Decision and Risk – Exercise 3

Question 1

Consider a random variable X N (µ, 1). It is desired to estimate the unknown parameter µ under loss function L(µ, a) = (µ a)2.  Let the decision rule be as follows: δ (X) = Σn   Xi = X.

(a) Compute the risk function R (µ, δ1).

Question 2

 Consider the following loss matrix.

  a1 a2 a3
θ1 £1000 £300 £4000
θ2 £1000 £5000 £3000
θ3 £300 £1000 £100
  • State for each action if it is admissible or inadmissible.
  • Find the minimax nonrandomized

Question 3

Consider a random variable X N (µ, σ2). It is desired to estimate the unknown parameter σ2 under loss function L(σ2, a) = (σ2a)2.  Let the decision rule be as follows: δ (X) =   1   Σn             X X 2 = S2 where X = Σn   Xi.

(a)  Computer the risk function R  σ2, δ2.

Decision and Risk – Exercise 4

Question 1

Let Xt denote weekly log-returns on asset A following normal distribution with mean 0.06 and standard deviation of 0.14, i.e.  X  i.i.d. N (0.06, 0.142).

  • Find the 95% VaR over a 1-week
  • Now find the 99% VaR over a 1-week
  • Repeat (a) and (b) over a 4-week

Question 2

 Show that the variance is not sub-additive, and hence is not coherent.

Question 3

Consider the following information on the hypothetical portfolio of £5,000 invested in two assets.  The information on each daily asset return is provided in the table below. It is assumed that these returns are jointly normally distributed.

 

Asset 1 Asset 2
Mean 0.03 0.02
Standard deviation 0.01 0.03
Portfolio weights 0.5 0.5

Portfolio value                                  £5,000

Correlation coefficient                        0.8

  • Compute the mean and the variance of the portfolio returns consisting of Asset 1 and Asset
  • Using results from part (i), compute the 95% 1-day Value-at-Risk (VaR) of the