Hello, i am a Master student and i need help to solve this mock exam for a financial course "Downside Risk". The topic is all about Capital Market Theory, Stochastical Dominance, Downside Risk Criteria, Lower Partial Moments, Value at Risk, Downside Portfolios, and VaR for Bonds.

Can you please provide me a detailed calculation with explanation?

Question $3$

You have given two lotteries, A and B$.$ Lottery A offers $50$$ with a probability of $20\%,100$$ with a probability of $20\%,150$$ with a probability of $40\%,200$$ with a probability of $10\%,$ and $250$$ with a probability of $10\%.$ Lottery B offers $75$$ with a probability of $20\%,$ and $225$$ with a probability of $80\%.$

Please select the right answer!

a$.$

B stochastically dominates lottery A at the second order

b$.$

Lottery A stochastically dominates lottery B at the first order

c$.$

Lottery B stochastically dominates lottery A at the first order

d$.$

Neither lottery stochastically dominates the other at the second order

e$.$

Lottery A stochastically dominates lottery B at the second order

Question $4$

Consider the cumulative distribution functions for assets A and B:

a$.$

A stochastically dominates B at the second order

b$.$

Neither asset stochastically dominates the other at the first order

c$.$

Neither asset stochastically dominates the other at the second order

d$.$

B stochastically dominates A at the first order

e$.$

A stochastically dominates B at the first order

Question $5$

Assume the stock market consists only of two stocks, A and B$,$ with the following properties based on yearly data: The expected return of stock A is $20\%,$ the expected return of stock B is $15\%.$ The volatility of stock A is $30\%,$ the volatility of stock B is $40\%.$ The covariance between the returns of stocks A and B is $0\mathrm{.}03.$ Assuming a target of $10\%,$ what is the weight of stock A$,$ when selecting the portfolio according to Roy's criterion?

a$.$

Somewhere between $48\%$ and $55\%$

b$.$

Somewhere between $78\%$ and $85\%$

c$.$

Somewhere between $88\%$ and $95\%$

d$.$

Somewhere between $38\%$ and $45\%$

e$.$

Somewhere between $68\%$ and $76\%$

Question $6$

Assume the stock market consists only of two stocks, A and B$,$ with the following properties based on yearly data: The expected return of stock A is $18\%,$ the expected return of stock B is $15\%.$ The volatility of stock A is $40\%,$ the volatility of stock B is $40\%.$ The covariance between the returns of stocks A and B is $0\mathrm{.}03.$ Assuming a downside probability of $30\%,$ what is the weight of stock A$,$ when selecting the portfolio according to Kataoka's criterion?

a$.$

Somewhere between $20\%$ and $28\%$

b$.$

Somewhere between $90\%$ and $98\%$

c$.$

Somewhere between $50\%$ and $58\%$

d$.$

Somewhere between $60\%$ and $68\%$

e$.$

Somewhere between $80\%$ and $88\%$

Question $7$

Assume the stock market consists only of two stocks, A and B$,$ with the following properties based on yearly data: The expected return of stock A is $25\%,$ the expected return of stock B is $25\%.$ The volatility of stock A is $30\%,$ the volatility of stock B is $40\%.$ The covariance between the returns of stocks A and B is $0\mathrm{.}03.$ Assuming a target of $2\%,$ what is the maximum downside premium, when selecting the portfolio according to Roy's criterion?

a$.$

Somewhere between $80\%$ and $90\%$

b$.$

Somewhere between $70\%$ and $80\%$

c$.$

Somewhere between $90\%$ and $100\%$

d$.$

Roy's criterion is not applicable since the target exceeds the expected return of the minimum$-$varianz portfolio

e$.$

Somewhere between $20\%$ and $30\%$