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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. Significance =1
Based on a dataset
Confidence level
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)

2. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Z = (Y - meany)/(stddev(y)/sqrt(n))
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. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Transformed to a unit variable - Mean = 0 Variance = 1
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

4. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Special type of pooled data in which the cross sectional unit is surveyed over time
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

5. Variance of X+Y
Var(X) + Var(Y)
Based on a dataset
(a^2)(variance(x)
Regression can be non - linear in variables but must be linear in parameters

6. Perfect multicollinearity
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
When one regressor is a perfect linear function of the other regressors
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Confidence set for two coefficients - two dimensional analog for the confidence interval

7. Test for unbiasedness
E(mean) = mean
Choose parameters that maximize the likelihood of what observations occurring
Easy to manipulate
For n>30 - sample mean is approximately normal

8. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Variance = (1/m) summation(u<n - i>^2)
Random walk (usually acceptable) - Constant volatility (unlikely)
i = ln(Si/Si - 1)

9. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Confidence level

10. Empirical frequency
When one regressor is a perfect linear function of the other regressors
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
Based on a dataset
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

11. Economical(elegant)
Only requires two parameters = mean and variance
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
E(XY) - E(X)E(Y)

12. Normal distribution
Sample mean will near the population mean as the sample size increases
Distribution with only two possible outcomes
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

13. Inverse transform method
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)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Nonlinearity

14. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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

15. Pooled data
i = ln(Si/Si - 1)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Yi = B0 + B1Xi + ui
Average return across assets on a given day

16. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

17. Standard error
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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)

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

19. Direction of OVB
Distribution with only two possible outcomes
Variance(x) + Variance(Y) + 2*covariance(XY)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
(a^2)(variance(x)

20. Law of Large Numbers
Low Frequency - High Severity events
Variance(X) + Variance(Y) - 2*covariance(XY)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample mean will near the population mean as the sample size increases

21. Variance of weighted scheme
Among all unbiased estimators - estimator with the smallest variance is efficient
Based on a dataset
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

22. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

23. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Expected value of the sample mean is the population mean

24. ESS
Does not depend on a prior event or information
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance = (1/m) summation(u<n - i>^2)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

25. Conditional probability functions
Confidence set for two coefficients - two dimensional analog for the confidence interval
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Average return across assets on a given day

26. Type II Error
E(mean) = mean
Use historical simulation approach but use the EWMA weighting system
We accept a hypothesis that should have been rejected
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)

27. Poisson distribution equations for mean variance and std deviation
Nonlinearity
When the sample size is large - the uncertainty about the value of the sample is very small
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

28. Four sampling distributions

29. Overall F - statistic
Special type of pooled data in which the cross sectional unit is surveyed over time
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Model dependent - Options with the same underlying assets may trade at different volatilities
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

30. Variance of X - Y assuming dependence
Mean = np - Variance = npq - Std dev = sqrt(npq)
Based on an equation - P(A) = # of A/total outcomes
Variance(X) + Variance(Y) - 2*covariance(XY)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

31. Heteroskedastic
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
More than one random variable

32. Deterministic Simulation
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Easy to manipulate
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

33. GARCH
Yi = B0 + B1Xi + ui
Based on an equation - P(A) = # of A/total outcomes
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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. 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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
P - value

35. SER
Only requires two parameters = mean and variance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

36. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Summation((xi - mean)^k)/n
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

37. Multivariate probability
Price/return tends to run towards a long - run level
More than one random variable
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Average return across assets on a given day

38. Single variable (univariate) probability
Distribution with only two possible outcomes
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Concerned with a single random variable (ex. Roll of a die)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

39. Sample correlation
Rxy = Sxy/(Sx*Sy)
More than one random variable
Average return across assets on a given day
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

40. Variance of sample mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(y)/n = variance of sample Y
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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

41. Continuously compounded return equation
i = ln(Si/Si - 1)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Rxy = Sxy/(Sx*Sy)

42. Central Limit Theorem
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Model dependent - Options with the same underlying assets may trade at different volatilities
For n>30 - sample mean is approximately normal

43. Persistence
Contains variables not explicit in model - Accounts for randomness
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
Among all unbiased estimators - estimator with the smallest variance is efficient
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

44. What does the OLS minimize?
We accept a hypothesis that should have been rejected
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
SSR

45. Cholesky factorization (decomposition)
Based on an equation - P(A) = # of A/total outcomes
Variance(y)/n = variance of sample Y
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

46. Block maxima
P(X=x - Y=y) = P(X=x) * P(Y=y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
(a^2)(variance(x)) + (b^2)(variance(y))

47. Importance sampling technique
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to sample along more important paths
We reject a hypothesis that is actually true

48. Discrete random variable
Independently and Identically Distributed
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sample mean will near the population mean as the sample size increases

49. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Summation((xi - mean)^k)/n

50. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Easy to manipulate
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one