(0 An investor has the utility function Ulw)--exp --exp(- (H) Determine whether the investor exhibits increasing,...

 

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(0 An investor has the utility function Ulw)--exp --exp(- (H) Determine whether the investor exhibits increasing, constant or decreasing absolute and relative risk aversion. The investor has an initial wealth of 1,000 and is offered a gamble with a payoff described by a random variable: +100 with probability 0.5 -50 with probability 0.5 Find the investor's certainty equivalent of this gamble. 3 Two assets are available to investors. Asset B is a risk-free investment that returns 1%, and the return on Asset A is given by: X= W 100 (a) -1% 3% Explain why Asset B must be second-order stochastically dominant over Asset A in terms of investors and utility functions. (b) probability 0.5 probability 0.5 Verify numerically the second-order stochastic dominance expressed in part (1). What can be said about dominance if Asset A offers instead a return of: -1% probability 0.5 RA- 4% probability 0.5 (-1% probability 0.5 1% probability 0.5 Can an asset that allows the possibility of a return less than 1% ever dominate Asset B? R₁- -4 5 An investor is trying to choose between the investments whose distributions of returns are described below: Investment A: Investment B: Investment C: Calculate the following for each investment: (i) expected return variance of return semi-variance expected shortfall below 12% shortfall probability below 15%. (iii) (iv) (v) 0.4 probability that it will return 10% 0.2 probability that it will return 15% 0.4 probability that it will return 20% 0.25 probability that it will return 10% 0.70 probability that it will return 15% 0.05 probability that it will return 40% A uniform distribution on the range 10% to 20% The annual rates of interest from a particular investment, in which part of an insurance company's funds is invested, are independently and identically distributed. Each year, the distribution of (1+i), where it is the rate of interest earned in year t, is log-normal with parameters and o². it has mean value 0.07 and standard deviation 0.02, the parameter = 0.06748 and o² = 0.0003493. (i) (ii) [1½] [31] [31] [21] [1] [Total 12] (a) (b) The insurance company has liabilities of £1m to meet in one year from now. It currently has assets of £950,000. Assets can either be invested in the risky investment described above or in an investment which has a guaranteed return of 5% per annum effective. Find, to two decimal places, the probability that the insurance company will be unable to meet its liabilities if: All assets are invested in the investment with the guaranteed return. 85% of assets are invested in the investment which does not have the guaranteed return and 15% of assets are invested in the asset with the guaranteed return. [7] Determine the variance of return from the portfolios in (i)(a) and (i)(b) above. [3] [Total 10] (0 An investor has the utility function Ulw)--exp --exp(- (H) Determine whether the investor exhibits increasing, constant or decreasing absolute and relative risk aversion. The investor has an initial wealth of 1,000 and is offered a gamble with a payoff described by a random variable: +100 with probability 0.5 -50 with probability 0.5 Find the investor's certainty equivalent of this gamble. 3 Two assets are available to investors. Asset B is a risk-free investment that returns 1%, and the return on Asset A is given by: X= W 100 (a) -1% 3% Explain why Asset B must be second-order stochastically dominant over Asset A in terms of investors and utility functions. (b) probability 0.5 probability 0.5 Verify numerically the second-order stochastic dominance expressed in part (1). What can be said about dominance if Asset A offers instead a return of: -1% probability 0.5 RA- 4% probability 0.5 (-1% probability 0.5 1% probability 0.5 Can an asset that allows the possibility of a return less than 1% ever dominate Asset B? R₁- -4 5 An investor is trying to choose between the investments whose distributions of returns are described below: Investment A: Investment B: Investment C: Calculate the following for each investment: (i) expected return variance of return semi-variance expected shortfall below 12% shortfall probability below 15%. (iii) (iv) (v) 0.4 probability that it will return 10% 0.2 probability that it will return 15% 0.4 probability that it will return 20% 0.25 probability that it will return 10% 0.70 probability that it will return 15% 0.05 probability that it will return 40% A uniform distribution on the range 10% to 20% The annual rates of interest from a particular investment, in which part of an insurance company's funds is invested, are independently and identically distributed. Each year, the distribution of (1+i), where it is the rate of interest earned in year t, is log-normal with parameters and o². it has mean value 0.07 and standard deviation 0.02, the parameter = 0.06748 and o² = 0.0003493. (i) (ii) [1½] [31] [31] [21] [1] [Total 12] (a) (b) The insurance company has liabilities of £1m to meet in one year from now. It currently has assets of £950,000. Assets can either be invested in the risky investment described above or in an investment which has a guaranteed return of 5% per annum effective. Find, to two decimal places, the probability that the insurance company will be unable to meet its liabilities if: All assets are invested in the investment with the guaranteed return. 85% of assets are invested in the investment which does not have the guaranteed return and 15% of assets are invested in the asset with the guaranteed return. [7] Determine the variance of return from the portfolios in (i)(a) and (i)(b) above. [3] [Total 10]

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Here are the stepbystep workings i The investor has a utility function Uw e100w This utility functio... View the full answer