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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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Answer 50 questions in 15 minutes.
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1. Critical z values
When the sample size is large - the uncertainty about the value of the sample is very small
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
95% = 1.65 99% = 2.33 For one - tailed tests

2. Non - parametric vs parametric calculation of VaR
Attempts to sample along more important paths
Transformed to a unit variable - Mean = 0 Variance = 1
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

3. Mean reversion in variance
Use historical simulation approach but use the EWMA weighting system
Variance reverts to a long run level
Least absolute deviations estimator - used when extreme outliers are not uncommon
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

4. Confidence interval for sample mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Confidence set for two coefficients - two dimensional analog for the confidence interval

5. Homoskedastic
Confidence level
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

6. Confidence interval (from t)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Has heavy tails

7. Homoskedastic only F - stat
Special type of pooled data in which the cross sectional unit is surveyed over time
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
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)

8. Joint probability functions
Probability that the random variables take on certain values simultaneously
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Average return across assets on a given day
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

9. 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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

10. Simulation models
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
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)
When one regressor is a perfect linear function of the other regressors

11. Variance of X+Y assuming dependence
Rxy = Sxy/(Sx*Sy)
Choose parameters that maximize the likelihood of what observations occurring
Variance(x) + Variance(Y) + 2*covariance(XY)
Expected value of the sample mean is the population mean

12. Importance sampling technique
Sampling distribution of sample means tend to be normal
Attempts to sample along more important paths
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

13. Key properties of linear regression
Variance reverts to a long run level
Regression can be non - linear in variables but must be linear in parameters
Combine to form distribution with leptokurtosis (heavy tails)
Peaks over threshold - Collects dataset in excess of some threshold

14. Mean(expected value)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Distribution with only two possible outcomes

15. POT
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Population denominator = n - Sample denominator = n - 1
Peaks over threshold - Collects dataset in excess of some threshold
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

16. Law of Large Numbers
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Sample mean will near the population mean as the sample size increases
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

17. Cholesky factorization (decomposition)
(a^2)(variance(x)
Does not depend on a prior event or information
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

18. Single variable (univariate) probability
Choose parameters that maximize the likelihood of what observations occurring
Summation((xi - mean)^k)/n
E(mean) = mean
Concerned with a single random variable (ex. Roll of a die)

19. F distribution
Returns over time for an individual asset
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Application of mathematical statistics to economic data to lend empirical support to models

20. Confidence ellipse
E(XY) - E(X)E(Y)
Variance(X) + Variance(Y) - 2*covariance(XY)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

21. Chi - squared 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
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

22. Time series data
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Nonlinearity
Returns over time for an individual asset

23. Econometrics
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Probability that the random variables take on certain values simultaneously
Application of mathematical statistics to economic data to lend empirical support to models

24. Adjusted R^2
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample mean will near the population mean as the sample size increases
Confidence level
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

25. Perfect multicollinearity
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Contains variables not explicit in model - Accounts for randomness
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
When one regressor is a perfect linear function of the other regressors

26. What does the OLS minimize?
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
SSR

27. Efficiency
Least absolute deviations estimator - used when extreme outliers are not uncommon
Rxy = Sxy/(Sx*Sy)
Among all unbiased estimators - estimator with the smallest variance is efficient
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

28. Reliability
Statement of the error or precision of an estimate
E(XY) - E(X)E(Y)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(X) + Variance(Y) - 2*covariance(XY)

29. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Use historical simulation approach but use the EWMA weighting system

30. GPD
Expected value of the sample mean is the population mean
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

31. Two assumptions of square root rule
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
Random walk (usually acceptable) - Constant volatility (unlikely)
Combine to form distribution with leptokurtosis (heavy tails)
Var(X) + Var(Y)

32. K - th moment
Probability that the random variables take on certain values simultaneously
Summation((xi - mean)^k)/n
Peaks over threshold - Collects dataset in excess of some threshold
Variance(x)

33. Limitations of R^2 (what an increase doesn't necessarily imply)

34. Sample correlation
Rxy = Sxy/(Sx*Sy)
Low Frequency - High Severity events
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

35. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
i = ln(Si/Si - 1)
For n>30 - sample mean is approximately normal
Z = (Y - meany)/(stddev(y)/sqrt(n))

36. Multivariate probability
P(X=x - Y=y) = P(X=x) * P(Y=y)
Probability that the random variables take on certain values simultaneously
More than one random variable
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

37. LFHS
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
Low Frequency - High Severity events
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

38. Least squares estimator(m)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(y)/n = variance of sample Y
Contains variables not explicit in model - Accounts for randomness

39. Discrete random variable
Nonlinearity
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

40. WLS
Among all unbiased estimators - estimator with the smallest variance is efficient
Mean = np - Variance = npq - Std dev = sqrt(npq)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Combine to form distribution with leptokurtosis (heavy tails)

41. Continuously compounded return equation
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
i = ln(Si/Si - 1)
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
Summation((xi - mean)^k)/n

42. Logistic distribution
Has heavy tails
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

43. Hybrid method for conditional volatility
Yi = B0 + B1Xi + ui
Use historical simulation approach but use the EWMA weighting system
Low Frequency - High Severity events
Distribution with only two possible outcomes

44. Block maxima
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

45. Mean reversion in asset dynamics
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
Peaks over threshold - Collects dataset in excess of some threshold
Price/return tends to run towards a long - run level
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

46. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
If variance of the conditional distribution of u(i) is not constant
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Rxy = Sxy/(Sx*Sy)

47. i.i.d.
P(Z>t)
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance = (1/m) summation(u<n - i>^2)
Independently and Identically Distributed

48. Bernouli Distribution
Distribution with only two possible outcomes
For n>30 - sample mean is approximately normal
Contains variables not explicit in model - Accounts for randomness
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

49. Sample mean
Expected value of the sample mean is the population mean
Based on an equation - P(A) = # of A/total outcomes
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

50. R^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
Special type of pooled data in which the cross sectional unit is surveyed over time
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Summation((xi - mean)^k)/n