2017CFA
二级培训项目
Quantitative
Methods
周琪
工作职称:金程教育金融研究院CFA/FRM高级培训师
教育背景:中央财经大学国际经济学学士、澳大利亚维多利亚大学金融风 险管理学学士
工作背景:学术功底深厚、培训经验丰富,曾任课AFP、CFP多年,参与教
学研究及授课,现为金程教育CFA/FRM双证培训老师,担任CFA项目学术
研发负责人,对CFA教学产品的研发工作负责,曾亲自参与中国工商银行总
行、中国银行总行、杭州联合银行等CFA、FRM培训项目。累计课时达400 0小时,课程清晰易懂,深受学员欢迎。
服务客户:中国工商银行、中国银行、中国建设银行、杭州联合银行、杭 州银行、国泰君安证券、深圳综合开发研究院、中国CFP标准委员会、太平
Topic Weightings in CFA Level II
Session NO. Content Weightings
Study Session 1-2 Ethics & Professional Standards 10-15
Study Session 3 Quantitative Methods 5-10
Study Session 4 Economic Analysis 5-10
Study Session 5-6 Financial Statement Analysis 15-20
Study Session 7-8 Corporate Finance 5-15
Study Session 9-11 Equity Analysis 15-25
Study Session 12-13 Fixed Income Analysis 10-20
Study Session 14 Derivative Investments 5-15
Framework
Quantitative Methods
SS3 Quantitative Methods for Valuation
• R9 Correlation and regression
• R10 Multiple regression and issues in regression analysis
• R11 Time-series analysis
• R12 Excerpt
from ’’Probabilistic
Approaches: Scenario
Analysis, Decision Trees, and
Reading
9
Framework
1. Scatter Plots
2. Covariance and Correlation
3. Interpretations of Correlation Coefficients 4. Significance Test of the Correlation
5. Limitations to Correlation Analysis 6. The Basics of Simple Linear Regression 7. Interpretation of regression coefficients 8. Standard Error of Estimate & Coefficient of
Determination (R2)
9. Analysis of Variance (ANOVA)
10. Regression coefficient confidence interval 11. Hypothesis Testing about the Regression
Scatter Plots
Sample Covariance and Correlation
Covariance:
Covariance measures how one random variable moves with
another random variable. ----It captures the linear relationship.
Covariance ranges from negative infinity to positive infinity
Interpretations of Correlation Coefficients
The correlation coefficient is a measure of linear association.
It is a simple number with no unit of measurement attached, so the correlation coefficient is much easier to explain than the covariance.
Correlation coefficient Interpretation
r = +1 perfect positive correlation 0 < r < +1 positive linear correlation
r = 0 no linear correlation
−1 < r < 0 negative linear correlation
Significance Test of the Correlation
Test whether the correlation between the population of two variables is equal to zero.
H0: ρ=0; HA: ρ≠0 (Two-tailed test)
Test statistic:
Decision rule: reject H0 if t>+t critical, or t<- t critical
Conclusion: the correlation between the population of two variables is significantly different from zero.
2 2
r-0
r n-2
t=
, df = n-2
1-r
1-r
n-2
Example
The covariance between X and Y is 16. The standard deviation of X is 4 and the standard deviation of Y is 8. The sample size is 20. Test the significance of the correlation coefficient at the 5% significance level.
Solution :
The sample correlation coefficient r = 16/(4×8) = 0.5. The t-statistic can be computed as:
The critical t-value for α=5%, two-tailed test with df=18 is 2.101.
Since the test statistic of 2.45 is larger than the critical value of
20 2
0.5 2.45
1 0.25
t
Limitations to Correlation Analysis
Outliers
Outliers represent a few extreme values for sample observations.
Relative to the rest of the sample data, the value of an outlier may be extraordinarily large or small.
Limitations to Correlation Analysis
Spurious correlation
Spurious correlation refers to the appearance of a causal linear relationship
when, in fact, there is no relation. Certain data items may be highly
correlated purely by chance. That is to say, there is no economic explanation
for the relationship, which would be considered a spurious correlation.
correlation between two variables that reflects chance relationships in a particular data set,
correlation induced by a calculation that mixes each of two variables with a third (two variables that are uncorrelated may be correlated if
Limitations to Correlation Analysis
Nonlinear relationships
Correlation only measures the linear relationship between two variables,
so it dose not capture strong nonlinear relationships between variables.
