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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Economical(elegant)
Summation((xi - mean)^k)/n
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Only requires two parameters = mean and variance
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

2. Square root rule
E(XY) - E(X)E(Y)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Average return across assets on a given day

3. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Summation((xi - mean)^k)/n
Probability that the random variables take on certain values simultaneously
Distribution with only two possible outcomes

4. What does the OLS minimize?
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
SSR
Easy to manipulate

5. Least squares estimator(m)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

6. Adjusted R^2
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Nonlinearity
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
SSR

7. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Average return across assets on a given day
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)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

8. Overall F - statistic
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

9. LAD
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Low Frequency - High Severity events
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

10. Confidence interval for sample mean
Transformed to a unit variable - Mean = 0 Variance = 1
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
We reject a hypothesis that is actually true

11. Normal distribution
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

12. Persistence
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

13. Law of Large Numbers
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sample mean will near the population mean as the sample size increases
More than one random variable
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

14. Variance - covariance approach for VaR of a portfolio
Variance(x) + Variance(Y) + 2*covariance(XY)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Among all unbiased estimators - estimator with the smallest variance is efficient

15. Lognormal
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

16. Type II Error
Statement of the error or precision of an estimate
We accept a hypothesis that should have been rejected
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

17. Empirical frequency
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
Based on a dataset
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Sample mean +/ - t*(stddev(s)/sqrt(n))

18. Two assumptions of square root rule
Z = (Y - meany)/(stddev(y)/sqrt(n))
Random walk (usually acceptable) - Constant volatility (unlikely)
Contains variables not explicit in model - Accounts for randomness
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

19. Single variable (univariate) probability
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
Population denominator = n - Sample denominator = n - 1
Concerned with a single random variable (ex. Roll of a die)
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

20. Simulating for VaR
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

21. Covariance
(a^2)(variance(x)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Based on a dataset
E(XY) - E(X)E(Y)

22. Four sampling distributions

23. Potential reasons for fat tails in return distributions
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
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Concerned with a single random variable (ex. Roll of a die)

24. Logistic distribution
Variance = (1/m) summation(u<n - i>^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Has heavy tails

25. Continuous random variable
Peaks over threshold - Collects dataset in excess of some threshold
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Price/return tends to run towards a long - run level

26. Variance of X+b
Regression can be non - linear in variables but must be linear in parameters
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Rxy = Sxy/(Sx*Sy)
Variance(x)

27. Variance of X+Y
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Var(X) + Var(Y)

28. SER
Regression can be non - linear in variables but must be linear in parameters
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 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
For n>30 - sample mean is approximately normal

29. Variance(discrete)
Attempts to sample along more important paths
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Least absolute deviations estimator - used when extreme outliers are not uncommon

30. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sample mean will near the population mean as the sample size increases

31. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
If variance of the conditional distribution of u(i) is not constant
Independently and Identically Distributed

32. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
When the sample size is large - the uncertainty about the value of the sample is very small
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Does not depend on a prior event or information

33. Stochastic error term
Peaks over threshold - Collects dataset in excess of some threshold
Use historical simulation approach but use the EWMA weighting system
Contains variables not explicit in model - Accounts for randomness
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

34. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
P - value
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

35. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(y)/n = variance of sample Y
Expected value of the sample mean is the population mean
Statement of the error or precision of an estimate

36. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Low Frequency - High Severity events

37. Exponential distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Statement of the error or precision of an estimate
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
When the sample size is large - the uncertainty about the value of the sample is very small

38. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

39. Antithetic variable technique
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

40. SER
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean of sampling distribution is the population mean
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

41. Reliability
Statement of the error or precision of an estimate
Mean = np - Variance = npq - Std dev = sqrt(npq)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
P(X=x - Y=y) = P(X=x) * P(Y=y)

42. Conditional probability functions
Variance reverts to a long run level
Expected value of the sample mean is the population mean
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)

43. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Confidence level
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

44. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean of sampling distribution is the population mean
If variance of the conditional distribution of u(i) is not constant
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

45. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Has heavy tails
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Confidence level

46. Difference between population and sample variance
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Population denominator = n - Sample denominator = n - 1
Variance(y)/n = variance of sample Y
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

47. Central Limit Theorem
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
For n>30 - sample mean is approximately normal

48. Poisson Distribution
Easy to manipulate
Mean of sampling distribution is the population mean
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

49. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Normal - Student's T - Chi - square - F distribution
We reject a hypothesis that is actually true

50. Joint probability functions
Peaks over threshold - Collects dataset in excess of some threshold
Probability that the random variables take on certain values simultaneously
Mean of sampling distribution is the population mean
Summation((xi - mean)^k)/n