Risk Assessment

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Chapter 25 Risk Assessment
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Introduction

• Risk assessment is the evaluation of
  distributions of outcomes, with a focus on the
  worse that might happen.
• Insurance companies, for example, are in the
  business of determining the likelihood of, and
  loss associated with, an insured event.

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Value at Risk

• Value at risk (VaR) is a way to perform risk
  assessment for complex portfolios.
• In general, computing value at risk means finding
  the value of a portfolio such that there is a
  specified probability that the portfolio will be
  worth at least this much over a given horizon.
• Regulators have proposed assessing capital at
  three times the 99 10-day VaR

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Value at Risk

• There are at least three uses of value at risk
• Regulators can use VaR to compute capital
  requirements for financial institutions.
• Managers can use VaR as an input in making
  risk-taking and risk-management decisions.
• Managers can also use VaR to assess the quality
  of the banks models (accuracy of the bank
  models predictions of its own losses).

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Value at risk for one stock

• Suppose is the dollar return on a portfolio
  over the horizon h, and f (x, h) is the
  distribution of returns.
• Define the value at risk of the portfolio as the
  return, xh(c), such that
• Suppose a portfolio consists of a single stock
  and we wish to compute value at risk over the
  horizon h.

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Value at risk for one stock

• If we pick a stock price and the
  distribution of the stock price after h periods,
  Sh, is lognormal, then
• (24.5)

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Example

• Assume you have 3M worth of stock, with an
  expected return of 15, a 30 volatility, and no
  dividend.
• You know (from cumulative Normal distribution
  tables) that the 95 lower tail corresponds to a
  z-value of -1.645
• Compute the value that your portfolio has a 95
  chance of exceeding in one week. Infer the VAR.

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Example - Solution

• The value of the 95 position is given by
• Hence we get
• Sh 2.8072 million
• And the VAR is 3 million - 2.8072 million
  0.1928 m

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Value at risk for one stock

• In practice, it is common to simplify the VaR
  calculation by assuming a normal return rather
  than a lognormal return. A normal approximation
  is
• (24.7)
• We could further simplify by ignoring the mean
• Mean is hard to estimate precisely.
• For short horizons, the mean is less important
  than the diffusion term in an Itô process.
• (24.8)
• Both equations become less reasonable as h grows.

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Example

• Assume you have 3M worth of stock, with an
  expected return of 15, a 30 volatility, and no
  dividend.
• You know (from cumulative Normal distribution
  tables) that the 95 lower tail corresponds to a
  z-value of -1.645
• Compute the value that your portfolio has a 95
  chance of exceeding in one week. Infer the VAR.

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Example - Solution

• Using the following formula
• We obtain
• S 3 ( 1.15(1/52) -1.645(.30)sqrt(1/52) )
• S 2.7947
• And the VAR is
• 3m - 2.7947m 0.2053m that we stand to
  loose.

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Two or more stocks

• When we consider a portfolio having two or more
  stocks, the distribution of the future portfolio
  value is the sum of lognormally distributed
  random variables.
• Since the distribution is no longer lognormal, we
  can use the normal approximation.

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Two or more stocks

• Let the annual mean of the return on stock i, ri,
  be ?i.
• The standard deviation of the return on stock i
  is ?i.
• The correlation between stocks i and j is ?ij.
• The dollar investment in stock i is Wi.
• The value of a portfolio containing n stocks is

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Two or more stocks

• If there are n assets, the VaR calculation
  requires that we specify the standard deviation
  for each stock, along with all pairwise
  correlations.
• The return on the portfolio over the horizon h,
  Rh, is
• Assuming normality, the annualized distribution
  of the portfolio return is
• (24.9)

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VaR for nonlinear portfolios

• If a portfolio contains options as well as
  stocks, it is more complicated to compute the
  distribution of returns.
• The sum of the lognormally distributed stock
  prices is not lognormal.
• The option price distribution is complicated.
• There are two approaches to handling
  nonlinearity
• Delta approximation We can create a linear
  approximation to the option price by using the
  option delta.
• Monte Carlo simulation We can value the option
  using an appropriate option pricing formula and
  then perform Monte Carlo simulation to obtain the
  return distribution.