For example, two variables could have a nonlinear relationship such as
Y= (1-X) 3 and the correlation coefficient would be close to zero, which is
The Basics of Simple Linear Regression
Linear regression allows you to use one variable to make predictions
about another
, test hypotheses about the relation between two
variables, and quantify the strength of the relationship between the
two variables.
Linear regression
assumes a linear relation between the dependent
and the independent variables.
The dependent variable is the variable whose variation is
explained by the independent variable. The dependent variable
is also refer to as
the explained variable, the endogenous
variable, or the predicted variable
.
The Basics of Simple Linear Regression
The simple linear regression model
Where
Yi = ith observation of the dependent variable, Y
Xi = ith observation of the independent variable, X
b0 = regression intercept term
b1 = regression slope coefficient
εi= the residual for the ith observation (also referred to as the disturbance term or error term)
n
i
X
b
b
Interpretation of regression coefficients
Interpretation of regression coefficients
The
estimated intercept coefficient
( ) is interpreted as the
value of Y when X is equal to zero.
The
estimated slope coefficient
( ) defines the sensitivity of
Y to a change in X .The estimated slope coefficient ( ) equals
covariance divided by variance of X.
Example
An estimated slope coefficient of 2 would indicate that the
dependent variable will change two units for every 1 unit change
in the independent variable.
The assumptions of the linear regression
The assumptions
A linear relationship exists between X and Y
X is not random, and the condition that X is uncorrelated with the error term can substitute the condition that X is not random.
The expected value of the error term is zero (i.e., E(εi)=0 )
The variance of the error term is constant (i.e., the error terms are homoskedastic)
The error term is uncorrelated across observations (i.e., E(εiεj)=0 for all i≠j)
Calculation of regression coefficients
Ordinary least squares (OLS)
OLS estimation is a process that estimates the population parameters Bi with corresponding values for bi that minimize the squared residuals (i.e., error terms).
the OLS sample coefficients are those that:
Example: calculate a regression coefficient
Bouvier Co. is a Canadian company that sells forestry products to several Pacific Rim customers. Bouvier’s sales are very sensitive to exchange rates. The following table shows recent annual sales (in millions of Canadian dollars) and the average exchange rate for the year (expressed as the units of foreign currency needed to buy one Canadian dollar).
Calculate the intercept and coefficient for an estimated linear
regression with the exchange rate as the independent variable and sales as the dependent variable.
Example: calculate a regression coefficient
The following table provides several useful calculations:
Example: calculate a regression coefficient
The sample mean of the exchange rate is:
The sample mean of sales is:
We want to estimate a regression equation of the form Yi = b0 + b1Xi +εi. The estimates of the slope coefficient and the intercept are
ANOVA Table
ANOVA Table
Standard error of estimate:
Coefficient of determination (R²)
explained variation=1-unexplained variation
Standard Error of Estimate
Standard Error of Estimate (SEE) measures the degree of variability of the
actual Y-values relative to the estimated Y-values from a regression equation.
SEE will be low (relative to total variability) if the relationship is very strong
and high if the relationship is weak.
The SEE gauges the “fit” of the regression line. The smaller the standard
error, the better the fit.
The SEE is the standard deviation of the error terms in the regression.
MSE
n
SSE
SEE
Coefficient Determination (R
2)
A measure of the “goodness of fit” of the regression. It is interpreted as a percentage of variation in the dependent variable explained by the independent variable. Its limits are 0≤R2≤1.
Example: R2 of 0.63 indicates that the variation of the independent
variable explains 63% of the variation in the dependent variable.
For simple linear regression, R² is equal to the squared correlation coefficient (i.e., R² = r² )
The Different between the R2 and Correlation Coefficient
The correlation coefficient indicates the sign of the relationship between two variables, whereas the coefficient of determination does not.
Example
An analyst ran a regression and got the following result:
Fill in the blanks of the ANOVA Table.
ANOVA Table df SS MSS
Regression 1 8000 ?
Error ? 2000 ?
Total 51 ? -
Coefficient t-statistic p-value
Intercept -0.5 -0.91 0.18
Regression coefficient confidence interval
Regression coefficient confidence interval
If the confidence interval at the desired level of significance dose not
include zero, the null is rejected, and the coefficient is said to be statistically different from zero.
is the standard error of the regression coefficient.
As SEE rises, also increases, and the confidence interval widens because SEE measures the variability of the data about the regression line, and the more variable the data, the less confidence there is in the regression model to estimate a coefficient.
