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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. ESS
Nonlinearity
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

2. Monte Carlo Simulations
Expected value of the sample mean is the population mean
For n>30 - sample mean is approximately normal
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

3. Variance of sample mean
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(y)/n = variance of sample Y
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

4. GPD
Choose parameters that maximize the likelihood of what observations occurring
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

5. 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
Combine to form distribution with leptokurtosis (heavy tails)
Special type of pooled data in which the cross sectional unit is surveyed over time
Application of mathematical statistics to economic data to lend empirical support to models

6. Lognormal
Model dependent - Options with the same underlying assets may trade at different volatilities
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
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
Does not depend on a prior event or information

7. Kurtosis
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Regression can be non - linear in variables but must be linear in parameters
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

8. i.i.d.
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Independently and Identically Distributed
Yi = B0 + B1Xi + ui
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)

9. Binomial distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E(XY) - E(X)E(Y)
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!)

10. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

11. P - value
(a^2)(variance(x)
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
P(Z>t)

12. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Easy to manipulate
Variance(x) + Variance(Y) + 2*covariance(XY)

13. Tractable
95% = 1.65 99% = 2.33 For one - tailed tests
Easy to manipulate
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

14. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Independently and Identically Distributed
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

15. Beta distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Does not depend on a prior event or information
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

16. Difference between population and sample 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!)
More than one random variable
We accept a hypothesis that should have been rejected
Population denominator = n - Sample denominator = n - 1

17. Cholesky factorization (decomposition)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Var(X) + Var(Y)
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

18. BLUE
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

19. Key properties of linear regression
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
Regression can be non - linear in variables but must be linear in parameters
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

20. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Regression can be non - linear in variables but must be linear in parameters
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Statement of the error or precision of an estimate

21. Exact significance level
P - value
Expected value of the sample mean is the population mean
Sampling distribution of sample means tend to be normal
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

22. Continuous random variable
Variance = (1/m) summation(u<n - i>^2)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Application of mathematical statistics to economic data to lend empirical support to models
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

23. Variance(discrete)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

24. Hybrid method for conditional volatility
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Use historical simulation approach but use the EWMA weighting system
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
When the sample size is large - the uncertainty about the value of the sample is very small

25. Chi - squared distribution
We accept a hypothesis that should have been rejected
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Expected value of the sample mean is the population mean
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

26. Poisson distribution equations for mean variance and std 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)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

27. Historical std dev
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
E(XY) - E(X)E(Y)

28. Normal distribution
SSR
Peaks over threshold - Collects dataset in excess of some threshold
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

29. Shortcomings of implied volatility
Normal - Student's T - Chi - square - F distribution
Attempts to sample along more important paths
Random walk (usually acceptable) - Constant volatility (unlikely)
Model dependent - Options with the same underlying assets may trade at different volatilities

30. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Application of mathematical statistics to economic data to lend empirical support to models
Easy to manipulate
Var(X) + Var(Y)

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

32. Continuously compounded return equation
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
i = ln(Si/Si - 1)
Returns over time for an individual asset
Statement of the error or precision of an estimate

33. Heteroskedastic
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
If variance of the conditional distribution of u(i) is not constant
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

34. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Peaks over threshold - Collects dataset in excess of some threshold
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Mean = np - Variance = npq - Std dev = sqrt(npq)

35. Pooled data
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

36. Implications of homoscedasticity
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
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
When one regressor is a perfect linear function of the other regressors
Use historical simulation approach but use the EWMA weighting system

37. Standard error for Monte Carlo replications
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
P - value

38. Variance of X - Y assuming dependence
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(y)/n = variance of sample Y
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(X) + Variance(Y) - 2*covariance(XY)

39. Cross - sectional
Average return across assets on a given day
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Application of mathematical statistics to economic data to lend empirical support to models
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

40. Test for unbiasedness
Variance(X) + Variance(Y) - 2*covariance(XY)
Among all unbiased estimators - estimator with the smallest variance is efficient
E(mean) = mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

41. Variance of weighted scheme
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

42. Confidence interval (from t)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(X) + Variance(Y) - 2*covariance(XY)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

43. Discrete random variable
Peaks over threshold - Collects dataset in excess of some threshold
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
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

44. Econometrics
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
Choose parameters that maximize the likelihood of what observations occurring
Application of mathematical statistics to economic data to lend empirical support to models
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

45. Reliability
Peaks over threshold - Collects dataset in excess of some threshold
Statement of the error or precision of an estimate
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance reverts to a long run level

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

47. Mean reversion
Nonlinearity
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
95% = 1.65 99% = 2.33 For one - tailed tests
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

48. Panel data (longitudinal or micropanel)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Special type of pooled data in which the cross sectional unit is surveyed over time
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

49. WLS
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(X) + Variance(Y) - 2*covariance(XY)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

50. Adjusted R^2
Variance(X) + Variance(Y) - 2*covariance(XY)
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
Mean of sampling distribution is the population mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2