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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. Confidence ellipse
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

2. Poisson distribution equations for mean variance and std deviation
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Based on an equation - P(A) = # of A/total outcomes

3. Confidence interval (from t)
E(XY) - E(X)E(Y)
Normal - Student's T - Chi - square - F distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Sample mean +/ - t*(stddev(s)/sqrt(n))

4. Confidence interval for sample mean
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
Normal - Student's T - Chi - square - F distribution
Statement of the error or precision of an estimate
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

5. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - 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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Application of mathematical statistics to economic data to lend empirical support to models

6. Pooled data
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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)
Easy to manipulate

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

8. Priori (classical) probability
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
Based on an equation - P(A) = # of A/total outcomes
Nonlinearity
Attempts to sample along more important paths

9. Hybrid method for conditional volatility
P - value
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Population denominator = n - Sample denominator = n - 1
Use historical simulation approach but use the EWMA weighting system

10. Key properties of linear regression
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Regression can be non - linear in variables but must be linear in parameters
If variance of the conditional distribution of u(i) is not constant
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

11. Regime - switching volatility model
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
If variance of the conditional distribution of u(i) is not constant
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

12. Shortcomings of implied volatility
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

13. Overall F - statistic
Probability that the random variables take on certain values simultaneously
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

14. Control variates technique
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
Easy to manipulate
Regression can be non - linear in variables but must be linear in parameters
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

15. Standard error
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)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

16. Persistence
Sample mean will near the population mean as the sample size increases
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
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

17. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Regression can be non - linear in variables but must be linear in parameters
Peaks over threshold - Collects dataset in excess of some threshold

18. Logistic distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Has heavy tails

19. Discrete representation of the GBM
Based on a dataset
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

20. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Based on a dataset
SSR

21. Unbiased
Rxy = Sxy/(Sx*Sy)
Yi = B0 + B1Xi + ui
For n>30 - sample mean is approximately normal
Mean of sampling distribution is the population mean

22. Time series data
Returns over time for an individual asset
Average return across assets on a given day
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Among all unbiased estimators - estimator with the smallest variance is efficient

23. Unconditional vs conditional distributions
Yi = B0 + B1Xi + ui
Confidence set for two coefficients - two dimensional analog for the confidence interval
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

24. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
95% = 1.65 99% = 2.33 For one - tailed tests
SSR
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

25. Marginal unconditional probability function
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
When one regressor is a perfect linear function of the other regressors
Does not depend on a prior event or information

26. Mean reversion in variance
Variance reverts to a long run level
Nonlinearity
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

27. Cholesky factorization (decomposition)
Based on an equation - P(A) = # of A/total outcomes
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
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

28. Simulation models
Easy to manipulate
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
When one regressor is a perfect linear function of the other regressors
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

29. Reliability
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Statement of the error or precision of an estimate
Only requires two parameters = mean and variance
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

30. Type I error
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
We reject a hypothesis that is actually true
Least absolute deviations estimator - used when extreme outliers are not uncommon
(a^2)(variance(x)

31. Historical std dev
Choose parameters that maximize the likelihood of what observations occurring
Application of mathematical statistics to economic data to lend empirical support to models
Based on an equation - P(A) = # of A/total outcomes
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

32. Significance =1
P - value
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Confidence level

33. Law of Large Numbers
Least absolute deviations estimator - used when extreme outliers are not uncommon
Concerned with a single random variable (ex. Roll of a die)
Confidence level
Sample mean will near the population mean as the sample size increases

34. Central Limit Theorem(CLT)
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
Sampling distribution of sample means tend to be normal
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
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

35. Hazard rate of exponentially distributed random variable
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Nonlinearity
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

36. EWMA
Easy to manipulate
(a^2)(variance(x)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
i = ln(Si/Si - 1)

37. Homoskedastic only F - stat
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
i = ln(Si/Si - 1)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

38. Consistent
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
When the sample size is large - the uncertainty about the value of the sample is very small
Has heavy tails
Model dependent - Options with the same underlying assets may trade at different volatilities

39. Sample correlation
SSR
95% = 1.65 99% = 2.33 For one - tailed tests
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Rxy = Sxy/(Sx*Sy)

40. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 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
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
Sampling distribution of sample means tend to be normal

41. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Contains variables not explicit in model - Accounts for randomness

42. Antithetic variable technique
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Only requires two parameters = mean and variance
Variance = (1/m) summation(u<n - i>^2)

43. Variance - covariance approach for VaR of a portfolio
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Rxy = Sxy/(Sx*Sy)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Sample mean will near the population mean as the sample size increases

44. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
SSR
Sample mean will near the population mean as the sample size increases
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

45. Cross - sectional
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Average return across assets on a given day
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

46. 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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(x)

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

48. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
When the sample size is large - the uncertainty about the value of the sample is very small

49. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(x)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

50. Monte Carlo Simulations
(a^2)(variance(x)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails