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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Variance - covariance approach for VaR of a portfolio
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

2. WLS
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

3. Mean reversion
Variance(X) + Variance(Y) - 2*covariance(XY)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

4. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Returns over time for an individual asset

5. Continuous random variable
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

6. Poisson Distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

7. F distribution
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
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)
Variance = (1/m) summation(u<n - i>^2)
Choose parameters that maximize the likelihood of what observations occurring

8. Mean reversion in asset dynamics
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance = (1/m) summation(u<n - i>^2)
Price/return tends to run towards a long - run level

9. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Distribution with only two possible outcomes
Z = (Y - meany)/(stddev(y)/sqrt(n))

10. Type II Error
We accept a hypothesis that should have been rejected
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

11. SER
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance(y)/n = variance of sample Y
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

12. Efficiency
Var(X) + Var(Y)
Among all unbiased estimators - estimator with the smallest variance is efficient
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
More than one random variable

13. LFHS
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Var(X) + Var(Y)
Low Frequency - High Severity events

14. Block maxima
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Mean = np - Variance = npq - Std dev = sqrt(npq)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

15. Perfect multicollinearity
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
Population denominator = n - Sample denominator = n - 1
When one regressor is a perfect linear function of the other regressors
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

16. Importance sampling technique
Sample mean +/ - t*(stddev(s)/sqrt(n))
Attempts to sample along more important paths
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)
Combine to form distribution with leptokurtosis (heavy tails)

17. GPD
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Use historical simulation approach but use the EWMA weighting system

18. Exact significance level
Normal - Student's T - Chi - square - F distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
P - value
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

19. Kurtosis
Among all unbiased estimators - estimator with the smallest variance is efficient
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))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

20. Marginal unconditional probability function
Does not depend on a prior event or information
Mean = np - Variance = npq - Std dev = sqrt(npq)
Var(X) + Var(Y)
Price/return tends to run towards a long - run level

21. Covariance
Easy to manipulate
E(XY) - E(X)E(Y)
95% = 1.65 99% = 2.33 For one - tailed tests
Transformed to a unit variable - Mean = 0 Variance = 1

22. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Price/return tends to run towards a long - run level
E(XY) - E(X)E(Y)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

23. Two drawbacks of moving average series
Only requires two parameters = mean and variance
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
SSR

24. Test for statistical independence
Nonlinearity
P(X=x - Y=y) = P(X=x) * P(Y=y)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

25. LAD
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Least absolute deviations estimator - used when extreme outliers are not uncommon
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

26. Variance of X+Y
Contains variables not explicit in model - Accounts for randomness
Var(X) + Var(Y)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

27. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Variance = (1/m) summation(u<n - i>^2)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

28. Poisson distribution equations for mean variance and std deviation
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

29. Sample mean
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)
Mean of sampling distribution is the population mean
Expected value of the sample mean is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

30. Variance of X+Y assuming dependence
Adjusted R^2 does not increase from addition of new independent variables -Adjusted R^2 = 1 - (n - 1)/(n - k - 1) * (SSR/TSS) = 1 - su^2/sy^2
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(x) + Variance(Y) + 2*covariance(XY)

31. Unconditional vs conditional distributions
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance reverts to a long run level
Variance(y)/n = variance of sample Y

32. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Average return across assets on a given day
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Returns over time for an individual asset

33. Single variable (univariate) probability
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
Concerned with a single random variable (ex. Roll of a die)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Has heavy tails

34. Multivariate Density Estimation (MDE)
Normal - Student's T - Chi - square - F distribution
Independently and Identically Distributed
Sample mean will near the population mean as the sample size increases
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

35. Hazard rate of exponentially distributed random variable
Independently and Identically Distributed
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Z = (Y - meany)/(stddev(y)/sqrt(n))
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

36. Tractable
E(XY) - E(X)E(Y)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Easy to manipulate
Variance(y)/n = variance of sample Y

37. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

38. Standard error
We accept a hypothesis that should have been rejected
Variance(y)/n = variance of sample Y
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

39. Empirical frequency
Probability that the random variables take on certain values simultaneously
Expected value of the sample mean is the population mean
Based on a dataset
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

40. Priori (classical) probability
Confidence set for two coefficients - two dimensional analog for the confidence interval
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Based on an equation - P(A) = # of A/total outcomes
Variance(X) + Variance(Y) - 2*covariance(XY)

41. Conditional probability functions
Expected value of the sample mean is the population mean
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
Choose parameters that maximize the likelihood of what observations occurring
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

42. Homoskedastic
Low Frequency - High Severity events
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

43. Variance of X - Y assuming dependence
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Independently and Identically Distributed
Variance(X) + Variance(Y) - 2*covariance(XY)

44. Simplified standard (un - weighted) variance
P - value
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
Variance = (1/m) summation(u<n - i>^2)
When the sample size is large - the uncertainty about the value of the sample is very small

45. Binomial distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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!)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Based on an equation - P(A) = # of A/total outcomes

46. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
(a^2)(variance(x)) + (b^2)(variance(y))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

47. Persistence
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
P(Z>t)
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

48. Simulating for VaR
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Does not depend on a prior event or information
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

49. Multivariate probability
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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!)
Z = (Y - meany)/(stddev(y)/sqrt(n))
More than one random variable

50. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Regression can be non - linear in variables but must be linear in parameters
We reject a hypothesis that is actually true
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)