Here Are 2 Questions And Each Has 2 Or 3 Parts

Here Are 2 Questions And Each Question Has 2 Or 3 Parts That I Need F

Here are 2 questions, and each question has 2 or 3 parts that require full and complete calculations and explanations. The assignment is due Thursday the 26th.

Paper For Above instruction

Question 20: Suppose the estimated linear probability model used by a financial institution to predict business loan applicant default probabilities is given by:

PD = 0.03X1 + 0.02X2 - 0.05X3 + error,

where X1 is the borrower’s debt/equity ratio, X2 is the volatility of borrower earnings, and X3 is the borrower’s profit ratio. For a particular loan applicant, X1 = 0.75, X2 = 0.25, and X3 = 0.10.

A. What is the projected probability of default for the borrower?

B. What is the projected probability of repayment if the debt/equity ratio is 2.5?

Question 32: The bond equivalent yields for U.S. Treasury and A-rated corporate bonds with maturities of 93 and 175 days are given as follows:

93 days: Treasury strip = 8.07%, A-rated corporate = 8.66%, Spread = 0.55%

175 days: Treasury strip = 8.11%, A-rated corporate = (not directly given, but related)

A. What are the implied forward rates for both an 82-day Treasury and an 82-day A-rated bond beginning in 93 days? Use daily compounding on a 365-day year basis.

B. What is the implied probability of default on A-rated bonds over the next 93 days? Over 175 days?

C. What is the implied default probability on an 82-day, A-rated bond to be issued in 93 days?

Solution to the Above Questions

Question 20: Default Probability Predictions

Let's analyze the model provided for the probability of default (PD):

PD = 0.03X1 + 0.02X2 - 0.05X3 + error

Given the applicant's data: X1 = 0.75, X2 = 0.25, X3 = 0.10.

A. Calculation of the Projected Probability of Default

Plugging the values into the linear model:

PD = 0.03(0.75) + 0.02(0.25) - 0.05(0.10)

PD = 0.0225 + 0.005 - 0.005

PD = 0.0225 + 0.005 - 0.005 = 0.0225

Thus, the projected probability of default for this borrower is approximately 2.25%.

B. Projected Probability of Repayment if Debt/Equity Ratio is 2.5

Assuming the same model, but replacing X1 with 2.5, while keeping X2 and X3 the same:

PD = 0.03(2.5) + 0.02(0.25) - 0.05(0.10)

PD = 0.075 + 0.005 - 0.005 = 0.075

Therefore, the projected probability of default increases to 7.5%.

Consequently, the probability of repayment is 1 - PD = 1 - 0.075 = 92.5%.

Question 32: Bond Yield and Default Probability Analysis

Given data:

  • 93 days Treasury yield: 8.07%
  • 93 days A-rated corporate yield: 8.66%
  • Spread over Treasury: 0.55%
  • 175 days Treasury yield: 8.11%

A. Implied Forward Rates for 82-day Bonds Starting in 93 Days

To find the implied forward rate (f) over the 82 days beginning in 93 days, we use the following relation based on compound interest:

(1 + R_{175})^{175/365} = (1 + R_{93})^{93/365} * (1 + f)^{82/365}

Where R_{93} = 8.07%, R_{175} = 8.11%, corresponding to the yields for the respective periods.

Step 1: Convert yields to decimal form:

  • R_{93} = 0.0807
  • R_{175} = 0.0811

Step 2: Calculate the accumulated growth over 93 days:

(1 + R_{93})^{93/365} = 1.0807^{93/365}

Calculating:

1.0807^{0.2548} ≈ e^{0.2548 ln(1.0807)} ≈ e^{0.2548 0.0775} ≈ e^{0.0197} ≈ 1.0199

Step 3: Similarly, for 175 days:

(1 + R_{175})^{175/365} = 1.0811^{0.4795} ≈ e^{0.4795 * 0.0780} ≈ e^{0.0374} ≈ 1.0381

Step 4: Now, solve for the forward rate f over the 82-day gap:

f = [(1 + R_{175})^{175/365} / (1 + R_{93})^{93/365}]^{1/(82/365)} - 1

f = (1.0381 / 1.0199)^{0.4795 / 0.2254} - 1 ≈ (1.0187)^{2.124} - 1

f ≈ e^{2.124 ln(1.0187)} - 1 ≈ e^{2.124 0.0185} - 1 ≈ e^{0.0393} - 1 ≈ 1.0401 - 1 = 0.0401

Thus, the implied forward rate is approximately 4.01% annually for the 82-day period starting in 93 days.

B. Implied Default Probabilities over Next 93 and 175 Days

To derive the implied probabilities of default (PD), we relate yield spreads to default risk, assuming a simple model where the spread equals the product of default probability and loss given default (LGD). For simplicity, assuming LGD is 100%, the probability over the period is approximately equal to the spread divided by 100%.

For 93 days:

Spread = 0.55% = 0.0055

Implied PD ≈ 0.0055 or 0.55%

For 175 days:

Similarly, the spread is approximately 0.55%, but since the actual yield spread for 175 days is not directly provided, we infer it based on the incremental spread over the longer period. If the spread for 175 days is consistent or slightly higher, say approximately 0.60%, then the implied PD would be approximately 0.006 or 0.60% over 175 days.

These calculations suggest the default probabilities over respective periods are less than 1%, indicating relatively low risk for A-rated bonds but increasing with time.

C. Default Probability on an 82-day A-rated Bond Issued in 93 Days

This involves calculating the implied PD over an 82-day period beginning 93 days from now, which we already approximated through the forward rate simplification. The key assumption is that the default risk over this short period is proportional to the forward rate's implied probability, scaled to the period length.

Implied PD over 82 days is roughly:

Forward rate approximation: 4.01% annually

Period length: 82/365 ≈ 0.2247 years

Therefore, implied PD ≈ 0.0401 * 0.2247 ≈ 0.009 or 0.9%

This reflects the probability of default for the bond issued in 93 days, due to the incremental risk implied by the forward rate for 82 days.

Conclusion

The analysis highlights how linear probability models can provide quick estimates for borrower default risks, with increased risk associated with higher debt/equity ratios. The bond yield analysis underscores the importance of forward rate calculations in understanding implied future rates and credit risk over specific periods. These models and calculations are fundamental in risk management, pricing, and investment decision-making within the financial industry.

References

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  • Mishkin, F. S. (2015). The Economics of Money, Banking, and Financial Markets. Pearson.
  • Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset. Wiley.
  • Bailey, D., & McGregor, G. (2018). Credit Risk Management. Routledge.
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  • Longstaff, F. A., & Rajan, A. (2008). An empirical analysis of the pricing of collateralized debt obligations. Journal of Financial Economics, 89(2), 249-273.
  • Altman, E. I. (2012). Corporate Default and Bankruptcy. The Journal of Risk and Insurance, 79(2), 437-440.

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