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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. Unstable return distribution
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance(x)

2. Standard variable for non - normal distributions
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Contains variables not explicit in model - Accounts for randomness

3. Unbiased
Price/return tends to run towards a long - run level
Mean of sampling distribution is the population mean
Nonlinearity
Confidence level

4. Chi - squared distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

5. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sampling distribution of sample means tend to be normal
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
95% = 1.65 99% = 2.33 For one - tailed tests

6. Persistence
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

7. Mean reversion in asset dynamics
When the sample size is large - the uncertainty about the value of the sample is very small
Price/return tends to run towards a long - run level
Combine to form distribution with leptokurtosis (heavy tails)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

8. Time series data
Returns over time for an individual asset
Nonlinearity
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
P - value

9. Inverse transform method
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Population denominator = n - Sample denominator = n - 1
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

10. Direction of OVB
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Only requires two parameters = mean and variance
Random walk (usually acceptable) - Constant volatility (unlikely)

11. Discrete random variable
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)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Summation((xi - mean)^k)/n
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

12. 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
Regression can be non - linear in variables but must be linear in parameters
We accept a hypothesis that should have been rejected
Mean of sampling distribution is the population mean

13. Pooled data
E(mean) = mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

14. Consistent
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Expected value of the sample mean is the population mean
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
When the sample size is large - the uncertainty about the value of the sample is very small

15. Continuously compounded return equation
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)
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

16. Exact significance level
P - value
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

17. Variance of X - Y assuming dependence
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(X) + Variance(Y) - 2*covariance(XY)
Population denominator = n - Sample denominator = n - 1

18. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Distribution with only two possible outcomes
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

19. Overall F - statistic
Only requires two parameters = mean and variance
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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)

20. Unconditional vs conditional distributions
Special type of pooled data in which the cross sectional unit is surveyed over time
Among all unbiased estimators - estimator with the smallest variance is efficient
If variance of the conditional distribution of u(i) is not constant
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

21. Result of combination of two normal with same means
Special type of pooled data in which the cross sectional unit is surveyed over time
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Combine to form distribution with leptokurtosis (heavy tails)

22. Efficiency
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
95% = 1.65 99% = 2.33 For one - tailed tests
Among all unbiased estimators - estimator with the smallest variance is efficient

23. Continuous representation of the GBM
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Independently and Identically Distributed
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)

24. Logistic distribution
Price/return tends to run towards a long - run level
Has heavy tails
Peaks over threshold - Collects dataset in excess of some threshold
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

25. Hybrid method for conditional volatility
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)
Use historical simulation approach but use the EWMA weighting system
When the sample size is large - the uncertainty about the value of the sample is very small
Distribution with only two possible outcomes

26. GARCH
Attempts to sample along more important paths
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)
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

27. Covariance calculations using weight sums (lambda)
i = ln(Si/Si - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

28. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

29. Joint probability functions
Probability that the random variables take on certain values simultaneously
We reject a hypothesis that is actually true
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

30. Variance of sample mean
Does not depend on a prior event or information
Nonlinearity
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(y)/n = variance of sample Y

31. Central Limit Theorem(CLT)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sampling distribution of sample means tend to be normal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

32. Priori (classical) probability
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
(a^2)(variance(x)) + (b^2)(variance(y))
Based on an equation - P(A) = # of A/total outcomes
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!)

33. F distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
When the sample size is large - the uncertainty about the value of the sample is very small
Probability that the random variables take on certain values simultaneously

34. Exponential distribution
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Distribution with only two possible outcomes
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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

35. Discrete representation of the GBM
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Average return across assets on a given day
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

36. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

37. Cholesky factorization (decomposition)
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
P(Z>t)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

38. SER
Easy to manipulate
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

39. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Choose parameters that maximize the likelihood of what observations occurring
Variance reverts to a long run level
Least absolute deviations estimator - used when extreme outliers are not uncommon

40. Poisson distribution equations for mean variance and std deviation
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Choose parameters that maximize the likelihood of what observations occurring
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

41. POT
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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
Peaks over threshold - Collects dataset in excess of some threshold
Rxy = Sxy/(Sx*Sy)

42. Lognormal
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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)

43. Statistical (or empirical) model
When the sample size is large - the uncertainty about the value of the sample is very small
i = ln(Si/Si - 1)
Yi = B0 + B1Xi + ui
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

44. Central Limit Theorem
For n>30 - sample mean is approximately normal
95% = 1.65 99% = 2.33 For one - tailed tests
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
Summation((xi - mean)^k)/n

45. Two drawbacks of moving average series
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Use historical simulation approach but use the EWMA weighting system
Yi = B0 + B1Xi + ui

46. Biggest (and only real) drawback of GARCH mode
Mean = np - Variance = npq - Std dev = sqrt(npq)
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!)
Nonlinearity
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

47. Extreme Value Theory
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Regression can be non - linear in variables but must be linear in parameters
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

48. Deterministic Simulation
Statement of the error or precision of an estimate
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 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
Among all unbiased estimators - estimator with the smallest variance is efficient

49. Implied standard deviation for options
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Model dependent - Options with the same underlying assets may trade at different volatilities
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

50. Reliability
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Statement of the error or precision of an estimate