Securitisation typically uses a bankruptcy-remote special purpose entity (SPE) to isolate collateral. This isolation allows higher leverage for the originator and often a lower funding cost; investors gain diversification, more stable cash flows, and often higher risk premia compared to similarly rated general-obligation bonds due to the complexity of collateralised structures.
Components of Credit Analysis of Secured Debt
1. Collateral pool: Credit analysis begins with the collateral. Homogeneity refers to similarity among assets in the pool. Granularity refers to the number and transparency of loans; a granular pool with many homogeneous loans allows summary statistics to be used, whereas a small, discrete pool requires loan-by-loan analysis.
Short-term, granular and homogeneous vehicles are evaluated using a statistical-based approach. Medium-term granular homogeneous pools can be evaluated using a portfolio-based approach because portfolio composition varies over time. Discrete and non-granular portfolios require loan-level analysis.
2. Servicer quality: Evaluate the servicer's ability to originate and service loans. Post-origination investors face operational and counterparty risk associated with the servicer. Past servicing performance is often informative.
3. Structure: Structural elements include tranching, overcollateralisation, excess spread and distribution waterfalls. Credit enhancement may be internal (tranches, overcollateralisation, excess servicing spread) or external (third-party guarantees such as banks or insurers).
Covered bonds are a special category: issued by financial institutions, covered bonds are senior, secured obligations backed by a collateral pool and by the issuer (investors have recourse to the issuer in addition to the cover pool). Covered bonds are common in Europe and elsewhere. The cover pool is dynamic: the sponsor must replace prepaid or nonperforming assets to maintain sufficient cash flows.
In case of sponsor payment failure, covered bonds may have soft or hard bullet structures. A soft bullet allows extension of maturity up to one year and avoids immediate acceleration; a hard bullet accelerates payments.
MODULE QUIZ 28.7
1.
Short-term granular and homogenous structured finance vehicles are most appropriately evaluated using a:
A. statistical-based approach.
B. portfolio-based approach.
C. loan level analysis.
2.
Medium-term granular and homogenous structured finance vehicles are most appropriately evaluated using a:
A. statistical-based approach.
B. portfolio-based approach.
C. loan level analysis.
3.
With respect to the servicer in a secured debt, investors most likely face:
A. operational and financial risk only.
B. financial and counterparty risk only.
C. operational and counterparty risk only.
4.
Covered bonds are most likely issued by a financial institution and:
A. are backed by the government.
B. are backed by the collateral pool only.
C. have recourse rights as well as backing of the collateral pool.
KEY CONCEPTS
LOS 28.a
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. Loss given default equals loss severity multiplied by exposure. Credit valuation adjustment (CVA) is the sum of the present values of expected losses for each period.
LOS 28.b
Credit scoring is used for individuals and small businesses; credit ratings are used for corporate and securitised debt and governments. Scores and ratings are ordinal; notching accounts for seniority and LGD differences across issues.
LOS 28.c
The change in the price of a bond resulting from credit migration depends on the bond's modified duration and the change in spread:
Δ%P = - (modified duration) × (Δ spread)
LOS 28.d
Structural models are based on balance sheet structure and option pricing insights; equity is treated as a call option on assets. A risky debt investment is equivalent to buying a risk-free bond and writing a put option. Reduced-form models statistically model default timing and treat default as an exogenous random variable.
LOS 28.e
credit spread = YTM (risky) - YTM (benchmark)
The value of a risky bond assuming no default is VND. VND can be computed by discounting cash flows using benchmark rates or via backward induction on an interest-rate tree.
LOS 28.f
Credit spreads change when investors update expectations for future PDs and recovery rates. Anticipation of recession typically implies higher PDs and lower recoveries, widening spreads.
LOS 28.g
The term structure of credit spreads shows spread vs maturity. It depends on credit quality, economic/financial conditions, supply/demand in bond markets and expected equity volatility.
LOS 28.h
Credit analysis of ABS focuses on the collateral pool, servicer quality, and structural protections (distribution waterfall, credit enhancement) rather than the issuer's entire balance sheet.
ANSWER KEY FOR MODULE QUIZZES
Module Quiz 28.1
1.
A Value of benchmark 5% annual pay, 5-year bond at 2% = N = 5, PMT = 5, FV = 100, I/Y = 2 → CPT PV = 114.14. Value of Barry Corp. bonds = 114.14 - 12.67 = $101.47. (LOS 28.a)
2.
B The hazard rate is the conditional probability of default and equals PD in the first year. In years after the first, PD < hazard rate because PS < 1. (LOS 28.a)
3.
C Probability of survival for 3 years with hazard 1.25%: PS = (1 - 0.0125)^3 = 0.9630 = 96.30%. (LOS 28.a)
4.
