Download JCh 4-6

Financial Risk Management
Zvi Wiener
mswiener@mscc.huji.ac.il
02-588-3049
RM
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
HUJI-03
Financial Risk Management
Following P. Jorion, Value at Risk, McGraw-Hill
Chapter 4
Measuring Financial Risk
RM
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
HUJI-03
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 3
Basic Statistics
Certainty and uncertainty
Probabilities, distribution, PDF, CDF
Mean, variance
Multivariable distributions
Covariance, correlation, beta
Quantile
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 4
A
100 km.
B
100 km/hr
50 km/hr
1 – 100
2 – 50
3 – 50
(100+50+50)/3 = 66.67 km/hr.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 5
1.
-2%
1.
+40%
2.
+1%
2.
+10%
3.
-1%
3.
-50%
4.
+1%
4.
+20%
0.98*1.01*0.99*1.01 =
0.9897
Zvi Wiener
1.4*1.1*0.5*1.2 =
0.924
VaR-PJorion-Ch 4-6
slide 6
Probabilities
Certainty
Uncertainty
Probabilities
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 7
Probabilities
Mean
Variance
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 8
Probabilities
0.3
30% 30%
0.2
0.1
10% 10%
20%
1
2
3
4
p
i
5
1
i
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 9
Probabilities
0.3
0.2
0.1
1
2
3
4
5

 dp  1
0
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 10
Probabilities
N
mean  X   X i pi
i 1
N
Variance   ( X )   ( X  X i ) pi
2
2
i 1
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 11
Probabilities

mean  X   Xdp


Variance   ( X )   ( X  X ) dp
2
2

Variance   ( X )
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 12
Sample Estimates
1
ˆ
X
N
N
X
i 1
i
N

1
ˆ
ˆ ( X ) 
X
 Xi

N  1 i 1
2

2
Sometimes one can use weights
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 13
Normal Distribution N(, )
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 14
Normal Distribution N(, )


Zvi Wiener
VaR-PJorion-Ch 4-6
slide 15
Normal Distribution
1%
quantile
Zvi Wiener

VaR-PJorion-Ch 4-6
slide 16
Lognormal Distribution
0.6
0.5
0.4
0.3
0.2
0.1
1
Zvi Wiener
2
VaR-PJorion-Ch 4-6
3
4
slide 17
Covariance
Shows how two random variables are connected
For example:
independent
move together
move in opposite directions
covariance(X,Y) =
Zvi Wiener
E X  X Y  Y 
VaR-PJorion-Ch 4-6
slide 18
Correlation
 XY
E X  X Y  Y 

 ( X )  (Y )
-1    1
=0
independent
=1
perfectly positively correlated
 = -1
perfectly negatively correlated
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 19
Properties
E (A  B)  E ( A)  E ( B)
 (A  B) 
2
  ( A)    ( B)  2Cov( A, B)
2
2
2
2
 (A  B) 
2
  ( A)    ( B)  2( A) ( B) 
2
Zvi Wiener
2
2
2
VaR-PJorion-Ch 4-6
slide 20
Time Aggregation
T   annualT
 T   annual T
Assuming normality
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 21
Time Aggregation
Assume that yearly parameters of CPI are:
mean = 5%, standard deviation (SD) = 2%.
Then daily mean and SD of CPI changes are:
1
d   y
 0.02%
250
1
d  y
 0.1265%
250
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 22
Portfolio
2(A+B) = 2(A) + 2(B) + 2(A)(B)

A
rf
B

Zvi Wiener
VaR-PJorion-Ch 4-6
slide 23
¥
¥$£
£¥
$¥
$£¥
$
Zvi Wiener
£$¥
£$
VaR-PJorion-Ch 4-6
£
slide 24
X  X  X  2 X 1 X 2Cos
2
12
2
1
2
2
      2 1 2 12
2
12
2
1
2
2
12 ~ Cos

