Probability Concept

FIN 3309

1. Properties of Probability function

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• ____________________________

2. 3 different types of probabilities

• ____________________________

• ____________________________

• ____________________________

3. You have a bag of m&ms with only 12 left. There is 3 red, 4 yellow, 5 blue. What is the probability that you will get a red m&m? What is the odds that you would choose red?

4. You are given the following probability distribution for the annual sales of ElStop Corporation:

Probability Distribution for ElStop Annual Sales

Probability Sales ($ Millions)

0.20 275

0.40 250

0.25 200

0.10 190

0.05 180

sum = 1.00

• Calculate the expected value of ElStop’s annual sales.

• Calculate the variance of ElStop’s annual sales.

• Calculate the standard deviation of ElStop’s annual sales.

5. An analyst developed two scenarios with respect to the recovery of $100,000 principal from defaulted loans is:

Scenario Probability of Scenario (%) Amount Recovered ($) Probability of Amount (%)

1 40 50,000 60

30,000 40

2 60 80,000 90

60,000 10

What is the amount of the expected recovery?

6. Calculate the covariance of the returns on Bedolf Corporation (RB) with the returns on Zedock Corporation (Rz), using the following data.

Probability Function of Bedolf and Zedock Returns

RZ = 15% RZ = 10% RZ = 5%

RB = 30% 0.25 0 0

RB = 15% 0 0.50 0

RB = 10% 0 0 0.25

Probability Concept

FIN 3309

1. Properties of Probability function

• ____________________________

• ____________________________

2. 3 different types of probabilities

• ____________________________

• ____________________________

• ____________________________

3. You have a bag of m&ms with only 12 left. There is 3 red, 4 yellow, 5 blue. What is the probability that you will get a red m&m? What is the odds that you would choose red?

4. You are given the following probability distribution for the annual sales of ElStop Corporation:

Probability Distribution for ElStop Annual Sales

Probability Sales ($ Millions)

0.20 275

0.40 250

0.25 200

0.10 190

0.05 180

sum = 1.00

• Calculate the expected value of ElStop’s annual sales.

• Calculate the variance of ElStop’s annual sales.

• Calculate the standard deviation of ElStop’s annual sales.

5. An analyst developed two scenarios with respect to the recovery of $100,000 principal from defaulted loans is:

Scenario Probability of Scenario (%) Amount Recovered ($) Probability of Amount (%)

1 40 50,000 60

30,000 40

2 60 80,000 90

60,000 10

What is the amount of the expected recovery?

6. Calculate the covariance of the returns on Bedolf Corporation (RB) with the returns on Zedock Corporation (Rz), using the following data.

Probability Function of Bedolf and Zedock Returns

RZ = 15% RZ = 10% RZ = 5%

RB = 30% 0.25 0 0

RB = 15% 0 0.50 0

RB = 10% 0 0 0.25

Stocks A and B have the following probability distributions of expected future returns:

1. Calculate the expected rate of return,  , for Stock B ( = 12.60%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

2. Calculate the standard deviation of expected returns, σ_A, for Stock A (σ_B = 21.04%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

   Now calculate the coefficient of variation for Stock B. Do not round intermediate calculations. Round your answer to two decimal places.

   Is it possible that most investors might regard Stock B as being less risky than Stock A?

   1. If Stock B is more highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be less risky in a portfolio sense.
   2. If Stock B is more highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   3. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.
   4. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   5. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.

   -Select-IIIIIIIVVItem 4

3. Assume the risk-free rate is 4.5%. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to four decimal places.

   Stock A:

   Stock B:

   Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b?

   1. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   2. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   3. In a stand-alone risk sense A is less risky than B. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.
   4. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   5. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.

   -Select-IIIIIIIVV

Stocks A and B have the following probability distributions of expected future returns:

1. Calculate the expected rate of return, , for Stock B ( = 12.10%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

2. Calculate the standard deviation of expected returns, σ_A, for Stock A (σ_B = 16.33%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

   Now calculate the coefficient of variation for Stock B. Do not round intermediate calculations. Round your answer to two decimal places.

   Is it possible that most investors might regard Stock B as being less risky than Stock A?

   1. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.
   2. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   3. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   4. If Stock B is more highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be less risky in a portfolio sense.
   5. If Stock B is more highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.

   -Select-IIIIIIIVVItem 4

3. Assume the risk-free rate is 2.5%. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to four decimal places.

   Stock A:

   Stock B:

   Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b?

   1. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   2. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   3. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   4. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   5. In a stand-alone risk sense A is less risky than B. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.

A firm can be worth $100 million (with 20% probability), $200 million (with 60% probability), or $300 million (with 20% probability). The firm has one senior bond outstanding, promising to pay $80 million. It also has one junior bond outstanding, promising to pay $70 million. The senior bond promises an interest rate of 5%. The junior bond promises an interest rate of 26%. If the firm’s projects require an appropriate cost of capital of 10%, then what is the firm’s levered equity cost of capital?

7.6

Stocks A and B have the following probability distributions of expected future returns:

1. Calculate the expected rate of return, , for Stock B ( = 12.30%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

2. Calculate the standard deviation of expected returns, σ_A, for Stock A (σ_B = 17.70%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

   Now calculate the coefficient of variation for Stock B. Do not round intermediate calculations. Round your answer to two decimal places.

   Is it possible that most investors might regard Stock B as being less risky than Stock A?

   1. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   2. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   3. If Stock B is more highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be less risky in a portfolio sense.
   4. If Stock B is more highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   5. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.

   -Select-IIIIIIIVVItem 4

3. Assume the risk-free rate is 3.5%. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to four decimal places.

   Stock A:

   Stock B:

   Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b?

   1. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   2. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   3. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   4. In a stand-alone risk sense A is less risky than B. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.
   5. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.

   -Select-IIIIIIIVVItem 7

Stocks A and B have the following probability distributions of expected future returns:

1. Calculate the expected rate of return, , for Stock B ( = 12.20%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

2. Calculate the standard deviation of expected returns, σ_A, for Stock A (σ_B = 17.82%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

   Now calculate the coefficient of variation for Stock B. Do not round intermediate calculations. Round your answer to two decimal places.

   Is it possible that most investors might regard Stock B as being less risky than Stock A?

   1. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   2. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   3. If Stock B is more highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be less risky in a portfolio sense.
   4. If Stock B is more highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   5. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.

   -Select-IIIIIIIVVItem 4

3. Assume the risk-free rate is 2.5%. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to four decimal places.

   Stock A:

   Stock B:

   Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b?

   1. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   2. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   3. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   4. In a stand-alone risk sense A is less risky than B. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.
   5. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.

   -Select-IIIIIIIVV

Using probability theory, what is the probability that a cross between theparents AaBbCcdd xAabbCCdd will produce an offspring with the genotype aabbccdd? Explain your work and draw whatever Punnett Squres are needed