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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Covariance
Confidence set for two coefficients - two dimensional analog for the confidence interval
E(XY) - E(X)E(Y)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

2. Efficiency
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
Peaks over threshold - Collects dataset in excess of some threshold
Among all unbiased estimators - estimator with the smallest variance is efficient
E(XY) - E(X)E(Y)

3. Least squares estimator(m)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Statement of the error or precision of an estimate
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

4. Persistence
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

5. Discrete random variable
Use historical simulation approach but use the EWMA weighting system
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(sample y) = (variance(y)/n)*(N - n/N - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

6. Consistent
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Low Frequency - High Severity events
When the sample size is large - the uncertainty about the value of the sample is very small
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

7. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Independently and Identically Distributed
We reject a hypothesis that is actually true

8. Variance of X+Y
SSR
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Var(X) + Var(Y)
Regression can be non - linear in variables but must be linear in parameters

9. Hazard rate of exponentially distributed random variable
When the sample size is large - the uncertainty about the value of the sample is very small
Distribution with only two possible outcomes
Sample mean will near the population mean as the sample size increases
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

10. Variance of sampling distribution of means when n<N
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Yi = B0 + B1Xi + ui
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Confidence level

11. Four sampling distributions

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12. Normal distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 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
Statement of the error or precision of an estimate
Summation((xi - mean)^k)/n

13. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Nonlinearity
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

14. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Variance(x) + Variance(Y) + 2*covariance(XY)
Concerned with a single random variable (ex. Roll of a die)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

15. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Statement of the error or precision of an estimate
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

16. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Use historical simulation approach but use the EWMA weighting system
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

17. Time series data
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Returns over time for an individual asset
Does not depend on a prior event or information

18. Unbiased
(a^2)(variance(x)) + (b^2)(variance(y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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 of sampling distribution is the population mean

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

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20. Lognormal
Price/return tends to run towards a long - run level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

21. Standard variable for non - normal distributions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Z = (Y - meany)/(stddev(y)/sqrt(n))

22. Direction of OVB
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P(Z>t)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

23. Binomial distribution equations for mean variance and std dev
i = ln(Si/Si - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)
P(Z>t)
Contains variables not explicit in model - Accounts for randomness

24. Panel data (longitudinal or micropanel)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Special type of pooled data in which the cross sectional unit is surveyed over time

25. Multivariate probability
Variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
More than one random variable
Based on an equation - P(A) = # of A/total outcomes

26. Implied standard deviation for options
Use historical simulation approach but use the EWMA weighting system
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard 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)
Low Frequency - High Severity events

27. Sample covariance
SSR
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Use historical simulation approach but use the EWMA weighting system
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

28. Unconditional vs conditional distributions
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

29. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Independently and Identically Distributed
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

30. Marginal unconditional probability function
Does not depend on a prior event or information
Random walk (usually acceptable) - Constant volatility (unlikely)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

31. Variance of sample mean
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)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
More than one random variable

32. Control variates technique
Easy to manipulate
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
(a^2)(variance(x)) + (b^2)(variance(y))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

33. Variance of X+Y assuming dependence
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Low Frequency - High Severity events
Variance(x) + Variance(Y) + 2*covariance(XY)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

34. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

35. Single variable (univariate) probability
Only requires two parameters = mean and variance
Summation((xi - mean)^k)/n
Concerned with a single random variable (ex. Roll of a die)
Z = (Y - meany)/(stddev(y)/sqrt(n))

36. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
95% = 1.65 99% = 2.33 For one - tailed tests
Only requires two parameters = mean and variance
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

37. Test for statistical independence
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
Distribution with only two possible outcomes
P(X=x - Y=y) = P(X=x) * P(Y=y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

38. Type I error
Variance = (1/m) summation(u<n - i>^2)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
We reject a hypothesis that is actually true
Variance(y)/n = variance of sample Y

39. GARCH
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
P(Z>t)
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)
SSR

40. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(X) + Variance(Y) - 2*covariance(XY)
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

41. Critical z values
Sampling distribution of sample means tend to be normal
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

42. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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

43. Variance of X - Y assuming dependence
Price/return tends to run towards a long - run level
Normal - Student's T - Chi - square - F distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
Distribution with only two possible outcomes

44. Logistic distribution
Regression can be non - linear in variables but must be linear in parameters
Has heavy tails
Low Frequency - High Severity events
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

45. EWMA
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!)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Special type of pooled data in which the cross sectional unit is surveyed over time
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

46. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Mean of sampling distribution is the population mean
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 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

47. Tractable
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Easy to manipulate
If variance of the conditional distribution of u(i) is not constant
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

48. Mean reversion in variance
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance reverts to a long run level
For n>30 - sample mean is approximately normal

49. Significance =1
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Confidence level
We accept a hypothesis that should have been rejected
Mean of sampling distribution is the population mean

50. Covariance calculations using weight sums (lambda)
Low Frequency - High Severity events
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

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FRM Foundations Of Risk Management Quantitative Methods

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