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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. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Among all unbiased estimators - estimator with the smallest variance is efficient
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

2. Cholesky factorization (decomposition)
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
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
Does not depend on a prior event or information
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

3. Variance of X+Y assuming dependence
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Variance(y)/n = variance of sample Y
Variance(x) + Variance(Y) + 2*covariance(XY)

4. Single variable (univariate) probability
Choose parameters that maximize the likelihood of what observations occurring
Concerned with a single random variable (ex. Roll of a die)
Distribution with only two possible outcomes
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

5. Implications of homoscedasticity
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Attempts to sample along more important paths
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

6. Joint probability functions
Based on a dataset
Probability that the random variables take on certain values simultaneously
Independently and Identically Distributed
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

7. Confidence interval (from t)
Distribution with only two possible outcomes
Sampling distribution of sample means tend to be normal
Sample mean +/ - t*(stddev(s)/sqrt(n))
Yi = B0 + B1Xi + ui

8. Four sampling distributions

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

10. Importance sampling technique
Attempts to sample along more important paths
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Application of mathematical statistics to economic data to lend empirical support to models

11. Type I error
We reject a hypothesis that is actually true
Easy to manipulate
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

12. ESS
Sampling distribution of sample means tend to be normal
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

13. R^2
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Statement of the error or precision of an estimate
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

14. Consistent
Model dependent - Options with the same underlying assets may trade at different volatilities
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
When the sample size is large - the uncertainty about the value of the sample is very small
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

15. Variance of weighted scheme
(a^2)(variance(x)
Population denominator = n - Sample denominator = n - 1
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

16. Lognormal
Application of mathematical statistics to economic data to lend empirical support to models
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(y)/n = variance of sample Y
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

17. Bootstrap method
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Population denominator = n - Sample denominator = n - 1
If variance of the conditional distribution of u(i) is not constant

18. Maximum likelihood method
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
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)
Choose parameters that maximize the likelihood of what observations occurring

19. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Price/return tends to run towards a long - run level
Sample mean +/ - t*(stddev(s)/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

20. GARCH
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Returns over time for an individual asset
Summation((xi - mean)^k)/n
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)

21. Confidence interval for sample mean
Statement of the error or precision of an estimate
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
i = ln(Si/Si - 1)

22. GPD
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Population denominator = n - Sample denominator = n - 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

23. P - value
P(Z>t)
Nonlinearity
Normal - Student's T - Chi - square - F distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

24. Deterministic Simulation
When one regressor is a perfect linear function of the other regressors
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Statement of the error or precision of an estimate
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

25. Statistical (or empirical) model
Mean of sampling distribution is the population mean
Based on an equation - P(A) = # of A/total outcomes
Yi = B0 + B1Xi + ui
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

26. Variance - covariance approach for VaR of a portfolio
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

27. Variance of aX + bY
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

28. Multivariate Density Estimation (MDE)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
(a^2)(variance(x)) + (b^2)(variance(y))

29. Unstable return distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Special type of pooled data in which the cross sectional unit is surveyed over time
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

30. GEV
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance = (1/m) summation(u<n - i>^2)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

31. Cross - sectional
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!)
Average return across assets on a given day
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

32. Standard error
Price/return tends to run towards a long - run level
Average return across assets on a given day
Expected value of the sample mean is the population mean
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

33. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
We accept a hypothesis that should have been rejected
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

34. Marginal unconditional probability function
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Does not depend on a prior event or information
95% = 1.65 99% = 2.33 For one - tailed tests
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

35. Variance of X+b
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(x)
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
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)

36. Law of Large Numbers
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample mean will near the population mean as the sample size increases
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

37. i.i.d.
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Independently and Identically Distributed

38. Variance of sample mean
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance(y)/n = variance of sample Y
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

39. Key properties of linear regression
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Regression can be non - linear in variables but must be linear in parameters
We accept a hypothesis that should have been rejected
Sampling distribution of sample means tend to be normal

40. 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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

41. SER
E(XY) - E(X)E(Y)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Price/return tends to run towards a long - run level
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

42. Non - parametric vs parametric calculation of VaR
Among all unbiased estimators - estimator with the smallest variance is efficient
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Confidence level
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

43. Sample covariance
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Yi = B0 + B1Xi + ui
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
P - value

44. Homoskedastic only F - stat
If variance of the conditional distribution of u(i) is not constant
Attempts to sample along more important paths
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sampling distribution of sample means tend to be normal

45. POT
Variance(X) + Variance(Y) - 2*covariance(XY)
Peaks over threshold - Collects dataset in excess of some threshold
Returns over time for an individual asset
If variance of the conditional distribution of u(i) is not constant

46. Expected future variance rate (t periods forward)
95% = 1.65 99% = 2.33 For one - tailed tests
P(Z>t)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When one regressor is a perfect linear function of the other regressors

47. Weibul distribution
Concerned with a single random variable (ex. Roll of a die)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Among all unbiased estimators - estimator with the smallest variance is efficient
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)

48. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance(X) + Variance(Y) - 2*covariance(XY)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

49. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
When one regressor is a perfect linear function of the other regressors
E(mean) = mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

50. BLUE
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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance reverts to a long run level