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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. K - th moment
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Summation((xi - mean)^k)/n
Sampling distribution of sample means tend to be normal
Only requires two parameters = mean and variance

2. Sample correlation
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
Rxy = Sxy/(Sx*Sy)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Transformed to a unit variable - Mean = 0 Variance = 1

3. Variance - covariance approach for VaR of a portfolio
Distribution with only two possible outcomes
95% = 1.65 99% = 2.33 For one - tailed tests
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

4. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Rxy = Sxy/(Sx*Sy)
When the sample size is large - the uncertainty about the value of the sample is very small
E(mean) = mean

5. Type II Error
Has heavy tails
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Peaks over threshold - Collects dataset in excess of some threshold
We accept a hypothesis that should have been rejected

6. Efficiency
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(x) + Variance(Y) + 2*covariance(XY)
Among all unbiased estimators - estimator with the smallest variance is efficient

7. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Independently and Identically Distributed
Sample mean will near the population mean as the sample size increases

8. F distribution
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
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
We accept a hypothesis that should have been rejected
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

9. Empirical frequency
Based on a dataset
Only requires two parameters = mean and variance
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

10. T distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Statement of the error or precision of an estimate
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

11. Key properties of linear regression
Mean of sampling distribution is the population mean
Among all unbiased estimators - estimator with the smallest variance is efficient
Regression can be non - linear in variables but must be linear in parameters
i = ln(Si/Si - 1)

12. Time series data
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
P(X=x - Y=y) = P(X=x) * P(Y=y)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Returns over time for an individual asset

13. Marginal unconditional probability function
Does not depend on a prior event or information
Concerned with a single random variable (ex. Roll of a die)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

14. Confidence interval for sample mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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

15. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E(XY) - E(X)E(Y)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Rxy = Sxy/(Sx*Sy)

16. Variance(discrete)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

17. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
i = ln(Si/Si - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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

18. What does the OLS minimize?
P(Z>t)
SSR
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
(a^2)(variance(x)

19. Beta distribution
Based on a dataset
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

20. Variance of X+Y
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Var(X) + Var(Y)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

21. Monte Carlo Simulations
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

22. Importance sampling technique
When the sample size is large - the uncertainty about the value of the sample is very small
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Attempts to sample along more important paths
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

23. Four sampling distributions

24. Implied standard deviation for options
Concerned with a single random variable (ex. Roll of a die)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Based on an equation - P(A) = # of A/total outcomes
(a^2)(variance(x)

25. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Average return across assets on a given day
Confidence level
Price/return tends to run towards a long - run level

26. Skewness
Combine to form distribution with leptokurtosis (heavy tails)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

27. Type I error
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
We reject a hypothesis that is actually true
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

28. Perfect multicollinearity
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
When one regressor is a perfect linear function of the other regressors

29. Heteroskedastic
Confidence level
E(XY) - E(X)E(Y)
If variance of the conditional distribution of u(i) is not constant
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. Homoskedastic only F - stat
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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 = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

31. Poisson distribution equations for mean variance and std deviation
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Independently and Identically Distributed

32. LAD
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Mean of sampling distribution is the population mean
Least absolute deviations estimator - used when extreme outliers are not uncommon

33. Two drawbacks of moving average series
Mean of sampling distribution is the population mean
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)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Population denominator = n - Sample denominator = n - 1
SSR
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

35. Deterministic Simulation
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
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
(a^2)(variance(x)
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

36. Tractable
Easy to manipulate
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Has heavy tails
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

37. Econometrics
Peaks over threshold - Collects dataset in excess of some threshold
Application of mathematical statistics to economic data to lend empirical support to models
Confidence set for two coefficients - two dimensional analog for the confidence interval
SSR

38. Confidence interval (from t)
Low Frequency - High Severity events
Sample mean +/ - t*(stddev(s)/sqrt(n))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

39. Logistic distribution
Has heavy tails
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Statement of the error or precision of an estimate
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

40. Adjusted R^2
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
When one regressor is a perfect linear function of the other regressors
Based on an equation - P(A) = # of A/total outcomes

41. Conditional probability functions
Nonlinearity
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)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

42. POT
Concerned with a single random variable (ex. Roll of a die)
Peaks over threshold - Collects dataset in excess of some threshold
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
More than one random variable

43. Variance of X - Y assuming dependence
Peaks over threshold - Collects dataset in excess of some threshold
Rxy = Sxy/(Sx*Sy)
Variance(X) + Variance(Y) - 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon

44. Potential reasons for fat tails in return distributions
More than one random variable
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
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

45. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Special type of pooled data in which the cross sectional unit is surveyed over time

46. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

47. Variance of sample mean
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Variance(y)/n = variance of sample Y
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

48. Exact significance level
P - value
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(x)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

49. Simplified standard (un - weighted) variance
Price/return tends to run towards a long - run level
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance = (1/m) summation(u<n - i>^2)
Nonlinearity

50. Variance of X+b
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(x)
Concerned with a single random variable (ex. Roll of a die)