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.

5% 95%

Choice of confidence level – 95%

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.

Investment returns

If a 95% confidence level is used to estimateValue at Risk for a monthly horizon;

losses greater than the Value at Risk estimateare 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:

r*0ValueAtRisk V (1 e )

ˆ ˆr* 1.645*

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

VAR = $18,101 $19,024 $28,891 $22,769 $16,223

Value at Risk estimates for monthly horizon using 95% confidence level December 31, 2001, V0 = $250,000

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 variance-covaraince matrix.

• Rp=W1R1+W2R2+W3R3+W4R4

• σp²=ΣWi σi²+Σρi,jWiWjσiσj

• VAR=V0*(1-exp(r*))

Variance & Covariance

Var-Covar Hshld Chips Trans oilHshld 0.0026 0.0026 0.0022 0.0013Chips 0.0026 0.0064 0.0034 0.0019Trans 0.0022 0.0034 0.0040 0.0018

Oil 0.0013 0.0019 0.0018 0.0029

Correlation

Correlation Hshld Chips Trans OilHshld 1 0.6313 0.6974 0.4556Chips 0.6313 1 0.6842 0.4350Trans 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

Household Chips Transportation OilMarginal Value

At Risk $0.07 $0.11 $0.09 $0.06Component

Value At Risk $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 byVAR

Measured byß

Stand-Alone RiskOr

Total Risk

Systematic Risk

Unsystematic Risk

Non-Diversifiable

Risk

Diversifiable Risk

Market Risk Company-Specific Risk

Dispersion of Returns –Variances and Standard Deviations

• Variance () Formula:

• 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.

2

1

2 )(

kkn

ii

Risk Measure - Beta (ß)• Beta (ß) formula:

• 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.

)(),(

m

mi

kVarkkCov

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 RisksRisk 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.)

Monthly Hshld Oil Chips Trans MktStd. Dev. 5.12% 5.42% 8.01% 6.30% 4.65%

Beta 0.94 0.78 1.42 1.12 1.00 VaR (%) -7.52% -7.91% -12.28% -9.55% -6.71%VaR ($) $18,100.83 $19,024.02 $28,890.92 $22,768.71 $16,223.08

Industries vs. Market

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.)

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.

Monthly Hshld Oil Chips Trans MktAverage 0.91% 1.01% 0.89% 0.81% 0.94%Std. Dev. 5.12% 5.42% 8.01% 6.30% 4.65% VaR (%) -7.52% -7.91% -12.28% -9.55% -6.71%VaR ($) $18,100.83 $19,024.02 $28,890.92 $22,768.71 $16,223.08

Monthly Rel_Hshld Rel_Oil Rel_Chips Rel_Trans MktAverage -0.04% 0.07% -0.05% -0.13% -Std. Dev. 2.69% 4.17% 4.90% 3.59% -

Relative VaR (%) -4.47% -6.79% -8.11% -6.04% -Relative VaR ($) $10,917.36 $16,417.18 $19,471.36 $14,646.77 -