Question

I know that VaR is not a coherent risk measure, but, how can i prove it?

Answer 1

coherent risk measure should satisfy 4 conditions :

1.) p(X+c) = p(X) - c

2.) p(lambda. X) = lambda . p(X)

3.) If X<=Y THEN p(X) >= p(Y)

4.) p(X+Y)<= p(X) +p(Y) : SUBADDITIVITY

Value–at–Risk2 (VaR) satisfies all but subadditivity

U SIGMA

X 4% 30%

Y 8% 30%

CORRELATION 0

p(X+Y)=42.2%

so p(X+Y)<= p(X) + p(Y)

i.e.

42.2%< 30%+ 30%

now

lets say

PD 4%

# OF BONDS IN q 95% VAR

1 0

2 1

3 1

4 1

8 1

10 2

so the risk of 2 bonds i.e.

p(X+Y)> p(X) + p(Y)

1 > 0 + 0

ie risk of two bonds combined in the portfolio is greater than the individual bonds. So it does not fulfill the subadditivity property.