Assume expected returns and deviations for all securities, as well as the risk-free rate for lending and borrowing, are known. Also, assume that the rate for borrowing and lending are the same. Will investors arrive at the same optimal risky portfolio?
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- Suppose Caroline is choosing how to allocate her portfolio between two asset classes: risk-free government bonds and a risky group of diversified stocks. The following table shows the risk and return associated with different combinations of stocks and bonds.CombinationFraction of Portfolio in Diversified StocksAverage Annual ReturnStandard Deviation of Portfolio Return (Risk)(Percent)(Percent)(Percent)A 0 1.50 0B 25 3.00 5C 50 4.50 10D 75 6.00 15E 100 7.50 20There is a relationship between the risk of Caroline's portfolio and its average annual return.Suppose Caroline currently allocates 75% of her portfolio to a diversified group of stocks and 25% of her portfolio to risk-free bonds; that is, she chooses combination D. She wants to reduce the level of risk associated with her portfolio from a standard deviation of 15 to a standard deviation of 5. In order to do so, she must do which of the following? Check all that apply. Sell some of her stocks and use the proceeds to purchase…ANSWER E PLEASE ONLY Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) / n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset.a) What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?b) Find A as a function of w. c) Does the investor put more or less of his portfolio into the risky assetas his wealth increases? d) Now find the share of wealth, α, invested in the risky asset. How doesα change with wealth? e) Calculate relative risk aversion for this investor. How does relativerisk aversion depend on wealth?E2
- Optimal Portfolio: Mean-Variance OptimizationIf you are a portfolio manager who predicted that the tension in Ukraine might spiral into a global economic problem back in December 2021. She decided to construct a portfolio that, she think, would outperform in a war scenario, or in a heightened war risk scenario. Please use the following ETFs:IAU: iShares Gold Trust ETFVDE: Vanguard Energy ETFXLB: Materials Sector SPDR ETFDBC: Invesco DB Commodity Index Tracking FundCQQQ: China Technology Index ETFConstraints:i. Use all ETF products. (Weight of each ETF>= 2% )ii. No ETF is to have more than 40% weight in portfolioObjective: Maximize Expected Return, Minimize volatility, ie. MaximizeSharpe RatioStep 1: Collect historical price/return data for the ETFs over Jan-2018 to Dec-21 period.Step 2: Assume the Average Historical Return is the Expected Return for each asset (strong assumption) and Historical Volatility is the Expected Volatility (strong assumption).Step 3: Present the var-cov…3. The risk free rate is 3%. The optimal risky portfolio has an expected return of 9% and standard deviation of 20%. Answer the following questions. (a) Assume the utility function of an investor is U = E(r) − 0.5Aσ2. What is condition of A to make the investors prefer the optimal risky portfolio than the risk free asset? (b) Assume the utility function of an investor is U = E(r) − 2.5σ2. What is the expected return and standard deviation of the investor’s optimal complete portfolio?Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) /n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset. Calculate relative risk aversion for this investor. How does relative risk aversion depend on wealth?
- Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) /n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset.1) What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?2) Find A as a function of w.Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) /n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset.a) What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?b) Find A as a function of w.please explain clearly
- ANSWER C AND D PLEASE ONLY Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) / n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset.a) What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?b) Find A as a function of w. c) Does the investor put more or less of his portfolio into the risky assetas his wealth increases? d) Now find the share of wealth, α, invested in the risky asset. How doesα change with wealth?Need qualitiey answer Define a rational risk aversive investor.1. Which of the following is INCORRECT? a All of a stock's risk could be unsystematic. b. A negative beta stock has an expected return less than the risk-free rate. c. Anticipated returns on any given stock are always greater than 0. d. Two assets with a correlation of -1 could be combined to create a portfolio with a standard deviation of zero (no risk). 2. Which of the following measures the total risk of a portfolio? a. Beta b. Standard Deviation c. Correlation Coefficient d. Alpha 3. Which of the following stocks have the highest systematic risk? a A stock with high correlation to the market and high returm volatility. b. A stock with low correlation to the market and a high return volatility. c A stock with high correlation to the market and a low return volatility. d. A stock with low correlation to the market and a low return volatility. 4. Which of the following companics have the lowest systematic risk? a A company that sells soups (Campbells), beta=0.60 b. A coffee company…