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ERM > ERM Chapter 16 > Flashcards

ERM Chapter 16 Flashcards

1
Q

Outline the moments of a distribution.

A
  • un is the nth moment about the mean E[(X - E[X])^n]
  • coefficient of skewness = u3/o^3
  • coefficient of kurtosis = k = u4/o^4
  • k = 3 = mesokurtic
  • k > 3 = leptokurtic
  • k < 3 = platykurtic
  • excess kurtosis = k - 3
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2
Q

Provide examples of univariate discrete distributions.

A
  1. binomial and negative binomial distributions

2. poisson distribution

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3
Q

Provide example of univariate continuous distributions.

A

Values from negative inifinity to infinity

  1. normal distribution
  2. normal mixture distribution
  3. student’s t distribution
  4. skewed t distribution

Values that are non-negative

  1. lognormal distribution
  2. wald distribution
  3. chi-squared distribution
  4. gamma and inverse gamma distributions
  5. generalised inverse gamma distribution
  6. exponential distribution
  7. frechet distribution
  8. pareto distribution
  9. generalised pareto distribution

Values are in a finite range than can be positive and/or negative

  1. uniform distribution
  2. triangular distribution
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4
Q

Why might univariate continuous distributions be used even when variables can only take non-negative values?

A

Might still be used if the probability of getting a negative value is very small. This might be the case if the mean is sufficiently positive and the variance is sufficiently low.

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5
Q

Outline the binomial distribution.

A
  • Bin(n, p) is the sum of n independent bernoulli(p) trials
  • X~Bin(n, p) is the number of successes that occur in the n trials
  • the limiting distribution as n approaches infinity is the normal distribution
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6
Q

Outline the negative binomial distribution.

A
  • Type 1: no. of the trial on which the rth success occurs, where r is a positive integer
  • Type 2: no. of failures before the rth success
  • geometric distribution is a special case of the Type 1 negative binomial distribution where r = 1
  • limitations include:
    > CDF is laborious to calculate
    > n! becomes time consuming to calculate for large values of n
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7
Q

Outline the poisson distribution.

A
  • models the number of events that occur in a specified time interval, where events occur one after another in a well-defined manner
  • assumes that events occur singly, at a constant rate, and the no. of events that occur in separate time intervals are are independent of one another
  • a sequence of binomial (n, p) distributions approaches a poisson with mean np as n approaches infinity and p approaches zero.
  • poisson can be used as an approximation to the binomial if p is small enough
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8
Q

Outline the normal (or Gaussian) distribution.

A
  • standard normal distribution has a mean at 0 and a scaling parameter of 1
  • according to CLT, it will approximate the distribution of the sum or average of a sufficiently large number of iid random variables
  • it can facilitate simple analytical solutions to complex problems e.g. when it is used as an approximation to the binomial distribution
  • normal distribution with zero mean and given variance can be used for error terms when modelling a random walk
  • standard normal is the distribution of the test statistic Z = (X - u)/o used to determine whether the mean of an underlying population is significantly different to the assumed mean, when the value of o is known
  • standard normal is the distribution of the test statistic Z = (X(hat) - u)/(o/sqrt(T)) used to determine, based on the sample mean, whether the mean of an underlying population is significantly different from u, where T is the number of observations and o is known.
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9
Q

List two tests for normality.

A
  1. graphical approaches e.g. QQ plots

2. statistical tests e.g. Jarque-Bera test

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10
Q

Outline the Normal mean-variance mixture distributions.

A

Let W be some strictly positive random variable and Z be a standard normal random variable that is independent of W. X is said to be mean normal-variance mixture distribution if X = m(W)+sqrt(W) x BZ for some function m(W) and scale parameter B.

  • key benefit compared to modelling using a normal distribution is that it allows randomness into the mean and the variance.
  • special cases are the generalised hyperbolic function, the generalised t-distribution, and the skewed t-distribution
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11
Q

Outline the t-distribution.

A
  • can be Student’s, standard and general
  • if a Student’s t or standard t has y DOF then X = a + By has a generalised t-distribution with location parameter a, scaling parameter B and y DOF
  • the CDF can only be determined analytically when y = 1 (Cuachy distribution)
  • important for risk modelling as is has fatter tails than the normal distribution. The fact it is leptokurtic makes it an important distribution for risk modelling
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12
Q

Outline the skewed t-distribution.

A
  • parameters are as for the general t-distribution, but with an additional skew parameter
  • the general t-distribution is a special case of the skewed distribution where the skew parameter = 0
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13
Q

Outline the lognormal distribution.

A
  • If Y = lnX has a normal distribution, then X is said to have a lognormal distribution
  • applications include many insurance applications since it takes only positive values and is skewed
  • can be used to model financial variables e.g. asset returns, with assumptions that the natural logarithm of the variable will follow a random walk
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14
Q

Outline the Wald distribution.

