June 2009, Questions 7/8/9
Question 7 (2 points)
During the period July 11, 1988 and July 10, 1998 a total of 2,256 daily return observations on the S&P 500 have been observed,
ranging from -6.87% to +5.12%. Consider the left tail of the distribution and apply extreme value theory with u=0.02, there were
in total 28 returns less than -2%. Furthermore use as parameters:
ξ=0 . 3232
, and
β=0 . 0055
.
a) Estimate the probability that the return will be less than 0.04.
b) What is the value of the one-day 99% VaR for a portfolio where $1 million is invested in the S&P 500?
Answer:
We assume that the distribution is symmetric!
(12.5)
Prob
(
v>x
)
=nu
n
(
1+x−u
ß
)
−1/¿¿
a)
Prob
(
v>0.04
)
=28
2256
(
1+0.3232 0.04−0.02
0.0055
)
−1/0.3232
=0.0012
Prob
(
v<0.04
)
=1−Prob
(
v>0.04
)
=1−0.001121=0.9988
b)
(12.6)
VaR=u+ß
❑⌈
(
n
n
u
(1−q)
)
−¿−1⌉¿
q = F(VaR)
VaR=0.02 +0.0055
0.3232 ⌈
(
2256
28 (1−0.99)
)
−0.3232
−1⌉=0.0212
For 1 Million $ is invested: 0.02121x1,000,000 = 21200