Hypothesis Testing about Regression Coefficient
Significance test for a regression coefficient
H0: b1=The hypothesized value(usually 0)
Test Statistic:
Decision rule: reject H0 if +t critical <t, or t<- t critical
Rejection of the null means that the slope coefficient is different from the hypothesized value of b1.
Predicted Value of the Dependent Variable
Predicted values are values of the dependent variable based on the
estimated regression coefficients and a prediction about the value of the independent variable.
Point estimate
Confidence interval estimate
Limitations of Regression Analysis
Regression relations change over time
This means that the estimation equation based on data from a specific time period may not be relevant for forecasts or predictions in another time period. This is referred to as parameter instability.
The usefulness will be limited if others are also aware of and act on the relationship.
Regression assumptions are violated
Reading
10
Framework
1. The Basics of Multiple Regression 2. Interpreting the Multiple Regression
Results
3. Hypothesis Testing about the Regression Coefficient
4. Regression Coefficient F-test 5. Coefficient of Determination (R2)
6. Analysis of Variance (ANOVA) 7. Dummy variables
8. Multiple Regression Assumptions 9. Multiple Regression Assumption
Violations
The Basics of Multiple Regression
Multiple regression is regression analysis with more than one independent variable
The multiple linear regression model
Xij = ith observation of the jth independent variable
N = number of observation
K = number of independent variables
Predicted value of the dependent variable
Interpreting the Multiple Regression Results
The intercept term is the value of the dependent variable when the
independent variables are all equal to zero.
Each slope coefficient is the estimated change in the dependent variable for
a one unit change in that independent variable, holding the other
independent variables constant. That’s why the slope coefficients in a
Multiple Regression Assumptions
The assumptions of the multiple linear regression
A linear relationship exists between the dependent and independent variables
The independent variables are not random ( OR X is not correlated with error terms). There is no exact linear relation between any two or more independent variables
The expected value of the error term is zero (i.e., E(εi)=0 )
The variance of the error term is constant (i.e., the error terms are homoskedastic)
The error term is uncorrelated across observations (i.e., E(εiεj)=0 for all i≠j)
Dummy variables
To use qualitative variables as independent variables in a regression
The qualitative variable can only take on two values, 0 and 1
If we want to distinguish between n categories, we need n−1 dummy
Dummy variables
Interpreting the coefficients
Example: EPSt = b0 + b1Q1t + b2Q2t + b3Q3t + ϵ
EPSt = a quarterly observation of earnings per share
The intercept term, represents the average value of EPS for the fourth quarter.
The slope coefficient on each dummy variable estimates the difference in earnings per share (on average) between the respective quarter (i.e., quarter 1, 2, or 3) and the omitted quarter (the fourth quarter in this case).
Analysis of Variance (ANOVA)
ANOVA Table
Standard error of estimate
Coefficient of determination (R²)
Adjusted R
2 R2 and adjusted R2
R2 by itself may not be a reliable measure of the explanatory power of
the multiple regression model. This is because R2 almost always
increases as variables are added to the model, even if the marginal contribution of the new variables is not statistically significant.
Hypothesis Testing about Regression Coefficient
Significance test for a regression coefficient
H0: bj=0
Test statistic: df = n-k-1
p-value: the smallest significance level for which the null hypothesis can be rejected
Reject H0 if p-value<α
Fail to reject H0 if p-value>α
Regression coefficient confidence interval
Regression Coefficient F-test
An F-statistic assesses how well the set of independent variables, as a group, explains the variation in the dependent variable.
An F-test is used to test whether at least one slope coefficient is significantly different from zero
Define hypothesis:
Regression Coefficient F-test
Decision rule
Reject H0 : if F (test-statistic) > F c (critical value)
Rejection of the null hypothesis at a stated level of significance indicates that at least one of the coefficients is significantly different than zero, which is interpreted to mean that at least one of the independent variables in the regression model makes a significant contribution on the explanation of the dependent variable.
The F-test here is always a one-tailed test.
Unbiased and consistent estimator
An unbiased estimator is one for which the expected value of the estimator
is equal to the parameter you are trying to estimate.
If not, called as unreliable.
A consistent estimator is one for which the accuracy of the parameter
Multiple Regression Assumption Violations
Heteroskedasticity 异方差
Heteroskedasticity refers to the situation that the variance of the error term is not constant (i.e., the error terms are not homoskedastic)
Unconditional heteroskedasticity occurs when the heteroskedasticity is not related to the level of the independent variables, which means that it dose not systematically increase or decrease with the change in the
value of the independent variables. It usually causes no major problems with the regression.
