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, δ_{1}).
Question 2
Consider the following loss matrix.
a_{1} | a_{2} | a_{3} | |
θ_{1} | £1000 | −£300 | £4000 |
θ_{2} | −£1000 | £5000 | £3000 |
θ_{3} | £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, σ^{2}). It is desired to estimate the unknown parameter σ^{2} under loss function L(σ^{2}, a) = (σ^{2} − a)^{2}. Let the decision rule be as follows: δ (X) = ^{1} Σn X − X 2 = S^{2} where X = Σn Xi.
(a) Computer the risk function R σ^{2}, δ_{2}.
Decision and Risk – Exercise 4
Question 1
Let X_{t} 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, 0.14^{2}).
- 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