John Zerolis
"Triangulating Risk",
Risk v.9 n.12, Dec. 1996
Zvi Wiener
2
VaR-PJorion-Ch 4-6
12
1
slide 25
Example
We will receive n dollars where
n is determined by a die.
What would be a fair price for
participation in this game?
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 26
Example 1
Score
Probability
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
Zvi Wiener
1 2 3 4 5 6
      3.5
6 6 6 6 6 6
Fair price is 3.5 NIS.
Assume that we can play
the game for 3 NIS only.
VaR-PJorion-Ch 4-6
slide 27
Example
If there is a pair of dice the
mean is doubled.
What is the probability to
gain $5?
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 28
Example
All combinations:
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
36 combinations with equal probabilities
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 29
Example
All combinations:
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
4 out of 36 give $5, probability = 1/9
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 30
Additional information:
the first die gives 4.
All combinations:
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
1 out of 9 give $5, probability = 1/9
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 31
Additional information:
the first die gives 4.
All combinations:
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
4 out of 24 give $5, probability = 1/6
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 32
Example 1
1
 16.67%
6
-2
Zvi Wiener
-1
0
1
VaR-PJorion-Ch 4-6
2
3
slide 33
Example 1
1
2
3
4
5
6
Zvi Wiener
1
2
3
4
5
6
7
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
VaR-PJorion-Ch 4-6
6
7
8
9
10
11
12
we pay
6 NIS.
slide 34
P&L
1
2
3
4
5
6
Zvi Wiener
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
VaR-PJorion-Ch 4-6
6
1
2
3
4
5
6
slide 35
Example 1 (2 cubes)
0.15
0.125
0.1
0.075
0.05
0.025
-4
Zvi Wiener
-3
-2
-1
0
1
VaR-PJorion-Ch 4-6
2
3
4
5
6
slide 36
Example 1 (5 cubes)
0.1
0.08
0.06
0.04
0.02
-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 101112131415
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 37
Random Variables
Values, probabilities.
Distribution function, cumulative probability.
Example: a die with 6 faces.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 38
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
F ( x )   f ( xi )
xi  x
x
If X is continuous then F ( x) 
dF ( x)
Note that f ( x) 
dx
Zvi Wiener
VaR-PJorion-Ch 4-6
 f (u)du

slide 39
Random Variables
Probability density function of a random
variable X has the following properties
f ( x)  0

1
 f (u )du

Zvi Wiener
VaR-PJorion-Ch 4-6
slide 40
Moments
Mean = Average = Expected value
  E( X ) 

xf
(
x
)
dx


Variance
  V (X ) 
2

 x  E ( X )
2
f ( x)dx

  S tan dard Deviation  Variance
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 41
Cov( X 1 , X 2 )  E X 1  EX 1  X 2  EX 2 
( X1, X 2 ) 
Cov( X 1 , X 2 )
 1 2
Skewness (non-symmetry)
Kurtosis (fat tails)
Zvi Wiener
Its meaning ...
 
 
1

3
1

VaR-PJorion-Ch 4-6
4

E  X  E X 

3
E  X  E  X 

4

slide 42
Main properties
E (a  bX )  a  bE ( X )
 (a  bX )  b ( X )
E( X 1  X 2 )  E( X 1 )  E( X 2 )
 ( X 1  X 2 )   ( X 1 )   ( X 2 )  2Cov( X 1 , X 2 )
2
Zvi Wiener
2
2
VaR-PJorion-Ch 4-6
slide 43
Portfolio of Random Variables
N
Y   wi X i  w X
T
i 1
N
E (Y )   p  w E ( X )  w  X   wi  i
T
T
i 1
N
N
 (Y )  w w   wi ij w j
2
T
i 1 j 1
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 44
Transformation of Random Variables
Consider a zero coupon bond
100
V 
T
(1  r )
If r=6% and T=10 years, V = $55.84,
we wish to estimate the probability that the
bond price falls below $50.
This corresponds to the yield 7.178%.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 45
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%
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 46
Quantile
Quantile (loss/profit x with probability c)
x
F ( x) 
 f (u)du  c

50% quantile is called median
Very useful in VaR definition.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 47
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 (5X+2Y)?
A. 11.0
B. 29.0
C. 29.4
D. 37.0
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 48
FRM-99, Question 11
    2  A B 
2
A
2
B
5  2  2  0.4  5  2  37
2
Zvi Wiener
2
VaR-PJorion-Ch 4-6
slide 49
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 50
FRM-99, Question 21

B 
Cov( A, B )
 A B
Cov( A, B)
 A
 2.89
  8.33
2
B
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 51
Uniform Distribution
Uniform distribution defined over a range of
2
values axb.
ab 2
(b  a)
E( X ) 
,  (X ) 
2
12
1
f ( x) 
, a xb
ba
xa
0,
x  a

F ( x)  
, a xb
b  a
bx
1,
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 52
Uniform Distribution
1
1
ba
a
Zvi Wiener
b
VaR-PJorion-Ch 4-6
slide 53
Normal Distribution
Is defined by its mean and variance.
f ( x) 

