HUJI03

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Financial Risk Management

• Zvi Wiener
• mswiener_at_mscc.huji.ac.il
• 02-588-3049

2
Financial Risk Management

• Following P. Jorion, Value at Risk, McGraw-Hill
• Chapter 4
• Measuring Financial Risk

3
Risks Measures

• Duration bonds, futures, fixed income
• Convexity bonds
• Beta diversified portfolio
• Sigma FX, undiversified portfolio
• Delta options
• Gamma options
• Risk is measured by short term volatility

4
Basic Statistics

• Certainty and uncertainty
• Probabilities, distribution, PDF, CDF
• Mean, variance
• Multivariable distributions
• Covariance, correlation, beta
• Quantile

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1 100 2 50 3 50
(1005050)/3 66.67 km/hr.
6

• 1. 40
• 2. 10
• 3. -50
• 4. 20

• 1. -2
• 2. 1
• 3. -1
• 4. 1

1.41.10.51.2 0.924
0.981.010.991.01 0.9897
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Probabilities

• Certainty
• Uncertainty
• Probabilities

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Probabilities

• Mean
• Variance

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Probabilities
0.3
0.2
0.1
1 2 3 4 5
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Probabilities
0.3
0.2
0.1
1 2 3 4 5
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Probabilities
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Probabilities
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Sample Estimates
Sometimes one can use weights
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Normal Distribution N(?, ?)
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Normal Distribution N(?, ?)
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Normal Distribution
quantile
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Lognormal Distribution
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Covariance

• Shows how two random variables are connected
• For example
• independent
• move together
• move in opposite directions
• covariance(X,Y)

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Correlation

• -1 ? ? ? 1
• ? 0 independent
• ? 1 perfectly positively correlated
• ? -1 perfectly negatively correlated

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Properties
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Time Aggregation
Assuming normality
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Time Aggregation

• Assume that yearly parameters of CPI are
• mean 5, standard deviation (SD) 2.
• Then daily mean and SD of CPI changes are

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Portfolio

• ?2(AB) ?2(A) ?2(B) 2?(A)?(B)?

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?12
?2
?
John Zerolis "Triangulating Risk", Risk v.9
n.12, Dec. 1996
?1
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Example

• We will receive n dollars where n is determined
  by a die.
• What would be a fair price for participation in
  this game?

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Example 1

• Score Probability
• 1 1/6
• 2 1/6
• 3 1/6
• 4 1/6
• 5 1/6
• 6 1/6

Fair price is 3.5 NIS. Assume that we can
play the game for 3 NIS only.
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Example

• If there is a pair of dice the mean is doubled.
• What is the probability to gain 5?

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Example
All combinations

• 1,1 2,1 3,1 4,1 5,1 6,1
• 1,2 2,2 3,2 4,2 5,2 6,2
• 1,3 2,3 3,3 4,3 5,3 6,3
• 1,4 2,4 3,4 4,4 5,4 6,4
• 1,5 2,5 3,5 4,5 5,5 6,5
• 1,6 2,6 3,6 4,6 5,6 6,6

36 combinations with equal probabilities
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Example
All combinations

• 1,1 2,1 3,1 4,1 5,1 6,1
• 1,2 2,2 3,2 4,2 5,2 6,2
• 1,3 2,3 3,3 4,3 5,3 6,3
• 1,4 2,4 3,4 4,4 5,4 6,4
• 1,5 2,5 3,5 4,5 5,5 6,5
• 1,6 2,6 3,6 4,6 5,6 6,6

4 out of 36 give 5, probability 1/9
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Additional information the first die gives 4.
All combinations
1,1 2,1 3,1 4,1 5,1 6,1 1,2 2,2 3,2 4,2 5,2 6,2 1,
3 2,3 3,3 4,3 5,3 6,3 1,4 2,4 3,4 4,4 5,4 6,4 1,5
2,5 3,5 4,5 5,5 6,5 1,6 2,6 3,6 4,6 5,6 6,6
1 out of 9 give 5, probability 1/9
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Additional information the first die gives ?4.
All combinations
1,1 2,1 3,1 4,1 5,1 6,1 1,2 2,2 3,2 4,2 5,2 6,2 1,
3 2,3 3,3 4,3 5,3 6,3 1,4 2,4 3,4 4,4 5,4 6,4 1,5
2,5 3,5 4,5 5,5 6,5 1,6 2,6 3,6 4,6 5,6 6,6
4 out of 24 give 5, probability 1/6
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Example 1
-2 -1 0 1 2 3
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Example 1