Conditional heteroskedasticity is heteroskedasticity, that is, variance of error term is related to the level of the independent variables.
Multiple Regression Assumption Violations
Effect of Heteroskedasticity on Regression Analysis
Not affect the consistency of regression parameter estimators
Consistency: the larger the number of sample, the lower probability of error.
The coefficient estimates (the ) are not affected.
The standard errors are usually unreliable estimates.
If the standard errors are too small, but the coefficient estimates themselves are not affected, the t-statistics will be too large and the null hypothesis of no statistical significance is rejected too often (一
类错误).
The opposite will be true if the standard errors are too large. (二类错
误)
The F-test is also unreliable.
ˆ
j
Multiple Regression Assumption Violations
Detecting Heteroskedasticity
Two methods to detect heteroskedasticity
residual scatter plots (residual vs. independent variable)
the Breusch-Pagen χ² test
H0: No heteroskedasticity, one-tailed test
Chi-square test: BP = n×Rresidual², df=k
注意:以误差项squred residuals和X做回归,Rresidual²是此回
归的决定系数
Decision rule: BP test statistic should be small (χ²分布表)
Multiple Regression Assumption Violations
Serial correlation (autocorrelation)序列相关,自相关
Serial correlation (autocorrelation) refers to the situation that the error terms are correlated with one another
Serial correlation is often found in time series data
Positive serial correlation exists when a positive regression error in one time period increases the probability of observing regression error for the next time period.
Negative serial correlation occurs when a positive error in one
Multiple Regression Assumption Violations
Effect of Serial correlation on Regression Analysis
Positive serial correlation → Type I error & F-test unreliable
Not affect the consistency of estimated regression coefficients.
Because of the tendency of the data to cluster together from observation to observation, positive serial correlation typically results in coefficient standard errors that are too small, which will cause the computed t-statistics to be larger.
Positive serial correlation is much more common in economic and financial data, so we focus our attention on its effects.
Multiple Regression Assumption Violations
Detecting Serial correlation
Two methods to detect serial correlation
residual scatter plots
the Durbin-Watson test
H0: No serial correlation
DW ≈ 2×(1−r) Decision rule
Multiple Regression Assumption Violations
Detecting Serial correlation
Two methods to detect serial correlation
residual scatter plots
the Durbin-Watson test
H0: No positive serial correlation
DW ≈ 2×(1−r) Decision rule
Inconclusive Fail to reject null hypothesis of no positive serial correlation
Reject H0, conclude positive serial
Multiple Regression Assumption Violations
Methods to Correct Serial correlation
adjusting the coefficient standard errors (e.g., Hansen method): the Hansen method also corrects for conditional heteroskedaticity.
The White-corrected standard errors are preferred if only heteroskedasticity is a problem.
Multiple Regression Assumption Violations
Multicollinearity
Multicollinearity refers to the situation that two or more
independent variables are highly correlated with each other
In practice, multicollinearity is often a matter of degree rather
than of absence or presence.
Two methods to detect multicollinearity
t-tests indicate that none of the individual coefficients is
significantly different than zero, while the F-test indicates
overall significance and the R² is high.
the absolute value of the sample correlation between any
Summary of assumption violations
Assumption
violation Impact Detection Solution
Conditional Heteroskeda
sticity
Type I /II error
① Residual scatter plots ② Breusch-Pagen χ²-test
(BP = n×R²)
①robust standard errors
(White-corrected standard errors)
② generalized least squares
Positive serial correlation
Type I error
① Residual scatter plots ② Durbin-Watson test
(DW≈2×(1−r))
①robust standard errors
(Hansen method)
②Improve the specification
Model Misspecification
There are three broad categories of model misspecification, or ways in which the regression model can be specified incorrectly, each with several
subcategories:
1. The functional form can be misspecified.
Important variables are omitted.
Variables should be transformed.
Data is improperly pooled.
2. Time series misspecification. (Explanatory variables are correlated with the error term in time series models.)
A lagged dependent variable is used as an independent variable with serially correlated errors.
A function of the dependent variable is used as an independent variable ("forecasting the past").
Qualitative Dependent Variables
Qualitative dependent variable is a dummy variable that takes on a value of either zero or one
Probit and logit model: Application of these models results in estimates of the probability that the event occurs (e.g., probability of default).
A probit model based on the normal distribution, while a logit model is based on the logistic distribution.
Both models must be estimated using maximum likelihood methods
(极大似然估计).
