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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. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
When one regressor is a perfect linear function of the other regressors
Sampling distribution of sample means tend to be normal

2. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Var(X) + Var(Y)
P - value

3. Central Limit Theorem
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
For n>30 - sample mean is approximately normal
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

4. Exponential distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

5. Extreme Value Theory
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
For n>30 - sample mean is approximately normal
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

6. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Yi = B0 + B1Xi + ui
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(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

7. Unconditional vs conditional distributions
P(X=x - Y=y) = P(X=x) * P(Y=y)
Nonlinearity
Transformed to a unit variable - Mean = 0 Variance = 1
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

8. 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
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!)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

9. Covariance calculations using weight sums (lambda)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Nonlinearity

10. Weibul distribution
Mean of sampling distribution is the population mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

11. Heteroskedastic
Only requires two parameters = mean and variance
If variance of the conditional distribution of u(i) is not constant
Population denominator = n - Sample denominator = n - 1
Mean = np - Variance = npq - Std dev = sqrt(npq)

12. Variance(discrete)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

13. Empirical frequency
Based on a dataset
Normal - Student's T - Chi - square - F distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

14. Logistic distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
Price/return tends to run towards a long - run level
P(X=x - Y=y) = P(X=x) * P(Y=y)
Has heavy tails

15. Economical(elegant)
Choose parameters that maximize the likelihood of what observations occurring
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Confidence level
Only requires two parameters = mean and variance

16. Gamma distribution
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Peaks over threshold - Collects dataset in excess of some threshold
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

17. Sample variance
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Based on a dataset
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

18. Significance =1
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Concerned with a single random variable (ex. Roll of a die)
Confidence level
Probability that the random variables take on certain values simultaneously

19. BLUE
Sampling distribution of sample means tend to be normal
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
P(Z>t)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

20. Difference between population and sample variance
Variance(x) + Variance(Y) + 2*covariance(XY)
Price/return tends to run towards a long - run level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Population denominator = n - Sample denominator = n - 1

21. Simulation models
When one regressor is a perfect linear function of the other regressors
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
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
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors

22. Mean reversion in variance
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance reverts to 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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

23. Continuous random variable
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

24. Two ways to calculate historical volatility
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)

25. Hazard rate of exponentially distributed random variable
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

26. Variance of weighted scheme
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
If variance of the conditional distribution of u(i) is not constant
Does not depend on a prior event or information
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

27. Mean reversion in asset dynamics
Least absolute deviations estimator - used when extreme outliers are not uncommon
Price/return tends to run towards a long - run level
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Sample mean will near the population mean as the sample size increases

28. Two drawbacks of moving average series
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Z = (Y - meany)/(stddev(y)/sqrt(n))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

29. Unbiased
Use historical simulation approach but use the EWMA weighting system
Mean of sampling distribution is the population mean
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Transformed to a unit variable - Mean = 0 Variance = 1

30. Discrete random variable
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(y)/n = variance of sample Y
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

31. GARCH
(a^2)(variance(x)) + (b^2)(variance(y))
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)
Low Frequency - High Severity events
Sample mean +/ - t*(stddev(s)/sqrt(n))

32. Deterministic Simulation
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Mean of sampling distribution is the population mean
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

33. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Expected value of the sample mean is the population mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

34. SER
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
E(mean) = mean

35. Continuously compounded return equation
i = ln(Si/Si - 1)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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!)

36. Block maxima
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

37. Priori (classical) probability
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
Normal - Student's T - Chi - square - F distribution
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
Based on an equation - P(A) = # of A/total outcomes

38. Test for unbiasedness
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Average return across assets on a given day
E(mean) = mean

39. Joint probability functions
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)
Probability that the random variables take on certain values simultaneously
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(y)/n = variance of sample Y

40. Chi - squared distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

41. T distribution
For n>30 - sample mean is approximately normal
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Yi = B0 + B1Xi + ui

42. Consistent
Regression can be non - linear in variables but must be linear in parameters
Variance(X) + Variance(Y) - 2*covariance(XY)
Only requires two parameters = mean and variance
When the sample size is large - the uncertainty about the value of the sample is very small

43. Persistence
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
We reject a hypothesis that is actually true
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

44. Binomial distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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!)
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

45. Implications of homoscedasticity
Based on a dataset
More than one random variable
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

46. Least squares estimator(m)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Confidence level
Model dependent - Options with the same underlying assets may trade at different volatilities

47. Panel data (longitudinal or micropanel)
Returns over time for an individual asset
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Special type of pooled data in which the cross sectional unit is surveyed over time

48. Confidence interval (from t)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Sample mean +/ - t*(stddev(s)/sqrt(n))
We accept a hypothesis that should have been rejected
Least absolute deviations estimator - used when extreme outliers are not uncommon

49. Variance of X+b
If variance of the conditional distribution of u(i) is not constant
Variance(x)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

50. Monte Carlo Simulations
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Var(X) + Var(Y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d