• 1 2 3 4 5 6 we pay
• 1 2 3 4 5 6 7 6 NIS.
• 2 3 4 5 6 7 8
• 3 4 5 6 7 8 9
• 4 5 6 7 8 9 10
• 5 6 7 8 9 10 11
• 6 7 8 9 10 11 12

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PL

• 1 2 3 4 5 6
• 1 -4 -3 -2 -1 0 1
• 2 -3 -2 -1 0 1 2
• 3 -2 -1 0 1 2 3
• 4 -1 0 1 2 3 4
• 5 0 1 2 3 4 5
• 6 1 2 3 4 5 6

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Example 1 (2 cubes)
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Example 1 (5 cubes)
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Random Variables

• Values, probabilities.
• Distribution function, cumulative probability.
• Example a die with 6 faces.

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Random Variables

• Distribution function of a random variable X
• F(x) P(X ? x) - the probability of x or less.
• If X is discrete then

If X is continuous then
Note that
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Random Variables

• Probability density function of a random variable
  X has the following properties

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Moments

• Mean Average Expected value

Variance
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Its meaning ...
Skewness (non-symmetry)
Kurtosis (fat tails)
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Main properties
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Portfolio of Random Variables
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Transformation of Random Variables

• Consider a zero coupon bond

If r6 and T10 years, V 55.84, we wish to
estimate the probability that the bond price
falls below 50. This corresponds to the yield
7.178.
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Example

• The probability of this event can be derived from
  the distribution of yields.
• Assume that yields change are normally
  distributed with mean zero and volatility 0.8.
• Then the probability of this change is 7.06

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Quantile

• Quantile (loss/profit x with probability c)

median
50 quantile is called
Very useful in VaR definition.
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FRM-99, Question 11

• X and Y are random variables each of which
  follows a standard normal distribution with
  cov(X,Y)0.4.
• What is the variance of (5X2Y)?
• A. 11.0
• B. 29.0
• C. 29.4
• D. 37.0

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FRM-99, Question 11
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FRM-99, Question 21

• The covariance between A and B is 5. The
  correlation between A and B is 0.5. If the
  variance of A is 12, what is the variance of B?
• A. 10.00
• B. 2.89
• C. 8.33
• D. 14.40

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FRM-99, Question 21
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Uniform Distribution

• Uniform distribution defined over a range of
  values a?x?b.

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Uniform Distribution
1
a b
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Normal Distribution

• Is defined by its mean and variance.

Cumulative is denoted by N(x).
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Normal Distribution
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Normal Distribution
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Normal Distribution

• symmetric around the mean
• mean median
• skewness 0
• kurtosis 3
• linear combination of normal is normal

99.99 99.90 99 97.72 97.5 95 90
84.13 50 3.715 3.09 2.326 2.000 1.96
1.645 1.282 1 0
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Central Limit Theorem

• The mean of n independent and identically
  distributed variables converges to a normal
  distribution as n increases.

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Lognormal Distribution

• The normal distribution is often used for rate of
  return.
• Y is lognormally distributed if XlnY is normally
  distributed. No negative values!

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Lognormal Distribution

• If r is the expected value of the lognormal
  variable X, the mean of the associated normal
  variable is r-0.5?2.

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Student t Distribution

• Arises in hypothesis testing, as it describes the
  distribution of the ratio of the estimated
  coefficient to its standard error. k - degrees of
  freedom.