These coefficients relate the independent variables to the likelihood of an event occurring, such as a merger, bankruptcy, or default.
Discriminant models yields a linear function, similar to a regression
Credit Analysis
Z
–
score
Z = 1.2 A + 1.4 B + 3.3 C + 0.6 D + 1.0 E
Where:
A = WC / TA
B = RE / TA
C = EBIT / TA
D = MV of Equity / BV of Debt
E = Revenue / TA
Reading
11
Framework
1. Trend Models
2. Autoregressive Models (AR)
3. Random Walks
4. Autoregressive Conditional Heteroskedasticity (ARCH)
5. Regression with More Than One Time Series
Trend Models
Linear trend model
yt=b0+b1t+εt
Same as linear regression, except for that the independent variable is
time t (t=1, 2, 3, ……)
Trend Models
Log-linear trend model
yt=e(b0+b1t)
Ln(yt )=b0+b1t+εt
Model the natural log of the series using a linear trend
Trend Models
Factors that Determine Which Model is Best
A linear trend model may be appropriate if the data points appear to be
equally distributed above and below the regression line (inflation rate data).
A log-linear model may be more appropriate if the data plots with a non-linear (curved) shape, then the residuals from a linear trend model will be persistently positive or negative for a period of time (stock
indices and stock prices).
Limitations of Trend Model
Usually the time series data exhibit serial correlation, which means that the model is not appropriate for the time series, causing inconsistent b0 and b1
Autoregressive Models (AR)
An autoregressive model uses past values of dependent variables as independent variables
AR(p) model
AR (p): AR model of order p (p indicates the number of lagged values that the autoregressive model will include).
For example, a model with two lags is referred to as a second-order autoregressive model or an AR (2) model.
0 1 1 2 2
...
t t t p t p t
Autoregressive Models (AR)
Forecasting With an Autoregressive Model
Chain rule of forecasting
A one-period-ahead forecast for an AR (1) model is determined in the following manner:
Likewise, a two-step-ahead forecast for an AR (1) model is calculated as:
1 0 1
t t
x
b
b x
2 0 1 1
t t
x
b
b x
Autoregressive Models (AR)
Forecasting With an Autoregressive Model, we should prove:
No autocorrelation
Covariance-stationary series
Autocorrelation
Autocorrelation in an AR model
Whenever we refer to autocorrelation without qualification, we mean autocorrelation of time series itself rather than autocorrelation of the error term .
Detecting autocorrelation in an AR model
Compute the autocorrelations of the residual
t-tests to see whether the residual autocorrelations differ significantly from 0,
If the residual autocorrelations differ significantly from 0, the model is not correctly specified, so we may need to modify it (e.g. seasonality) Correction: add lagged values
Autocorrelation
Seasonality – a special question
Time series shows regular patterns of movement within the year
The seasonal autocorrelation of the residual will differ significantly from
0
We should uses a seasonal lag in an AR model
Example
Suppose we decide to use an autoregressive model with a seasonal lag because of the seasonal autocorrelation in the previous problem. We are modeling quarterly data, so we estimate Equation:
(ln Salest – ln Salest–1) = b0 + b1(ln Salest–1 – ln Salest–2) + b2(ln Salest–4 – ln Salest–5) + εt.
Using the information in Table 1, determine if the model is correctly specified.
Table 1
Table 1.Log Differenced Sales
Coefficient Standard Error t-Statistic
Intercept 0.0121 0.0053 2.3055 Lag 1 –0.0839 0.0958 –0.8757 Lag 4 0.6292 0.0958 6.5693
Autocorrelations of the Residual
Lag Autocorrelation Standard Error t-Statistic
Regression Statistics
Example
Answer
At the 0.05 significance level, with 68 observations and three parameters, this model has 65 degrees of freedom. The critical value of the t-statistic needed to reject the null hypothesis is thus about 2.0. The absolute value of the t-statistic for each
autocorrelation is below 0.60 (less than 2.0), so we cannot reject the null hypothesis that each autocorrelation is not significantly different from 0. We have determined that the model is correctly specified.
If sales grew by 1 percent last quarter and by 2 percent four quarters ago, then the model predicts that sales growth this
quarter will be 0.0121 – 0.0839 ln(1.01) + 0.6292 ln(1.02) = e0.02372 –
Covariance-stationary
Covariance-stationary series
Statistical inference based on OLS estimates for a lagged time series model assumes that the time series is covariance stationary.
