Download QA2 Probability

Financial Risk Management
Zvi Wiener
Following
P. Jorion, Financial Risk Manager Handbook
http://pluto.huji.ac.il/~mswiener/zvi.html
FRM
972-2-588-3049
Chapter 2
Quantitative Analysis
Fundamentals of Probability
Following P. Jorion 2001
Financial Risk Manager Handbook
http://pluto.huji.ac.il/~mswiener/zvi.html
FRM
972-2-588-3049
Random Variables
Values, probabilities.
Distribution function, cumulative probability.
Example: a die with 6 faces.
Ch. 2, Handbook
Zvi Wiener
slide 3
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
Ch. 2, Handbook
Zvi Wiener
 f (u)du

slide 4
Random Variables
Probability density function of a random
variable X has the following properties
f ( x)  0

1
 f (u )du

Ch. 2, Handbook
Zvi Wiener
slide 5
Multivariate Distribution Functions
Joint distribution function
F12 ( x1 , x2 )  P( X 1  x1 , X 2  x2 )
F12 ( x1 , x 2 ) 
x1 x2

f12 (u1 , u 2 )du1 du 2
  
Joint density - f12(u1,u2)
Ch. 2, Handbook
Zvi Wiener
slide 6
Independent variables
f12 (u1 , u2 )  f1 (u1 )  f 2 (u2 )
F12 (u1 , u2 )  F1 (u1 )  F2 (u2 )
Credit exposure in a swap depends on two random
variables: default and exposure.
If the two variables are independent one can
construct the distribution of the credit loss easily.
Ch. 2, Handbook
Zvi Wiener
slide 7
Conditioning
Marginal density

f1 ( x1 ) 
f
12
( x1 , u 2 )du 2

Conditional density
f12 ( x1 , x2 )
f12 ( x1 x2 ) 
f 2 ( x2 )
Ch. 2, Handbook
Zvi Wiener
slide 8
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
Ch. 2, Handbook
Zvi Wiener
slide 9
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)
Ch. 2, Handbook
Its meaning ...
 
 
Zvi Wiener
1

3
1

4

E  X  E X 
3

E  X  E  X 
slide 10

4

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
Ch. 2, Handbook
2
2
Zvi Wiener
slide 11
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
Ch. 2, Handbook
Zvi Wiener
slide 12
Portfolio of Random Variables
 (Y ) 
2
  11  12

w1 , w2 ,, wN  
 N 11  N 2
Ch. 2, Handbook
Zvi Wiener
 w1 
  1N   
  w2 
  
  NN   
 wN 
slide 13
Product of Random Variables
Credit loss derives from the product of the
probability of default and the loss given default.
E( X 1 X 2 )  E( X 1 ) E( X 2 )  Cov( X 1 , X 2 )
When X1 and X2 are independent
E( X 1 X 2 )  E( X 1 ) E( X 2 )
Ch. 2, Handbook
Zvi Wiener
slide 14
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%.
Ch. 2, Handbook
Zvi Wiener
slide 15
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%
Ch. 2, Handbook
Zvi Wiener
slide 16
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.
Ch. 2, Handbook
Zvi Wiener
slide 17
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
Ch. 2, Handbook
Zvi Wiener
slide 18
FRM-99, Question 11
    2  A B 
2
A
2
B
5  2  2  0.4  5  2  37
2
Ch. 2, Handbook
2
Zvi Wiener
slide 19
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
Ch. 2, Handbook
Zvi Wiener
slide 20
FRM-99, Question 21

B 
Cov( A, B )
 A B
Cov( A, B)
 A
 2.89
  8.33
2
B
Ch. 2, Handbook
Zvi Wiener
slide 21
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,
Ch. 2, Handbook
Zvi Wiener
slide 22
Uniform Distribution
1
1
ba
a
Ch. 2, Handbook
b
Zvi Wiener
slide 23
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).
Ch. 2, Handbook
Zvi Wiener
slide 24
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
Ch. 2, Handbook
-2
-1
1
Zvi Wiener
2
3
slide 25
Normal Distribution
1
0.8
0.6
0.4
0.2
-3
Ch. 2, Handbook
-2
-1
1
Zvi Wiener
2
3
slide 26
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
Ch. 2, Handbook
Zvi Wiener
50
0
slide 27
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 

Ch. 2, Handbook
Zvi Wiener
slide 28
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
Ch. 2, Handbook
Zvi Wiener
2
2
slide 29
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
Ch. 2, Handbook
1
1.5
Zvi Wiener
2
2.5
3
slide 30
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
Ch. 2, Handbook
Zvi Wiener
slide 31
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.
Ch. 2, Handbook
Zvi Wiener
slide 32
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
Ch. 2, Handbook
Zvi Wiener
slide 33
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%
Ch. 2, Handbook
Zvi Wiener
slide 34
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
Ch. 2, Handbook
Zvi Wiener
slide 35
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
Ch. 2, Handbook
Zvi Wiener
slide 36
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
Ch. 2, Handbook
Zvi Wiener
slide 37
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
Ch. 2, Handbook
Zvi Wiener
slide 38
FRM-98, Question 10
E[ X ]  e
Ch. 2, Handbook

2
2
e
Zvi Wiener
0.2 2
0
2
 1.02
slide 39
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
Ch. 2, Handbook
Zvi Wiener
slide 40
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
Ch. 2, Handbook
Zvi Wiener
slide 41
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.
Ch. 2, Handbook
Zvi Wiener
slide 42
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
Ch. 2, Handbook
Zvi Wiener
slide 43