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Student t Distribution

• As k increases t-distribution tends to the normal
  one.
• This distribution is symmetrical with mean zero
  and variance (kgt2)

The t-distribution is fatter than the normal one.
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Binomial Distribution

• Discrete random variable with density function

For large n it can be approximated by a normal.
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FRM-99, Question 12

• For a standard normal distribution, what is the
  approximate area under the cumulative
  distribution function between the values -1 and
  1?
• A. 50
• B. 66
• C. 75
• D. 95

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FRM-99, Question 13

• What is the kurtosis of a normal distribution?
• A. 0
• B. can not be determined, since it depends on the
  variance of the particular normal distribution.
• C. 2
• D. 3

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FRM-99, Question 16

• If a distribution with the same variance as a
  normal distribution has kurtosis greater than 3,
  which of the following is TRUE?
• A. It has fatter tails than normal distribution
• B. It has thinner tails than normal distribution
• C. It has the same tail fatness as normal
• D. can not be determined from the information
  provided

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FRM-99, Question 5

• Which of the following statements best
  characterizes the relationship between normal and
  lognormal distributions?
• A. The lognormal distribution is logarithm of the
  normal distribution.
• B. If ln(X) is lognormally distributed, then X is
  normally distributed.
• C. If X is lognormally distributed, then ln(X) is
  normally distributed.
• D. The two distributions have nothing in common

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FRM-98, Question 10

• For a lognormal variable x, we know that ln(x)
  has a normal distribution with a mean of zero and
  a standard deviation of 0.2, what is the expected
  value of x?
• A. 0.98
• B. 1.00
• C. 1.02
• D. 1.20

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FRM-98, Question 10
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FRM-98, Question 16

• Which of the following statements are true?
• I. The sum of normal variables is also normal
• II. The product of normal variables is normal
• III. The sum of lognormal variables is lognormal
• IV. The product of lognormal variables is
  lognormal
• A. I and II
• B. II and III
• C. III and IV
• D. I and IV

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FRM-99, Question 22

• Which of the following exhibits positively skewed
  distribution?
• I. Normal distribution
• II. Lognormal distribution
• III. The returns of being short a put option
• IV. The returns of being long a call option
• A. II only
• B. III only
• C. II and IV only
• D. I, III and IV only

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FRM-99, Question 22

• C. The lognormal distribution has a long right
  tail, since the left tail is cut off at zero.
  Long positions in options have limited downsize,
  but large potential upside, hence a positive
  skewness.

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FRM-99, Question 3

• It is often said that distributions of returns
  from financial instruments are leptokurtotic.
  For such distributions, which of the following
  comparisons with a normal distribution of the
  same mean and variance MUST hold?
• A. The skew of the leptokurtotic distribution is
  greater
• B. The kurtosis of the leptokurtotic distribution
  is greater
• C. The skew of the leptokurtotic distribution is
  smaller
• D. The kurtosis of the leptokurtotic distribution
  is smaller

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Financial Risk Management

• Following P. Jorion, Value at Risk, McGraw-Hill
• Chapter 5
• Computing Value at Risk

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(No Transcript)
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Breakfast
2 4 5 7 9 11 13 15
50 50
Lunch
50 50
? 11 ? ??
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Correlation ?1
Breakfast

• 2 4
• 5 7 9
• 11 13 15

50 50
Lunch
50 50
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Correlation ?-1
Breakfast

• 2 4
• 5 7 9
• 11 13 15

50 50
Lunch
50 50
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Correlation ?0
Breakfast

• 2 4
• 5 7 9
• 11 13 15

50 50
Lunch
50 50
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How to measure VaR

• Historical Simulations
• Variance-Covariance
• Monte Carlo
• Analytical Methods
• Parametric versus non-parametric approaches

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Historical Simulations

• Fix current portfolio.
• Pretend that market changes are similar to those
  observed in the past.
• Calculate PL (profit-loss).
• Find the lowest quantile.

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Example
Assume we have 1 and our main currency is
SHEKEL. Today 14.30. Historical data
PL 0.215 0 -0.112 0.052

• 4.00
• 4.20
• 4.20
• 4.10
• 4.15

4.304.20/4.00 4.515 4.304.20/4.20
4.30 4.304.10/4.20 4.198 4.304.15/4.10 4.352
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USD NIS 2003 100 -120 2004 200
100 2005 -300 -20 2006 20 30
today
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today
Changes in IR
USD 1 1 1 1 NIS 1 0
-1 -1
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Returns
year
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VaR
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Variance Covariance

• Means and covariances of market factors
• Mean and standard deviation of the portfolio
• Delta or Delta-Gamma approximation
• VaR1 ?P 2.33 ?P
• Based on the normality assumption!

