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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. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Sample mean will near the population mean as the sample size increases
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

2. Economical(elegant)
Only requires two parameters = mean and variance
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Confidence set for two coefficients - two dimensional analog for the confidence interval

3. SER
E(XY) - E(X)E(Y)
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
i = ln(Si/Si - 1)

4. What does the OLS minimize?
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
SSR
(a^2)(variance(x)

5. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(mean) = mean
Use historical simulation approach but use the EWMA weighting system
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

6. Unbiased
Summation((xi - mean)^k)/n
95% = 1.65 99% = 2.33 For one - tailed tests
When the sample size is large - the uncertainty about the value of the sample is very small
Mean of sampling distribution is the population mean

7. Hybrid method for conditional volatility
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Use historical simulation approach but use the EWMA weighting system
Probability that the random variables take on certain values simultaneously

8. Time series data
Regression can be non - linear in variables but must be linear in parameters
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Returns over time for an individual asset
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

9. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

10. Adjusted R^2
Based on a dataset
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Independently and Identically Distributed
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

11. Pooled data
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

12. Mean reversion in variance
(a^2)(variance(x)) + (b^2)(variance(y))
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
Variance reverts to a long run level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

13. Continuous representation of the GBM
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

14. Tractable
We reject a hypothesis that is actually true
Easy to manipulate
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Use historical simulation approach but use the EWMA weighting system

15. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Random walk (usually acceptable) - Constant volatility (unlikely)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Peaks over threshold - Collects dataset in excess of some threshold

16. Econometrics
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
E(mean) = mean
Application of mathematical statistics to economic data to lend empirical support to models

17. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
(a^2)(variance(x)
Easy to manipulate
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

18. Perfect multicollinearity
P - value
Distribution with only two possible outcomes
When one regressor is a perfect linear function of the other regressors
Only requires two parameters = mean and variance

19. Variance - covariance approach for VaR of a portfolio
Only requires two parameters = mean and variance
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

20. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Concerned with a single random variable (ex. Roll of a die)

21. Normal distribution
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
Average return across assets on a given day
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

22. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

23. Type I error
E(mean) = mean
We reject a hypothesis that is actually true
Probability that the random variables take on certain values simultaneously
Peaks over threshold - Collects dataset in excess of some threshold

24. Cholesky factorization (decomposition)
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
Based on an equation - P(A) = # of A/total outcomes
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
Returns over time for an individual asset

25. Variance of X - Y assuming dependence
Low Frequency - High Severity events
Based on a dataset
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(X) + Variance(Y) - 2*covariance(XY)

26. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Price/return tends to run towards a long - run level
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Transformed to a unit variable - Mean = 0 Variance = 1

27. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Random walk (usually acceptable) - Constant volatility (unlikely)

28. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

29. Covariance calculations using weight sums (lambda)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Population denominator = n - Sample denominator = n - 1
Summation((xi - mean)^k)/n

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

31. Poisson Distribution
Application of mathematical statistics to economic data to lend empirical support to models
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

32. Standard error for Monte Carlo replications
Nonlinearity
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

33. Four sampling distributions

34. Discrete random variable
Independently and Identically Distributed
Variance = (1/m) summation(u<n - i>^2)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

35. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
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

36. P - value
P(Z>t)
Random walk (usually acceptable) - Constant volatility (unlikely)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

37. Skewness
Peaks over threshold - Collects dataset in excess of some threshold
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
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
Low Frequency - High Severity events

38. Continuous random variable
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
Random walk (usually acceptable) - Constant volatility (unlikely)
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

39. Gamma distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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

40. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Variance(x)
Among all unbiased estimators - estimator with the smallest variance is efficient
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

41. Logistic distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Confidence level
Has heavy tails
Use historical simulation approach but use the EWMA weighting system

42. Confidence interval for 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)
Special type of pooled data in which the cross sectional unit is surveyed over time
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

43. Monte Carlo Simulations
Combine to form distribution with leptokurtosis (heavy tails)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Peaks over threshold - Collects dataset in excess of some threshold

44. GPD
Contains variables not explicit in model - Accounts for randomness
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
i = ln(Si/Si - 1)

45. Hazard rate of exponentially distributed random variable
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
Application of mathematical statistics to economic data to lend empirical support to models
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

46. Mean reversion in asset dynamics
Sampling distribution of sample means tend to be normal
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Low Frequency - High Severity events
Price/return tends to run towards a long - run level

47. Marginal unconditional probability function
When one regressor is a perfect linear function of the other regressors
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Does not depend on a prior event or information
Rxy = Sxy/(Sx*Sy)

48. Extreme Value Theory
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Price/return tends to run towards a long - run level
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

49. Poisson distribution equations for mean variance and std deviation
P(Z>t)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Confidence level
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

50. Continuously compounded return equation
Based on an equation - P(A) = # of A/total outcomes
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
i = ln(Si/Si - 1)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail