Suppose The Estimated Linear Probability Model Used By An FI

Suppose the estimated linear probability model used by an FI to predict

Suppose the estimated linear probability model used by an FI to predict

Question 20: Suppose the estimated linear probability model used by a financial institution (FI) 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.

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

- Treasury strip 93 days: 8.07%

- Treasury strip 175 days: 8.11%

- A-rated corporate 93 days: 8.42%

- A-rated corporate 175 days: 8.66%

- Spread for 93 days: 0.35%

- Spread for 175 days: 0.55%

Paper For Above instruction

Introduction

Financial institutions rely heavily on predictive models and market data to evaluate risk and make informed decisions. Linear probability models (LPM) are often utilized to estimate the likelihood of borrower default based on specific financial indicators. Understanding how to interpret these models and extract meaningful probabilities is vital for risk management. Additionally, bond yields and spreads provide insight into the expectations of market participants regarding future interest rates, potential defaults, and economic outlook. This paper thoroughly analyzes the provided models and data, performing complete calculations and explanations for each part of the questions, with reference to financial theories and empirical evidence.

Part 1: Predicting Borrower Default Probability

Part A: Calculating the Projected Probability of Default (PD)

The given linear probability model states:

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

For the specific borrower, X1 = 0.75, X2 = 0.25, and X3 = 0.10. Ignoring the error term which represents model residuals, the projected probability of default (PD) is calculated as follows:

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


= 0.0225 + 0.005 - 0.005


= 0.0225

This results in a projected PD of approximately 0.0225, or 2.25%, indicating a low probability of default for this borrower based on the model.

Part B: Estimating Probability of Repayment with a Different Debt/Equity Ratio X1=2.5

Using the same model, but now with X1 = 2.5, X2 = 0.25, and X3 = 0.10, the projected PD becomes:

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


= 0.075 + 0.005 - 0.005


= 0.075

Thus, the probability of default is approximately 7.5%. To find the probability of repayment (PR), we subtract the PD from 1:

PR = 1 - PD = 1 - 0.075 = 0.925 or 92.5%

Therefore, if the debt/equity ratio increases to 2.5, the likelihood that the borrower repays the loan rises to 92.5%, contingent upon the model's assumptions and the linearity of the relationship.

Part 2: Bond Yields, Forward Rates, and Default Probabilities

Part A: Calculating Implied Forward Rates

Background

The implied forward rate can be derived from bond yields, reflecting market expectations of future interest rates. Using the bond yields and maturities provided, daily compounding yields over a 365-day year facilitate calculation.

Calculation of Forward Rate for an 82-day period starting in 93 days for Treasury Strips

The yield for the 93-day treasury strip is 8.07%, and for the 175-day treasury strip is 8.11%. The forward rate (f) for the period from day 93 to day 175 (which is 82 days) is given by:

f = [ (1 + y_{total})^{T_{total}} / (1 + y_{initial})^{T_{initial}} ]^{1/(T_{final} - T_{initial})} - 1

Expressed with daily compounding:

f = [ (1 + 0.0811/100)^{175/365} / (1 + 0.0807/100)^{93/365} ]^{365/82} - 1

Calculations proceed as follows:

First, convert yields to decimal form:


Y_{93} = 0.0807


Y_{175} = 0.0811


Calculate accumulations:


A = (1 + Y_{175})^{175/365} ≈ (1 + 0.0811)^{0.47945} ≈ 1.0811^{0.47945}


B = (1 + Y_{93})^{93/365} ≈ (1 + 0.0807)^{0.25479} ≈ 1.0807^{0.25479}


Numerical calculation:


A ≈ e^{0.47945  ln(1.0811)} ≈ e^{0.47945  0.0779} ≈ e^{0.0373} ≈ 1.0379


B ≈ e^{0.25479  ln(1.0807)} ≈ e^{0.25479  0.0775} ≈ e^{0.0197} ≈ 1.0199


Forward rate:


f = A / B^{(175 - 93)/365} - 1 = 1.0379 / 1.0199^{82/365} - 1


Calculate exponent:


82/365 ≈ 0.2247


Calculate denominator:


1.0199^{0.2247} ≈ e^{0.2247  ln(1.0199)} ≈ e^{0.2247  0.0197} ≈ e^{0.0044} ≈ 1.0044


Final:


f ≈ 1.0379 / 1.0044 - 1 ≈ 1.0333 - 1 = 0.0333 or 3.33% annualized forward rate over the 82-day period.


Similarly, for the A-rated corporate bonds, yields of 8.42% for 93 days and 8.66% for 175 days lead to a forward rate calculation following the same methodology, resulting in an implied forward rate of approximately 3.88% annually.


Part B: Implied Default Probabilities


Over the Next 93 Days and 175 Days


The spread over the Treasury yield reflects the market's compensation for default risk. The spread difference indicates increased risk for A-rated bonds. The implied probability of default (PD) can be approximated from the spread as follows:


PD ≈ Spread / (1 + Yield) * (Time in years)




For 93 days (about 0.2548 years), with a spread of 0.35% (0.0035), and Treasury yield of 8.07%:


PD_{93} ≈ 0.0035 / (1 + 0.0807)  0.2548 ≈ 0.0035 / 1.0807  0.2548 ≈ 0.00082 or 0.082%




Similarly, for 175 days (0.4795 years), spread of 0.55% (0.0055), and yield 8.11%:


PD_{175} ≈ 0.0055 / 1.0811 * 0.4795 ≈ 0.00243 or 0.243%




These estimates suggest relatively low default probabilities over both periods, consistent with credit ratings and market expectations.


Part C: Default Probability for an 82-day Bond to be Issued in 93 days


This forward-looking default probability captures the market's expectation of default risk for the new bond issued in 93 days, over the next 82 days. Using the previously calculated forward rate (approximately 3.33% annualized), the probability of default during this period can be inferred with a hazard rate approximation, or directly from market spreads if available.


Assuming the spread over the forward rate reflects the default probability, the implied risk over 82 days (0.2247 years) is approximately:


Default probability ≈ Spread / (1 + Forward rate) * Time


≈ 0.0055 / 1.0333 * 0.2247


≈ 0.0012 or 0.12%




This indicates a very low implied default risk for the bond issued in 93 days over its initial 82 days, reinforcing that market-anticipated default probabilities remain low within this short time horizon.


Conclusion


In summary, the analysis demonstrates the application of linear probability models to estimate default risks, complemented by bond yield calculations to infer market expectations about future interest rates and credit risk. The low default probabilities derived from spreads and forward rates align with the high credit ratings of the instruments analyzed. These tools are crucial for financial decision-making, enabling risk managers and investors to quantify uncertainties and develop appropriate strategies.


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