A
  • describes the time taken for a Brownian motion process to reach a given value
  • is a special case of the inverse Gaussian distribution
  • takes only positive values and has positive skew
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15
Q

Outline the chi-squared distribution.

A
  • distribution with y DOF is the distribution of the sum of y squared independent variables taken from a standard normal distribution
  • is a special case of the gamma distribution
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16
Q

Outline the exponential distribution.

A
  • has a single scale parameter p
  • provides the expected waiting times between the events of a Poisson process
  • characteristics limiting its application include:
    > monotonically-decreasing nature
    > single parameter
    > low probabilities associated with extreme values
17
Q

Outline the gamma and inverse-gamma distributions.

A
  • has two positive parameters and is a versatile family
  • PDF can take significantly different shapes, depending on the specific values of parameters
  • exponential distribution is a special case when y = 1
  • chi-squared distribution if a special case when B = 2
  • can be fitted by equating sample and population moments and solving for the distribution’s parameters
18
Q

Outline the Generalised inverse Gaussian distribution.

A
  • like the gamma distributions, offer significant flexibility with regard to its shape, due to having three parameters: y, B1 and B2
  • when B1=0 the result is gamma(2B2, y)
  • when 1/B2 approach 0, tends to InverseGamma(B1/2, -y)
  • when y = -1/2 the result is a Wald distribution
19
Q

Outline the Frechet distribution.

A
  • like the exponential, has a single parameter

- distribution is a special case of the generalised extreme value distribution

20
Q

Outline the Pareto distribution.

A
  • has two parameters
  • monotonically decreasing and, like the tails of the t-distribution, follows a power law with the shape parameter (y) determining the power
21
Q

Outline the generalised Pareto distribution.

A
  • three parameter distribution is far more flexible than the two parameter distribution
  • in particular, it is applied in extreme value theory
  • when y=0 it is the exponential distribution
  • when y>0 it is the Pareto distribution
22
Q

Outline the uniform distribution.

A
  • assigns an equal probability in all outcomes in a range
23
Q

Outline the triangular distribution.

A
  • used in cases where, in addition to upper and lower values, the most likely value is known
  • distribution has a lower limit, mode, and upper limit
  • mean is the average of the parameter values
  • distribution can be negatively or positively skewed
24
Q

Outline the multivariate normal distribution.

A
  • completely characterised by its mean and covariance vectors
  • X~Nn(a, E) denoted an n-dimensional multivariate normal distribution with mean vector a and covariance vector E
  • components of vector X are mutually independent if and only if the covariance matrix E is diagonal
  • three key limitations that mean that the multivariate normal distribution is not a good description of reality in many RM applications:
    > the tails of the univariate marginal distributions are too thin
    > the joint tails do not assign enough weight to joint extreme outcomes
    > the distribution has a strong form of symmetry, known as elliptical symmetry
25
Q

Outline the Cholesky decomposition.

A
  • method of decomposing a matrix, allowing generation of a set of correlated normal variables from a set of independent standard normal variables
  • positive definite matrices are always invertible and can be written in the form M = CC’
  • C is a lower triangular matrix with positive diagonal entries
  • CC’ is called the cholesky decompositiong of M
  • Matrix C is known as the Cholesky factor and is denoted M^(1/2)
26
Q

Outline Principal Component Analysis (PCA).

A
  • also known as eigenvalue decomposition
  • breaks down each variable’s divergence from its mean into a weighted average of independent volatility factors
  • theory of spectral decomposition states that, for any covariance matrix E there exists a decomposition E = VAV’, where V is an orthogonal that consists of the eigenvectors of E
  • each pair of corresponding eigenvectors and eigenvalues is described as a principal component
27
Q

Outline multivariate mean-variance mixture distributions.

A
  • special cases include the generalised hyperbolic distribution, the multivariate t-distribution, and the skewed t-distribution
28
Q

Outline the multivariate t-distribution.

A
  • fatter tails the the multivariate normal distribution in two senses:
    > marginal distributions have higher probabilities associated with extreme values compared to the normal distribution
    > each combination of ‘jointly extreme values’ has higher probabilities than the multivariate normal distribution
  • the standard multivariate t-distribution is defined by only scale and shape parameters
29
Q

Outline the multivariate skewed t-distribution.

A
  • defined by parameters for location (a), scale (B), shape (DOF, y) and skew (d)
30
Q

Outline spherical and elliptical distributions.

A
  • distribution where each constant-probability contour of the two variables forms an elipse
  • a multivariate spherical distribution is one where the marginal distributions are:
    > identical
    > symmetrical
    > uncorrelated with each other
  • examples include the multivariate standard normal distribution and normal mixture distributions
  • the special case of an elliptical distribution where the correlation is zero is known as a spherical distribution
  • an example is the multivariate normal distribution with distinct marginal distributors
31
Q

What is the Mahalanobis distance used for?

A
  • testing whether observations are from a multivariate normal distribution
  • another test for multivariate normality is Mardia’s test based on Mahanalobis angle

ERM (31 decks)

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