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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. Stochastic error term
E(mean) = mean
Contains variables not explicit in model - Accounts for randomness
P - value
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

2. Tractable
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Easy to manipulate
We accept a hypothesis that should have been rejected

3. Square root rule
(a^2)(variance(x)) + (b^2)(variance(y))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

4. Multivariate probability
More than one random variable
Variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities
Mean of sampling distribution is the population mean

5. Non - parametric vs parametric calculation of VaR
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

6. R^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Summation((xi - mean)^k)/n
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

7. Regime - switching volatility model
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

8. Continuous representation of the GBM
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Random walk (usually acceptable) - Constant volatility (unlikely)
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)
Contains variables not explicit in model - Accounts for randomness

9. WLS
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample mean will near the population mean as the sample size increases
Confidence level

10. Hybrid method for conditional volatility
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Use historical simulation approach but use the EWMA weighting system
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

11. Marginal unconditional probability function
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
More than one random variable

12. Variance of X+Y assuming dependence
Only requires two parameters = mean and variance
Variance(x) + Variance(Y) + 2*covariance(XY)
Confidence level
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

13. Adjusted R^2
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
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Returns over time for a combination of assets (combination of time series and cross - sectional data)

14. 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)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

15. Mean(expected value)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance(X) + Variance(Y) - 2*covariance(XY)
Among all unbiased estimators - estimator with the smallest variance is efficient
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

16. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
i = ln(Si/Si - 1)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

17. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Special type of pooled data in which the cross sectional unit is surveyed over time
Combine to form distribution with leptokurtosis (heavy tails)

18. Pooled data
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
P(Z>t)

19. Covariance
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
E(XY) - E(X)E(Y)
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

20. Joint probability functions
Confidence set for two coefficients - two dimensional analog for the confidence interval
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Probability that the random variables take on certain values simultaneously
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

21. Binomial distribution
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Distribution with only two possible outcomes
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!)
(a^2)(variance(x)) + (b^2)(variance(y))

22. Bernouli Distribution
Variance(y)/n = variance of sample Y
Distribution with only two possible outcomes
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)

23. Key properties of linear regression
Peaks over threshold - Collects dataset in excess of some threshold
Transformed to a unit variable - Mean = 0 Variance = 1
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Regression can be non - linear in variables but must be linear in parameters

24. Importance sampling technique
Attempts to sample along more important paths
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Peaks over threshold - Collects dataset in excess of some threshold
Sample mean will near the population mean as the sample size increases

25. Implied standard deviation for options
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
(a^2)(variance(x)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(y)/n = variance of sample Y

26. Kurtosis
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Has heavy tails
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

27. Lognormal
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Mean of sampling distribution is the population 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
Among all unbiased estimators - estimator with the smallest variance is efficient

28. Extreme Value Theory
P - value
Model dependent - Options with the same underlying assets may trade at different volatilities
Sampling distribution of sample means tend to be normal
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

29. What does the OLS minimize?
Price/return tends to run towards a long - run level
SSR
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

30. Standard error
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

31. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Yi = B0 + B1Xi + ui
P - value
Distribution with only two possible outcomes

32. GEV
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Sample mean will near the population mean as the sample size increases

33. Exact significance level
P - value
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
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
Easy to manipulate

34. Normal distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance reverts to a long run level
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

35. Variance - covariance approach for VaR of a portfolio
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Based on a dataset
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

36. Priori (classical) probability
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
Distribution with only two possible outcomes
Based on an equation - P(A) = # of A/total outcomes
Returns over time for a combination of assets (combination of time series and cross - sectional data)

37. Two drawbacks of moving average series
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

38. LFHS
Low Frequency - High Severity events
Distribution with only two possible outcomes
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
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

39. Direction of OVB
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

40. Sample mean
Expected value of the sample mean is the population mean
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

41. ESS
Confidence set for two coefficients - two dimensional analog for the confidence interval
SSR
Model dependent - Options with the same underlying assets may trade at different volatilities
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

42. Reliability
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Statement of the error or precision of an estimate
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

43. Persistence
E(mean) = 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
Sample mean will near the population mean as the sample size increases
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

44. Inverse transform method
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
P - value

45. Continuous random variable
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Regression can be non - linear in variables but must be linear in parameters
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

46. Simulation models
Expected value of the sample mean is the population mean
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Only requires two parameters = mean and variance

47. Biggest (and only real) drawback of GARCH mode
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
Nonlinearity
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

48. Type I error
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
We reject a hypothesis that is actually true
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Mean = np - Variance = npq - Std dev = sqrt(npq)

49. Variance(discrete)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Rxy = Sxy/(Sx*Sy)
Low Frequency - High Severity events
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

50. Variance of sample mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
If variance of the conditional distribution of u(i) is not constant
Variance = (1/m) summation(u<n - i>^2)
Variance(y)/n = variance of sample Y