Financial Risk Management
Quantitative Analysis Fundamentals of Probability Following P. Jorion, Financial Risk Management Chapter 2
Daniel HERLEMONT
Random Variables Values, probabilities. Distribution function, cumulative probability. Example: a die with 6 faces.
Daniel HERLEMONT
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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
F ( x) =
∑ f (x ) i
xi ≤ x
x
If X is continuous then F ( x) =
dF ( x) Note that f ( x) = dx
∫ f (u )du −∞
Daniel HERLEMONT
Random Variables
Probability density function of a random variable X has the following properties
f ( x) ≥ 0 ∞
1=
∫ f (u )du
−∞
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Probability Density and Cumulative Functions
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Multivariate Distribution Functions Joint distribution function
F12 ( x1 , x 2 ) = P ( X 1 ≤ x1 , X 2 ≤ x 2 ) x1 x2
F12 ( x1 , x 2 ) =
∫∫f
12
(u1 , u 2 ) du1 du 2
− ∞− ∞
Joint density - f12(u1,u2) Daniel HERLEMONT
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Independent variables
f 12 (u1 , u 2 ) = f 1 (u1 ) × f 2 (u 2 ) F12 (u1 , u 2 ) = F1 (u1 ) × F2 (u 2 ) 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. Daniel HERLEMONT
Conditioning Marginal density ∞
∫f
f1 ( x1 ) =
12
( x1 , u 2 )du 2
−∞
Conditional density
f 1− 2 ( x1 x 2 ) =
f 12 ( x1 , x 2 ) f 2 ( x2 )
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Moments Mean = Average = Expected value ∞
µ = E( X ) =
∫ xf ( x)dx
−∞
Variance ∞ 2
σ = V (X ) =
2 ( ) x − E ( X ) f ( x)dx ∫
−∞
σ = S tan dard Deviation = Variance Daniel HERLEMONT
Probabilities
Mean Variance
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Probabilities
0.3 30% 30%
0.2 0.1
10% 10%
20% 1
2
3
∑p
4
5
=1
i
i Daniel HERLEMONT
Probabilities
0.3 0.2 0.1 1
2
3
4
∫ dp
=1
5
∞
0 Daniel HERLEMONT
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Cov ( X 1 , X 2 ) = E [( X 1 − E [ X 1 ])( X 2 − E [ X 2 ])]
ρ(X1, X 2 ) =
Cov ( X 1 , X 2 )
σ 1σ 2
Skewness (non-symmetry)
Kurtosis (fat tails)
Its meaning ...
γ =
δ =
1
σ
3
1
σ4
[
E ( X − E [ X ])
[
3
E ( X − E [ X ])
]
4
]
Daniel HERLEMONT
Main properties
E (a + bX ) = a + bE ( X )
σ (a + bX ) = bσ ( X ) E( X 1 + X 2 ) = E( X 1 ) + E( X 2 )
σ 2 ( X1 + X 2 ) = σ 2 ( X1 ) + σ 2 ( X 2 ) + 2Cov( X1 , X 2 ) Daniel HERLEMONT
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Portfolio of Random Variables
N
Y = ∑ wi X i = wT X i =1
N
E (Y ) = µ p = w E ( X ) = w µ X = ∑ wi µ i T
T
i =1 N
N
σ (Y ) = w Σw = ∑∑ wiσ ij w j 2
T
i =1 j =1 Daniel HERLEMONT
Portfolio of Random Variables
σ 2 (Y ) = σ 11 σ 12 [w1 , w2 ,K, wN ] M σ N 11 σ N 2
w1 K σ 1N w2 M K σ NN wN
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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 )
Daniel HERLEMONT
Transformation of Random Variables Consider a zero coupon bond
V =
100 (1 + r ) T
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%.
Daniel HERLEMONT
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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) x
F ( x) =
∫ f (u )du = c −∞
50% quantile is called
median
Very useful in VaR definition.
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Quantile
1%
quantile
µ
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VAR
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Uniform Distribution Uniform distribution defined over a range of values a≤ ≤x≤ ≤b.
a+b (b − a) 2 2 E( X ) = , σ (X ) = 2 12
f ( x) =
1 , a≤ x≤b b−a
x≤a 0, x − a F ( x) = , a≤ x≤b b a − b≤x 1, Daniel HERLEMONT
Uniform Distribution
1 1 b−a
a
b
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Normal Distribution Is defined by its mean and variance.
f ( x) =
1
−
e
σ 2π
( x − µ )2 2σ 2
E( X ) = µ, σ 2 ( X ) = σ 2 Cumulative is denoted by N(x).
Daniel HERLEMONT
Normal Distribution 68% of events lie between -1 and 1
0.4
0.3
95% of events lie between -2 and 2
0.2
0.1
-3
-2
-1
1
2
3
Daniel HERLEMONT
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Normal Distribution 1 0.8 0.6 0.4 0.2
-3
-2
-1
1
2
3
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Normal Distribution symmetric around the mean mean = median skewness = 0 kurtosis = 3 linear combination of normal is normal
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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 σ2 X → N µ , n
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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
µ+
1
σx 2π
−
e
(ln( x ) − µ ) 2 2σ 2
σ2 2
2
, σ 2 ( X ) = e 2 µ + 2σ − e 2 µ +σ
2
E (Y ) = E (ln X ) = µ , σ 2 (Y ) = σ 2 (ln X ) = σ 2 Daniel HERLEMONT
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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-0.5σ σ2.
0.6 0.5 0.4 0.3 0.2 0.1
0.5
1
1.5
2
2.5
3
Daniel HERLEMONT
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 Γ 1 2 1 f ( x) = k +1 k kπ 2 Γ x 2 1 + 2 ∞ k Γ(k ) = ∫ x k −1e − x dx 0 Daniel HERLEMONT
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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 (k>2)
σ 2 ( x) =
k k −2
The t-distribution is fatter than the normal one. Daniel HERLEMONT
Binomial Distribution Discrete random variable with density function:
n f ( x ) = p x (1 − p ) n − x , x = 0,1,.K, n x
E ( X ) = pn, σ 2 ( X ) = p(1 − p)n For large n it can be approximated by a normal.
z=
x − pn p(1 − p)n
~ N (0,1)
Daniel HERLEMONT
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FRM Exam questions
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FRM questions
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