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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. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
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/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Probability that the random variables take on certain values simultaneously

2. Cross - sectional
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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!)
Average return across assets on a given day
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

3. Expected future variance rate (t periods forward)
We accept a hypothesis that should have been rejected
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Use historical simulation approach but use the EWMA weighting system
E(mean) = mean

4. Efficiency
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Among all unbiased estimators - estimator with the smallest variance is efficient
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
Probability that the random variables take on certain values simultaneously

5. Difference between population and sample variance
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Population denominator = n - Sample denominator = n - 1
Summation((xi - mean)^k)/n
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

6. SER
Population denominator = n - Sample denominator = n - 1
Rxy = Sxy/(Sx*Sy)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

7. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Application of mathematical statistics to economic data to lend empirical support to models

8. Covariance
E(XY) - E(X)E(Y)
Variance = (1/m) summation(u<n - i>^2)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We reject a hypothesis that is actually true

9. Simplified standard (un - weighted) variance
Expected value of the sample mean is the population mean
Variance = (1/m) summation(u<n - i>^2)
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)
Variance(y)/n = variance of sample Y

10. Empirical frequency
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sample mean will near the population mean as the sample size increases
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
Based on a dataset

11. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Easy to manipulate

12. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Variance(X) + Variance(Y) - 2*covariance(XY)
Rxy = Sxy/(Sx*Sy)
Expected value of the sample mean is the population mean

13. Law of Large Numbers
Var(X) + Var(Y)
Choose parameters that maximize the likelihood of what observations occurring
Sample mean will near the population mean as the sample size increases
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

14. K - th moment
Var(X) + Var(Y)
Summation((xi - mean)^k)/n
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 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

15. Normal distribution
P - value
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
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(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

16. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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())
Easy to manipulate

17. Non - parametric vs parametric calculation of VaR
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Least absolute deviations estimator - used when extreme outliers are not uncommon
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

18. Unstable return distribution
If variance of the conditional distribution of u(i) is not constant
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Regression can be non - linear in variables but must be linear in parameters
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

19. Mean(expected value)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Mean of sampling distribution is the population mean

20. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

21. Exponential distribution
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
We accept a hypothesis that should have been rejected
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

22. Poisson distribution equations for mean variance and std deviation
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample mean +/ - t*(stddev(s)/sqrt(n))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

23. 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
Based on an equation - P(A) = # of A/total outcomes
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

24. Implications of homoscedasticity
Rxy = Sxy/(Sx*Sy)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Transformed to a unit variable - Mean = 0 Variance = 1
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

25. Single variable (univariate) probability
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Concerned with a single random variable (ex. Roll of a die)
Summation((xi - mean)^k)/n

26. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

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

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28. Test for statistical independence
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
P(X=x - Y=y) = P(X=x) * P(Y=y)
Concerned with a single random variable (ex. Roll of a die)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

29. Conditional probability functions
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Special type of pooled data in which the cross sectional unit is surveyed over time

30. Reliability
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Statement of the error or precision of an estimate

31. Tractable
Attempts to sample along more important paths
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

32. Variance of sample mean
Variance(y)/n = variance of sample Y
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
P(Z>t)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

33. Hazard rate of exponentially distributed random variable
Least absolute deviations estimator - used when extreme outliers are not uncommon
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)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

34. Gamma distribution
Concerned with a single random variable (ex. Roll of a die)
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(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 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

35. Four sampling distributions

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36. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Returns over time for an individual asset
Independently and Identically Distributed

37. Central Limit Theorem(CLT)
Does not depend on a prior event or information
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sampling distribution of sample means tend to be normal
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)

38. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

39. GPD
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

40. Kurtosis
Average return across assets on a given day
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
We reject a hypothesis that is actually true
More than one random variable

41. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Nonlinearity
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

42. Sample correlation
Confidence set for two coefficients - two dimensional analog for the confidence interval
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Rxy = Sxy/(Sx*Sy)

43. Variance of X+Y
Var(X) + Var(Y)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
We accept a hypothesis that should have been rejected
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

44. R^2
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Confidence level
We accept a hypothesis that should have been rejected
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

45. Confidence interval (from t)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Has heavy tails
Sample mean +/ - t*(stddev(s)/sqrt(n))

46. Homoskedastic only F - stat
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Transformed to a unit variable - Mean = 0 Variance = 1
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

47. Multivariate Density Estimation (MDE)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Confidence level

48. Mean reversion in variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance reverts to a long run level
Among all unbiased estimators - estimator with the smallest variance is efficient
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

49. Panel data (longitudinal or micropanel)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
For n>30 - sample mean is approximately normal
Special type of pooled data in which the cross sectional unit is surveyed over time
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

50. Chi - squared distribution
95% = 1.65 99% = 2.33 For one - tailed tests
i = ln(Si/Si - 1)
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail