Chapter 25
Risk Assessment
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.
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
Value at Risk

There are at least three uses of value at risk:
1.
Regulators can use VaR to compute capital requirements for
financial institutions.
2.
Managers can use VaR as an input in making risk-taking and
risk-management decisions.
3.
Managers can also use VaR to assess the quality of the bank’s
models (accuracy of the bank models’ predictions of its own
losses).
Value at risk for one stock
 Suppose ~
xh 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
Prob( xh (c)  xh )  c
 Suppose a portfolio consists of a single stock and we wish to compute
value at risk over the horizon h.
Value at risk for one stock
 If we pick a stock price S h and the distribution of the stock price after
h periods, Sh, is lognormal, then
S h (c )  S 0 e
(  0.5 2 ) h  h N 1 ( c )
(24.5)
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.
Example - Solution
 The value of the “95% position” is given by:
S h  S0 e
(.1500.5(0.3)2 )(1/ 52)  1/ 52 ( 1.645)
 Hence we get:
 Sh = $ 2.8072 million
 And the VAR is: $ 3 million - $ 2.8072 million = $ 0.1928 m
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:
Sh  S0 (1  h  z h )
(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.
Sh  S0 (1  z h )
 Both equations become less reasonable as h grows.
(24.8)
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.
Example - Solution
 Using the following formula:
Sh  S0 (1  h  z h )
 We obtain:

S = 3 ( 1+.15(1/52) -1.645(.30)*sqrt(1/52) )
S = 2.8033
 And the VAR is:

$ 3m - $ 2.8033m = $ 0.1967m that we stand to loose.
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.
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
W  i 1Wi
n
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
Rh 
1
W

n
r Wi
i 1 i .h
Assuming normality, the annualized distribution of the portfolio return
is
1
Rh ~ N 
W
i1 iWi,
n
1
n
n




W
W
2 i 1  j 1 i j ij i j 
W

(24.9)
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:
1.
Delta approximation: We can create a linear approximation
to the option price by using the option delta.
2.
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.
Delta approximation
~
 If the return on stock i is  i, we can approximate the return on the option as
 i~i , 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
1
Rp 
W

n
i 1
 i Si (w i  N i  i )
(24.10)
The term wi + Nii measures the exposure to stock i.
 The variance of the return is
1
n
n
  2 i 1  j 1 Si S j (w i  N i  i )(w j  N j  j ) i j ij
W
2
p
 With this mean and variance, we can mimic the n-stock analysis.
(24.11)
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.
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.
Monte Carlo simulation
 First, we randomly draw a set of z ~ N(0,1), and construct the stock
price as
S h  S0e
(  0.5 2 ) h  h z
(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.
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.
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.
Monte Carlo simulation
 Example 2: Histogram of values of a portfolio that contains a written
put and call having different, correlated underlying stocks.
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.
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.
Historical volatility
 Historical volatility estimates are typically used in VaR calculations.
 Here are two rolling n-day measures of volatility:
a.
1 n 2
Vt 
r

i 1 t i
n
(24.18)
where rt is the daily continuously compounded return on day t.
b.


i 1
(
1


)

 rt 2i
Vˆt  i 1  n
  (1   ) j 1 
 j 1

n
(24.19)
Historical volatility
 V is the standard estimator for volatility.

Vˆ 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.
Historical volatility
 Volatility estimates for IBM and the S&P 500 using the two measures: