Chapter 12: Value at Risk and Expected Shortfall
12.13.
Suppose that each of two investments has a 4% chance of a loss of $10 million, a 2% chance of a
loss of $1 million, and a 94% chance of a profit of $1 million. They are independent of each
other.
(a) What is the VaR for one of the investments when the confidence level is 95%?
(b) What is the expected shortfall when the confidence level is 95%?
(c) What is the VaR for a portfolio consisting of the two investments when the confidence level is
95%?
(d) What is the expected shortfall for a portfolio consisting of the two investments when the
confidence level is 95%?
(e) Show that, in this example, VaR does not satisfy the subadditivity condition whereas expected
shortfall does.
(a) A loss of $1 million extends from the 94 percentile point of the loss distribution to the 96
(b) The expected shortfall for one of the investments is the expected loss conditional that the loss
(c) For a portfolio consisting of the two investments there is a 0.04 × 0.04 = 0.0016 chance that
(d) The expected shortfall for the portfolio consisting of the two investments is the expected loss
(e) VaR does not satisfy the subadditivity condition because 9 > 1 + 1. However, expected
12.14.
Suppose that daily changes for a portfolio have first-order correlation with correlation
parameter 0.12. The 10-day VaR, calculated by multiplying the one-day VaR by
10
, is $2
million. What is a better estimate of the VaR that takes account of autocorrelation?
The correct multiplier for the variance is
The estimate of VaR should be increased to
2 12.417 /10´
= 2.229
12.16.
The change in the value of a portfolio in three months is normally distributed with a mean of
$500,000 and a standard deviation of $3 million. Calculate the VaR and ES for a confidence
level of 99.5% and a time horizon of three months.
The loss has a mean of −500 and a standard deviation of 3000. Also, N−1(0.995) =2.576. The
–
176,9
005.02
3000500
2/57 6.2
2
e
The expected loss conditional that it is in the 0.5% tail of the distribution is $9.176 million.
12.17
The probability that the loss from a portfolio will be greater than $10 million in one month is
estimated to be 5%.
(a) What is the one-month 99% VaR assuming the change in value of the portfolio is normally
distributed with zero mean?
(b) What is the one-month 99% VaR assuming that the power law applies with =3?
(a) The 5% tail of the distribution starts 1.645 standard deviations from the mean. The
(b) We know that Prob (v > x) = Kx−3. Hence 0.05=K×10−3. so that K=50. To find the 99%