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FRM Foundations Of Risk Management Quantitative Methods

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1. Two assumptions of square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Random walk (usually acceptable) - Constant volatility (unlikely)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

2. Key properties of linear regression
Sample mean will near the population mean as the sample size increases
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Confidence set for two coefficients - two dimensional analog for the confidence interval
Regression can be non - linear in variables but must be linear in parameters

3. F distribution
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia
Variance reverts to a long run level
i = ln(Si/Si - 1)
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

4. Poisson Distribution
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

5. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Independently and Identically Distributed
Returns over time for an individual asset
We reject a hypothesis that is actually true

6. Covariance calculations using weight sums (lambda)
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)
95% = 1.65 99% = 2.33 For one - tailed tests
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

7. Two requirements of OVB
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Special type of pooled data in which the cross sectional unit is surveyed over time
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sample mean +/ - t*(stddev(s)/sqrt(n))

8. Discrete representation of the GBM
Has heavy tails
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
i = ln(Si/Si - 1)

9. Direction of OVB
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

10. SER
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test

11. Beta distribution
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sample mean +/ - t*(stddev(s)/sqrt(n))
E(mean) = mean
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

12. Variance of aX
(a^2)(variance(x)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
E(XY) - E(X)E(Y)
For n>30 - sample mean is approximately normal

13. Variance of aX + bY
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
(a^2)(variance(x)) + (b^2)(variance(y))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

14. Variance of weighted scheme
95% = 1.65 99% = 2.33 For one - tailed tests
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Average return across assets on a given day

15. Multivariate probability
Sample mean +/ - t*(stddev(s)/sqrt(n))
P(X=x - Y=y) = P(X=x) * P(Y=y)
Has heavy tails
More than one random variable

16. Regime - switching volatility model
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!)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Nonlinearity

17. Homoskedastic
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(mean) = mean
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

18. Exact significance level
P - value
We reject a hypothesis that is actually true
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

19. Perfect multicollinearity
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Refers to whether distribution is symmetrical - Sigma^3 = E[(x - mean)^3]/sigma^3 - Positive skew = mean>median>mode - Negative skew = mean<median<mode - if zero - all are equal - Function of the third moment
We accept a hypothesis that should have been rejected
When one regressor is a perfect linear function of the other regressors

20. Square root rule
(a^2)(variance(x)
Variance(y)/n = variance of sample Y
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Among all unbiased estimators - estimator with the smallest variance is efficient

21. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Combine to form distribution with leptokurtosis (heavy tails)
Summation((xi - mean)^k)/n
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

22. Two ways to calculate historical volatility
P(Z>t)
Returns over time for an individual asset
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

23. Multivariate Density Estimation (MDE)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
More than one random variable
Average return across assets on a given day

24. Cross - sectional
More than one random variable
Average return across assets on a given day
Variance(y)/n = variance of sample Y
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

25. Persistence
Average return across assets on a given day
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia
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
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

26. POT
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Rxy = Sxy/(Sx*Sy)
Peaks over threshold - Collects dataset in excess of some threshold
Use historical simulation approach but use the EWMA weighting system

27. Standard error
Combine to form distribution with leptokurtosis (heavy tails)
Variance = (1/m) summation(u<n - i>^2)
SSR
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

28. Variance of X+b
Sample mean will near the population mean as the sample size increases
Variance(x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

29. EWMA
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Var(X) + Var(Y)
Summation((xi - mean)^k)/n

30. Antithetic variable technique
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)
Variance = (1/m) summation(u<n - i>^2)
Average return across assets on a given day

31. Sample correlation
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test
Rxy = Sxy/(Sx*Sy)

32. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Does not depend on a prior event or information

33. Time series data
Expected value of the sample mean is the population mean
Concerned with a single random variable (ex. Roll of a die)
Returns over time for an individual asset
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)

34. Variance of sample mean
(a^2)(variance(x)
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(y)/n = variance of sample Y
Low Frequency - High Severity events

35. Sample covariance
95% = 1.65 99% = 2.33 For one - tailed tests
For n>30 - sample mean is approximately normal
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
If variance of the conditional distribution of u(i) is not constant

36. Priori (classical) probability
Variance(y)/n = variance of sample Y
Sample mean +/ - t*(stddev(s)/sqrt(n))
Based on an equation - P(A) = # of A/total outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

37. WLS
Easy to manipulate
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
P(X=x - Y=y) = P(X=x) * P(Y=y)

38. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
P - value
Probability that the random variables take on certain values simultaneously
Sample mean +/ - t*(stddev(s)/sqrt(n))

39. Variance(discrete)
P(X=x - Y=y) = P(X=x) * P(Y=y)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Application of mathematical statistics to economic data to lend empirical support to models

40. Extreme Value Theory
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)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Only requires two parameters = mean and variance
Var(X) + Var(Y)

41. Unbiased
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Mean of sampling distribution is the population mean
Least absolute deviations estimator - used when extreme outliers are not uncommon
Average return across assets on a given day

42. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Returns over time for an individual asset
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Population denominator = n - Sample denominator = n - 1

43. Result of combination of two normal with same means
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Normal - Student's T - Chi - square - F distribution
Returns over time for an individual asset
Combine to form distribution with leptokurtosis (heavy tails)

44. Four sampling distributions

45. Confidence interval (from t)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sample mean +/ - t*(stddev(s)/sqrt(n))
Z = (Y - meany)/(stddev(y)/sqrt(n))

46. Inverse transform method
Sample mean +/ - t*(stddev(s)/sqrt(n))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Peaks over threshold - Collects dataset in excess of some threshold

47. Simulating for VaR
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Mean of sampling distribution is the population mean
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

48. Mean reversion in asset dynamics
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Price/return tends to run towards a long - run level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

49. Type I error
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
We reject a hypothesis that is actually true
P(X=x - Y=y) = P(X=x) * P(Y=y)

50. Empirical frequency
Based on a dataset
95% = 1.65 99% = 2.33 For one - tailed tests
Sample mean will near the population mean as the sample size increases
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

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FRM Foundations Of Risk Management Quantitative Methods

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