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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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance = (1/m) summation(u<n - i>^2)
Yi = B0 + B1Xi + ui

2. Mean reversion in variance
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)
Variance reverts to a long run level
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)

3. Biggest (and only real) drawback of GARCH mode
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Mean = np - Variance = npq - Std dev = sqrt(npq)
Nonlinearity
Based on an equation - P(A) = # of A/total outcomes

4. Standard normal distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Transformed to a unit variable - Mean = 0 Variance = 1

5. Kurtosis
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Summation((xi - mean)^k)/n
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

6. POT
Peaks over threshold - Collects dataset in excess of some threshold
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

7. Non - parametric vs parametric calculation of VaR
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
SSR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

8. Persistence
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
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
Application of mathematical statistics to economic data to lend empirical support to models
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

9. Variance of X - Y assuming dependence
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance(X) + Variance(Y) - 2*covariance(XY)

10. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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

11. Mean reversion in asset dynamics
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Price/return tends to run towards 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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

12. Discrete representation of the GBM
For n>30 - sample mean is approximately normal
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Easy to manipulate

13. Maximum likelihood method
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
We accept a hypothesis that should have been rejected
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
Choose parameters that maximize the likelihood of what observations occurring

14. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
E(mean) = mean

15. Unstable return distribution
Average return across assets on a given day
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

16. Simulation models
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

17. Type I error
We reject a hypothesis that is actually true
Special type of pooled data in which the cross sectional unit is surveyed over time
Based on an equation - P(A) = # of A/total outcomes
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

18. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(x)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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

19. Potential reasons for fat tails in return distributions
E(mean) = mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

20. Skewness
P(Z>t)
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 = np - Variance = npq - Std dev = sqrt(npq)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

21. Block maxima
Population denominator = n - Sample denominator = n - 1
Special type of pooled data in which the cross sectional unit is surveyed over time
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Combine to form distribution with leptokurtosis (heavy tails)

22. Heteroskedastic
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
If variance of the conditional distribution of u(i) is not constant
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

23. LFHS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Normal - Student's T - Chi - square - F distribution
Sample mean will near the population mean as the sample size increases
Low Frequency - High Severity events

24. Variance(discrete)
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
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)
For n>30 - sample mean is approximately normal

25. Reliability
Statement of the error or precision of an estimate
Sample mean +/ - t*(stddev(s)/sqrt(n))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Use historical simulation approach but use the EWMA weighting system

26. Confidence ellipse
More than one random variable
Variance(X) + Variance(Y) - 2*covariance(XY)
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

27. Shortcomings of implied volatility
Rxy = Sxy/(Sx*Sy)
Average return across assets on a given day
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Model dependent - Options with the same underlying assets may trade at different volatilities

28. Logistic distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Has heavy tails
Only requires two parameters = mean and variance

29. Variance of X+b
Variance(x)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Special type of pooled data in which the cross sectional unit is surveyed over time
Sample mean will near the population mean as the sample size increases

30. Unconditional vs conditional distributions
Random walk (usually acceptable) - Constant volatility (unlikely)
Average return across assets on a given day
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

31. Variance of X+Y assuming dependence
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Variance(x) + Variance(Y) + 2*covariance(XY)
For n>30 - sample mean is approximately normal

32. Cross - sectional
More than one random variable
Average return across assets on a given day
Returns over time for an individual asset
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

33. Direction of OVB
(a^2)(variance(x)
Normal - Student's T - Chi - square - F distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Has heavy tails

34. Importance sampling technique
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Attempts to sample along more important paths
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

35. Pooled data
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)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

36. Multivariate Density Estimation (MDE)
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
Variance(y)/n = variance of sample Y
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)

37. Poisson distribution equations for mean variance and std deviation
Probability that the random variables take on certain values simultaneously
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Based on a dataset
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

38. Four sampling distributions

39. Sample correlation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
E(XY) - E(X)E(Y)
Rxy = Sxy/(Sx*Sy)

40. WLS
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
Only requires two parameters = mean and variance
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

41. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Summation((xi - mean)^k)/n
Z = (Y - meany)/(stddev(y)/sqrt(n))

42. Law of Large Numbers
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Sampling distribution of sample means tend to be normal
i = ln(Si/Si - 1)
Sample mean will near the population mean as the sample size increases

43. Bernouli Distribution
Variance(x)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Distribution with only two possible outcomes
Variance(y)/n = variance of sample Y

44. i.i.d.
Independently and Identically Distributed
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

45. Sample variance
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
E(mean) = mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

46. Historical std dev
We accept a hypothesis that should have been rejected
i = ln(Si/Si - 1)
Based on an equation - P(A) = # of A/total outcomes
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

47. Variance of sample mean
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sample mean will near the population mean as the sample size increases
Variance(y)/n = variance of sample Y

48. Control variates technique
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Has heavy tails
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

49. Perfect multicollinearity
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Easy to manipulate
When one regressor is a perfect linear function of the other regressors
P(Z>t)

50. Time series data
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Returns over time for an individual asset
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