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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
SSR
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

2. EWMA
Mean = np - Variance = npq - Std dev = sqrt(npq)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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)
Easy to manipulate

3. Law of Large Numbers
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Special type of pooled data in which the cross sectional unit is surveyed over time
Random walk (usually acceptable) - Constant volatility (unlikely)
Sample mean will near the population mean as the sample size increases

4. Priori (classical) probability
Probability that the random variables take on certain values simultaneously
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
Expected value of the sample mean is the population mean
Based on an equation - P(A) = # of A/total outcomes

5. Kurtosis
Variance reverts to a long run level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

6. Covariance
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
E(XY) - E(X)E(Y)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Model dependent - Options with the same underlying assets may trade at different volatilities

7. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(y)/n = variance of sample Y

8. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

9. Variance of X+Y
SSR
Nonlinearity
Sampling distribution of sample means tend to be normal
Var(X) + Var(Y)

10. Sample mean
Expected value of the sample mean is the population mean
When the sample size is large - the uncertainty about the value of the sample is very small
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

11. 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
Has heavy tails
Least absolute deviations estimator - used when extreme outliers are not uncommon

12. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

13. Logistic distribution
Has heavy tails
Distribution with only two possible outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Confidence set for two coefficients - two dimensional analog for the confidence interval

14. Poisson distribution equations for mean variance and std deviation
Summation((xi - mean)^k)/n
Confidence set for two coefficients - two dimensional analog for the confidence interval
Has heavy tails
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

15. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
We reject a hypothesis that is actually true
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

16. LFHS
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Low Frequency - High Severity events

17. Mean reversion
Variance(y)/n = variance of sample Y
If variance of the conditional distribution of u(i) is not constant
Does not depend on a prior event or information
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

18. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

19. Test for unbiasedness
Expected value of the sample mean is the population mean
E(mean) = mean
Contains variables not explicit in model - Accounts for randomness
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

20. Lognormal
Contains variables not explicit in model - Accounts for randomness
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
Does not depend on a prior event or information
P - value

21. Unconditional vs conditional distributions
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

22. Cross - sectional
Average return across assets on a given day
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
Based on a dataset
When one regressor is a perfect linear function of the other regressors

23. ESS
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Distribution with only two possible outcomes
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

24. Direction of OVB
Independently and Identically Distributed
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Returns over time for an individual asset

25. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Regression can be non - linear in variables but must be linear in parameters
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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)

26. Control variates technique
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 increase accuracy by reducing sample variance instead of increasing sample size
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

27. Binomial distribution equations for mean variance and std dev
Choose parameters that maximize the likelihood of what observations occurring
Mean = np - Variance = npq - Std dev = sqrt(npq)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

28. Mean reversion in asset dynamics
Confidence set for two coefficients - two dimensional analog for the confidence interval
Price/return tends to run towards a long - run level
Regression can be non - linear in variables but must be linear in parameters
Rxy = Sxy/(Sx*Sy)

29. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
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
For n>30 - sample mean is approximately normal
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!)

30. Hybrid method for conditional volatility
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Use historical simulation approach but use the EWMA weighting system
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

31. Variance of sampling distribution of means when n<N
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)
Rxy = Sxy/(Sx*Sy)
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

32. Econometrics
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Application of mathematical statistics to economic data to lend empirical support to models

33. Hazard rate of exponentially distributed random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(x)

34. Homoskedastic
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Only requires two parameters = mean and variance
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
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

35. Confidence ellipse
Independently and Identically Distributed
Confidence set for two coefficients - two dimensional analog for the confidence interval
Yi = B0 + B1Xi + ui
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

36. Covariance calculations using weight sums (lambda)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(y)/n = variance of sample Y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

37. Variance of X+b
Variance(x)
When the sample size is large - the uncertainty about the value of the sample is very small
95% = 1.65 99% = 2.33 For one - tailed tests
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

38. Standard error for Monte Carlo replications
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Var(X) + Var(Y)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

39. Gamma distribution
Low Frequency - High Severity events
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
Population denominator = n - Sample denominator = n - 1
E(mean) = mean

40. Conditional probability functions
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Expected value of the sample mean is the population mean

41. Chi - squared distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Application of mathematical statistics to economic data to lend empirical support to models
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

42. POT
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)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Peaks over threshold - Collects dataset in excess of some threshold
Probability that the random variables take on certain values simultaneously

43. Test for statistical independence
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Has heavy tails
P(X=x - Y=y) = P(X=x) * P(Y=y)
Summation((xi - mean)^k)/n

44. Binomial distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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!)
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. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Independently and Identically Distributed

46. Four sampling distributions

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47. Sample correlation
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Rxy = Sxy/(Sx*Sy)
Variance(X) + Variance(Y) - 2*covariance(XY)
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)

48. Continuous representation of the GBM
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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)
Normal - Student's T - Chi - square - F distribution
Variance(X) + Variance(Y) - 2*covariance(XY)

49. Consistent
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(x) + Variance(Y) + 2*covariance(XY)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

50. Stochastic error term
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Contains variables not explicit in model - Accounts for randomness
Variance(x)