Risk Management and
Value at Risk
Lecture
20
Phil Davies
The Process of Risk Management
What are the risks?
How should the risks be measured?
What are the exposures to the risks?
What should be done about the risks identified?
Nothing
Create capital cushion
Hedge
Insurance
Diversify
Limit
Risk is Multidimensional
Portfolio
Concentration
Risk
Transaction Risk
Counterparty
Risk
Issuer Risk
Trading Risk
Gap Risk
Equity Risk
Interest Rate Risk
Currency Risk
Commodity Risk
Financial
Risks
Operational
Risk
Reputational
Risk
Business and
strategic risks
Market Risk
Credit Risk
00pecific
Risk00/b>
General
Market
Risk
Issue Risk
* For more details, see Chapter-1, 00isk Management00by Crouhy, Galai and Mark
Other Types of Risk
Operational
Hazard
Physical
Strategic
Capital / resource allocation
Industry / competitors
Technological
Databases
Security
Confidential information
Stakeholder
Legal
Compliance
Regulatory
Human capital
Retention
Training
Reputational
Risk Interactions
We should not look at risks in isolation.
Focusing on market risk may
lead to exposure to credit risk or liquidity risk.
We should try to understand the
correlations between risks.
000the present practice
of modeling market risk separately from credit risk, a simplification
made for expediency, is certainly questionable in times of extraordinary
market stress. Under extreme conditions, discontinuous jumps in market
valuations raise the specter of insolvency, and market risk becomes
indistinct from credit risk.00/font>
Speech by Alan Greenspan at the 36th annual conference on bank structure and competition of the Federal Reserve Bank of Chicago, Chicago, Illinois, May 4, 2000.
How Do We Measure Risk?
What risk measure should we be
concerned about?
Systematic risk?
Volatility?
In most cases, investment managers get hurt by negative returns 00downside risk.
Large losses can lead investors to shift funds to other managers.
Investment managers who perform poorly get fired.
Hedge funds that perform poorly
are liquidated.
Since we care about downside risk, we must control that risk.
We need a measure of downside risk.
Value at Risk (VaR)
In early 1990s, JPMorgan created
a 00orst case scenario00measure for their trading portfolio called
Value-at-Risk (VaR).
VaR
measures the dollar loss in value that will be exceeded with a given
probability, p, over the next
n days.
For financial institutions n is
typically 1 day and p is either 5% or 1%.
VaR summarizes exposure to downside risk under typical market conditions.
Calculating VaR
Parametric approach:
Choose a distribution, estimate its parameters.
Compute VaR analytically
Historical Approach:
Let the data tell us the distribution.
The historical approach uses past returns to estimate VaR.
No distributional assumptions are necessary
Concern: It only works if past
is a good predictor of future.
Simulation Approach:
Assume returns have same distribution as in the past.
Compute VaR through simulation using past data.
Monte Carlo Simulations are beyond the scope of this class
A random variable follows the standard normal distribution when:
It is normally distributed with an expected value of zero, and standard deviation equal to one.
The probability that a random variable
following a standard normal distribution takes on a value of less than
-1.65 (-2.33) is 5% (1%)
Let average daily returns be denoted by m, and the standard deviation of daily returns by s. The VaR is:
5% 1 day VaR = (m 001.65s) x Portfolio
Value.
With daily VaRs, the mean is always
ignored.
VaR using the Normal Distribution
An Example Using the Normal Distribution
Fidelity Magellen is an actively
managed fund with a net asset value of $47.34bn.
We want to calculate its 1 day
VaR at 5% level using daily data since 2004 for any estimates that we
require.
VaR using the normal distribution
The expected return over one day for Fidelity is 0.0387%
The standard deviation of daily returns is 0.8387%.
The 1 day 5% VaR is 1.65 x 0.008387 x $47.34bn.
VaR = $0.6551bn
Interpretation:
There is a 5% chance the firm will lose more than $655m tomorrow.