1
 2
e
( x )2
2 2
E( X )  ,  ( X )  
2
2
Cumulative is denoted by N(x).
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 54
Normal Distribution
66% of events lie
between -1 and 1
0.4
0.3
95% of events lie
between -2 and 2
0.2
0.1
-3
Zvi Wiener
-2
-1
1
VaR-PJorion-Ch 4-6
2
3
slide 55
Normal Distribution
1
0.8
0.6
0.4
0.2
-3
Zvi Wiener
-2
-1
1
VaR-PJorion-Ch 4-6
2
3
slide 56
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
3.715 3.09 2.326 2.000 1.96 1.645 1.282 1
Zvi Wiener
VaR-PJorion-Ch 4-6
50
0
slide 57
Central Limit Theorem
The mean of n independent and identically
distributed variables converges to a normal
distribution as n increases.
1 n
X   Xi
n i 1
 2 

X  N   ,
n 

Zvi Wiener
VaR-PJorion-Ch 4-6
slide 58
Lognormal Distribution
The normal distribution is often used for rate
of return.
Y is lognormally distributed if X=lnY is
normally distributed. No negative values!
f ( x) 
E( X )  e

2
2
1
x 2

(ln(x )   ) 2
e
,  (X )  e
2
2 2
2   2 2
e
2   2
E (Y )  E (ln X )   ,  (Y )   (ln X )  
2
Zvi Wiener
VaR-PJorion-Ch 4-6
2
2
slide 59
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.
0.6
0.5
0.4
0.3
0.2
0.1
0.5
Zvi Wiener
1
1.5
2
VaR-PJorion-Ch 4-6
2.5
3
slide 60
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.
 k 1


1
2  1

f ( x) 
k 1
k
 
k
 

x2  2
1  
2

k 
k 1  x

(k )   x e dx
0
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 61
Student t Distribution
As k increases t-distribution tends to the
normal one.
This distribution is symmetrical with mean
zero and variance (k>2)
k
 ( x) 
k 2
2
The t-distribution is fatter than the normal one.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 62
Binomial Distribution
Discrete random variable with density function:
n x
n x


f ( x)    p (1  p) , x  0,1,., n
 x
E ( X )  pn,  ( X )  p(1  p)n
2
For large n it can be approximated by a normal.
x  pn
z
~ N (0,1)
p(1  p)n
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 63
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%
Error!
C. 75%
D. 95%
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 64
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 65
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 66
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 67
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 68
FRM-98, Question 10
E[ X ]  e
Zvi Wiener

2
2
e
0.2 2
0
2
VaR-PJorion-Ch 4-6
 1.02
slide 69
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 70
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 71
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.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 72
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 73
Financial Risk Management
Following P. Jorion, Value at Risk, McGraw-Hill
Chapter 5
Computing Value at Risk
RM
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
HUJI-03
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 75
Breakfast
Lunch
$2
$4
$5
$7
$9
50%
$11
$13
$15
50%
50%
 = $11
Zvi Wiener
VaR-PJorion-Ch 4-6
50%
 = ??
slide 76
Correlation =+1
Breakfast
$5
$2
$4
$7
$9
Lunch
$11
$13
$15
50%
50%
 = $11
Zvi Wiener
VaR-PJorion-Ch 4-6
50%
50%
 = $4
slide 77
Correlation =-1
Breakfast
$2
$4
$5
$7
$9
50%
$11
$13
$15
50%
50%
50%
Lunch
 = $11
Zvi Wiener
VaR-PJorion-Ch 4-6
 = $2
slide 78
Correlation =0
Breakfast
$2
$4
$5
$7
$9
50%
$11
$13
$15
50%
50%
50%
Lunch
 = $11
Zvi Wiener
VaR-PJorion-Ch 4-6
 = $3.16
slide 79
How to measure VaR
Historical Simulations
Variance-Covariance
Monte Carlo
Analytical Methods
Parametric versus non-parametric approaches
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 80
Historical Simulations
Fix current portfolio.
Pretend that market changes are
similar to those observed in the past.
Calculate P&L (profit-loss).
Find the lowest quantile.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 81
Example
Assume we have $1 and our main currency is
SHEKEL. Today $1=4.30.
Historical data:
P&L
4.00
4.20
4.30*4.20/4.00 = 4.515
0.215
4.20
4.30*4.20/4.20 = 4.30
0
4.10
4.30*4.10/4.20 = 4.198
-0.112
4.15
4.30*4.15/4.10 = 4.352
0.052
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 82
USD
NIS
2003
100
-120
2004
200
100
2005
-300
-20
2006
20
30
today
Zvi Wiener
100
200
 300
20



2
3
1  0.06 (1  0.061) (1  0.062) (1  0.063) 4
 120
100
 20
30



2
3
1  0.1 (1  0.11)
(1  0.12)
(1  0.13) 4
VaR-PJorion-Ch 4-6
slide 83
today
Changes
in IR
100
200
 300
20



