Portfolio HW

1. Consider the Arrows portfolio model with one risky asset and one risk-free asset.

Let the initial wealth be $1,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%, 10%; 0.55, 0.45).

Write the probability distribution of final wealth if the investment in the risky asset is $300. Denote this portfolio as P(300).

2. Consider the Arrows portfolio model with one risky asset and one risk-free asset.

Let the initial wealth be $1,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%, 10%; 0.55, 0.45).

Write the probability distribution of final wealth if the investment in the risky asset is $500. Denote this portfolio as P(500).

3. The von Newmann - Morgenstern utility function of an individual is: v(w) = (w)^1/2, where w is wealth.

Using the portfolios of the previous questions:

Calculate the expected utility of v with respect to P(300):

Ev(300) = ________ (round the number to four decimals)

Calculate the expected utility of v with respect to P(500): Ev(500) = _______ (round the number to four decimals)

Given that Ev(300) ________ (> , <) Ev(500), individual v prefers the portfolio described by (P(300) / P(500)).

4. Consider the Arrows portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of two individuals are: u(w) = ln w, where w represents wealth, and ln natural log; and v(w) = (w)^1/2.

Let the initial wealth be $1,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%, 10%; 0.55, 0.45).

Under the assumed values of the exogenous variables, the optimal investment in the risky asset of investors u and v are $262.50, and $525.59 respectively.

Why does the individual v invest more in the risky asset?

5. Consider the Arrows portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of two individuals are: u(w) = ln w, where w represents wealth and ln natural log; and c(w) = w (0.001) (w)². Let the initial wealth be $200, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%, 10%; 0.55, 0.45). Under these parameters values the optimal investment in the risky asset of individuals u and c are $52.5 and $72.32 respectively. If initial wealth increases to $250, the optimal investment of u is now $65.63, and that of c is $59.23.

The investment of individual u increased, while the investment of individual c decreased.

How do you explain that contrasting behavior?

6. Consider the Arrows portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of a risk averse individual is: u(w) = ln w, where w represents wealth and ln natural log.

Let the initial wealth be w⁰ = $1,200, the interest rate of the risk-free asset 5%, and the probability distribution of the random return X = (1%, 10%; 0.55, 0.45).

Using the expected utility function, the optimal amount invested in the risky asset is * = $ _________ ; and the amount that should be invested in the risk free asset is w⁰ - * = $__________ .

7. Consider the Portfolio Model with two risky assets X and Y whose random returns are x and y respectively. The final wealth is defined as: W = (1 + x) + (w₀ - ) (1 + y), where w₀ is the initial wealth, is the amount invested in asset X, and (w₀ - ) is the amount invested in Y. The return of these assets is independent and identically distributed (IID). The probability distribution or each risky return is: (r₁, r₂ ; , 1 - ). Suppose that r₁ = 2%, r₂ = 6%, and that = 1 - = . Also assume that w₀ = $5,000.

   Write the probability distribution of final wealth of the concentrated portfolio, when all wealth (w₀) is invested in asset X. Denote this portfolio as P1

   8. Consider the Portfolio Model with two risky assets X and Y whose random returns are x and y respectively. The final wealth is defined as: W = (1 + x) + (w₀ - ) (1 + y), where w₀ is the initial wealth, is the amount invested in asset X, and (w₀ - ) is the amount invested in Y. The return of these assets is independent and identically distributed (IID). The probability distribution or each risky return is: (r₁, r₂ ; , 1 - ). Suppose that r₁ = 2%, r₂ = 6%, and that = 1 - = . Also assume that w₀ = $5,000.

      Write the probability distribution of portfolio 2 (P2), a diversified portfolio, when the individual invests $2,000 of wealth in asset X and $3,000 in asset Y. Denote this portfolio as P2

      9. Consider the Portfolio Model with two risky assets X and Y whose random returns are x and y respectively. The final wealth is defined as: W = (1 + x) + (w₀ - ) (1 + y), where w₀ is the initial wealth, is the amount invested in asset X, and (w₀ - ) is the amount invested in Y. The return of these assets is independent and identically distributed (IID). The probability distribution or each risky return is: (r₁, r₂ ; , 1 - ). Suppose that r₁ = 2%, r₂ = 6%, and that = 1 - = . Also assume that w₀ = $5,000.

         Write the probability distribution of portfolio 3 (P3), the fully-diversified portfolio, when the individual invests $2,500 of wealth in asset X and $2,500 in asset Y. Denote this portfolio as P3

         10. Calculate the expected values of the various portfolios defined above:

         EP1 = _______ ,

         EP2 = [EP], and

         EP3 = [EP].

         11. Using the RS (Rotschild-Stiglitz) criteria, compare the degree of risk of portfolios P1 and P3, as defined in the previous questions. Tell which portfolio is riskier and why is that so.
         12. Using the RS (Rotschild-Stiglitz) criteria, compare the degree of risk of portfolios P2 and P3, as defined in the previous questions. Tell which portfolio is riskier and why is that so.
         13. Consider an investor with utility function u(w) = (w)^1/2. Using this utility function the expected utility of P1 is Eu(P1) = ______(use four decimals); and the expected utility of P3 is Eu(P3) = ______ (use four decimals).

         According to your calculations of the expected utility of the different portfolios, investor u(w) prefers portfolio _____ (P1 / P3), to portfolio ______ (P1 / P3).