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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. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sampling distribution of sample means tend to be normal

2. Bootstrap method
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

3. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Distribution with only two possible outcomes
Variance(y)/n = variance of sample Y
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

4. Extreme Value Theory
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
We reject a hypothesis that is actually true
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Only requires two parameters = mean and variance

5. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

6. Variance of aX + bY
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Sampling distribution of sample means tend to be normal
(a^2)(variance(x)) + (b^2)(variance(y))
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

7. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Mean = np - Variance = npq - Std dev = sqrt(npq)
Application of mathematical statistics to economic data to lend empirical support to models

8. Mean reversion in variance
Special type of pooled data in which the cross sectional unit is surveyed over time
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*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
Variance reverts to a long run level

9. 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
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

10. Weibul distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
We reject a hypothesis that is actually true
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

11. Type I error
We reject a hypothesis that is actually true
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Easy to manipulate

12. Time series data
Random walk (usually acceptable) - Constant volatility (unlikely)
Returns over time for an individual asset
Application of mathematical statistics to economic data to lend empirical support to models
For n>30 - sample mean is approximately normal

13. What does the OLS minimize?
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SSR
Returns over time for an individual asset
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

14. Skewness
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

15. Binomial distribution equations for mean variance and std dev
Sample mean +/ - t*(stddev(s)/sqrt(n))
Returns over time for an individual asset
We reject a hypothesis that is actually true
Mean = np - Variance = npq - Std dev = sqrt(npq)

16. Multivariate probability
Statement of the error or precision of an estimate
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
More than one random variable

17. 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
Attempts to sample along more important paths
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
Combine to form distribution with leptokurtosis (heavy tails)

18. Perfect multicollinearity
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Only requires two parameters = mean and variance
Returns over time for a combination of assets (combination of time series and cross - sectional data)
When one regressor is a perfect linear function of the other regressors

19. P - value
E(mean) = 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)
P(Z>t)
Probability that the random variables take on certain values simultaneously

20. Sample variance
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Contains variables not explicit in model - Accounts for randomness
Model dependent - Options with the same underlying assets may trade at different volatilities

21. Statistical (or empirical) model
For n>30 - sample mean is approximately normal
Sample mean +/ - t*(stddev(s)/sqrt(n))
Yi = B0 + B1Xi + ui
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

22. Kurtosis
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance = (1/m) summation(u<n - i>^2)
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. Law of Large Numbers
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
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
Sample mean will near the population mean as the sample size increases
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

24. Variance of X+Y assuming dependence
(a^2)(variance(x)
Yi = B0 + B1Xi + ui
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(x) + Variance(Y) + 2*covariance(XY)

25. F distribution
Nonlinearity
Normal - Student's T - Chi - square - F distribution
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

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

27. Antithetic variable technique
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Independently and Identically Distributed
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Model dependent - Options with the same underlying assets may trade at different volatilities

28. Two assumptions of square root rule
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Random walk (usually acceptable) - Constant volatility (unlikely)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

29. 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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

30. Logistic distribution
Has heavy tails
Use historical simulation approach but use the EWMA weighting system
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sample mean will near the population mean as the sample size increases

31. K - th 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
Summation((xi - mean)^k)/n
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Peaks over threshold - Collects dataset in excess of some threshold

32. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Concerned with a single random variable (ex. Roll of a die)

33. POT
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Peaks over threshold - Collects dataset in excess of some threshold
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Independently and Identically Distributed

34. Multivariate Density Estimation (MDE)
We accept a hypothesis that should have been rejected
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
More than one random variable
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

35. Mean(expected value)
Has heavy tails
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

36. 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
When the sample size is large - the uncertainty about the value of the sample is very small
Concerned with a single random variable (ex. Roll of a die)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

37. Hybrid method for conditional volatility
Variance = (1/m) summation(u<n - i>^2)
Use historical simulation approach but use the EWMA weighting system
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

38. Joint probability functions
Normal - Student's T - Chi - square - F distribution
(a^2)(variance(x)
Probability that the random variables take on certain values simultaneously
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

39. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Concerned with a single random variable (ex. Roll of a die)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

40. Exponential distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Average return across assets on a given day
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

41. Deterministic Simulation
Mean of sampling distribution is the population mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Normal - Student's T - Chi - square - F 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

42. Empirical frequency
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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)
Based on a dataset
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

43. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
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)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Low Frequency - High Severity events

44. Confidence interval (from t)
P - value
Peaks over threshold - Collects dataset in excess of some threshold
Sample mean +/ - t*(stddev(s)/sqrt(n))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

45. Four sampling distributions

46. Variance of X+Y
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Var(X) + Var(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)

47. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Z = (Y - meany)/(stddev(y)/sqrt(n))
Special type of pooled data in which the cross sectional unit is surveyed over time
Sample mean +/ - t*(stddev(s)/sqrt(n))

48. GPD
Use historical simulation approach but use the EWMA weighting system
Has heavy tails
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Yi = B0 + B1Xi + ui

49. Mean reversion in asset dynamics
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
Price/return tends to run towards a long - run level
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

50. Continuously compounded return equation
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
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