If the distribution of returns remains constant over time Fidelity Magellen will lose more than $655m 5 days out of 100.
VaR Using the Historical Approach
Use a time series of data for the
asset or portfolio.
Calculate returns.
Sort the returns from the lowest to the highest.
Find the 5th percentile.
If you have 100 daily returns,
it will be observation 5.
VaR is calculated by multiplying the 5th percentile return by the portfolio/asset value.
The Historical Approach in Excel
In Excel we do not need to sort the returns data to find the 5th percentile. The PERCENTILE function will calculate the 5th percentile directly:
=PERCENTILE(H3:H982,0.05)
Given the 5th percentile value we can calculate the historical 5% 1 day VaR:
= -0.01362 x $47.34bn
= $0.64477bn
Comparing The Two Approaches
The parametric 5% 1 day VaR is $655m, while the historic 5% 1 day VaR is $645m.
Both approaches provide similar answers.
This
is not always the case.
Large differences between parametric and historic VaRs can arise if asset returns are not normally distributed.
- Some asset distributions have
fat tails (Excess Kurtosis) and may be skewed to the left or right.
How do we deal with derivatives?
The return of derivatives are
not generally normal. The payoffs are also non-linear.
Using historical data assumes that the past is a good predictor of the future.
The Portfolio Problem
Suppose now we have many different
assets with normally distributed returns. Portfolio variance is:
To estimate portfolio variance,
we need estimates of the variance-covariance matrix for all portfolio
assets.
In 1994, JP Morgan made a method to estimate VaR, called Riskmetrics漏, publicly available through a risk management consulting company.
For information, see Riskmetrics.com
Riskmetrics provides data on volatilities and correlation coefficients for more than 750,000 assets.
Does Anyone Really Use VaR?
Goldman Sachs 10K report for 2006 (Available on Class Website)
00e use historical data
to estimate our VaR and, to better reflect current asset volatilities,
we generally weight historical data to give greater importance to more
recent observations.00/font>
The table below is the 1 day 5%
VaR for Goldman Sachs for the last 3 years:
Over the past 3 years Goldman has
increased its risk levels by 50%. Equity risks have more than doubled
in the past year.
The risk management section of
the 10K statement is pages 88 00104. Goldman Sachs provide a good
discussion of the issues and problems involved in assessing the risk
of their firm.
Goldman Sachs001 Day VaR
How good is the VaR measure?
As part of Goldman Sachs00overall risk control process, daily trading net revenues are compared with VaR calculated as of the end of the prior business day. Trading losses incurred on a single day exceeded their 5% one-day VaR on three occasions during 2006.
Problems with Goldman00 VaR
Goldman Sachs highlight several important issues related to VaR estimation:
Shortfalls on a single day can
exceed reported VaR by significant amounts.
Shortfalls can also accumulate
over a longer time horizon such as a number of consecutive trading days.
Given its reliance on historical data, VaR is most effective in estimating risk exposures in markets in which there are no sudden fundamental changes or shifts in market conditions.
An inherent limitation of
VaR is that the distribution of past changes in market risk factors
may not produce accurate predictions of future market risk.
Different VaR methodologies and distributional assumptions could produce a materially different VaR.
VaR calculated for a one-day time horizon does not fully capture the market risk of positions that cannot be liquidated or offset with hedges within one day.
VaR in Reality
When you choose a risk measure,
you are likely to decrease the risks you measure but increase the risks
you do not measure!
Risk managers are concerned about black holes
Large losses that risk measures do not pick up.
VaR creates its own black holes
00low probability payoffs with huge losses do not affect VaR.
For example: The recent events
in the credit markets that have greatly affected US Banks involved 23
standard deviation movements in returns. Such an event should only occur
once every 1000 years or so. VaR would not capture this type of risk.
When 10Ks for US Banks are published
for 2007 it will be very interesting to see how their VaR risk measures
performed.
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