Objectives:

1
Risk Measurement Using Value at Risk

• Objectives
• Discuss the rationale for risk-measurement and
  the definition of Value at Risk (VaR).
• How to implement VaR when portfolio returns are
  assumed to be normally distributed.
• The use of historical back simulation to
  calculate VaR.
• Understand how VaR can be applied to set banks
  risk-based capital requirements.

2
Measuring Downside Risk

• Recall that risk management can add value to a
  financial intermediary (FI) because it mitigates
  dead-weight costs of financial distress and/or
  bankruptcy and it also reduces the value of taxes
  paid.
• To implement a FIs risk management program, FI
  managers need to be able to measure the FIs
  risk. A popular approach to measuring the
  downside risk of a portfolio position in assets
  and liabilities which was developed by J.P.
  Morgan (now J.P. Morgan Chase) is called Value at
  Risk (VaR).
• VaR quantifies the size and probability of a
  portfolio loss. The portfolio can be the entire
  FIs assets and liabilities. In this case, VaR
  measures a loss to the FIs net worth or capital.
  VaR is used to set risk-based capital
  requirements for large international banks under
  the Basel II Capital Accord.

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• A risk-measurement approach, such as VaR, can
  also be applied to the portfolio position of a
  single trader or single type of asset (e.g., FX
  or fixed-income). In this context, it can be
  used to
• set limits on a traders portfolio positions.
• decide what FI activities provide the best
  trade-off of risk for expected return.
• evaluate a trader or activitys performance based
  on their risk exposure. Compensation can be
  determined not only by profits earned but also by
  VaR.
• For a given probability, p, and a given future
  investment horizon, h days, VaR is defined as the
  loss in value that has a probability p of being
  exceeded over the next h days, assuming that the
  portfolio position is not changed over the
  investment horizon.

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• Example Suppose that the loss in portfolio value
  that has a one percent probability of being
  exceeded over the next 10 days is estimated to be
  1 million. Then, 1 million is the portfolios
  VaR for p 1, and h10 days.
• VaR depends on the firms or portfolios
  distribution of value, which, in turn, depends on
  the firms or portfolios assets, liabilities,
  and derivative positions.
• We can graphically illustrate VaR using the
  probability distribution for a portfolios
  return. For example, suppose that over some
  given time horizon, h days, a particular
  portfolios return is estimated to have the
  following cumulative probability distribution
  function.

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Cumulative Distribution Function of Portfolio
Return
Probability
Probability that return lt p
p, return
-0.23

• We see that there is a 5 probability of the
  portfolio returning less than -23. If, say, the
  portfolios initial value is 1 m, then VaR(p5,
  h days) 0.23x1m 230,000.

6
VaR with Normally Distributed Portfolio Returns

• One approach to computing VaR is to assume that
  the portfolios returns are normally distributed.
  Let the portfolios random rate of return over a
  period of h days be . Its
  probability density function is

Probability density function
5 of area under curve
1 of area under curve
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• This implies that there is a 5 probability of a
  return less than
• and a 1 probability of a
  return less than
• VaR is usually calculated over a measurement
  horizon of a small number of days. For this
  short horizon, a portfolios standard deviation
  is typically much greater than its expected
  return. Hence, the practice is to ignore the
  expected return and set
• If this is done, then we have
• VaR(p5 , h days) 1.65x? x(Portfolio Value)
• VaR(p1 , h days) 2.33x? x(Portfolio Value)
• Example a portfolios 1 day standard deviation
  is 10, and its initial value is 1m, then
  VaR(p1, h1 day) 2.33x(0.10)x1m 233,000.

8

• What should be the time horizon (h days) over
  which to calculate VaR? If a FI can measure its
  risk and change it once a day, a one-day VaR is
  most useful. This would be relevant when a FIs
  portfolio consists of liquid securities that can
  be bought or sold quickly.
• However, if a FI holds a portfolio of illiquid
  assets that cannot be sold quickly, a longer
  horizon would be relevant. The FI should choose
  the VaRs h to be the number of days over which
  it could change its portfolio.
• If the return on a portfolio is estimated to have
  a one-day standard deviation of ?, then, assuming
  the portfolios composition stays the same over h
  days, its h-day standard deviation can be
  estimated as

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• Example A bank holds a 20 m. portfolio of
  syndicated loans that would likely take 5 days to
  arrange for a sale. The daily standard deviation
  of the portfolios value is 0.3 . Therefore
• Suppose a portfolio consists of n different
  assets. Its standard deviation depends on the
  standard deviations and correlations of the
  individual assets composing the portfolio.
• Let ?i be the proportion of the portfolios total
  value that is invested in asset i, and let ?i
  asset is standard deviation of return. Further,
  let ?ij be the correlation between the returns on
  asset i and asset j. Then the portfolio returns
  variance is