B expected loss = exposure × (1 - recovery rate) × PD = LGD × PD. (LOS 28.a)
5.
A CVA = price of risk-free bond - price of risky bond. CVA is the sum of PVs of expected loss. (LOS 28.a)
6.
A If Abramson's estimated hazard rate < risk-neutral PD implied by the market, the market has priced in greater credit risk and the bond is likely undervalued. (LOS 28.a)
7.
A Given market price, an increase in assumed recovery rate implies a higher risk-neutral PD to justify the same price. Therefore risk-neutral PD and assumed recovery rate are positively correlated. (LOS 28.a)
Module Quiz 28.2
1.
C Exposure in Year 2 = 100 / 1.03 = 97.09; recovery cash flow = exposure × recovery rate = 97.09 × 0.45 = 43.69. PV = -89.49, N = 2, FV = 43.69 → CPT I/Y = -30.13%. (LOS 28.a)
2.
B Bond Q has the lowest expected loss and hence the least credit risk. (LOS 28.a)
3.
C Value of comparable risk-free bond at 2.50%: N = 3, PMT = 5, FV = 100, I/Y = 2.50 → CPT PV = 107.14 (rounded). Sum of PV of expected loss column = CVA = 2.41. Risky bond value = 107.41 - 2.41 = $104.73 > market price $103 → bond is undervalued. (LOS 28.a)
Module Quiz 28.3
1.
C Credit ratings incorporate probability of default and LGD due to notching for subordinated debt. (LOS 28.b)
2.
A Credit scores and ratings are ordinal measures - a higher rating implies better credit but differences are not proportional. (LOS 28.b)
3.
C Notching accounts for LGD differences across classes of debt by same issuer (higher LGD for lower seniority). (LOS 28.b)
4.
A Change in spread from A to BBB is 0.60% - 0.49% = +0.11%. Percent change in price = -9.20 × 0.11% = -1.01%. (LOS 28.c)
Module Quiz 28.4
1.
A Under the structural model, PD is endogenous: it is the probability that future asset value falls below the default barrier. It is not estimated by regression in the model specification. (LOS 28.d)
2.
A Risky debt is equivalent to a risk-free bond and a short put option on firm assets with strike equal to face value of debt. (LOS 28.d)
3.
C Structural model disadvantages include: assumption that company assets trade and sensitivity to off-balance sheet financing. They do explain why default occurs (unlike RF models). (LOS 28.d)
4.
C Default intensity is the short-period probability of default and varies over the life of a bond as company and economic conditions change. It is not constant. (LOS 28.d)
Module Quiz 28.5
1.
B The completed tree yields VND = [(101.09 + 98.36) / 2 + 3.50] / 1.025 = $100.70. (LOS 28.e)
2.
A Year 2 expected exposure = (104 / 1.025) + 4 = $105.46. LGD_Year2 = exposure × (1 - recovery) = 105.46 × 0.4 = 42.19. expected loss = 42.19 × PD (0.0099) = $0.418. PV = expected loss × DF = 0.418 × 0.9518 = $0.40. CVA (sum) = 0.42 + 0.40 + 0.38 = $1.20 (rounded). (LOS 28.e)
3.
A Using the 3-year par rate: FV = 100, N = 3, PMT = 4, I/Y = 2.50 → CPT PV = 104.28 (VND). Value of corporate = VND - CVA = 104.28 - 1.88 = 102.40. Since market price = 103.53 > model value 102.40, the bond is overvalued. (LOS 28.e)
Module Quiz 28.6
1.
B Following the procedure for each maturity with coupon 4% and benchmark par rates, the computed risky yields decline with maturity → downward sloping credit spread curve. (LOS 28.f)
2.
A Credit spreads are positively related to probability of default and loss severity (which equals 1 - recovery rate). (LOS 28.g)
3.
B Highest rated sectors tend to have flat or slightly upward sloping credit curves. (LOS 28.g)
4.
C Flat credit curves indicate stable expectations about future PDs and recovery rates. (LOS 28.g)
5.
C The shape of the credit curve is determined principally by sector quality, market supply/demand, company value model inputs and financial conditions; the swap rate curve is not a primary determinant (though swaps may be used as benchmark when Treasury data is thin). (LOS 28.g)
Module Quiz 28.7
1.
A Short-term granular and homogeneous structures are evaluated using a statistical-based approach. (LOS 28.h)
2.
B Medium-term granular and homogeneous obligations are evaluated using a portfolio-based approach. (LOS 28.h)
3.
C After origination, investors in secured debt face operational and counterparty risk associated with the servicer. (LOS 28.h)
4.
C Covered bonds are senior, secured bonds backed by the collateral pool and by the issuer (i.e., investors have recourse rights). (LOS 28.h)
