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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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1. Continuous random variable
When one regressor is a perfect linear function of the other regressors
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sampling distribution of sample means tend to be normal
Price/return tends to run towards a long - run level

2. Efficiency
Distribution with only two possible outcomes
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean will near the population mean as the sample size increases

3. 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
When the sample size is large - the uncertainty about the value of the sample is very small
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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)

4. Skewness
Peaks over threshold - Collects dataset in excess of some threshold
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
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

5. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Var(X) + Var(Y)

6. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Based on an equation - P(A) = # of A/total outcomes

7. Mean reversion in asset dynamics
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Price/return tends to run towards a long - run level
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
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

8. Persistence
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Use historical simulation approach but use the EWMA weighting system
(a^2)(variance(x)
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

9. SER
Regression can be non - linear in variables but must be linear in parameters
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Rxy = Sxy/(Sx*Sy)

10. Variance of X+Y
Normal - Student's T - Chi - square - F distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
Var(X) + Var(Y)
Based on a dataset

11. Expected future variance rate (t periods forward)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Population denominator = n - Sample denominator = n - 1
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

12. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Among all unbiased estimators - estimator with the smallest variance is efficient
Population denominator = n - Sample denominator = n - 1

13. Heteroskedastic
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
If variance of the conditional distribution of u(i) is not constant
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)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

14. Variance of sample mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(y)/n = variance of sample Y
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

15. Chi - squared distribution
Variance reverts to a long run level
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
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

16. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
For n>30 - sample mean is approximately normal
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance(y)/n = variance of sample Y

17. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Random walk (usually acceptable) - Constant volatility (unlikely)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

18. EWMA
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Returns over time for an individual asset
Only requires two parameters = mean and variance

19. Confidence interval (from t)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
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

20. Kurtosis
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Yi = B0 + B1Xi + ui
Variance(x)

21. Hazard rate of exponentially distributed random variable
Variance(X) + Variance(Y) - 2*covariance(XY)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Peaks over threshold - Collects dataset in excess of some threshold

22. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Independently and Identically Distributed
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

23. Binomial distribution equations for mean variance and std dev
Easy to manipulate
Mean = np - Variance = npq - Std dev = sqrt(npq)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Normal - Student's T - Chi - square - F distribution

24. Variance of aX
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
(a^2)(variance(x)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

25. Least squares estimator(m)
Based on an equation - P(A) = # of A/total outcomes
95% = 1.65 99% = 2.33 For one - tailed tests
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Peaks over threshold - Collects dataset in excess of some threshold

26. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Application of mathematical statistics to economic data to lend empirical support to models
Price/return tends to run towards a long - run level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

27. Mean reversion in variance
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance(x)
Population denominator = n - Sample denominator = n - 1
Variance reverts to a long run level

28. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Regression can be non - linear in variables but must be linear in parameters
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

29. Binomial distribution
Statement of the error or precision of an estimate
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!)
If variance of the conditional distribution of u(i) is not constant
Least absolute deviations estimator - used when extreme outliers are not uncommon

30. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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)
Statement of the error or precision of an estimate
Model dependent - Options with the same underlying assets may trade at different volatilities

31. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Mean of sampling distribution is the population mean
Model dependent - Options with the same underlying assets may trade at different volatilities

32. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Low Frequency - High Severity events
Peaks over threshold - Collects dataset in excess of some threshold

33. GEV
Among all unbiased estimators - estimator with the smallest variance is efficient
Based on a dataset
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

34. Cholesky factorization (decomposition)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(y)/n = variance of sample Y

35. GARCH
(a^2)(variance(x)
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)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
More than one random variable

36. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Z = (Y - meany)/(stddev(y)/sqrt(n))
Mean of sampling distribution is the population mean

37. Control variates technique
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
If variance of the conditional distribution of u(i) is not constant
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Choose parameters that maximize the likelihood of what observations occurring

38. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

39. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
[1/(n - 1)]*summation((Xi - X)(Yi - 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

40. Antithetic variable technique
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
More than one random variable
Z = (Y - meany)/(stddev(y)/sqrt(n))

41. Confidence interval for sample mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
P(X=x - Y=y) = P(X=x) * P(Y=y)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Statement of the error or precision of an estimate

42. Sample mean
Expected value of the sample mean is the population mean
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Only requires two parameters = mean and variance

43. K - th moment
Summation((xi - mean)^k)/n
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

44. Law of Large Numbers
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sampling distribution of sample means tend to be normal
Sample mean will near the population mean as the sample size increases
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

45. Critical z values
Confidence set for two coefficients - two dimensional analog for the confidence interval
Price/return tends to run towards a long - run level
95% = 1.65 99% = 2.33 For one - tailed tests
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

46. Sample correlation
(a^2)(variance(x)) + (b^2)(variance(y))
Rxy = Sxy/(Sx*Sy)
Based on a dataset
Probability that the random variables take on certain values simultaneously

47. Direction of OVB
Average return across assets on a given day
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
When the sample size is large - the uncertainty about the value of the sample is very small
Only requires two parameters = mean and variance

48. ESS
Var(X) + Var(Y)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

49. Sample covariance
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

50. Empirical frequency
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
Returns over time for an individual asset
Based on a dataset
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