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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Key properties of linear regression
Has heavy tails
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Regression can be non - linear in variables but must be linear in parameters

2. Variance of weighted scheme
Sample mean will near the population mean as the sample size increases
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Contains variables not explicit in model - Accounts for randomness

3. Variance of X+Y
Peaks over threshold - Collects dataset in excess of some threshold
Expected value of the sample mean is the population mean
Var(X) + Var(Y)
Has heavy tails

4. Consistent
Regression can be non - linear in variables but must be linear in parameters
When the sample size is large - the uncertainty about the value of the sample is very small
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance = (1/m) summation(u<n - i>^2)

5. POT
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Peaks over threshold - Collects dataset in excess of some threshold
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Returns over time for an individual asset

6. LFHS
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Low Frequency - High Severity events
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

7. EWMA
Summation((xi - mean)^k)/n
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Combine to form distribution with leptokurtosis (heavy tails)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

8. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Mean of sampling distribution is the population mean

9. Central Limit Theorem
For n>30 - sample mean is approximately normal
Price/return tends to run towards a long - run level
Attempts to sample along more important paths
Variance(X) + Variance(Y) - 2*covariance(XY)

10. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Probability that the random variables take on certain values simultaneously

11. Expected future variance rate (t periods forward)
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Combine to form distribution with leptokurtosis (heavy tails)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

12. Implied standard deviation for options
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

13. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Contains variables not explicit in model - Accounts for randomness
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Mean = np - Variance = npq - Std dev = sqrt(npq)

14. Maximum likelihood method
P(X=x - Y=y) = P(X=x) * P(Y=y)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Choose parameters that maximize the likelihood of what observations occurring

15. Time series data
Returns over time for an individual asset
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
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!)

16. ESS
95% = 1.65 99% = 2.33 For one - tailed tests
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

17. Chi - squared distribution
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
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
Use historical simulation approach but use the EWMA weighting system
P - value

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

19. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
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)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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. Unstable return 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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

21. Regime - switching volatility model
Distribution with only two possible outcomes
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Special type of pooled data in which the cross sectional unit is surveyed over time
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

22. Potential reasons for fat tails in return distributions
Among all unbiased estimators - estimator with the smallest variance is efficient
We accept a hypothesis that should have been rejected
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

23. Variance of aX
Only requires two parameters = mean and variance
(a^2)(variance(x)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

24. Multivariate probability
Special type of pooled data in which the cross sectional unit is surveyed over time
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
More than one random variable
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

25. SER
SSR
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
Has heavy tails

26. Priori (classical) probability
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Based on an equation - P(A) = # of A/total outcomes
Independently and Identically Distributed
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

27. Skewness
Summation((xi - mean)^k)/n
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(X) + Variance(Y) - 2*covariance(XY)
Based on a dataset

28. Simulation models
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
Special type of pooled data in which the cross sectional unit is surveyed over time
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

29. Binomial distribution
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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!)
Does not depend on a prior event or information

30. Continuous representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

31. Importance sampling technique
Attempts to sample along more important paths
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sample mean will near the population mean as the sample size increases

32. Mean reversion in variance
Variance reverts to a long run level
Only requires two parameters = mean and variance
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
Nonlinearity

33. Mean reversion
Regression can be non - linear in variables but must be linear in parameters
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Mean = np - Variance = npq - Std dev = sqrt(npq)

34. What does the OLS minimize?
SSR
Based on a dataset
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

35. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Based on a dataset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

36. SER
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
Var(X) + Var(Y)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

37. R^2
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

38. Result of combination of two normal with same means
Distribution with only two possible outcomes
Easy to manipulate
Combine to form distribution with leptokurtosis (heavy tails)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

39. Persistence
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Among all unbiased estimators - estimator with the smallest variance is efficient
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
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

40. Binomial distribution equations for mean variance and std dev
Contains variables not explicit in model - Accounts for randomness
Rxy = Sxy/(Sx*Sy)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

41. Variance of sampling distribution of means when n<N
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Returns over time for an individual asset
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

42. Significance =1
Attempts to sample along more important paths
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Only requires two parameters = mean and variance
Confidence level

43. Bootstrap method
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Var(X) + Var(Y)
Random walk (usually acceptable) - Constant volatility (unlikely)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

44. Shortcomings of implied volatility
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Among all unbiased estimators - estimator with the smallest variance is efficient
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

45. Critical z values
(a^2)(variance(x)) + (b^2)(variance(y))
95% = 1.65 99% = 2.33 For one - tailed tests
Expected value of the sample mean is the population mean
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)

46. Joint probability functions
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
Probability that the random variables take on certain values simultaneously
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

47. Kurtosis
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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)

48. Gamma distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

49. Test for statistical independence
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)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

50. Test for unbiasedness
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
E(mean) = mean
Distribution with only two possible outcomes