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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. Shortcomings of implied volatility
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
We accept a hypothesis that should have been rejected
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
Model dependent - Options with the same underlying assets may trade at different volatilities

2. Inverse transform method
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(x)
Concerned with a single random variable (ex. Roll of a die)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

3. Sample mean
Model dependent - Options with the same underlying assets may trade at different volatilities
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Expected value of the sample mean is the population mean
i = ln(Si/Si - 1)

4. Four sampling distributions
Normal - Student's T - Chi - square - F distribution
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(y)/n = variance of sample Y
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

5. Homoskedastic
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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 = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Based on a dataset

6. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Low Frequency - High Severity events
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

7. Continuously compounded return equation
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
Based on an equation - P(A) = # of A/total outcomes
i = ln(Si/Si - 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

8. Unbiased
Mean of sampling distribution is the population mean
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

9. 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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

10. Maximum likelihood method
Model dependent - Options with the same underlying assets may trade at different volatilities
Choose parameters that maximize the likelihood of what observations occurring
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
E(mean) = mean

11. Critical z values
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
95% = 1.65 99% = 2.33 For one - tailed tests
Sample mean +/ - t*(stddev(s)/sqrt(n))

12. Test for unbiasedness
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
E(mean) = mean
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
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)

13. Confidence interval (from t)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Based on an equation - P(A) = # of A/total outcomes

14. Priori (classical) probability
P - value
Model dependent - Options with the same underlying assets may trade at different volatilities
Based on an equation - P(A) = # of A/total outcomes
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

15. Type I error
We reject a hypothesis that is actually true
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
E(mean) = mean
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

16. POT
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
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)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

17. Continuous representation of the GBM
Sampling distribution of sample means tend to be normal
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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)
Probability that the random variables take on certain values simultaneously

18. Two ways to calculate historical volatility
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Var(X) + Var(Y)

19. Extending the HS approach for computing value of a portfolio
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Model dependent - Options with the same underlying assets may trade at different volatilities
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

20. Adjusted R^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)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
For n>30 - sample mean is approximately normal

21. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P(Z>t)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

22. Kurtosis
If variance of the conditional distribution of u(i) is not constant
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
P(Z>t)

23. Perfect multicollinearity
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
When one regressor is a perfect linear function of the other regressors
Variance = (1/m) summation(u<n - i>^2)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

24. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
We accept a hypothesis that should have been rejected
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
More than one random variable

25. Bernouli Distribution
More than one random variable
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sampling distribution of sample means tend to be normal
Distribution with only two possible outcomes

26. 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
Statement of the error or precision of an estimate
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

27. Variance of X+Y
Var(X) + Var(Y)
Mean of sampling distribution is the population mean
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Average return across assets on a given day

28. Variance of X+Y assuming dependence
95% = 1.65 99% = 2.33 For one - tailed tests
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
Variance(x) + Variance(Y) + 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon

29. Least squares estimator(m)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Only requires two parameters = mean and variance
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

30. EWMA
If variance of the conditional distribution of u(i) is not constant
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Easy to manipulate
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

31. Type II Error
Sample mean will near the population mean as the sample size increases
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Summation((xi - mean)^k)/n
We accept a hypothesis that should have been rejected

32. Multivariate probability
More than one random variable
We reject a hypothesis that is actually true
Mean of sampling distribution is the population mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

33. Poisson Distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
i = ln(Si/Si - 1)

34. R^2
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

35. Skewness
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
Concerned with a single random variable (ex. Roll of a die)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

36. Consistent
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
When the sample size is large - the uncertainty about the value of the sample is very small

37. Importance sampling technique
Attempts to sample along more important paths
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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

38. Unconditional vs conditional distributions
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

39. Discrete representation of the GBM
Based on an equation - P(A) = # of A/total outcomes
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

40. Normal distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

41. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
When one regressor is a perfect linear function of the other regressors
Statement of the error or precision of an estimate
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

42. Joint probability functions
(a^2)(variance(x)
Probability that the random variables take on certain values simultaneously
We accept a hypothesis that should have been rejected
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

43. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Average return across assets on a given day
Sample mean will near the population mean as the sample size increases
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

44. P - value
Variance(X) + Variance(Y) - 2*covariance(XY)
P(Z>t)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

45. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Does not depend on a prior event or information
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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)

46. Sample correlation
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Rxy = Sxy/(Sx*Sy)
Among all unbiased estimators - estimator with the smallest variance is efficient
Mean = np - Variance = npq - Std dev = sqrt(npq)

47. Control variates technique
P - value
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

48. Difference between population and sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
Population denominator = n - Sample denominator = n - 1
Z = (Y - meany)/(stddev(y)/sqrt(n))

49. Central Limit Theorem(CLT)
Attempts to sample along more important paths
Sampling distribution of sample means tend to be normal
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

50. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

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FRM Foundations Of Risk Management Quantitative Methods

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