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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. Bernouli Distribution
Concerned with a single random variable (ex. Roll of a die)
Sample mean +/ - t*(stddev(s)/sqrt(n))
When the sample size is large - the uncertainty about the value of the sample is very small
Distribution with only two possible outcomes

2. Two assumptions of square root rule
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Random walk (usually acceptable) - Constant volatility (unlikely)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

3. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E(XY) - E(X)E(Y)

4. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Yi = B0 + B1Xi + ui
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Choose parameters that maximize the likelihood of what observations occurring

5. Discrete representation of the GBM
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
(a^2)(variance(x)
We accept a hypothesis that should have been rejected
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

6. Implications of homoscedasticity
Normal - Student's T - Chi - square - F distribution
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(x)
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

7. Sample correlation
Rxy = Sxy/(Sx*Sy)
If variance of the conditional distribution of u(i) is not constant
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

8. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

9. Cholesky factorization (decomposition)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
We reject a hypothesis that is actually true
Random walk (usually acceptable) - Constant volatility (unlikely)
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

10. Continuously compounded return equation
i = ln(Si/Si - 1)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
Mean of sampling distribution is the population mean

11. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

12. Shortcomings of implied volatility
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Nonlinearity
Model dependent - Options with the same underlying assets may trade at different volatilities
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

13. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
When one regressor is a perfect linear function of the other regressors
i = ln(Si/Si - 1)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

14. K - th moment
F(x) = (1/stddev(x)sqrt(2pi))e^ - (x - mean)^2/(2variance) - skew = 0 - Parsimony = only requires mean and variance - Summation stability = combination of two normal distributions is a normal distribution - Kurtosis = 3
Sampling distribution of sample means tend to be normal
Summation((xi - mean)^k)/n
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

15. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
i = ln(Si/Si - 1)
Sampling distribution of sample means tend to be normal

16. Lognormal
Population denominator = n - Sample denominator = n - 1
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
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

17. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Based on a dataset
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We reject a hypothesis that is actually true

18. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sampling distribution of sample means tend to be normal
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

19. GARCH
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)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Distribution with only two possible outcomes

20. 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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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 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

21. Four sampling distributions

22. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Summation((xi - mean)^k)/n

23. Efficiency
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
If variance of the conditional distribution of u(i) is not constant
Among all unbiased estimators - estimator with the smallest variance is efficient

24. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
For n>30 - sample mean is approximately normal
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

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

26. Variance of aX
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Does not depend on a prior event or information
(a^2)(variance(x)

27. Sample variance
Sample mean will near the population mean as the sample size increases
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
E(mean) = mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

28. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Low Frequency - High Severity events
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

29. Gamma distribution
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
Statement of the error or precision of an estimate
E(XY) - E(X)E(Y)

30. Tractable
Easy to manipulate
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Summation((xi - mean)^k)/n
Variance(x) + Variance(Y) + 2*covariance(XY)

31. Normal distribution
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
F(x) = (1/stddev(x)sqrt(2pi))e^ - (x - mean)^2/(2variance) - skew = 0 - Parsimony = only requires mean and variance - Summation stability = combination of two normal distributions is a normal distribution - Kurtosis = 3
When one regressor is a perfect linear function of the other regressors
When the sample size is large - the uncertainty about the value of the sample is very small

32. Maximum likelihood method
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)
Choose parameters that maximize the likelihood of what observations occurring
Special type of pooled data in which the cross sectional unit is surveyed over time
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

33. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

34. Test for unbiasedness
E(mean) = 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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

35. Central Limit Theorem(CLT)
Confidence level
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Sampling distribution of sample means tend to be normal

36. Exact significance level
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
P - value
Distribution with only two possible outcomes
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

37. Homoskedastic only F - stat
Only requires two parameters = mean and variance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Attempts to sample along more important paths
Based on an equation - P(A) = # of A/total outcomes

38. LFHS
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Low Frequency - High Severity events
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

39. Two requirements of OVB
Summation((xi - mean)^k)/n
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Var(X) + Var(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

40. Significance =1
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Choose parameters that maximize the likelihood of what observations occurring
Confidence level

41. POT
Has heavy tails
Among all unbiased estimators - estimator with the smallest variance is efficient
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Peaks over threshold - Collects dataset in excess of some threshold

42. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

43. GPD
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
Confidence level
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

44. Reliability
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
Statement of the error or precision of an estimate
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
When one regressor is a perfect linear function of the other regressors

45. Multivariate Density Estimation (MDE)
Price/return tends to run towards a long - run level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

46. Variance of weighted scheme
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
(a^2)(variance(x)) + (b^2)(variance(y))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

47. Implied standard deviation for options
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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)

48. Multivariate probability
E(mean) = mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
More than one random variable
Peaks over threshold - Collects dataset in excess of some threshold

49. Standard error
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

50. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Application of mathematical statistics to economic data to lend empirical support to models
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Among all unbiased estimators - estimator with the smallest variance is efficient