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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. Mean reversion in variance
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
P - value
Variance reverts to a long run level
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

2. Significance =1
Normal - Student's T - Chi - square - F distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(X) + Variance(Y) - 2*covariance(XY)
Confidence level

3. Sample covariance
Among all unbiased estimators - estimator with the smallest variance is efficient
Var(X) + Var(Y)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
i = ln(Si/Si - 1)

4. Joint probability functions
Probability that the random variables take on certain values simultaneously
Returns over time for an individual asset
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Confidence set for two coefficients - two dimensional analog for the confidence interval

5. Standard variable for non - normal distributions
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))
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)

6. Lognormal
P - value
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
Nonlinearity
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

7. Marginal unconditional probability function
Normal - Student's T - Chi - square - F distribution
Does not depend on a prior event or information
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

8. Biggest (and only real) drawback of GARCH mode
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Nonlinearity
Z = (Y - meany)/(stddev(y)/sqrt(n))

9. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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)
(a^2)(variance(x)) + (b^2)(variance(y))
P(X=x - Y=y) = P(X=x) * P(Y=y)

10. Normal distribution
More than one random variable
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

11. Empirical frequency
E(mean) = mean
Based on a dataset
Variance(y)/n = variance of sample Y
Independently and Identically Distributed

12. Central Limit Theorem(CLT)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sampling distribution of sample means tend to be normal
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
Sample mean will near the population mean as the sample size increases

13. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Average return across assets on a given day
Mean = np - Variance = npq - Std dev = sqrt(npq)
Choose parameters that maximize the likelihood of what observations occurring

14. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

15. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

16. Non - parametric vs parametric calculation of VaR
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(y)/n = variance of sample Y
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

17. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Nonlinearity

18. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

19. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

20. Poisson Distribution
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
Attempts to sample along more important paths
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

21. Key properties of linear regression
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
95% = 1.65 99% = 2.33 For one - tailed tests
Summation((xi - mean)^k)/n
Regression can be non - linear in variables but must be linear in parameters

22. Multivariate probability
Confidence level
More than one random variable
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Regression can be non - linear in variables but must be linear in parameters

23. Perfect multicollinearity
Among all unbiased estimators - estimator with the smallest variance is efficient
When one regressor is a perfect linear function of the other regressors
P - value
Least absolute deviations estimator - used when extreme outliers are not uncommon

24. Pooled data
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
SSR
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Returns over time for a combination of assets (combination of time series and cross - sectional data)

25. Multivariate Density Estimation (MDE)
Easy to manipulate
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Based on a dataset

26. Exponential distribution
Rxy = Sxy/(Sx*Sy)
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
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

27. Variance of X+b
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
When one regressor is a perfect linear function of the other regressors
Variance(x)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

28. Standard normal distribution
Statement of the error or precision of an estimate
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Transformed to a unit variable - Mean = 0 Variance = 1

29. Four sampling distributions

30. Two requirements of OVB
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

31. Confidence interval for sample mean
(a^2)(variance(x)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean = np - Variance = npq - Std dev = sqrt(npq)

32. Central Limit Theorem
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
For n>30 - sample mean is approximately normal
Mean of sampling distribution is the population mean

33. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Low Frequency - High Severity events
Based on an equation - P(A) = # of A/total outcomes
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)

34. Bootstrap method
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

35. Unbiased
Mean of sampling distribution is the population mean
Contains variables not explicit in model - Accounts for randomness
Z = (Y - meany)/(stddev(y)/sqrt(n))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

36. Critical z values
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Only requires two parameters = mean and variance
95% = 1.65 99% = 2.33 For one - tailed tests
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

37. Mean reversion
Variance reverts to a long run level
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(x)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

38. Variance of X+Y
For n>30 - sample mean is approximately normal
Application of mathematical statistics to economic data to lend empirical support to models
Confidence level
Var(X) + Var(Y)

39. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Population denominator = n - Sample denominator = n - 1
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
P(X=x - Y=y) = P(X=x) * P(Y=y)

40. Importance sampling technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Attempts to sample along more important paths
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)
P(X=x - Y=y) = P(X=x) * P(Y=y)

41. SER
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Var(X) + Var(Y)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

42. Sample correlation
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
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
Rxy = Sxy/(Sx*Sy)

43. Inverse transform method
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

44. Tractable
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Easy to manipulate
Concerned with a single random variable (ex. Roll of a die)

45. Skewness
Random walk (usually acceptable) - Constant volatility (unlikely)
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

46. POT
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
For n>30 - sample mean is approximately normal
Based on a dataset
Peaks over threshold - Collects dataset in excess of some threshold

47. Sample mean
Contains variables not explicit in model - Accounts for randomness
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Expected value of the sample mean is the population mean
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

48. Gamma distribution
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/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared
We reject a hypothesis that is actually true
Based on a dataset

49. Potential reasons for fat tails in return distributions
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sampling distribution of sample means tend to be normal
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
(a^2)(variance(x)) + (b^2)(variance(y))

50. Variance of weighted scheme
Independently and Identically Distributed
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Low Frequency - High Severity events
For n>30 - sample mean is approximately normal