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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)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Contains variables not explicit in model - Accounts for randomness

2. Cross - sectional
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Average return across assets on a given day

3. Bootstrap method
P - value
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Statement of the error or precision of an estimate
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

4. Exponential distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Expected value of the sample mean is the population mean
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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

5. Monte Carlo Simulations
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Use historical simulation approach but use the EWMA weighting system

6. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
P(X=x - Y=y) = P(X=x) * P(Y=y)

7. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
We reject a hypothesis that is actually true
Low Frequency - High Severity events
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

8. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Based on an equation - P(A) = # of A/total outcomes
More than one random variable
Combine to form distribution with leptokurtosis (heavy tails)

9. What does the OLS minimize?
Has heavy tails
SSR
If variance of the conditional distribution of u(i) is not constant
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

10. Biggest (and only real) drawback of GARCH mode
When one regressor is a perfect linear function of the other regressors
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
Nonlinearity
E(mean) = mean

11. Regime - switching volatility model
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
Concerned with a single random variable (ex. Roll of a die)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

12. Central Limit Theorem
For n>30 - sample mean is approximately normal
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
More than one random variable
Does not depend on a prior event or information

13. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Low Frequency - High Severity events
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!)
Variance(X) + Variance(Y) - 2*covariance(XY)

14. Priori (classical) probability
Use historical simulation approach but use the EWMA weighting system
When one regressor is a perfect linear function of the other regressors
Based on an equation - P(A) = # of A/total outcomes
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

15. Beta distribution
P - value
Variance reverts to a long run level
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

16. Hazard rate of exponentially distributed random variable
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Confidence set for two coefficients - two dimensional analog for the confidence interval

17. Economical(elegant)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Only requires two parameters = mean and variance
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
E(XY) - E(X)E(Y)

18. POT
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance reverts to a long run level
When one regressor is a perfect linear function of the other regressors
Peaks over threshold - Collects dataset in excess of some threshold

19. Heteroskedastic
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
(a^2)(variance(x)
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

20. Multivariate Density Estimation (MDE)
Easy to manipulate
Probability that the random variables take on certain values simultaneously
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

21. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
(a^2)(variance(x)
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

22. 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)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

23. Multivariate probability
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Confidence set for two coefficients - two dimensional analog for the confidence interval
More than one random variable

24. Direction of OVB
More than one random variable
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Least absolute deviations estimator - used when extreme outliers are not uncommon

25. 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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

26. LAD
E(mean) = mean
Least absolute deviations estimator - used when extreme outliers are not uncommon
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

27. Exact significance level
Summation((xi - mean)^k)/n
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
P - value

28. SER
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Mean of sampling distribution 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

29. Confidence interval (from t)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sample mean +/ - t*(stddev(s)/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

30. Binomial distribution
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!)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

31. Bernouli Distribution
Based on an equation - P(A) = # of A/total outcomes
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Distribution with only two possible outcomes
When the sample size is large - the uncertainty about the value of the sample is very small

32. Sample covariance
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

33. Sample variance
Contains variables not explicit in model - Accounts for randomness
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

34. Deterministic Simulation
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

35. Continuous random variable
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Mean of sampling distribution is the population mean
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
i = ln(Si/Si - 1)

36. Perfect multicollinearity
E(XY) - E(X)E(Y)
When one regressor is a perfect linear function of the other regressors
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

37. Mean reversion in variance
Variance reverts to a long run level
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
Low Frequency - High Severity events
Variance(x)

38. Hybrid method for conditional volatility
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
When the sample size is large - the uncertainty about the value of the sample is very small
Use historical simulation approach but use the EWMA weighting system
Average return across assets on a given day

39. Antithetic variable technique
(a^2)(variance(x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Population denominator = n - Sample denominator = n - 1

40. Continuously compounded return equation
i = ln(Si/Si - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Attempts to sample along more important paths
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

41. Significance =1
P - value
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
If variance of the conditional distribution of u(i) is not constant
Confidence level

42. Efficiency
Variance(y)/n = variance of sample Y
Among all unbiased estimators - estimator with the smallest variance is efficient
Average return across assets on a given day
Variance reverts to a long run level

43. Adjusted R^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
Variance(x) + Variance(Y) + 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
E(XY) - E(X)E(Y)

44. Variance(discrete)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Random walk (usually acceptable) - Constant volatility (unlikely)

45. Maximum likelihood method
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Choose parameters that maximize the likelihood of what observations occurring
Random walk (usually acceptable) - Constant volatility (unlikely)
When the sample size is large - the uncertainty about the value of the sample is very small

46. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Var(X) + Var(Y)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

47. Overall F - statistic
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Low Frequency - High Severity events
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

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

49. Joint probability functions
Probability that the random variables take on certain values simultaneously
(a^2)(variance(x)
Has heavy tails
Contains variables not explicit in model - Accounts for randomness

50. Two ways to calculate historical volatility
P(Z>t)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y