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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. Square root rule
Does not depend on a prior event or information
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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)

2. What does the OLS minimize?
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
SSR
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

3. Implied standard deviation for options
Rxy = Sxy/(Sx*Sy)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Concerned with a single random variable (ex. Roll of a die)

4. Normal distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Based on an equation - P(A) = # of A/total outcomes
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

5. Standard error
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(x) + Variance(Y) + 2*covariance(XY)

6. Test for statistical independence
Yi = B0 + B1Xi + ui
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Based on an equation - P(A) = # of A/total outcomes

7. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

8. Overall F - statistic
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Special type of pooled data in which the cross sectional unit is surveyed over time
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

9. Discrete random variable
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
i = ln(Si/Si - 1)

10. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))

11. Variance of sampling distribution of means when n<N
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
i = ln(Si/Si - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

12. Poisson distribution equations for mean variance and std deviation
Rxy = Sxy/(Sx*Sy)
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
Mean of sampling distribution is the population mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

13. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

14. Persistence
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

15. Sample variance
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
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!)
Probability that the random variables take on certain values simultaneously
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

16. Binomial distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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!)
P(X=x - Y=y) = P(X=x) * P(Y=y)

17. Consistent
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
When the sample size is large - the uncertainty about the value of the sample is very small
Easy to manipulate
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

18. Test for unbiasedness
Variance reverts to a long run level
E(mean) = mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Yi = B0 + B1Xi + ui

19. Exponential distribution
Based on an equation - P(A) = # of A/total outcomes
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

20. Covariance
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E(XY) - E(X)E(Y)
Has heavy tails
Does not depend on a prior event or information

21. Lognormal
Population denominator = n - Sample denominator = n - 1
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

22. Continuous representation of the GBM
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

23. Law of Large Numbers
Combine to form distribution with leptokurtosis (heavy tails)
Based on a dataset
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

24. Control variates technique
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

25. K - th moment
Summation((xi - mean)^k)/n
95% = 1.65 99% = 2.33 For one - tailed tests
Average return across assets on a given day
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

26. Adjusted R^2
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^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
Var(X) + Var(Y)

27. Stochastic error term
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

28. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
When one regressor is a perfect linear function of the other regressors
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
If variance of the conditional distribution of u(i) is not constant

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

30. Variance of X+b
Use historical simulation approach but use the EWMA weighting system
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x)

31. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence set for two coefficients - two dimensional analog for the confidence interval

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

33. 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
P - value
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 = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

34. Central Limit Theorem(CLT)
Distribution with only two possible outcomes
Yi = B0 + B1Xi + ui
Sampling distribution of sample means tend to be normal
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

35. Sample mean
Expected value of the sample mean is the population mean
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

36. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
If variance of the conditional distribution of u(i) is not constant
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
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)

37. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
i = ln(Si/Si - 1)
Variance(y)/n = variance of sample Y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

38. ESS
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sample mean will near the population mean as the sample size increases
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

39. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
We reject a hypothesis that is actually true
Variance(y)/n = variance of sample Y

40. Simulation models
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
E(mean) = mean
Expected value of the sample mean is the population mean
Application of mathematical statistics to economic data to lend empirical support to models

41. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Easy to manipulate
Choose parameters that maximize the likelihood of what observations occurring
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

42. Critical z values
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Peaks over threshold - Collects dataset in excess of some threshold
95% = 1.65 99% = 2.33 For one - tailed tests

43. Deterministic Simulation
Transformed to a unit variable - Mean = 0 Variance = 1
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.
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

44. Type I error
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
We reject a hypothesis that is actually true
Returns over time for a combination of assets (combination of time series and cross - sectional data)
E(mean) = mean

45. Conditional probability functions
E(mean) = mean
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

46. Variance of aX + bY
P - value
(a^2)(variance(x)) + (b^2)(variance(y))
Least absolute deviations estimator - used when extreme outliers are not uncommon
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!)

47. Variance of X+Y assuming dependence
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(x) + Variance(Y) + 2*covariance(XY)
Special type of pooled data in which the cross sectional unit is surveyed over time
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

48. Simplified standard (un - weighted) variance
Independently and Identically Distributed
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Variance = (1/m) summation(u<n - i>^2)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

49. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P(Z>t)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(x) + Variance(Y) + 2*covariance(XY)

50. Joint probability functions
Returns over time for an individual asset
We accept a hypothesis that should have been rejected
Probability that the random variables take on certain values simultaneously
Confidence set for two coefficients - two dimensional analog for the confidence interval