3. Consider an asset whose return follows the probability density function f(x). The mean of the distribution is u. a) Write down a formula for the Value at Risk for the asset, with (100-p)% confidence level. [1 mark] b) Write down a formula for the downside semi-variance of the return on the asset. [1 mark] c) State one argument for and three arguments against using semi-variance as a risk measure. [4 marks] d) A gardener has a rare orchid plant which flowers only once a year. The flowers produced by the plant could be of any shade of pink but red flowers are considered bad outcomes by the gardener. The number of red flowers the orchid produces follows a Poisson distribution with a mean and variance of 8. Note that a discrete random variable X is said to have a Poisson distribution with parameter 2 > 0 if, for k = 0,1,2, ..., the probability mass function of X is given by ake-2 P(X = k) = with E(X) = Var(X) = 1. k! i) Determine the Value at Risk for the number of red flowers produced over the flowering year with a 90% confidence level. [5 marks] ii) Determine the gardener's expected shortfall over the flowering year with a 90% confidence level. [5 marks] 3. Consider an asset whose return follows the probability density function f(x). The mean of the distribution is u. a) Write down a formula for the Value at Risk for the asset, with (100-p)% confidence level. [1 mark] b) Write down a formula for the downside semi-variance of the return on the asset. [1 mark] c) State one argument for and three arguments against using semi-variance as a risk measure. [4 marks] d) A gardener has a rare orchid plant which flowers only once a year. The flowers produced by the plant could be of any shade of pink but red flowers are considered bad outcomes by the gardener. The number of red flowers the orchid produces follows a Poisson distribution with a mean and variance of 8. Note that a discrete random variable X is said to have a Poisson distribution with parameter 2 > 0 if, for k = 0,1,2, ..., the probability mass function of X is given by ake-2 P(X = k) = with E(X) = Var(X) = 1. k! i) Determine the Value at Risk for the number of red flowers produced over the flowering year with a 90% confidence level. [5 marks] ii) Determine the gardener's expected shortfall over the flowering year with a 90% confidence level. [5 marks]