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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Chi - squared distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Observe sample variance and compare it to hypothetical population variance (sample variance/population variance)(n - 1) = chi - squared - Non - negative and skewed right - approaches zero as n increases - mean = k where k = degrees of freedom - Varia
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

2. Maximum likelihood method
Observe sample variance and compare it to hypothetical population variance (sample variance/population variance)(n - 1) = chi - squared - Non - negative and skewed right - approaches zero as n increases - mean = k where k = degrees of freedom - Varia
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Choose parameters that maximize the likelihood of what observations occurring

3. Standard variable for non - normal distributions
Distribution with only two possible outcomes
Contains variables not explicit in model - Accounts for randomness
Variance(x)
Z = (Y - meany)/(stddev(y)/sqrt(n))

4. Deterministic Simulation
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P(X=x - Y=y) = P(X=x) * P(Y=y)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

5. Kurtosis
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

6. Weibul distribution
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Nonlinearity
Variance = (1/m) summation(u<n - i>^2)

7. Tractable
Easy to manipulate
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
Normal - Student's T - Chi - square - F distribution
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

8. Hybrid method for conditional volatility
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Use historical simulation approach but use the EWMA weighting system
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Summation((xi - mean)^k)/n

9. Heteroskedastic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)
When one regressor is a perfect linear function of the other regressors
If variance of the conditional distribution of u(i) is not constant

10. Continuously compounded return equation
Confidence set for two coefficients - two dimensional analog for the confidence interval
i = ln(Si/Si - 1)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

11. Simulation models
Flexible and postulate stochastic process or resample historical data - Full valuation on target date - More prone to model risk - Slow and loses precision due to sampling variation
(a^2)(variance(x)
Probability that the random variables take on certain values simultaneously
Low Frequency - High Severity events

12. Variance(discrete)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
More than one random variable
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

13. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

14. Exponential distribution
We reject a hypothesis that is actually true
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
P - value
Random walk (usually acceptable) - Constant volatility (unlikely)

15. Variance of sample mean
More than one random variable
Variance(y)/n = variance of sample Y
Expected value of the sample mean is the population mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

16. Reliability
Statement of the error or precision of an estimate
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Based on an equation - P(A) = # of A/total outcomes
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

17. T distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
P(Z>t)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

18. Confidence ellipse
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Confidence set for two coefficients - two dimensional analog for the confidence interval
Easy to manipulate
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

19. GPD
Based on a dataset
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
P - value
i = ln(Si/Si - 1)

20. Bootstrap method
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for an individual asset
Mean of sampling distribution is the population mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

21. Sample correlation
Sample mean will near the population mean as the sample size increases
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related
Rxy = Sxy/(Sx*Sy)
Confidence level

22. Result of combination of two normal with same means
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Least absolute deviations estimator - used when extreme outliers are not uncommon
Combine to form distribution with leptokurtosis (heavy tails)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

23. Hazard rate of exponentially distributed random variable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
We reject a hypothesis that is actually true
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Easy to manipulate

24. Monte Carlo Simulations
Based on an equation - P(A) = # of A/total outcomes
Variance = (1/m) summation(u<n - i>^2)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

25. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
We accept a hypothesis that should have been rejected
Summation((xi - mean)^k)/n
Has heavy tails

26. Single variable (univariate) probability
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Concerned with a single random variable (ex. Roll of a die)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Price/return tends to run towards a long - run level

27. Persistence
Probability that the random variables take on certain values simultaneously
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
In EWMA - the lambda parameter - In GARCH(1 -1) - sum of alpha and beta - Higher persistence implies slow decay toward the long - run average variance

28. K - th moment
P(X=x - Y=y) = P(X=x) * P(Y=y)
Generalized Auto Regressive Conditional Heteroscedasticity model - GARCH(1 -1) is the weighted sum of a long term variance (weight=gamma) - the most recent squared return (weight=alpha) and the most recent variance (weight=beta)
Summation((xi - mean)^k)/n
E(mean) = mean

29. Unconditional vs conditional distributions
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
OLS estimators are unbiased - consistent - and normal regardless of homo or heterskedasticity - OLS estimates are efficient - Can use homoscedasticity - only variance formula - OLS is BLUE
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

30. Extending the HS approach for computing value of a portfolio

31. Importance sampling technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Attempts to sample along more important paths
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability that the random variables take on certain values simultaneously

32. Discrete representation of the GBM
Regression can be non - linear in variables but must be linear in parameters
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

33. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When one regressor is a perfect linear function of the other regressors
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

34. Econometrics
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Application of mathematical statistics to economic data to lend empirical support to models
Least absolute deviations estimator - used when extreme outliers are not uncommon
Normal - Student's T - Chi - square - F distribution

35. Priori (classical) probability
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Contains variables not explicit in model - Accounts for randomness
Based on an equation - P(A) = # of A/total outcomes
When the sample size is large - the uncertainty about the value of the sample is very small

36. Two assumptions of square root rule
Normal - Student's T - Chi - square - F distribution
Variance = (1/m) summation(u<n - i>^2)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Random walk (usually acceptable) - Constant volatility (unlikely)

37. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
95% = 1.65 99% = 2.33 For one - tailed tests
Among all unbiased estimators - estimator with the smallest variance is efficient
E(mean) = mean

38. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Confidence level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

39. Expected future variance rate (t periods forward)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

40. Mean reversion in variance
95% = 1.65 99% = 2.33 For one - tailed tests
Observe sample variance and compare it to hypothetical population variance (sample variance/population variance)(n - 1) = chi - squared - Non - negative and skewed right - approaches zero as n increases - mean = k where k = degrees of freedom - Varia
Transformed to a unit variable - Mean = 0 Variance = 1
Variance reverts to a long run level

41. Standard error
E(mean) = mean
(a^2)(variance(x)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
F(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared

42. Lognormal
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
In EWMA - the lambda parameter - In GARCH(1 -1) - sum of alpha and beta - Higher persistence implies slow decay toward the long - run average variance
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

43. P - value
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P(Z>t)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

44. Block maxima
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)
Among all unbiased estimators - estimator with the smallest variance is efficient
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

45. Variance of sampling distribution of means when n<N
Expected value of the sample mean is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Low Frequency - High Severity events

46. Conditional probability functions
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Summation((xi - mean)^k)/n

47. Simplified standard (un - weighted) variance
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance = (1/m) summation(u<n - i>^2)
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
Model dependent - Options with the same underlying assets may trade at different volatilities

48. R^2
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)
More than one random variable
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Expected value of the sample mean is the population mean

49. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Expected value of the sample mean is the population mean
Contains variables not explicit in model - Accounts for randomness

50. Poisson Distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Easy to manipulate