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Variance-Covariance
?-2.33?
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Monte Carlo
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Monte Carlo

• Distribution of market factors
• Simulation of a large number of events
• PL for each scenario
• Order the results
• VaR lowest quantile

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Monte Carlo Simulation
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Weights

• Since old observations can be less relevant,
  there is a technique that assigns decreasing
  weights to older observations. Typically the
  decrease is exponential.
• See RiskMetrics Technical Document for details.

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Stock Portfolio

• Single risk factor or multiple factors
• Degree of diversification
• Tracking error
• Rare events

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Bond Portfolio

• Duration
• Convexity
• Partial duration
• Key rate duration
• OAS, OAD
• Principal component analysis

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Options and other derivatives

• Greeks
• Full valuation
• Credit and legal aspects
• Collateral as a cushion
• Hedging strategies
• Liquidity aspects

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Credit Portfolio

• rating, scoring
• credit derivatives
• reinsurance
• probability of default
• recovery ratio

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Credit Rating and Default Rates

• Rating Default frequency
• 1 year 10 years
• Aaa 0.02 1.49
• Aa 0.05 3.24
• A 0.09 5.65
• Baa 0.17 10.50
• Ba 0.77 21.24
• B 2.32 37.98

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Returns

• Past spot rates S0, S1, S2,, St.
• We need to estimate St1.
• Random variable

Alternatively we can do
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Independent returns

• A very important question is whether a sequence
  of observations can be viewed as independent.
• If so, one could assume that it is drawn from a
  known distribution and then one can estimate
  parameters.
• In an efficient market returns on traded assets
  are independent.

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Random Walk

• We could consider that the observations rt are
  independent draws from the same distribution N(?,
  ?2). They are called i.i.d. independently and
  identically distributed.
• An extension of this model is a non-stationary
  environment.
• Often fat tails are observed.

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Time Aggregation
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Time Aggregation
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FRM-99, Question 4

• Random walk assumes that returns from one time
  period are statistically independent from another
  period. This implies
• A. Returns on 2 time periods can not be equal.
• B. Returns on 2 time periods are uncorrelated.
• C. Knowledge of the returns from one period does
  not help in predicting returns from another
  period
• D. Both b and c.

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FRM-99, Question 14

• Suppose returns are uncorrelated over time. You
  are given that the volatility over 2 days is
  1.2. What is the volatility over 20 days?
• A. 0.38
• B. 1.2
• C. 3.79
• D. 12.0

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FRM-99, Question 14
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FRM-98, Question 7

• Assume an asset price variance increases linearly
  with time. Suppose the expected asset price
  volatility for the next 2 months is 15
  (annualized), and for the 1 month that follows,
  the expected volatility is 35 (annualized).
  What is the average expected volatility over the
  next 3 months?
• A. 22
• B. 24
• C. 25
• D. 35

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FRM-98, Question 7
108
Financial Risk Management

• Following P. Jorion, Value at Risk, McGraw-Hill
• Chapter 6
• Backtesting VaR Models

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Backtesting

• Verification of Risk Management models.
• Comparison if the models forecast VaR with the
  actual outcome - PL.
• Exception occurs when actual loss exceeds VaR.
• After exception - explanation and action.

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Backtesting
OK increasing k intervention

• Green zone - up to 4 exceptions
• Yellow zone - 5-9 exceptions
• Red zone - 10 exceptions or more

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Probability of Multiple Exceptions

• Each period the probability of exception is 1,
  then after 250 business days the probability that
  there will be 0 exceptions is

General formula of binomial distribution is
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The End
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FRM-00, Question 93

• A fund manages an equity portfolio worth 50M
  with a beta of 1.8. Assume that there exists an
  index call option contract with a delta of 0.623
  and a value of 0.5M. How many options contracts
  are needed to hedge the portfolio?
• A. 169
• B. 289
• C. 306
• D. 321

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FRM-00, Question 93

• The optimal hedge ratio is
• N -1.8?50,000,000/(0.623?500,000)289

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VaR system
Risk factors
Portfolio
Historical data
positions
Model
Mapping
Distribution of risk factors
VaR method
Exposures
VaR