## QUESTION

Chapter 10

34.The following is a schedule of historical defaults (yearly and cumulative) experienced by an FI manager on a portfolio of commercial and mortgage loans.

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Complete the blank spaces in the table. What are the probabilities that each type of loan will not be in default after five years? What is the measured difference between the cumulative default (-mortality) rates for commercial and mortgage loans after four years?

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Years after Issuance

Loan Type 1 Year 2 Years 3 Years 4 Years 5 Years

Commercial:

Annual default 0.00% _____ 0.50% _____ 0.30%

Cumulative default _____ 0.10% _____ 0.80% _____

Mortgage:

Annual default 0.10% 0.25% 0.60% _____ 0.80%

Cumulative default _____ _____ _____ 1.64% _____

Complete the blank spaces in the table.

What are the probabilities that each type of loan will not be in default after five years?

What is the measured difference between the cumulative default (-mortality) rates for commercial and mortgage loans after four years?

Chapter 11

12. A bank vice president is attempting to rank, in terms of the risk-reward trade-off, the loan portfolios of three loan officers. Information on the portfolios is noted below. How would you rank the three portfolios?

Portfolio Expected

Return Standard Deviation

A 10% 8%

B 12 9

C 11 10

13.Suppose that an FI holds two loans with the following characteristics.

Loan Xi Annual Spread between Loan Rate and FI’s Cost of FundsAnnual Fees Lossto FI Given Default Expected Default Frequency

1 0.45 5.5% 2.25% 30% 3.5% ρ12 = −0.15

2 0.55 3.5 1.75 20 1.0

Chapter 15

4. Follow Bank has a $1 million position in a five-year, zero-coupon bond with a face value of $1,402,552. The bond is trading at a yield to maturity of 7.00 percent. The historical mean change in daily yields is 0.0 percent and the standard deviation is 12 basis points.

A. What is the modified duration of the bond?

B. What is the maximum adverse daily yield move given that we desire no more than a 1 percent chance that yield changes will be higher than this maximum?

C. What is the price volatility of this bond?

D. What is the daily earnings at risk for this bond?

## ANSWER

Chapter 10:

- The missing values in the table can be filled as follows:

Years after Issuance

Loan Type 1 Year 2 Years 3 Years 4 Years 5 Years

Commercial:

Annual default 0.00% 0.50% 0.30% 0.50% 0.30%

Cumulative default 0.00% 0.10% 0.40% 0.80% 1.10%

Mortgage:

Annual default 0.10% 0.25% 0.60% 0.40% 0.80%

Cumulative default 0.10% 0.35% 0.95% 1.64% 2.44%

The probabilities that each type of loan will not be in default after five years can be calculated by subtracting the cumulative default rates from 100%:

Commercial Loan: 100% – 1.10% = 98.90% Mortgage Loan: 100% – 2.44% = 97.56%

The measured difference between the cumulative default rates for commercial and mortgage loans after four years can be calculated by subtracting the commercial cumulative default rate from the mortgage cumulative default rate:

Difference = 1.64% – 0.80% = 0.84%

Chapter 11:

- To rank the portfolios based on the risk-reward trade-off, we can consider the portfolios’ risk-return ratios, which are calculated by dividing the expected return by the standard deviation.

Portfolio Risk-Return Ratio

A 1.25 (10% / 8%) B 1.33 (12% / 9%) C 1.10 (11% / 10%)

Based on the risk-return ratios, the portfolios can be ranked as follows:

- Portfolio B (highest risk-return ratio)
- Portfolio A
- Portfolio C (lowest risk-return ratio)

Chapter 15:

- The answers to the questions are as follows:

- The modified duration of the bond can be calculated using the formula:

Modified Duration = Macaulay Duration / (1 + Yield to Maturity)

Since it is a zero-coupon bond, the Macaulay Duration is equal to the bond’s maturity, which is 5 years. The yield to maturity is 7.00%.

Modified Duration = 5 / (1 + 0.07) = 4.67

- To determine the maximum adverse daily yield move, we need to consider the bond’s standard deviation and the desired probability of yield changes being higher than the maximum move. Let’s assume the desired probability is 1%.

Using statistical tables or software, we can find the corresponding z-score for a 1% probability, which is approximately 2.33. We multiply this by the standard deviation of daily yield changes (0.12%) to find the maximum adverse daily yield move:

Maximum Adverse Daily Yield Move = 2.33 * 0.12% = 0.28%

- The price volatility of the bond, also known as the percentage price change for a given change in yield, can be calculated using the modified duration:

Price Volatility = Modified Duration * Change in Yield

Since the modified duration is 4.67 and we want to find the price volatility for a 1% change in yield:

Price Volatility = 4.67 * 1% = 4.67%

- The daily earnings at risk for the bond can be calculated by multiplying the price volatility by the position in the bond:

Daily Earnings at Risk = Price Volatility * Position

Assuming the position is $1 million:

Daily Earnings at Risk = 4.67% * $1,000,000 = $46,700

Please note that these calculations are based on the information provided and the formulas used in financial risk analysis.