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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. Weibul distribution
(a^2)(variance(x)
Distribution with only two possible outcomes
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

2. Two assumptions of square root rule
Yi = B0 + B1Xi + ui
Random walk (usually acceptable) - Constant volatility (unlikely)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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!)

3. Single variable (univariate) probability
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Easy to manipulate
Concerned with a single random variable (ex. Roll of a die)
Choose parameters that maximize the likelihood of what observations occurring

4. Importance sampling technique
Attempts to sample along more important paths
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

5. T distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Rxy = Sxy/(Sx*Sy)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

6. R^2
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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)

7. WLS
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
SSR
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

8. Variance of X+Y
Var(X) + Var(Y)
Probability that the random variables take on certain values simultaneously
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

9. Normal distribution
Sample mean will near the population mean as the sample size increases
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

10. Beta distribution
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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)

11. Bernouli Distribution
Yi = B0 + B1Xi + ui
Distribution with only two possible outcomes
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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)

12. Binomial distribution
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
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!)
Var(X) + Var(Y)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

13. Extreme Value Theory
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
(a^2)(variance(x)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

14. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(XY) - E(X)E(Y)
Does not depend on a prior event or information
(a^2)(variance(x)

15. GPD
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Regression can be non - linear in variables but must be linear in parameters

16. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

17. What does the OLS minimize?
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
SSR

18. Homoskedastic only F - stat
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 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

19. LFHS
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Distribution with only two possible outcomes
Low Frequency - High Severity events
Population denominator = n - Sample denominator = n - 1

20. 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
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
Nonlinearity
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. Skewness
i = ln(Si/Si - 1)
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
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

22. Persistence
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Does not depend on a prior event or information
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

23. Homoskedastic
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/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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

24. Hybrid method for conditional volatility
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Nonlinearity
Use historical simulation approach but use the EWMA weighting system

25. POT
Peaks over threshold - Collects dataset in excess of some threshold
For n>30 - sample mean is approximately normal
Mean of sampling distribution is the population mean
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!)

26. Confidence interval for sample mean
Combine to form distribution with leptokurtosis (heavy tails)
We accept a hypothesis that should have been rejected
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Var(X) + Var(Y)

27. Marginal unconditional probability function
P(X=x - Y=y) = P(X=x) * P(Y=y)
More than one random variable
Does not depend on a prior event or information
Variance(x)

28. Economical(elegant)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
[1/(n - 1)]*summation((Xi - X)(Yi - 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
Only requires two parameters = mean and variance

29. Two ways to calculate historical volatility
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
(a^2)(variance(x)) + (b^2)(variance(y))
Peaks over threshold - Collects dataset in excess of some threshold
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

30. Reliability
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Based on a dataset
Statement of the error or precision of an estimate
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

31. Monte Carlo Simulations
Variance(X) + Variance(Y) - 2*covariance(XY)
Has heavy tails
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

32. Limitations of R^2 (what an increase doesn't necessarily imply)

33. SER
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Distribution with only two possible outcomes
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
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

34. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Choose parameters that maximize the likelihood of what observations occurring
P(X=x - Y=y) = P(X=x) * P(Y=y)
Concerned with a single random variable (ex. Roll of a die)

35. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Combine to form distribution with leptokurtosis (heavy tails)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Transformed to a unit variable - Mean = 0 Variance = 1

36. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
i = ln(Si/Si - 1)
More than one random variable
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

37. Exact significance level
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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)
P - value

38. Continuous representation of the GBM
Z = (Y - meany)/(stddev(y)/sqrt(n))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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!)
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)

39. Least squares estimator(m)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Has heavy tails
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

40. Exponential distribution
Returns over time for an individual asset
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

41. Confidence interval (from t)
Independently and Identically Distributed
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

42. Expected future variance rate (t periods forward)
We reject a hypothesis that is actually true
Population denominator = n - Sample denominator = n - 1
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

43. Variance of X+b
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(x)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

44. Antithetic variable technique
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E(XY) - E(X)E(Y)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Choose parameters that maximize the likelihood of what observations occurring

45. Variance - covariance approach for VaR of a portfolio
We reject a hypothesis that is actually true
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Has heavy tails
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

46. Continuously compounded return equation
Use historical simulation approach but use the EWMA weighting system
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
(a^2)(variance(x)
i = ln(Si/Si - 1)

47. Regime - switching volatility model
Z = (Y - meany)/(stddev(y)/sqrt(n))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Regression can be non - linear in variables but must be linear in parameters
Only requires two parameters = mean and variance

48. Law of Large Numbers
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
If variance of the conditional distribution of u(i) is not constant
Sample mean will near the population mean as the sample size increases
(a^2)(variance(x)

49. Mean reversion in variance
Price/return tends to run towards a long - run level
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Variance reverts to a long run level

50. Lognormal
Contains variables not explicit in model - Accounts for randomness
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications