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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. Discrete random variable
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(X) + Variance(Y) - 2*covariance(XY)
P(X=x - Y=y) = P(X=x) * P(Y=y)

2. Mean reversion
Independently and Identically Distributed
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Contains variables not explicit in model - Accounts for randomness
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

3. Covariance
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Probability that the random variables take on certain values simultaneously
Summation((xi - mean)^k)/n
E(XY) - E(X)E(Y)

4. Single variable (univariate) probability
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(XY) - E(X)E(Y)
Concerned with a single random variable (ex. Roll of a die)

5. Time series data
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Returns over time for an individual asset
Does not depend on a prior event or information

6. Reliability
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Statement of the error or precision of an estimate
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 reverts to a long run level

7. Binomial distribution
Only requires two parameters = mean and variance
i = ln(Si/Si - 1)
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!)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

8. Simulating for VaR
Nonlinearity
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

9. GEV
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Contains variables not explicit in model - Accounts for randomness
Yi = B0 + B1Xi + ui
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

10. BLUE
Variance(y)/n = variance of sample 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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

11. Statistical (or empirical) model
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Yi = B0 + B1Xi + ui
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
We reject a hypothesis that is actually true

12. Mean reversion in asset dynamics
(a^2)(variance(x)
Does not depend on a prior event or information
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Price/return tends to run towards a long - run level

13. Poisson Distribution
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
95% = 1.65 99% = 2.33 For one - tailed tests
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

14. Least squares estimator(m)
Among all unbiased estimators - estimator with the smallest variance is efficient
Random walk (usually acceptable) - Constant volatility (unlikely)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

15. Biggest (and only real) drawback of GARCH mode
We accept a hypothesis that should have been rejected
Nonlinearity
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Rxy = Sxy/(Sx*Sy)

16. 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)
Price/return tends to run towards a long - run level
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"

17. Normal distribution
Special type of pooled data in which the cross sectional unit is surveyed over time
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

18. Central Limit Theorem
Price/return tends to run towards a long - run level
Transformed to a unit variable - Mean = 0 Variance = 1
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
For n>30 - sample mean is approximately normal

19. WLS
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

20. Gamma distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Price/return tends to run towards a long - run level
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
Does not depend on a prior event or information

21. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance = (1/m) summation(u<n - i>^2)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Random walk (usually acceptable) - Constant volatility (unlikely)

22. Variance(discrete)
Sampling distribution of sample means tend to be normal
(a^2)(variance(x)
Probability that the random variables take on certain values simultaneously
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

23. Consistent
P(X=x - Y=y) = P(X=x) * P(Y=y)
Confidence set for two coefficients - two dimensional analog for the confidence interval
When the sample size is large - the uncertainty about the value of the sample is very small
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

24. Marginal unconditional probability function
For n>30 - sample mean is approximately normal
Does not depend on a prior event or information
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

25. Simplified standard (un - weighted) variance
Attempts to sample along more important paths
Variance = (1/m) summation(u<n - i>^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)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

26. F distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When one regressor is a perfect linear function of the other regressors
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

27. Historical std dev
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
We reject a hypothesis that is actually true
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

28. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sample mean will near the population mean as the sample size increases
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

29. GARCH
(a^2)(variance(x)) + (b^2)(variance(y))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Expected value of the sample mean is the population mean
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)

30. Direction of OVB
Variance reverts to a long run level
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(x)
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

31. Continuous 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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

32. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance reverts to a long run level

33. Control variates technique
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance = (1/m) summation(u<n - i>^2)

34. LAD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

35. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

36. Critical z values
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
95% = 1.65 99% = 2.33 For one - tailed tests

37. Simulation models
Population denominator = n - Sample denominator = n - 1
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
Variance(x)
Mean = np - Variance = npq - Std dev = sqrt(npq)

38. Overall F - statistic
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Price/return tends to run towards a long - run level
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Transformed to a unit variable - Mean = 0 Variance = 1

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

40. Persistence
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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

41. Central Limit Theorem(CLT)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sampling distribution of sample means tend to be normal

42. POT
Peaks over threshold - Collects dataset in excess of some threshold
Random walk (usually acceptable) - Constant volatility (unlikely)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

43. 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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Var(X) + Var(Y)
Choose parameters that maximize the likelihood of what observations occurring

44. Heteroskedastic
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
If variance of the conditional distribution of u(i) is not constant
Has heavy tails

45. Deterministic Simulation
Confidence level
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 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
Var(X) + Var(Y)

46. 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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(x) + Variance(Y) + 2*covariance(XY)

47. Lognormal
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean +/ - t*(stddev(s)/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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

48. Sample correlation
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Rxy = Sxy/(Sx*Sy)
We reject a hypothesis that is actually true
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

49. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Only requires two parameters = mean and variance
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

50. P - value
More than one random variable
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Statement of the error or precision of an estimate
P(Z>t)