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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. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
E(mean) = mean
Application of mathematical statistics to economic data to lend empirical support to models
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

2. Variance of X+Y
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Least absolute deviations estimator - used when extreme outliers are not uncommon
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Var(X) + Var(Y)

3. Sample variance
P(X=x - Y=y) = P(X=x) * P(Y=y)
Model dependent - Options with the same underlying assets may trade at different volatilities
Based on an equation - P(A) = # of A/total outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

4. Variance of X+b
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(x)
Combine to form distribution with leptokurtosis (heavy tails)

5. Variance of aX
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
For n>30 - sample mean is approximately normal
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
(a^2)(variance(x)

6. Variance(discrete)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

7. Persistence
Application of mathematical statistics to economic data to lend empirical support to models
Attempts to sample along more important paths
Variance = (1/m) summation(u<n - i>^2)
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

8. Variance of aX + bY
Mean of sampling distribution is the population 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
(a^2)(variance(x)) + (b^2)(variance(y))
Low Frequency - High Severity events

9. Heteroskedastic
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(X) + Variance(Y) - 2*covariance(XY)
If variance of the conditional distribution of u(i) is not constant
For n>30 - sample mean is approximately normal

10. Unconditional vs conditional distributions
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
If variance of the conditional distribution of u(i) is not constant
Probability that the random variables take on certain values simultaneously

11. Law of Large Numbers
Least absolute deviations estimator - used when extreme outliers are not uncommon
Sample mean will near the population mean as the sample size increases
Transformed to a unit variable - Mean = 0 Variance = 1
We accept a hypothesis that should have been rejected

12. Variance of X+Y assuming dependence
Independently and Identically Distributed
Distribution with only two possible outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(x) + Variance(Y) + 2*covariance(XY)

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

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14. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Regression can be non - linear in variables but must be linear in parameters
Confidence level

15. Confidence interval for sample mean
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Normal - Student's T - Chi - square - F distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

16. Key properties of linear regression
Sample mean +/ - t*(stddev(s)/sqrt(n))
We reject a hypothesis that is actually true
Regression can be non - linear in variables but must be linear in parameters
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

17. Implied standard deviation for options
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Among all unbiased estimators - estimator with the smallest variance is efficient
Contains variables not explicit in model - Accounts for randomness
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

18. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Expected value of the sample mean is the population mean
Sample mean will near the population mean as the sample size increases

19. Covariance
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
E(XY) - E(X)E(Y)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Contains variables not explicit in model - Accounts for randomness

20. Central Limit Theorem
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
For n>30 - sample mean is approximately normal
Use historical simulation approach but use the EWMA weighting system
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

21. Consistent
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
We reject a hypothesis that is actually true
When the sample size is large - the uncertainty about the value of the sample is very small
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

22. Two requirements of OVB
Confidence level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

23. Mean reversion in variance
Variance reverts to a long run level
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(y)/n = variance of sample Y

24. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Average return across assets on a given day
P - value
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)

25. Overall F - statistic
Random walk (usually acceptable) - Constant volatility (unlikely)
Population denominator = n - Sample denominator = n - 1
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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)

26. Confidence interval (from t)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

27. Perfect multicollinearity
Variance = (1/m) summation(u<n - i>^2)
More than one random variable
When one regressor is a perfect linear function of the other regressors
Has heavy tails

28. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Distribution with only two possible outcomes
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

29. LFHS
Low Frequency - High Severity events
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

30. Sample mean
Expected value of the sample mean is the population mean
Variance(X) + Variance(Y) - 2*covariance(XY)
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)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

31. Non - parametric vs parametric calculation of VaR
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(x) + Variance(Y) + 2*covariance(XY)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

32. BLUE
Z = (Y - meany)/(stddev(y)/sqrt(n))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
SSR
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

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

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34. Reliability
Statement of the error or precision of an estimate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Special type of pooled data in which the cross sectional unit is surveyed over time
Returns over time for a combination of assets (combination of time series and cross - sectional data)

35. Panel data (longitudinal or micropanel)
Application of mathematical statistics to economic data to lend empirical support to models
Peaks over threshold - Collects dataset in excess of some threshold
Special type of pooled data in which the cross sectional unit is surveyed over time
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

36. Confidence ellipse
Attempts to sample along more important paths
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
95% = 1.65 99% = 2.33 For one - tailed tests
Confidence set for two coefficients - two dimensional analog for the confidence interval

37. Joint probability functions
Probability that the random variables take on certain values simultaneously
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Easy to manipulate
Confidence set for two coefficients - two dimensional analog for the confidence interval

38. Lognormal
SSR
P - value
Concerned with a single random variable (ex. Roll of a die)
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

39. Beta distribution
Summation((xi - mean)^k)/n
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

40. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Combine to form distribution with leptokurtosis (heavy tails)
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!)

41. Economical(elegant)
Summation((xi - mean)^k)/n
Only requires two parameters = mean and variance
Among all unbiased estimators - estimator with the smallest variance is efficient
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

42. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
For n>30 - sample mean is approximately normal
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

43. Bernouli Distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Distribution with only two possible outcomes
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
SSR

44. Poisson distribution equations for mean variance and std deviation
i = ln(Si/Si - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

45. Unbiased
Mean of sampling distribution is the population mean
Concerned with a single random variable (ex. Roll of a die)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Returns over time for an individual asset

46. Two ways to calculate historical volatility
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean of sampling distribution is the population mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

47. Simulation models
P - value
Sampling distribution of sample means tend to be normal
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

48. Normal distribution
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
95% = 1.65 99% = 2.33 For one - tailed tests
Var(X) + Var(Y)

49. Square root rule
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)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

50. Tractable
Average return across assets on a given day
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
Easy to manipulate
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