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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. Homoskedastic only F - stat
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
For n>30 - sample mean is approximately normal
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

2. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Least absolute deviations estimator - used when extreme outliers are not uncommon
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)
Choose parameters that maximize the likelihood of what observations occurring

3. Tractable
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Easy to manipulate
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

4. Exact significance level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
P - value
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

5. Chi - squared distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Z = (Y - meany)/(stddev(y)/sqrt(n))

6. Variance - covariance approach for VaR of a portfolio
We reject a hypothesis that is actually true
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

7. Historical std dev
When the sample size is large - the uncertainty about the value of the sample is very small
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Average return across assets on a given day
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

8. Beta distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Easy to manipulate
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

9. Stochastic error term
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
E(mean) = mean
Contains variables not explicit in model - Accounts for randomness
Least absolute deviations estimator - used when extreme outliers are not uncommon

10. Gamma distribution
i = ln(Si/Si - 1)
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Peaks over threshold - Collects dataset in excess of some threshold

11. Multivariate Density Estimation (MDE)
Transformed to a unit variable - Mean = 0 Variance = 1
(a^2)(variance(x)) + (b^2)(variance(y))
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(sample y) = (variance(y)/n)*(N - n/N - 1)

12. Type I error
We reject a hypothesis that is actually true
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
E(mean) = mean
Concerned with a single random variable (ex. Roll of a die)

13. Central Limit Theorem(CLT)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
For n>30 - sample mean is approximately normal
Sampling distribution of sample means tend to be normal

14. Two requirements of OVB
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

15. Cross - sectional
We reject a hypothesis that is actually true
Expected value of the sample mean is the population mean
Average return across assets on a given day
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

16. Logistic distribution
Has heavy tails
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

17. Panel data (longitudinal or micropanel)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Least absolute deviations estimator - used when extreme outliers are not uncommon
Special type of pooled data in which the cross sectional unit is surveyed over time

18. POT
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
(a^2)(variance(x)
Peaks over threshold - Collects dataset in excess of some threshold
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

19. Confidence ellipse
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Price/return tends to run towards a long - run level
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Confidence set for two coefficients - two dimensional analog for the confidence interval

20. Law of Large Numbers
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample mean will near the population mean as the sample size increases
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

21. Mean reversion in asset dynamics
When one regressor is a perfect linear function of the other regressors
Price/return tends to run towards a long - run level
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)
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

22. Result of combination of two normal with same means
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
Combine to form distribution with leptokurtosis (heavy tails)
Choose parameters that maximize the likelihood of what observations occurring
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

23. Mean(expected value)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Attempts to sample along more important paths
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

24. Non - parametric vs parametric calculation of VaR
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)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Application of mathematical statistics to economic data to lend empirical support to models

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

26. Standard normal distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Transformed to a unit variable - Mean = 0 Variance = 1

27. Efficiency
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
For n>30 - sample mean is approximately normal
Among all unbiased estimators - estimator with the smallest variance is efficient
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

28. 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)
Choose parameters that maximize the likelihood of what observations occurring
Attempts to sample along more important paths
Special type of pooled data in which the cross sectional unit is surveyed over time

29. Antithetic variable technique
(a^2)(variance(x)) + (b^2)(variance(y))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

30. Heteroskedastic
Nonlinearity
If variance of the conditional distribution of u(i) is not constant
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(x)

31. Simulation models
Combine to form distribution with leptokurtosis (heavy tails)
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

32. Inverse transform method
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent 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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

33. Exponential distribution
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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
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)

34. Unstable return distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Sample mean +/ - t*(stddev(s)/sqrt(n))
We accept a hypothesis that should have been rejected

35. Covariance calculations using weight sums (lambda)
Has heavy tails
Z = (Y - meany)/(stddev(y)/sqrt(n))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

36. Consistent
Sample mean +/ - t*(stddev(s)/sqrt(n))
When the sample size is large - the uncertainty about the value of the sample is very small
Special type of pooled data in which the cross sectional unit is surveyed over time
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

37. Critical z values
(a^2)(variance(x)
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
95% = 1.65 99% = 2.33 For one - tailed tests
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

38. Covariance
E(XY) - E(X)E(Y)
Variance(X) + Variance(Y) - 2*covariance(XY)
Summation((xi - mean)^k)/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

39. Pooled data
Concerned with a single random variable (ex. Roll of a die)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance reverts to a long run level
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

40. Implied standard deviation for options
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
We reject a hypothesis that is actually true
Peaks over threshold - Collects dataset in excess of some threshold
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

41. Deterministic Simulation
Variance = (1/m) summation(u<n - i>^2)
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

42. Variance of X+b
Variance(x)
Statement of the error or precision of an estimate
Contains variables not explicit in model - Accounts for randomness
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

43. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Among all unbiased estimators - estimator with the smallest variance is efficient
95% = 1.65 99% = 2.33 For one - tailed tests
P(Z>t)

44. Variance of X+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)
Var(X) + Var(Y)
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
If variance of the conditional distribution of u(i) is not constant

45. Mean reversion in variance
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
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
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"

46. Marginal unconditional probability function
Does not depend on a prior event or information
Choose parameters that maximize the likelihood of what observations occurring
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

47. Variance of X+Y assuming dependence
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(x) + Variance(Y) + 2*covariance(XY)
If variance of the conditional distribution of u(i) is not constant
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

48. Central Limit Theorem
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))
Mean of sampling distribution is the population mean
For n>30 - sample mean is approximately normal

49. LAD
P - value
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Least absolute deviations estimator - used when extreme outliers are not uncommon

50. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Price/return tends to run towards a long - run level
Population denominator = n - Sample denominator = n - 1