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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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Match each statement with the correct term.
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1. Exact significance level
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
P - value
Attempts to sample along more important paths
P(X=x - Y=y) = P(X=x) * P(Y=y)

2. Sample covariance
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
95% = 1.65 99% = 2.33 For one - tailed tests

3. Discrete random variable
Confidence level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Yi = B0 + B1Xi + ui
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

4. 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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
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

5. Implied standard deviation for options
SSR
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
Sampling distribution of sample means tend to be normal
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

6. Continuous random variable
Sample mean will near the population mean as the sample size increases
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
P - value

7. Key properties of linear regression
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Regression can be non - linear in variables but must be linear in parameters

8. Reliability
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Statement of the error or precision of an estimate
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

9. SER
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
Independently and Identically Distributed
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Easy to manipulate

10. Implications of homoscedasticity
Var(X) + Var(Y)
E(XY) - E(X)E(Y)
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
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)

11. Variance of aX + bY
E(mean) = mean
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
(a^2)(variance(x)) + (b^2)(variance(y))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

12. Test for statistical independence
(a^2)(variance(x)) + (b^2)(variance(y))
Choose parameters that maximize the likelihood of what observations occurring
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P(X=x - Y=y) = P(X=x) * P(Y=y)

13. Maximum likelihood method
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Choose parameters that maximize the likelihood of what observations occurring

14. Priori (classical) probability
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Based on an equation - P(A) = # of A/total outcomes
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

15. R^2
Choose parameters that maximize the likelihood of what observations occurring
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

16. LFHS
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
Mean of sampling distribution is the population mean
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
Low Frequency - High Severity events

17. Pooled data
P(X=x - Y=y) = P(X=x) * P(Y=y)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

18. Skewness
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
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)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

19. Mean reversion
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Normal - Student's T - Chi - square - F distribution

20. Least squares estimator(m)
95% = 1.65 99% = 2.33 For one - tailed tests
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

21. Type II Error
If variance of the conditional distribution of u(i) is not constant
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Distribution with only two possible outcomes
We accept a hypothesis that should have been rejected

22. Potential reasons for fat tails in return distributions
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Expected value of the sample mean is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

23. Panel data (longitudinal or micropanel)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Special type of pooled data in which the cross sectional unit is surveyed over time
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

24. Extending the HS approach for computing value of a portfolio

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25. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

26. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Population denominator = n - Sample denominator = n - 1
Z = (Y - meany)/(stddev(y)/sqrt(n))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

27. Homoskedastic only F - stat
Sample mean +/ - t*(stddev(s)/sqrt(n))
Easy to manipulate
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

28. Variance of X+Y
Var(X) + Var(Y)
Transformed to a unit variable - Mean = 0 Variance = 1
Z = (Y - meany)/(stddev(y)/sqrt(n))
Attempts to sample along more important paths

29. Lognormal
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance(x)
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)

30. Poisson distribution equations for mean variance and std deviation
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Normal - Student's T - Chi - square - F distribution

31. Bootstrap method
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

32. Mean(expected value)
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
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

33. Multivariate Density Estimation (MDE)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

34. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Rxy = Sxy/(Sx*Sy)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

35. GPD
Only requires two parameters = mean and variance
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

36. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

37. Stochastic error term
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Contains variables not explicit in model - Accounts for randomness

38. Economical(elegant)
Based on an equation - P(A) = # of A/total outcomes
For n>30 - sample mean is approximately normal
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
Only requires two parameters = mean and variance

39. T distribution
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
i = ln(Si/Si - 1)

40. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

41. Gamma distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

42. Cross - sectional
Average return across assets on a given day
Does not depend on a prior event or information
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance(y)/n = variance of sample Y

43. Consistent
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(X) + Variance(Y) - 2*covariance(XY)
When the sample size is large - the uncertainty about the value of the sample is very small
Probability that the random variables take on certain values simultaneously

44. Binomial distribution
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!)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Choose parameters that maximize the likelihood of what observations occurring

45. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sampling distribution of sample means tend to be normal
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

46. Mean reversion in asset dynamics
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
Summation((xi - mean)^k)/n
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

47. Two ways to calculate historical volatility
Variance reverts to a long run level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Concerned with a single random variable (ex. Roll of a die)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

48. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Expected value of the sample mean is the population mean
Rxy = Sxy/(Sx*Sy)

49. Simplified standard (un - weighted) variance
Only requires two parameters = mean and variance
Variance = (1/m) summation(u<n - i>^2)
Var(X) + Var(Y)
Price/return tends to run towards a long - run level

50. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Use historical simulation approach but use the EWMA weighting system
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Regression can be non - linear in variables but must be linear in parameters