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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 X - Y assuming dependence
Combine to form distribution with leptokurtosis (heavy tails)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)

2. Exponential distribution
E(mean) = mean
Regression can be non - linear in variables but must be linear in parameters
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

3. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
i = ln(Si/Si - 1)
Variance = (1/m) summation(u<n - i>^2)
Based on a dataset

4. Continuously compounded return equation
i = ln(Si/Si - 1)
Yi = B0 + B1Xi + ui
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
More than one random variable

5. SER
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

6. Normal distribution
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
We reject a hypothesis that is actually true
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

7. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
We reject a hypothesis that is actually true
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

8. Weibul distribution
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Choose parameters that maximize the likelihood of what observations occurring

9. Variance of X+b
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

10. GARCH
Among all unbiased estimators - estimator with the smallest variance is efficient
Confidence level
When one regressor is a perfect linear function of the other regressors
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)

11. SER
Variance(X) + Variance(Y) - 2*covariance(XY)
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 = (1/m) summation(u<n - i>^2)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

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

13. LFHS
Confidence set for two coefficients - two dimensional analog for the confidence interval
Average return across assets on a given day
Low Frequency - High Severity events
Statement of the error or precision of an estimate

14. Gamma distribution
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 = (1/m) summation(u<n - i>^2)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

15. Importance sampling technique
Expected value of the sample mean is the population mean
Attempts to sample along more important paths
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(X) + Variance(Y) - 2*covariance(XY)

16. Tractable
Does not depend on a prior event or information
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!)
Easy to manipulate
Least absolute deviations estimator - used when extreme outliers are not uncommon

17. Regime - switching volatility model
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Model dependent - Options with the same underlying assets may trade at different volatilities
Transformed to a unit variable - Mean = 0 Variance = 1
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

18. 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)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

19. F distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
When the sample size is large - the uncertainty about the value of the sample is very small
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

20. Mean reversion in asset dynamics
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Price/return tends to run towards a long - run level
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

21. Hazard rate of exponentially distributed random variable
Probability that the random variables take on certain values simultaneously
SSR
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
95% = 1.65 99% = 2.33 For one - tailed tests

22. Covariance calculations using weight sums (lambda)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 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

23. Standard error for Monte Carlo replications
Easy to manipulate
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
When one regressor is a perfect linear function of the other regressors

24. What does the OLS minimize?
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
SSR
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)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

25. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
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

26. Central Limit Theorem
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
For n>30 - sample mean is approximately normal
Normal - Student's T - Chi - square - F distribution
Probability that the random variables take on certain values simultaneously

27. Antithetic variable technique
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
Variance = (1/m) summation(u<n - i>^2)
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

28. Pooled data
Summation((xi - mean)^k)/n
Returns over time for a combination of assets (combination of time series and cross - sectional data)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

29. Mean reversion in variance
E(XY) - E(X)E(Y)
Variance reverts to a long run level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

30. Continuous random variable
Special type of pooled data in which the cross sectional unit is surveyed over time
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

31. Mean reversion
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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

32. Inverse transform method
Has heavy tails
Sample mean will near the population mean as the sample size increases
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

33. Shortcomings of implied volatility
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Model dependent - Options with the same underlying assets may trade at different volatilities

34. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Based on a dataset

35. Beta distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

36. Least squares estimator(m)
Based on an equation - P(A) = # of A/total outcomes
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

37. Two assumptions of square root rule
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Among all unbiased estimators - estimator with the smallest variance is efficient
Random walk (usually acceptable) - Constant volatility (unlikely)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

38. Efficiency
Statement of the error or precision of an estimate
Among all unbiased estimators - estimator with the smallest variance is efficient
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P - value

39. Block maxima
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
Statement of the error or precision of an estimate
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

40. 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
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
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

41. Variance of X+Y
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Var(X) + Var(Y)

42. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Probability that the random variables take on certain values simultaneously
Normal - Student's T - Chi - square - F distribution
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

43. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Confidence level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Model dependent - Options with the same underlying assets may trade at different volatilities

44. Simulating for VaR
Returns over time for an individual asset
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

45. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Mean = np - Variance = npq - Std dev = sqrt(npq)
i = ln(Si/Si - 1)

46. Kurtosis
Variance(y)/n = variance of sample Y
Contains variables not explicit in model - Accounts for randomness
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Expected value of the sample mean is the population mean

47. Test for unbiasedness
E(mean) = mean
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
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
We accept a hypothesis that should have been rejected

48. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Mean = np - Variance = npq - Std dev = sqrt(npq)
Yi = B0 + B1Xi + ui

49. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
95% = 1.65 99% = 2.33 For one - tailed tests
Low Frequency - High Severity events

50. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
E(mean) = mean
When the sample size is large - the uncertainty about the value of the sample is very small