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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. Tractable
Easy to manipulate
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Returns over time for an individual asset
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

2. Persistence
Application of mathematical statistics to economic data to lend empirical support to models
Expected value of the sample mean is the population mean
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
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

3. Standard error for Monte Carlo replications
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
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
Expected value of the sample mean is the population mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

4. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
E(XY) - E(X)E(Y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
(a^2)(variance(x)) + (b^2)(variance(y))

5. Hybrid method for conditional volatility
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Use historical simulation approach but use the EWMA weighting system
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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. Biggest (and only real) drawback of GARCH mode
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
(a^2)(variance(x)) + (b^2)(variance(y))
Nonlinearity

7. Sample correlation
Only requires two parameters = mean and variance
Rxy = Sxy/(Sx*Sy)
Transformed to a unit variable - Mean = 0 Variance = 1
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

8. Central Limit Theorem
For n>30 - sample mean is approximately normal
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

9. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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!)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

10. Variance of aX
Summation((xi - mean)^k)/n
(a^2)(variance(x)
i = ln(Si/Si - 1)
Z = (Y - meany)/(stddev(y)/sqrt(n))

11. Law of Large Numbers
Mean of sampling distribution is the population mean
Sample mean will near the population mean as the sample size increases
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

12. Single variable (univariate) probability
Application of mathematical statistics to economic data to lend empirical support to models
Concerned with a single random variable (ex. Roll of a die)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

13. Difference between population and sample 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
Does not depend on a prior event or information
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Population denominator = n - Sample denominator = n - 1

14. Central Limit Theorem(CLT)
Regression can be non - linear in variables but must be linear in parameters
Sampling distribution of sample means tend to be normal
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

15. Confidence interval (from t)
P - value
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

16. Multivariate probability
When one regressor is a perfect linear function of the other regressors
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
More than one random variable
Concerned with a single random variable (ex. Roll of a die)

17. Critical z values
(a^2)(variance(x)
95% = 1.65 99% = 2.33 For one - tailed tests
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

18. Mean reversion in variance
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
Variance reverts to a long run level
Probability that the random variables take on certain values simultaneously
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

19. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Among all unbiased estimators - estimator with the smallest variance is efficient

20. Unconditional vs conditional distributions
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

21. Potential reasons for fat tails in return distributions
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

22. Monte Carlo Simulations
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Distribution with only two possible outcomes
Population denominator = n - Sample denominator = n - 1
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

23. Pooled data
SSR
(a^2)(variance(x)) + (b^2)(variance(y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

24. Continuously compounded return equation
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
(a^2)(variance(x)
Normal - Student's T - Chi - square - F distribution
i = ln(Si/Si - 1)

25. Four sampling distributions

26. Variance of X+Y assuming dependence
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(x) + Variance(Y) + 2*covariance(XY)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Normal - Student's T - Chi - square - F distribution

27. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Normal - Student's T - Chi - square - F distribution

28. Covariance calculations using weight sums (lambda)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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.
Population denominator = n - Sample denominator = n - 1

29. Priori (classical) probability
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Statement of the error or precision of an estimate
Based on an equation - P(A) = # of A/total outcomes
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

30. Variance of sampling distribution of means when n<N
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

31. Economical(elegant)
Only requires two parameters = mean and variance
Low Frequency - High Severity events
Among all unbiased estimators - estimator with the smallest variance is efficient
P(X=x - Y=y) = P(X=x) * P(Y=y)

32. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Easy to manipulate
95% = 1.65 99% = 2.33 For one - tailed tests
Yi = B0 + B1Xi + ui

33. Multivariate Density Estimation (MDE)
Var(X) + Var(Y)
Variance(x)
Application of mathematical statistics to economic data to lend empirical support to models
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

34. Binomial distribution
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
SSR

35. Covariance
E(XY) - E(X)E(Y)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

36. Discrete random variable
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Contains variables not explicit in model - Accounts for randomness
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

37. POT
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Peaks over threshold - Collects dataset in excess of some threshold

38. Standard normal distribution
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
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
Random walk (usually acceptable) - Constant volatility (unlikely)
Transformed to a unit variable - Mean = 0 Variance = 1

39. Deterministic Simulation
95% = 1.65 99% = 2.33 For one - tailed tests
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

40. Unstable return distribution
Choose parameters that maximize the likelihood of what observations occurring
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
P(X=x - Y=y) = P(X=x) * P(Y=y)

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

42. Simulation models
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
We reject a hypothesis that is actually true
Low Frequency - High Severity events

43. Confidence ellipse
Sample mean +/ - t*(stddev(s)/sqrt(n))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Confidence set for two coefficients - two dimensional analog for the confidence interval
Only requires two parameters = mean and variance

44. Variance of sample mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Normal - Student's T - Chi - square - F distribution
Variance(y)/n = variance of sample 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)

45. T distribution
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)
Based on an equation - P(A) = # of A/total outcomes
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

46. Econometrics
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
Application of mathematical statistics to economic data to lend empirical support to models
Based on a dataset
Mean of sampling distribution is the population mean

47. F distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Price/return tends to run towards a long - run level
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

48. Weibul distribution
Application of mathematical statistics to economic data to lend empirical support to models
Concerned with a single random variable (ex. Roll of a die)
(a^2)(variance(x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

49. Logistic distribution
Mean of sampling distribution is the population mean
Has heavy tails
Population denominator = n - Sample denominator = n - 1
Sample mean will near the population mean as the sample size increases

50. Extending the HS approach for computing value of a portfolio