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FRM Foundations Of Risk Management Quantitative Methods

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1. SER
Variance(y)/n = variance of sample Y
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and 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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

2. Variance of sample mean
Variance(y)/n = variance of sample Y
Special type of pooled data in which the cross sectional unit is surveyed over time
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

3. Multivariate probability
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
More than one random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

4. Unconditional vs conditional distributions
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Confidence level

5. Variance of X+Y assuming dependence
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance(x) + Variance(Y) + 2*covariance(XY)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

6. Persistence
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
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(x) + Variance(Y) + 2*covariance(XY)
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)

7. Sample mean
Among all unbiased estimators - estimator with the smallest variance is efficient
(a^2)(variance(x)
Expected value of the sample mean is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

8. ESS
Application of mathematical statistics to economic data to lend empirical support to models
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Average return across assets on a given day
Easy to manipulate

9. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
(a^2)(variance(x)
Statement of the error or precision of an estimate

10. Two ways to calculate historical volatility
Regression can be non - linear in variables but must be linear in parameters
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
P - value

11. Shortcomings of implied volatility
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Model dependent - Options with the same underlying assets may trade at different volatilities

12. Variance of X+b
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
Variance(x)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

13. Simulation models
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Variance(x) + Variance(Y) + 2*covariance(XY)

14. Economical(elegant)
Choose parameters that maximize the likelihood of what observations occurring
Only requires two parameters = mean and variance
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Has heavy tails

15. Time series data
Returns over time for an individual asset
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

16. Beta distribution
Summation((xi - mean)^k)/n
SSR
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

17. Exact significance level
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance reverts to a long run level
Sampling distribution of sample means tend to be normal
P - value

18. Variance of aX
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
(a^2)(variance(x)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

19. Variance of X+Y
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Var(X) + Var(Y)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

20. Inverse transform method
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sampling distribution of sample means tend to be normal
Based on an equation - P(A) = # of A/total outcomes

21. Reliability
(a^2)(variance(x)
Statement of the error or precision of an estimate
95% = 1.65 99% = 2.33 For one - tailed tests
Only requires two parameters = mean and variance

22. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sample mean will near the population mean as the sample size increases
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on a dataset

23. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

24. Hazard rate of exponentially distributed random variable
Var(X) + Var(Y)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Summation((xi - mean)^k)/n
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

25. Panel data (longitudinal or micropanel)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Summation((xi - mean)^k)/n
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Special type of pooled data in which the cross sectional unit is surveyed over time

26. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

27. Confidence interval (from t)
Confidence level
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
E(mean) = mean
Sample mean +/ - t*(stddev(s)/sqrt(n))

28. Heteroskedastic
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
If variance of the conditional distribution of u(i) is not constant
Confidence level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

29. Statistical (or empirical) model
Confidence level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Yi = B0 + B1Xi + ui
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

30. Stochastic error term
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
Mean = lambda - Variance = lambda - Std dev = sqrt(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)
Contains variables not explicit in model - Accounts for randomness

31. Perfect multicollinearity
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
Distribution with only two possible outcomes
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
When one regressor is a perfect linear function of the other regressors

32. Unbiased
Mean = np - Variance = npq - Std dev = sqrt(npq)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

33. GARCH
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(sample y) = (variance(y)/n)*(N - n/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)
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

34. Implied standard deviation for options
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Nonlinearity

35. Two requirements of OVB
Has heavy tails
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sample mean +/ - t*(stddev(s)/sqrt(n))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

36. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
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
Choose parameters that maximize the likelihood of what observations occurring

37. Binomial distribution equations for mean variance and std dev
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Mean = np - Variance = npq - Std dev = sqrt(npq)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

38. 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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Confidence set for two coefficients - two dimensional analog for the confidence interval

39. Pooled data
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Var(X) + Var(Y)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

40. BLUE
If variance of the conditional distribution of u(i) is not constant
We reject a hypothesis that is actually true
Nonlinearity
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

41. Efficiency
Concerned with a single random variable (ex. Roll of a die)
Among all unbiased estimators - estimator with the smallest variance is efficient
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Rxy = Sxy/(Sx*Sy)

42. Key properties of linear regression
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(x)
Regression can be non - linear in variables but must be linear in parameters
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

43. Sample correlation
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Rxy = Sxy/(Sx*Sy)
P(Z>t)

44. Test for unbiasedness
Sample mean will near the population mean as the sample size increases
When one regressor is a perfect linear function of the other regressors
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
E(mean) = mean

45. Sample covariance
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

46. Unstable return distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

47. Homoskedastic
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Yi = B0 + B1Xi + ui
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

48. Adjusted R^2
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
Z = (Y - meany)/(stddev(y)/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
Attempts to sample along more important paths

49. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
i = ln(Si/Si - 1)
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
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

50. Covariance calculations using weight sums (lambda)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Sample mean will near the population mean as the sample size increases
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

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