Three conditions for covariance stationary
Constant and finite expected value of the time series
Constant and finite variance of the time series
Constant and finite covariance with leading or lagged values
Stationary in the past does not guarantee stationary in the future
Covariance-stationary
Mean reversion
A time series exhibits mean reversion if it has a tendency to move towards its mean
For an AR(1) model, the mean reverting level is:
Covariance-stationary
Instability of regression coefficients
Financial and economic relationships are dynamic
Models estimated with shorter time series are usually more stable than
those with longer time series
Random Walks
Random walk
Random walk without a drift
Simple random walk: xt =xt-1+εt (b0=0 and b1=1)
The best forecast of xt is xt-1
Random walk with a drift
xt=b0+xt-1+εt (b0≠0, b1=1)
The time series is expected to increase/decrease by a constant amount
Features
A random walk has an undefined mean reverting level
A time series must have a finite mean reverting level to be covariance stationary
Unit root test
The unit root test of nonstationarity
The time series is said to have a unit root if the lag coefficient is equal to one
A common t-test of the hypothesis that b1=1 is invalid to test the unit root, however, it is not often the case.
Dickey-Fuller test (DF test) to test the unit root
Start with an AR(1) model xt=b0+b1 xt-1+εt
Subtract xt-1 from both sides xt-xt-1 =b0+(b1 –1)xt-1+εt
xt-xt-1 =b0+gxt-1+εt
Unit root correction
If a time series appears to have a unit root
One method that is often successful is to first-difference the time series
(as discussed previously) and try to model the first-differenced series as
an autoregressive time series.
First differencing
Define yt as yt = xt - xt-1 =εt
This is an AR(1) model yt = b0 + b1 yt-1 +εt ,whereb0=b1=0
Autoregressive Conditional Heteroskedasticity
Heteroskedasticity refers to the situation that the variance of the error term
is not constant.
Test whether a time series is ARCH(1)
If the coefficient a1 is significantly different from 0, the time series is
ARCH(1), If a time-series model has ARCH(1) errors, then the variance of
the errors in period t + 1 can be predicted in period t.
If ARCH exists,
多元回归中用BP test
2 2
0 1 1
t
a
a
tu
tCompare forecasting power with RMSE
Comparing forecasting model performance
In-sample forecasts are within the range of data (i.e., time period) used to estimate the model, which for a time series is known as the sample or test period.
Out-of-sample forecasts are made outside. In other words, we compare how accurate a model is in forecasting the y variable value for a time period outside the period used to develop the model.
Regression with More Than One Time Series
In linear regression, if any time series contains a unit root, OLS may be invalid
Use DF tests for each of the time series to detect unit root, we will have 3 possible scenarios
None of the time series has a unit root: we can use multiple regression
At least one time series has a unit root while at least one time series does not: we cannot use multiple regression
Each time series has a unit root: we need to establish whether the time series are cointegrated.
If conintegrated, can estimate the long-term relation between the two series (but may not be the best model of the short-term
Regression with More Than One Time Series
Use the Dickey-Fuller Engle-Granger test (DF-EG test) to test the
cointegration
H0: no cointegration Ha: cointegration
If we cannot reject the null, we cannot use multiple regression
Steps in Time-Series Forecasting
画出散点图,判断序列是否有趋势
Does series have a trend?
No
a linear trend
指数趋势
an exponential trend
判断是否有季节性因素
Seasonality?
Steps in Time-Series Forecasting
Is series Covariance Stationary?
以差额法重新组建序列
Take First Differences
Reading
12
Framework
1. Simulation
Simulation
Steps in Simulation
Determine “probabilistic” variables
Define probability distributions for these variables
Historical data
Cross sectional data
Statistical distribution and parameters
Check for correlation across variables
Simulation
Advantage of using simulation in decision making
Better input estimation
A distribution for expected value rather than a point estimate
Simulations with Constraints
Book value constraints
Regulatory capital restrictions
Financial service firms
Negative book value for equity
Earnings and cash flow constraints
Either internally or externally imposed
Market value constraints
Simulation
Issues in using simulation
GIGO
Real data may not fit distributions
Non-stationary distributions
Comparing the Approaches
Choose scenario analysis, decision trees, or simulations
Selective versus full risk analysis
Type of risk
Discrete risk vs. Continuous risk
Concurrent risk vs. Sequential risk
Correlation across risk
Correlated risks are difficult to model in decision trees Risk type and Probabilistic Approaches
Discrete/ Continuous
Correlated/ Independent
Sequential/
Concurrent Risk approach Discrete Correlated Sequential decision trees
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