Value at Risk
• Concepts, data, industry estimates
– Adam Hoppes
– Moses Chao
• Portfolio applications
– Cathy Li
– Muthu Ramanujam
• Comparison to volatility and beta
– John Summers
– Wei-Hao Tseng
Value at Risk:
The objective of this presentation is to introduce an
alternative way of looking at risk.
Data Description:
Monthly returns from 1970 to 2001 were collected on the following
industries and the total market(NYMEX, AMEX, NASDAQ).
Industries:
SIC Codes:
– Households
(2047-3995)
– Chip Manufactures
(3622-3812)
– Transportation
(4000-4789)
– Oil
(1399-2999)
Source: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/
• Each company is assigned to an industry portfolio based on their SIC
code. Value weighted monthly returns are used for the analysis.
Description of Value at Risk:
• Definition:
– Value at Risk is an estimate of the worst possible loss an
investment could realize over a given time horizon, under normal
market conditions (defined by a given level of confidence).
– To estimate Value at Risk a confidence level must be specified.
Choice of confidence level – 95%
5%
95%
Investment returns
Normal market conditions – the returns that account for
95% of the distribution of possible outcomes.
Abnormal market conditions – the returns that account for
the other 5% of the possible outcomes.
If a 95% confidence level is used to estimate
Value at Risk for a monthly horizon;
losses greater than the Value at Risk estimate
are expected to occur one in twenty months
(5%).
Illustrate Value at Risk:
• Step 1: Transform simple monthly stock returns into continuously
compounded stock returns.
Note: Technically, log stock returns are “more likely” to be normally distributed.
• Step 2: Choose a level of confidence.
– 90%, 95%, 99%, etc.
– Banks are required to report Value at Risk estimated with a 99%
level of confidence to determine regulatory capital requirements.
• Step 3: Compute Value at Risk from sample estimates of  and .
– For example, the largest likely loss in the household industry over
the next month under normal market conditions with a 95% level
of confidence is: $18,000.
Note: It is possible to realize a loss greater than $18,000.
Other Common Interpretations of
Value at Risk:
• “an attempt to provide a single number for senior management
summarizing the total risk in a portfolio of assets”
– Hull, OF&OD
• “an estimate, with a given degree of confidence, of how much one can
lose from one’s portfolio over a given time horizon”
– Wilmott, PWOQF
Conclusions:
• Value at Risk can be used as a stand alone risk measure or be applied
to a portfolio of assets.
• Value at Risk is a dollar value risk measure, as opposed to the other
measurements of risk in the financial industry such as: beta and
standard deviation.
• “We are X percent certain that we will not lose more than V dollars in
the next N days.” – Hull
Value at Risk:
ValueAtRisk  V0 (1  e )
r*
r*  1.645 * ˆ  ˆ
Value at Risk estimates for monthly horizon
using 95% confidence level December 31,
2001, V0 = $250,000
Household
Oil
Chips
Trans
Market
 =
0.0091
0.0101
0.0089
0.0081
0.0094
 =
0.0512
0.0542
0.0801
0.063
0.0465
$18,101
$19,024
$28,891
$22,769
$16,223
VAR =
Stand-alone estimates of Value at risk for each investment.
Value at Risk
Part Π
Value at Risk(Portfolio)
•
•
•
•
One month
Parametric method-normal distribution
Current market value of portfolio $1M
Confidence level 95%
Portfolio
• Equally weighted portfolio; household,
chips, transportation and oil.
• Sample used to estimate variancecovaraince matrix.
• Rp=W1R1+W2R2+W3R3+W4R4
• σp²=ΣW σ ²+Σρ W W σ σ
• VAR=V *(1-exp(r*))
i
i
0
i,j
i
j
i
j
Variance & Covariance
Var-Covar Hshld
Hshld
0.0026
Chips
0.0026
Trans
0.0022
Oil
0.0013
Chips
0.0026
0.0064
0.0034
0.0019
Trans
0.0022
0.0034
0.0040
0.0018
oil
0.0013
0.0019
0.0018
0.0029
Correlation
Correlation
Hshld
Chips
Trans
Oil
Hshld
1
0.6313
0.6974
0.4556
Chips
0.6313
1
0.6842
0.4350
Trans
0.6974
0.6842
1
0.5177
Oil
0.4556
0.4350
0.5177
1
Portfolio benefits
• Value At Risk for the portfolio - $72,750
• Sum of stand-alone Value at Risks - 88,784
• Benefits due to diversification – $16,034
Marginal and Component Value at Risk
Marginal Value
At Risk
Component
Value At Risk
Household
Chips
Transportation
Oil
$0.07
$0.11
$0.09
$0.06
$17,411
$28,628
$22,834
$15,705
Value at Risk
Part ΠI
Measures of Risk
• Standard Deviation ()
• Beta (ß)
• Value at Risk (VaR)
Measured by
VAR
Stand-Alone Risk
Or
Total Risk
Measured by
ß