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Delta approximation

• If the return on stock i is , we can
  approximate the return on the option as
  , where ?i is the option delta. Let Ni be the
  number of options and wi the number of shares of
  the stock.
• The expected return on the stock and option
  portfolio over the horizon h is then
• (24.10)
• The term ?i Ni?i measures the exposure to
  stock i.
• The variance of the return is
• (24.11)
• With this mean and variance, we can mimic the
  n-stock analysis.

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Monte Carlo simulation

• Monte Carlo simulation works well in situations
  where we need a two-tailed approach to VaR (e.g.,
  straddle).
• Simulation produces the distribution of portfolio
  values.
• To use Monte Carlo simulation,
• We randomly draw a set of stock prices.
• Once we have the portfolio values corresponding
  to each draw of random prices, we sort the
  resulting portfolio values in ascending order.
• The 5 lower tail of portfolio values, for
  example, is used to compute the 95 value at risk.

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Monte Carlo simulation

• Example 1
• Consider the 1-week 95 value at risk of an
  at-the-money written straddle on 100,000 shares
  of a single stock.
• Assume that S 100, K 100, ? 30, r 8,
  t 30 days, and ? 0.
• The initial value of the straddle is ?685,776.

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Monte Carlo simulation

• First, we randomly draw a set of z N(0,1), and
  construct the stock price as
• (24.13)
• Next, we compute the Black-Scholes call and put
  prices using each stock price, which gives us a
  distribution of straddle values.
• We then sort the resulting straddle values in
  ascending order.
• The 5 value is used to compute the 95 value at
  risk.

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Monte Carlo simulation

• Histogram of values resulting from 100,000 random
  simulations of the value of the straddle
• The 95 value at risk is ?943,028 ? (?685,776)
  ?257,252.

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Monte Carlo simulation

• Note that the value of the portfolio never
  exceeds ?597,000.
• If a call and put are written on the same stock,
  stock price moves can never induce the two to
  appreciate together. The same effect limits a
  loss.
• When options are written on different stocks, it
  is possible for both to gain or lose
  simultaneously. As a result, the distribution of
  prices has a greater variance and increased value
  at risk.

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Monte Carlo simulation

• Example 2 Histogram of values of a portfolio
  that contains a written put and call having
  different, correlated underlying stocks.

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Estimating volatility

• Volatility is the key input in any VaR
  calculation.
• In most examples, return volatility is assumed to
  be constant and returns are independent over
  time.
• Assessing return correlation is complicated.
• Over horizons as short as a day, returns may be
  negatively correlated due to factors such as
  bid-ask bounce.
• With commodities, return independence is not
  reasonable for long horizons due to supply and
  demand responses.

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Implied volatility

• Option prices can be used to estimate implied
  volatility, which is a forward-looking measure of
  volatility.
• However, implied volatility is not readily
  available for some underlying assets.
• For VaR calculations, we also need to know
  correlations between assets, for which there is
  no measure comparable to implied volatility.
• A conceptual problem is which volatility to use
  when different options give different implied
  volatilities.

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Historical volatility

• Historical volatility estimates are typically
  used in VaR calculations.
• Here are two rolling n-day measures of
  volatility
• a.
• (24.18)
• where rt is the daily continuously compounded
  return on day t.
• b.
• (24.19)

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Historical volatility

• V is the standard estimator for volatility.
• is an estimate that gives more weight to
  recent returns and less to older returns.
• This estimator is called an exponential weighted
  moving average (EWMA) estimate of volatility.
• In both measures, we ignore the mean, which is
  small for daily returns.

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Historical volatility

• Volatility estimates for IBM and the SP 500
  using the two measures