Numerade

Questions asked

ANSWERED

Danielle Fairburn

Numerade educator

(b) Suppose there are only two industries in a market; Industry A and Industry B, with the following properties. egin{tabular}{lcc} & Average Return & Variance \ hline Industry A & 0.08 & 0.2 \ Industry B & 0.13 & 0.4 \ hline end{tabular} Where the estimates above are based on the annual average value weighted returns for the 2 industry portfolios. The risk-free rate is 5 percent, and Industry A and Industry B have correlation of -0.625 . Determine the expected return and standard deviation of the optimal portfolio in this market(i.e. portfolio that maximizes the Sharpe ratio). [8 marks]

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ANSWERED

Andreas Papavassiliou

Numerade educator

1. An investor makes decisions based on the utility function U(w) = w - 6w^2, where w is the investor's wealth in millions of rand (Rm). (a) Demonstrate that the investor has both increasing absolute and relative risk aversion. [3 marks] The investor has R50,000 to invest over a 1-year period and has no other wealth. They have three options: i. Invest in a risk-free account. There will be no change in the value of the investment over 1 year. ii. Invest in an asset that will give a 60% return over 1 year with probability 0.2, a 20% return with probability 0.7 and a -40% return with probability 0.1. iii. Invest in an asset that will give a 30% return with probability 0.5 and a 20% return with probability 0.5. The investor makes no allowance for discounting when making investment decisions. The investor must invest the whole R50,000 in a single option. (b) Determine which option the investor should choose to maximise their expected utility at the end of the year. [6 marks] (c) Comment on why the investor could not use U(w) to choose from the above options if their initial wealth was R65,000. [3 marks] [Total: 12 marks]

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INSTANT ANSWER

In a market in which the Arbitrage Pricing Theory (APT) model holds, the expected return is given by: \[ E\left[R_{i}\right]=\lambda_{0}+\lambda_{1} b_{i, 1}+\lambda_{2} b_{i, 2} \cdots \] (b) Define all the terms in the equation. [2 marks] Assume the risk-free rate \( r_{f}=0.04 \). Consider two well diversified portfolios \( P_{i} \) with the following features in a two factor model: UCT- 2019 Question 5 continues on the next page... ECO 4053S-makeup Page 4 of 4 \[ \begin{array}{cll} & P_{1} & P_{2} \\ E\left(R_{i}\right) & 15.50 \% & 11.95 \% \\ b_{i, 1} & (\mathrm{a}) & (\mathrm{b}) \\ b_{i, 2} & 1.5 & 0.7 \end{array} \] (c) Determine the values (a) and (b) for \( \lambda_{1}=0.05 \) and \( \lambda_{2}=0.06 \). [3 marks] [Total 10 marks]

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ANSWERED

Shu Naito

Numerade educator

4. Consider a market in which the Capital Asset Pricing Model (CAPM) holds. (a) Write down the equation of the Security Market Line(SML), defining all the notation you use. Explain how one can use the SML to determine over and/or under-valued securities. [3 marks] In this market, the risk-free rate of interest is ( 9.44 % ) per annum. There are only two risky assets in the market with the following attributes. egin{tabular}{lccc|ccc} hline multicolumn{3}{c}{ Rate of return (per annum) } & multicolumn{3}{c}{ Variance/Covariance Matrix } \ hline State & Probability & Asset 1 & Asset 2 & & Asset 1 & Asset 2 \ 1 & 0.2 & ( 10.00 % ) & ( 11.00 % ) & Asset 1 & 0.00142 & 0.00379 \ 2 & 0.3 & ( 15.00 % ) & ( 30.00 % ) & Asset 2 & 0.00379 & 0.01146 \ 3 & 0.1 & ( 18.00 % ) & ( 25.00 % ) & & & \ 4 & 0.4 & ( 20.00 % ) & ( 40.00 % ) & & & \ hline end{tabular} (b) Determine the weight of each asset in the market portfolio to be consistent with ( eta_{1} ) ( =0.46, eta_{2}=1.36 ). [3 marks] (c) Calculate the Market Price of Risk. [2 marks] (d) Outline the main shortcomings of the CAPM. [8 marks] [Total: 16 marks]

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ANSWERED

Andreas Papavassiliou

Numerade educator

(a) Define an 'efficient portfolio' in the context of mean-variance portfolio theory.. [2 marks] (b) Suppose that there are two risky assets: A and B, with the following properties. egin{tabular}{lcc} & Expected Return & Variance \ hline Asset A & 0.08 & 0.2 \ Asset B & 0.13 & 0.4 \ hline end{tabular} The risk-free rate is 5 percent, and assets A and B have correlation of -0.625 . Determine the expected return and standard deviation of the portfolio of A and B that maximizes the Sharpe ratio (i.e. the optimal portfolio). [ 10 marks]

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ANSWERED

Ivan Kochetkov

Numerade educator

3. (a) Define an efficient portfolio in the context of mean-variance portfolio theory.. [2 marks] (b) Suppose that there are two risky assets: A and B, with the following properties. Expected Return Variance Asset A 0.08 Asset B 0.13 0.2 0.4 The risk-free rate is 5 percent, and assets A and B have correlation of-0.625. Determine the expected return and standard deviation of the portfolio of A and B that maximizes the Sharpe ratio (i.e. the optimal portfolio). [ 10 marks]

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INSTANT ANSWER

Consider the following investments whose payo s, in millions of rand, have the following probability distribution(s): Investment 1 Payo Investment 2 Probability Payo Probability R1m R7m R12m 0.25 0.50 0.25 R3m R5m R8m 0.33 0.33 0.34 (c) If the only available choice is 100% of your wealth in Investment 1 or 100% in Investment 2 and you choose on the basis of mean only, which investment is preferred. [3 marks] (d) Comparethetwoinvestments using the second-order stochastic dominance criterion. [5 marks] (e) Explain the practical limitation(s) of the concept of stochastic dominance in modelling investor choice. [2 marks]

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INSTANT ANSWER

An investor makes decisions using a quadratic utility function, U(w) = a+bw+cw2 (a) Write down the absolute and relative risk aversion for this utility function. [ 3 marks] The investor currently has wealth of R100, and using her utility function U(100) = 610 The investor is o ered a gamble with a pro t of R20 with probability p, and a loss of R20 with probability (1 p) She will accept this gamble only if p 055 (b) Explain what this implies about the investors risk aversion. The investor accepts the gamble and wins. She now has wealth of R120. The investor is o ered the same gamble again, with a pro t of R20 with probability p, and a loss of R20 with probability (1 p) Based on her new wealth, she will now accept this gamble only if p 05625. (c) Determine ab and c. [1 mark] [6 marks] (d) Determine the maximumwealthfor which the function U(w) satis es the requirement of non-satiation.

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Temeeka D.