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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. Unconditional vs conditional distributions
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
E(mean) = mean

2. Two ways to calculate historical volatility
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance = (1/m) summation(u<n - i>^2)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

3. Homoskedastic only F - stat
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

4. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Transformed to a unit variable - Mean = 0 Variance = 1

5. GPD
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

6. GEV
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 Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

7. Deterministic Simulation
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
Low Frequency - High Severity events
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
Contains variables not explicit in model - Accounts for randomness

8. Exponential distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Distribution with only two possible outcomes
Independently and Identically Distributed

9. Simplified standard (un - weighted) variance
Population denominator = n - Sample denominator = n - 1
Variance = (1/m) summation(u<n - i>^2)
When one regressor is a perfect linear function of the other regressors
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

10. Economical(elegant)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Only requires two parameters = mean and variance
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!)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

11. Mean reversion in asset dynamics
(a^2)(variance(x)) + (b^2)(variance(y))
Price/return tends to run towards a long - run level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

12. Antithetic variable technique
Easy to manipulate
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
For n>30 - sample mean is approximately normal
Nonlinearity

13. Sample correlation
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Rxy = Sxy/(Sx*Sy)
Does not depend on a prior event or information
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

14. Conditional probability functions
Average return across assets on a given day
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(X) + Variance(Y) - 2*covariance(XY)

15. Implications of homoscedasticity
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
Application of mathematical statistics to economic data to lend empirical support to models
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

16. Shortcomings of implied volatility
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Model dependent - Options with the same underlying assets may trade at different volatilities
When one regressor is a perfect linear function of the other regressors
We accept a hypothesis that should have been rejected

17. Empirical frequency
Based on a dataset
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

18. Overall F - statistic
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance = (1/m) summation(u<n - i>^2)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

19. P - value
P(Z>t)
If variance of the conditional distribution of u(i) is not constant
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sample mean will near the population mean as the sample size increases

20. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Returns over time for an individual asset
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

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

22. Mean reversion in variance
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)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance reverts to a long run level
Independently and Identically Distributed

23. Multivariate Density Estimation (MDE)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

24. Expected future variance rate (t periods forward)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

25. Central Limit Theorem
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
For n>30 - sample mean is approximately normal
Only requires two parameters = mean and variance
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

26. Poisson Distribution
Peaks over threshold - Collects dataset in excess of some threshold
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
(a^2)(variance(x)) + (b^2)(variance(y))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

27. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(x)
(a^2)(variance(x)) + (b^2)(variance(y))

28. Reliability
Application of mathematical statistics to economic data to lend empirical support to models
E(XY) - E(X)E(Y)
Statement of the error or precision of an estimate
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

29. 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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

30. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Mean = np - Variance = npq - Std dev = sqrt(npq)
Model dependent - Options with the same underlying assets may trade at different volatilities
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

31. What does the OLS minimize?
When the sample size is large - the uncertainty about the value of the sample is very small
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
SSR
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

32. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
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)
Contains variables not explicit in model - Accounts for randomness
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

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

34. GARCH
Distribution with only two possible outcomes
Does not depend on a prior event or information
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)
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

35. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
Expected value of the sample mean is the population mean
Yi = B0 + B1Xi + ui

36. Beta distribution
E(mean) = mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sample mean will near the population mean as the sample size increases
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

37. Four sampling distributions

38. Block maxima
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Transformed to a unit variable - Mean = 0 Variance = 1
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

39. Two requirements of OVB
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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!)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

40. Variance of sample mean
Only requires two parameters = mean and variance
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(y)/n = variance of sample Y
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)

41. Discrete representation of the GBM
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

42. Discrete random variable
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Special type of pooled data in which the cross sectional unit is surveyed over time
i = ln(Si/Si - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

43. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Mean = np - Variance = npq - Std dev = sqrt(npq)
E(XY) - E(X)E(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

44. Marginal unconditional probability function
Application of mathematical statistics to economic data to lend empirical support to models
Yi = B0 + B1Xi + ui
Does not depend on a prior event or information
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

45. Confidence ellipse
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Confidence set for two coefficients - two dimensional analog for the confidence interval
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

46. Potential reasons for fat tails in return distributions
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample mean +/ - t*(stddev(s)/sqrt(n))

47. LFHS
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Low Frequency - High Severity events
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

48. Confidence interval for sample mean
We reject a hypothesis that is actually true
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

49. Covariance
E(XY) - E(X)E(Y)
Average return across assets on a given day
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

50. Efficiency
Returns over time for an individual asset
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Among all unbiased estimators - estimator with the smallest variance is efficient
P - value