Slide MGT305 Slide08

Value at Risk (VaR)
Chapter IX

Definition of VaR
• VaR is an attempt to provide a single
number that summarizes the total portfolio
risk.
• When using VaR, we are interested in
making a statement of the following form:
“We are X percent certain that we will not
lose more than V dollars at time T.”

Definition of VaR
• The variable V is the VaR of the portfolio. It is a

function of two parameters: the time horizon, T,
and the confidence level, X percent.
• It is the loss level during a time period of length

T that we are X% certain will not be exceeded.
• VaR can be calculated from either the probability

of gains or losses during time T.

Definition of VaR
• When the distribution of gains is used,
VaR is equal to minus the gain at the (100
– X)th percentile of the distribution.
• When the distribution of losses is used,
VaR is equal to the loss at the Xth
percentile of the distribution.

Calculation of VaR
• Suppose the gain from a portfolio during six

months is normally distributed with a mean of $2
million and a standard deviation of $10 million.
From the properties of normal distribution, the
one-percentile point of this distribution is 2 2.33*10, or -$21.3 million. The VaR of the
portfolio with horizon of six months and
confidence level of 99% is therefore $21.3 million.

Calculation of VaR
• Suppose that for one year project all outcomes
between a loss of $50 million and a gain of $50
million are considered equally likely. In this case the
loss from the project has a uniform distribution
extending from -$50 million to +$50 million. There is
a 1% chance there will be a loss greater than $49
million. The VaR with one year time horizon and a
99% confidence level is therefore $49 million.