10

• Example A portfolio has three assets held in
  proportions ?1 0.2, ?2 0.5, and ?3 0.3.
  The assets h-day standard deviations are ?1
  0.3, ?2 0.2, and ?3 0.4. Their correlations
  are ?12 0.1, ?13 0.6, ?23 -0.1. The
  portfolios h-day return variance is then

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• Therefore, the portfolio returns standard
  deviation is
• If the three-asset portfolio was initially worth
  50m, then, for example, VaR(p5, h days)
  1.65x0.188 x50 m 15.5 m.
• Under the approach outlined thus far,
  implementing VaR for a portfolio of many
  different assets requires estimates of each asset
  returns standard deviation and the correlations
  between all of the assets returns.
• A consulting firm, RiskMetrics, provides daily
  estimates of standard deviations and correlations
  for different types of assets in many different
  countries. Of course, an individual FI could
  compute these estimates on its own using
  historical data.

12
VaR Using Back Simulation

• Rather than assuming a portfolios return is
  normally distributed, other methods to computing
  VaR are used.
• The historic or back simulation approach uses the
  historical returns on the individual assets
  contained in the FIs current portfolio. It then
  simulates what would have been the losses on the
  current portfolio if this portfolio had been held
  during the historical period, say the last 250 or
  500 trading days.
• Specifically, this approach involves the
  following steps
• Consider each of the asset/liability positions in
  the FIs current portfolio. Suppose, as before,
  that there are n different assets and ?i is the
  proportion of the portfolios total value that is
  invested in asset i. Obtain data on the returns
  of these assets and liabilities for, say, the
  last 500 trading days.

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• Let rit is the return on asset/liability i on
  prior day t. Then if the portfolio had been held
  on that day, its return would have been
• Next, rank the portfolio returns, Rt, for
  previous t 1, , 500 days from the lowest
  return to the highest. Let R5 be the fifth worst
  return over the last 500 days and let R25 be the
  25th worst return over the last 500 days. Most
  likely, both of these returns are negative
  (losses).
• Then we would compute VaR as

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• A similar procedure using historical returns
  over, say, h 5 day intervals could be used to
  calculate VaR(p, h5 days).
• The advantage of this historical simulation
  approach is that it uses the sample frequency
  (histogram) of actual returns. There is evidence
  that empirical distributions of asset returns
  display large losses more frequently than would
  be predicted by the thin-tailed normal
  distribution, and the back simulation method
  could account for this.
• One disadvantage is that the historical period
  may not be representative of the near future. It
  may have been an unusually quiet (low volatility)
  period, and volatility is now likely to be
  greater. One correction for this is to give more
  recent observations a greater weight.

15
Risk-Based Capital Requirements Using VaR

• VaR is the basis for setting minimum capital (net
  worth) requirements for large international
  banks.
• In 1996, an amendment to the 1988 Basel Capital
  Accord created a rule for bank capital
  requirements to cover their liquid securities
  (non-loan) portfolios, so-called trading
  accounts
• Required capital for day t1
• where SRt is additional capital to cover
  idiosyncratic risks. The terms in brackets are
  the banks current VaR estimate and an average of
  VaR estimates over the last 60 days.
• The multiplier St depends on the accuracy of the
  banks VaR model.

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• St is computed by back-testing the banks VaR
  model estimates over the last 250 days. If the
  banks daily trading portfolio losses exceeded
  VaR(p1, h 1 day) on x days over the last 250
  days, then
• Thus, a bank with a less accurate internal VaR
  model has a higher multiplier, St , and must have
  more capital.

17

• In June of 2004, central bank governors and the
  heads of bank supervisory authorities in the
  Group of Ten (G10) countries announced agreement
  on a framework for revised risk-based capital
  standards, commonly known as Basel II.
• Basel II will take effect at the beginning of
  2007. This New Basel Capital Accord consists
  of three pillars
• 1. Minimum bank capital requirements.
• 2. Bank supervisor review of capital adequacy.
• 3. Public disclosure by banks of their financial
  conditions.
• Large international banks are expected to use
  Basel IIs Internal Ratings Based (IRB) Approach
  to setting minimum capital. It requires a bank
  to hold capital so that, based on the riskiness
  of its on- and off- balance sheet assets and
  liabilities, the banks probability of solvency
  over a one-year horizon exceeds 99.9 .

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• Hence, to calculate a banks minimum capital, one
  treats its total assets and liabilities as a
  single portfolio whose initial value equals the
  banks initial capital (net worth).
• Defining end-of-year solvency as capital that
  exceeds zero, the banks minimum capital is equal
  to its overall portfolio VaR(p0.1, h 365
  days).
• The risk of the banks assets and liabilities can
  be evaluated using risk estimates specified in
  the Basel II Accord (Foundation IRB Approach) or
  using risk estimates based on the banks own
  models (Advanced IRB Approach).