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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. Cross - sectional
Statement of the error or precision of an estimate
Average return across assets on a given day
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Concerned with a single random variable (ex. Roll of a die)

2. Time series data
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E(mean) = 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
Returns over time for an individual asset

3. Variance of aX
(a^2)(variance(x)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(y)/n = variance of sample Y
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

4. Statistical (or empirical) model
Combine to form distribution with leptokurtosis (heavy tails)
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)
Based on an equation - P(A) = # of A/total outcomes
Yi = B0 + B1Xi + ui

5. SER
Only requires two parameters = mean and variance
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

6. Multivariate Density Estimation (MDE)
Expected value of the sample mean is the population mean
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Attempts to sample along more important paths

7. Variance of X+b
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Price/return tends to run towards a long - run level
Variance(x)
Average return across assets on a given day

8. Gamma distribution
(a^2)(variance(x)) + (b^2)(variance(y))
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Z = (Y - meany)/(stddev(y)/sqrt(n))

9. Sample mean
Expected value of the sample mean is the population mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Confidence level
Returns over time for a combination of assets (combination of time series and cross - sectional data)

10. Priori (classical) probability
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Based on an equation - P(A) = # of A/total outcomes
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

11. Joint probability functions
Population denominator = n - Sample denominator = n - 1
Probability that the random variables take on certain values simultaneously
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)

12. Test for statistical independence
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance reverts to a long run level
P(X=x - Y=y) = P(X=x) * P(Y=y)
Sample mean +/ - t*(stddev(s)/sqrt(n))

13. GARCH
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)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Contains variables not explicit in model - Accounts for randomness

14. Mean reversion
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

15. Kurtosis
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Rxy = Sxy/(Sx*Sy)
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!)

16. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
When the sample size is large - the uncertainty about the value of the sample is very small

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

18. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

19. Tractable
For n>30 - sample mean is approximately normal
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Easy to manipulate

20. Control variates technique
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
When the sample size is large - the uncertainty about the value of the sample is very small
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

21. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Returns over time for an individual asset
Least absolute deviations estimator - used when extreme outliers are not uncommon
Rxy = Sxy/(Sx*Sy)

22. Covariance
E(XY) - E(X)E(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Transformed to a unit variable - Mean = 0 Variance = 1
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

23. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Statement of the error or precision of an estimate

24. Type II Error
(a^2)(variance(x)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Z = (Y - meany)/(stddev(y)/sqrt(n))
We accept a hypothesis that should have been rejected

25. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

26. Direction of OVB
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
When one regressor is a perfect linear function of the other regressors
Z = (Y - meany)/(stddev(y)/sqrt(n))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

27. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

28. Poisson Distribution
SSR
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Confidence set for two coefficients - two dimensional analog for the confidence interval

29. Continuous representation of the GBM
Independently and Identically Distributed
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)
P - value

30. Standard variable for non - normal distributions
(a^2)(variance(x)) + (b^2)(variance(y))
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
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

31. Sample covariance
[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
Variance(x) + Variance(Y) + 2*covariance(XY)
i = ln(Si/Si - 1)

32. Variance of X+Y
E(mean) = mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Var(X) + Var(Y)

33. Potential reasons for fat tails in return distributions
Returns over time for an individual asset
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
We accept a hypothesis that should have been rejected

34. GPD
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
For n>30 - sample mean is approximately normal
Choose parameters that maximize the likelihood of what observations occurring
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

35. Standard error
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

36. Expected future variance rate (t periods forward)
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!)
We accept a hypothesis that should have been rejected
Average return across assets on a given day
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

37. Econometrics
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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)
Application of mathematical statistics to economic data to lend empirical support to models
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

38. Monte Carlo Simulations
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Based on a dataset

39. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Statement of the error or precision of an estimate
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)

40. 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
95% = 1.65 99% = 2.33 For one - tailed tests
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

41. Reliability
Sample mean +/ - t*(stddev(s)/sqrt(n))
Statement of the error or precision of an estimate
Confidence set for two coefficients - two dimensional analog for the confidence interval
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

42. Importance sampling technique
Nonlinearity
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sample mean will near the population mean as the sample size increases
Attempts to sample along more important paths

43. R^2
Application of mathematical statistics to economic data to lend empirical support to models
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

44. Hazard rate of exponentially distributed random variable
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Transformed to a unit variable - Mean = 0 Variance = 1
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean of sampling distribution is the population mean

45. Consistent
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
When the sample size is large - the uncertainty about the value of the sample is very small
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Attempts to sample along more important paths

46. Type I error
We reject a hypothesis that is actually true
Has heavy tails
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

47. LFHS
P - value
Low Frequency - High Severity events
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
Concerned with a single random variable (ex. Roll of a die)

48. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
For n>30 - sample mean is approximately normal

49. Four sampling distributions

50. Confidence interval for sample mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)