Systematic
Risk

Unsystematic
Risk

NonDiversifiable
Risk

Diversifiable
Risk

CompanySpecific Risk
 Market Risk
Dispersion of Returns –
Variances and Standard Deviations
• Variance (2) Formula:

n
2

  ( ki  k ) 2
i 1
• Variance and Standard Deviation are measures of
total (or stand-alone) risk.
• The larger the variance (or Std. Dev.), the lower
the probability that actual returns will be close to
the expected return.
Risk Measure - Beta (ß)
• Beta (ß) formula:
Cov( ki , k m )

Var ( k m )
• Beta measures the portfolio’s systematic risk, that
is, the degree to which its return is correlated with
the return on the market as a whole.
• Stock with high beta (ß>1) is more volatile than
the market taken as a whole.
Risk Measures – Value at Risk (VaR)
• VaR is a measure of risk based on a
probability of loss and a specific time
horizon.
• VaR translates portfolio volatility into a
dollar value.
• Measure of Total Risk) rather than
Systematic (or Non-Diversifiable Risk)
measured by Beta.
Advantages of VaR
• VaR provides an measure of total risk.
• VaR is an easy number to understand and explain
to clients.
• VaR translates portfolio volatility into a dollar
value.
• VaR is useful for monitoring and controlling risk
within the portfolio.
Advantages of VaR (Cont.)
• VaR can measure the risk of many types of
financial securities (i.e., stocks, bonds,
commodities, foreign exchange, off-balance-sheet
derivatives such as futures, forwards, swaps, and
options, and etc.)
• As a tool, VaR is very useful for comparing a
portfolio with the market portfolio (S&P500).
VaR vs. Traditional Risks
Risk measures for four industries:
(Note: One-month VaR with 95% confidence level, mean monthly return
and standard deviation of return calculated using monthly
observations from 1970- 2001, N = 384.)
Industries vs. Market
Monthly
Std. Dev.
Beta
VaR (%)
VaR ($)
Hshld
5.12%
0.94
-7.52%
$18,100.83
Oil
5.42%
0.78
-7.91%
$19,024.02
Chips
8.01%
1.42
-12.28%
$28,890.92
Trans
6.30%
1.12
-9.55%
$22,768.71
Mkt
4.65%
1.00
-6.71%
$16,223.08
Relative VaR
• Relative VaR measures the risk of
underperformance relative to a pre-defined
benchmark.
• Relative VaR is calculated from a time series of
the difference in monthly logarithmic returns of an
investment minus the logarithmic return of the
benchmark portfolio.
Relative VaR (Cont.)
Monthly
Average
Std. Dev.
VaR (%)
VaR ($)
Hshld
0.91%
5.12%
-7.52%
$18,100.83
Oil
1.01%
5.42%
-7.91%
$19,024.02
Chips
0.89%
8.01%
-12.28%
$28,890.92
Monthly
Average
Std. Dev.
Relative VaR (%)
Relative VaR ($)
Rel_Hshld
-0.04%
2.69%
-4.47%
$10,917.36
Rel_Oil
0.07%
4.17%
-6.79%
$16,417.18
Rel_Chips
-0.05%
4.90%
-8.11%
$19,471.36
Trans
Mkt
0.81%
0.94%
6.30%
4.65%
-9.55%
-6.71%
$22,768.71 $16,223.08
Rel_Trans
-0.13%
3.59%
-6.04%
$14,646.77
Mkt
-
Note: One-month VaR and Relative VaR with 95% confidence level,
mean monthly return and standard deviation of return calculated using
monthly observations from 1970- 2001, N = 384.