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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. Adjusted R^2
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
(a^2)(variance(x)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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

2. Importance sampling technique
Mean = np - Variance = npq - Std dev = sqrt(npq)
Attempts to sample along more important paths
Expected value of the sample mean is the population mean
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

3. T distribution
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Among all unbiased estimators - estimator with the smallest variance is efficient
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

4. Antithetic variable technique
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(x) + Variance(Y) + 2*covariance(XY)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Sampling distribution of sample means tend to be normal

5. Standard variable for non - normal distributions
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Z = (Y - meany)/(stddev(y)/sqrt(n))
Rxy = Sxy/(Sx*Sy)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

6. Kurtosis
Returns over time for an individual asset
Variance(x) + Variance(Y) + 2*covariance(XY)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

7. SER
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

8. Implied standard deviation for options
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)
Use historical simulation approach but use the EWMA weighting system
Variance reverts to a long run level

9. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Peaks over threshold - Collects dataset in excess of some threshold
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

10. Mean reversion in asset dynamics
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Price/return tends to run towards a long - run level
Combine to form distribution with leptokurtosis (heavy tails)
Mean of sampling distribution is the population mean

11. Two assumptions of square root rule
Independently and Identically Distributed
Random walk (usually acceptable) - Constant volatility (unlikely)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
E(mean) = mean

12. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
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
Confidence level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

13. Unstable return distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

14. Exponential distribution
Peaks over threshold - Collects dataset in excess of some threshold
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Special type of pooled data in which the cross sectional unit is surveyed over time

15. Critical z values
i = ln(Si/Si - 1)
Normal - Student's T - Chi - square - F distribution
95% = 1.65 99% = 2.33 For one - tailed tests
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

16. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Regression can be non - linear in variables but must be linear in parameters
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

17. Variance(discrete)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

18. Central Limit Theorem
Normal - Student's T - Chi - square - F distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
For n>30 - sample mean is approximately normal
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

19. Economical(elegant)
Variance = (1/m) summation(u<n - i>^2)
(a^2)(variance(x)
Only requires two parameters = mean and variance
(a^2)(variance(x)) + (b^2)(variance(y))

20. F distribution
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
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
Variance = (1/m) summation(u<n - i>^2)
We accept a hypothesis that should have been rejected

21. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

22. 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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

23. Binomial distribution equations for mean variance and std dev
95% = 1.65 99% = 2.33 For one - tailed tests
When one regressor is a perfect linear function of the other regressors
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Mean = np - Variance = npq - Std dev = sqrt(npq)

24. Cross - sectional
Sample mean will near the population mean as the sample size increases
Yi = B0 + B1Xi + ui
Average return across assets on a given day
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

25. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

26. Poisson Distribution
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

27. Central Limit Theorem(CLT)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sampling distribution of sample means tend to be normal
Only requires two parameters = mean and variance

28. Lognormal
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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!)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

29. P - value
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
Contains variables not explicit in model - Accounts for randomness
P(Z>t)
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

30. Continuous representation of the GBM
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)
Low Frequency - High Severity events
Expected value of the sample mean is the population mean
Variance reverts to a long run level

31. Variance of sampling distribution of means when n<N
Nonlinearity
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Transformed to a unit variable - Mean = 0 Variance = 1

32. Discrete representation of the GBM
Sample mean will near the population mean as the sample size increases
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance = (1/m) summation(u<n - i>^2)

33. R^2
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

34. Chi - squared distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Special type of pooled data in which the cross sectional unit is surveyed over time

35. Variance - covariance approach for VaR of a portfolio
More than one random variable
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
We reject a hypothesis that is actually true

36. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Expected value of the sample mean is the population mean

37. Shortcomings of implied volatility
(a^2)(variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

38. Confidence interval for sample mean
Does not depend on a prior event or information
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

39. Expected future variance rate (t periods forward)
Easy to manipulate
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

40. Variance of aX + bY
Easy to manipulate
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
(a^2)(variance(x)) + (b^2)(variance(y))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

41. Sample correlation
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Rxy = Sxy/(Sx*Sy)
Use historical simulation approach but use the EWMA weighting system

42. Perfect multicollinearity
i = ln(Si/Si - 1)
P(Z>t)
When one regressor is a perfect linear function of the other regressors
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

43. Confidence interval (from t)
Probability that the random variables take on certain values simultaneously
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample mean +/ - t*(stddev(s)/sqrt(n))
For n>30 - sample mean is approximately normal

44. 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)
Based on an equation - P(A) = # of A/total outcomes
Least absolute deviations estimator - used when extreme outliers are not uncommon
P(X=x - Y=y) = P(X=x) * P(Y=y)

45. Bootstrap method
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
i = ln(Si/Si - 1)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

46. GPD
Only requires two parameters = mean and variance
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 Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

47. Covariance
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E(XY) - E(X)E(Y)

48. EWMA
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Yi = B0 + B1Xi + ui
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Expected value of the sample mean is the population mean

49. Confidence ellipse
Only requires two parameters = mean and variance
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
Random walk (usually acceptable) - Constant volatility (unlikely)

50. Sample variance
Variance(x)
i = ln(Si/Si - 1)
95% = 1.65 99% = 2.33 For one - tailed tests
Sample variance = (1/(k - 1))Summation(Yi - mean)^2