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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. Variance of weighted scheme
Independently and Identically Distributed
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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

2. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance = (1/m) summation(u<n - i>^2)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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

3. Law of Large Numbers
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Sample mean will near the population mean as the sample size increases
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

4. R^2
Expected value of the sample mean is the population mean
We accept a hypothesis that should have been rejected
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance(x) + Variance(Y) + 2*covariance(XY)

5. Mean(expected value)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
We accept a hypothesis that should have been rejected
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

6. Poisson Distribution
Use historical simulation approach but use the EWMA weighting system
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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

7. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
We reject a hypothesis that is actually true
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(X=x - Y=y) = P(X=x) * P(Y=y)

8. Sample covariance
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
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!)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

9. Square root rule
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
P(X=x - Y=y) = P(X=x) * P(Y=y)

10. Type II Error
We accept a hypothesis that should have been rejected
When the sample size is large - the uncertainty about the value of the sample is very small
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

11. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
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

12. Implications of homoscedasticity
Easy to manipulate
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
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

13. Statistical (or empirical) model
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Yi = B0 + B1Xi + ui
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

14. Two ways to calculate historical volatility
Population denominator = n - Sample denominator = n - 1
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Var(X) + Var(Y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

15. Panel data (longitudinal or micropanel)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sampling distribution of sample means tend to be normal
Special type of pooled data in which the cross sectional unit is surveyed over time
Peaks over threshold - Collects dataset in excess of some threshold

16. Block maxima
Returns over time for an individual asset
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Variance(x)

17. Implied standard deviation for options
P(X=x - Y=y) = P(X=x) * P(Y=y)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

18. Monte Carlo Simulations
P(Z>t)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

19. Adjusted R^2
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

20. 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!)
Independently and Identically Distributed
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Regression can be non - linear in variables but must be linear in parameters

21. Multivariate probability
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
More than one random variable
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

22. Test for unbiasedness
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Only requires two parameters = mean and variance
When the sample size is large - the uncertainty about the value of the sample is very small
E(mean) = mean

23. LFHS
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
Low Frequency - High Severity events
95% = 1.65 99% = 2.33 For one - tailed tests
Rxy = Sxy/(Sx*Sy)

24. Exact significance level
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P - value
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

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

26. Control variates technique
Nonlinearity
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

27. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Low Frequency - High Severity events
Statement of the error or precision of an estimate
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

28. Gamma distribution
If variance of the conditional distribution of u(i) is not constant
When one regressor is a perfect linear function of the other regressors
Distribution with only two possible outcomes
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

29. EWMA
Based on an equation - P(A) = # of A/total outcomes
Among all unbiased estimators - estimator with the smallest variance is efficient
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Nonlinearity

30. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
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
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 = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

31. Hazard rate of exponentially distributed random variable
i = ln(Si/Si - 1)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many 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)

32. Reliability
Variance(x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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!)
Statement of the error or precision of an estimate

33. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Does not depend on a prior event or information
Price/return tends to run towards a long - run level
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

34. Confidence interval for sample mean
Z = (Y - meany)/(stddev(y)/sqrt(n))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

35. POT
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Peaks over threshold - Collects dataset in excess of some threshold

36. F distribution
Use historical simulation approach but use the EWMA weighting system
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Price/return tends to run towards a long - run level

37. Simplified standard (un - weighted) variance
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance = (1/m) summation(u<n - i>^2)
Variance(y)/n = variance of sample Y
Mean = np - Variance = npq - Std dev = sqrt(npq)

38. Logistic distribution
Has heavy tails
(a^2)(variance(x)) + (b^2)(variance(y))
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

39. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Sample mean will near the population mean as the sample size increases
Variance(x) + Variance(Y) + 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))

40. Four sampling distributions

41. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

42. Conditional probability functions
Regression can be non - linear in variables but must be linear in parameters
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
We reject a hypothesis that is actually true

43. Cross - sectional
Average return across assets on a given day
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

44. Pooled data
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)
Easy to manipulate
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

45. Variance of X - Y assuming dependence
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Combine to form distribution with leptokurtosis (heavy tails)
Variance(X) + Variance(Y) - 2*covariance(XY)
Price/return tends to run towards a long - run level

46. Unbiased
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean of sampling distribution is the population mean

47. Tractable
Var(X) + Var(Y)
E(XY) - E(X)E(Y)
Yi = B0 + B1Xi + ui
Easy to manipulate

48. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
More than one random variable
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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)

49. Variance of aX
(a^2)(variance(x)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Least absolute deviations estimator - used when extreme outliers are not uncommon
We reject a hypothesis that is actually true

50. Chi - squared distribution
When the sample size is large - the uncertainty about the value of the sample is very small
i = ln(Si/Si - 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
Sample mean +/ - t*(stddev(s)/sqrt(n))