(a) (5pts). Given a position P&L X of a portfolio with X N(10; 22), consider :

VaR(X) = minfa j P(-X a) 1 - g (5)

Find the Value-at-Risk of X with = 90%.

(b) (15pts). Consider a risk measure for a P&L Y

1-(Y ) = inf

>0 1 ln E(e-Y ) ; (6)

where E(e-Y ) < 1.

2

(i) Show that (Y ) is coherent.

(ii) Compute (X), where X N(10; 22), as defined previously. (Hint: For normal distribution, E(e-Y ) = e-+ 12 22 for Y N(; 2))