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1. A fund manager is attempting to construct a diversified portfolio, which contains stocks, bonds and commodity indices. Collecting and analysing monthly data for each index for the past 5 years has revealed the following information:
Annualised expected asset returns Index Stocks Bonds Commodities Return 11.7% 7.9% 10.2% Standard deviations and correlations† of stock, bond and commodity index returns Stocks Bonds Commodities Stocks 38.7% -0.52 -0.34 Bonds 5.4% 0.24 Commodities 21.2% † Entries above diagonal are correlations; diagonal elements are standard deviations of asset returns
a) Find the return and variance of a portfolio of equally weighted investments in the stock, bond and commodity indices.
b) Find appropriate weights for a portfolio of stocks, bonds and commodities, which minimise the variance of the portfolio and verify minimality.
c) Find appropriate weights for a portfolio of stocks and bonds, which minimise the variance of the portfolio.
2. Function shapes and properties a) For the function 𝑓𝑓(𝑥𝑥) = (2𝑥𝑥 − 7)3 , find the stationary values and determine their nature. Evaluate the function at the stationary values and sketch the function. b) On the interval [-5, 15] sketch the graph of function 𝑓𝑓(𝑥𝑥) with the following properties: i. 𝑓𝑓′′′ is strictly positive and 𝑓𝑓′′(9) = 0; ii. 𝑓𝑓′ (6) = 𝑓𝑓′ (10) = 0; iii. (0, 10), (6, 15), and (10, 0) are on the graph;
3. Consider the following family of utility functions, indexed by parameter 𝛾𝛾, 𝑈𝑈𝛾𝛾(𝑥𝑥) = (1 + 𝑥𝑥/𝛾𝛾)1−𝛾𝛾 − 1 1/𝛾𝛾 − 1 .
a) Compute 𝑈𝑈𝛾𝛾 ′ (0),𝑈𝑈𝛾𝛾 ′′(0) and determine the coefficient of absolute risk aversion at 0. b) For fixed x compute the limit lim 𝛾𝛾→1 𝑈𝑈𝛾𝛾(𝑥𝑥), by writing (1 + 𝑥𝑥/𝛾𝛾)1−𝛾𝛾 = 𝑒𝑒(1−𝛾𝛾) ln(1+𝑥𝑥/𝛾𝛾) , (1) and using the Taylor expansion 𝑒𝑒𝑦𝑦 = 1 + 𝑦𝑦 + 𝑂𝑂(𝑦𝑦2) with 𝑦𝑦 = (1 − 𝛾𝛾) ln(1 + 𝑥𝑥/𝛾𝛾). c) For fixed x compute the limit lim 𝛾𝛾→∞ 𝑈𝑈𝛾𝛾(𝑥𝑥), once again using the transformation (111), but this time use the Taylor expansion ln(1 + 𝑦𝑦) = 𝑦𝑦 + 𝑂𝑂(𝑦𝑦2) with 𝑦𝑦 = 𝑥𝑥/𝛾𝛾.
4. A monopolistic firm has the following demand and cost functions for each of its two products x and y: 𝑄𝑄𝑥𝑥 = 54 − 𝑃𝑃𝑥𝑥 − 𝑃𝑃𝑦𝑦 𝑄𝑄𝑦𝑦 = 81 − 2𝑃𝑃𝑥𝑥 − 𝑃𝑃𝑦𝑦 𝐶𝐶 = 2𝑄𝑄𝑥𝑥 2 + 𝑄𝑄𝑦𝑦 2 + 10, where 𝑃𝑃𝑥𝑥, 𝑃𝑃𝑦𝑦 are the prices and 𝑄𝑄𝑥𝑥,𝑄𝑄𝑦𝑦 are the quantities of the goods x and y, respectively. a) Invert the demand function to obtain price as a function of quantity. b) Find the output levels 𝑄𝑄𝑥𝑥, 𝑄𝑄𝑦𝑦 that optimise total profit and confirm that the optimal point is a unique maximum (note that for each product the corresponding revenue is 𝑅𝑅 = 𝑃𝑃𝑃𝑃). c) Find also the profit-maximizing prices and the maximum profit.
5. A company wishes to design a rectangular container whose volume is 12m3 . The construction technology requires the base to be made from a material costing EUR 30 per square meter while for the sides and top it is acceptable to use material costing EUR 10 per square meter. Find the design dimensions that minimize the total material cost of the container. Be sure to specify which side is the base.
6. The marketing department of a beauty product company estimates that the demand q (in thousands of units per year) for its shaving oil is related to the unit price p through the demand equation 𝑞𝑞 = 20 𝑒𝑒−𝑝𝑝. Current price of the product is £3.50 per unit. Because of increasing price of ingredients, the price per unit is expected to rise by 10p per year for the foreseeable future. At what rate are the revenues changing (in £ per year)? Use differentiation to compute the answer.