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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. Expected future variance rate (t periods forward)
Variance(x) + Variance(Y) + 2*covariance(XY)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

2. Poisson Distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
E(XY) - E(X)E(Y)
P - value
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

3. BLUE
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Use historical simulation approach but use the EWMA weighting system

4. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(X) + Variance(Y) - 2*covariance(XY)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Probability that the random variables take on certain values simultaneously

5. Economical(elegant)
Only requires two parameters = mean and variance
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

6. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

7. Mean reversion in asset dynamics
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
Among all unbiased estimators - estimator with the smallest variance is efficient

8. Confidence ellipse
(a^2)(variance(x)
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

9. Inverse transform method
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
P - value
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

10. Logistic distribution
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Has heavy tails

11. SER
For n>30 - sample mean is approximately normal
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We reject a hypothesis that is actually true
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

12. Unstable return distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We accept a hypothesis that should have been rejected
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

13. Significance =1
Confidence level
Z = (Y - meany)/(stddev(y)/sqrt(n))
Var(X) + Var(Y)
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

14. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance = (1/m) summation(u<n - i>^2)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

15. Exact significance level
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
P - value
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

16. POT
i = ln(Si/Si - 1)
Peaks over threshold - Collects dataset in excess of some threshold
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

17. Standard normal distribution
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
Transformed to a unit variable - Mean = 0 Variance = 1
P - value
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

18. Homoskedastic only F - stat
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
E(XY) - E(X)E(Y)

19. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Transformed to a unit variable - Mean = 0 Variance = 1

20. Variance of aX
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
(a^2)(variance(x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

21. What does the OLS minimize?
Variance(x)
SSR
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

22. Unbiased
If variance of the conditional distribution of u(i) is not constant
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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)
Mean of sampling distribution is the population mean

23. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
For n>30 - sample mean is approximately normal

24. Maximum likelihood method
Transformed to a unit variable - Mean = 0 Variance = 1
Choose parameters that maximize the likelihood of what observations occurring
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

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

26. Variance(discrete)
Contains variables not explicit in model - Accounts for randomness
Application of mathematical statistics to economic data to lend empirical support to models
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

27. Variance of X+Y assuming dependence
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Population denominator = n - Sample denominator = n - 1
Variance(x) + Variance(Y) + 2*covariance(XY)

28. Central Limit Theorem(CLT)
Mean of sampling distribution is the population mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
When the sample size is large - the uncertainty about the value of the sample is very small
Sampling distribution of sample means tend to be normal

29. Mean(expected value)
Based on a dataset
Variance(y)/n = variance of sample Y
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Least absolute deviations estimator - used when extreme outliers are not uncommon

30. Sample correlation
Rxy = Sxy/(Sx*Sy)
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 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
Least absolute deviations estimator - used when extreme outliers are not uncommon

31. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Application of mathematical statistics to economic data to lend empirical support to models

32. T distribution
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Average return across assets on a given day
Easy to manipulate
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

33. Block maxima
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

34. Binomial distribution
Population denominator = n - Sample denominator = n - 1
Variance(X) + Variance(Y) - 2*covariance(XY)
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
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!)

35. Hazard rate of exponentially distributed random variable
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

36. Control variates technique
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
Price/return tends to run towards a long - run level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

37. Discrete random variable
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

38. Antithetic variable technique
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

39. Continuous random variable
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Random walk (usually acceptable) - Constant volatility (unlikely)

40. Simplified standard (un - weighted) variance
Has heavy tails
Contains variables not explicit in model - Accounts for randomness
Variance = (1/m) summation(u<n - i>^2)
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

41. Sample mean
Application of mathematical statistics to economic data to lend empirical support to models
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Expected value of the sample mean is the population mean
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

42. Continuous representation of the GBM
E(mean) = mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(sample y) = (variance(y)/n)*(N - n/N - 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)

43. Key properties of linear regression
Confidence set for two coefficients - two dimensional analog for the confidence interval
Regression can be non - linear in variables but must be linear in parameters
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
Application of mathematical statistics to economic data to lend empirical support to models

44. Continuously compounded return equation
Transformed to a unit variable - Mean = 0 Variance = 1
i = ln(Si/Si - 1)
Based on a dataset
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

45. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Among all unbiased estimators - estimator with the smallest variance is efficient
Independently and Identically Distributed
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

46. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

47. Importance sampling technique
Attempts to sample along more important paths
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Among all unbiased estimators - estimator with the smallest variance is efficient

48. GPD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

49. Standard variable for non - normal distributions
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

50. Discrete representation of the GBM
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Has heavy tails