2
3
1  0.06 (1  0.061) (1  0.062) (1  0.063) 4
 120
100
 20
30



2
3
1  0.1 (1  0.11)
(1  0.12)
(1  0.13) 4
USD:
NIS:
+1% +1%
+1% 0%
+1%
-1%
+1%
-1%
100
200
 300
20



2
3
1  0.07 (1  0.071) (1  0.072) (1  0.073) 4
 120
100
 20
30



2
3
1  0.11 (1  0.11)
(1  0.11)
(1  0.12) 4
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 84
Returns
year
1% of worst cases
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 85
VaR
1
0.8
0.6
0.4
VaR1%
1%
0.2
Profit/Loss
-3
Zvi Wiener
-2
-1
VaR-PJorion-Ch 4-6
1
2
3
slide 86
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!
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 87
Variance-Covariance VaR1%  V  2.33 V
1%
2.33
-2.33
Zvi Wiener

VaR-PJorion-Ch 4-6
slide 88
Monte Carlo
1
0.5
-1
0.5
-0.5
1
-0.5
-1
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 89
Monte Carlo
Distribution of market factors
Simulation of a large number of events
P&L for each scenario
Order the results
VaR = lowest quantile
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 90
Monte Carlo Simulation
15
10
5
10
20
30
40
-5
-10
-15
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 91
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.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 92
Stock Portfolio
Single risk factor or multiple factors
Degree of diversification
Tracking error
Rare events
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 93
Bond Portfolio
Duration
Convexity
Partial duration
Key rate duration
OAS, OAD
Principal component analysis
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 94
Options and other derivatives
Greeks
Full valuation
Credit and legal aspects
Collateral as a cushion
Hedging strategies
Liquidity aspects
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 95
Credit Portfolio
rating, scoring
credit derivatives
reinsurance
probability of default
recovery ratio
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 96
Credit Rating and Default Rates
Rating
Aaa
Aa
A
Baa
Ba
B
Zvi Wiener
Default frequency
1 year
10 years
0.02%
1.49%
0.05%
3.24%
0.09%
5.65%
0.17%
10.50%
0.77%
21.24%
2.32%
37.98%
VaR-PJorion-Ch 4-6
slide 97
Returns
Past spot rates S0, S1, S2,…, St.
We need to estimate St+1.
Random variable
S t  S t 1
rt 
S t 1
 St 

Alternatively we can do Rt  ln 
 S t 1 
 St 
 S t  S t 1 
  ln 1 
  ln 1  rt   rt
Rt  ln 
S t 1 
 S t 1 

Zvi Wiener
VaR-PJorion-Ch 4-6
slide 98
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.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 99
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.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 100
Time Aggregation
 S2 
 S 2 S1 
 S1 
 S2 
  ln    ln    R01  R12
R02  ln    ln 
 S1 
 S0 
 S1 S 0 
 S0 
E ( R02 )  E ( R01 )  E ( R12 )
 ( R02 )   ( R01 )   ( R12 )  2Cov( R01 , R12 )
2
2
2
E ( R02 )  2 E ( R01 )
 ( R02 )  2 ( R01 )
2
Zvi Wiener
2
VaR-PJorion-Ch 4-6
slide 101
Time Aggregation
E ( RT )  E ( R1 )T
 ( RT )   ( R1 )T
2
2
 ( RT )   ( R1 ) T
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 102
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.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 103
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%
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 104
FRM-99, Question 14
 ( R20 )  10 ( R10 )
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 105
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%
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 106
FRM-98, Question 7
 13        0.15  0.15  0.35
2
1
2
2
 av 
Zvi Wiener
2
3
 13
3
2
2
2
 0.236  24%
VaR-PJorion-Ch 4-6
slide 107
Financial Risk Management
Following P. Jorion, Value at Risk, McGraw-Hill
Chapter 6
Backtesting VaR Models
RM
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
HUJI-03
Backtesting
Verification of Risk Management models.
Comparison if the model’s forecast VaR with
the actual outcome - P&L.
Exception occurs when actual loss exceeds
VaR.
After exception - explanation and action.
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 109
Backtesting
Green zone - up to 4 exceptions
OK
Yellow zone - 5-9 exceptions
increasing k
Red zone - 10 exceptions or more
intervention
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 110
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
250!
0
250
0.01  0.99  0.081
0!250!
General formula of binomial distribution is
n!
x
n x
p  (1  p)
x!(n  x)!
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 111
The End
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 112
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
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 113
FRM-00, Question 93
The optimal hedge ratio is
N = -1.8$50,000,000/(0.623$500,000)=289
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 114
VaR system
Risk factors
Portfolio
Historical data
positions
Model
Mapping
Distribution of
risk factors
VaR
method
Exposures
VaR
Zvi Wiener
VaR-PJorion-Ch 4-6
slide 115