Introduction to CFA Level II Formula Sheet Equations and Latex CodeQUANTITATIVE and ECONOMICS
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In this blog, we will summarize the latex code for equations and formula sheet of CFA Level II exam, Formula Sheet Equations and Latex Code. Topics include QUANTITATIVE METHODS, such as LINEAR REGRESSION, MULTIPLE REGRESSION, TIME SERIES ANALYSIS, Machine Learning, ECONOMICS, Exchange rate, Covered interest rate parity, Uncovered interest rate parity, Relative purchasing power parity, Fisher and international Fisher effects, FX carry trade, MundellFleming model, ECONOMIC GROWTH, Growth accounting equation, Labor productivity growth accounting equation, Classical growth model (Malthusian model), Neoclassical growth model (Solow’s model), Endogenous growth model, Convergence, ECONOMICS OF REGULATION. The data source of this blog is summarized from WILEY’S CFA PROGRAM LEVEL II quicksheet. Tag: CFA,CFA II,AI Courses
 1.QUANTITATIVE METHODS
 LINEAR REGRESSIONStandard Error of the Estimate
 LINEAR REGRESSIONPrediction Interval
 LINEAR REGRESSIONPrediction Interval
 MULTIPLE REGRESSION
 MULTIPLE REGRESSIONHypothesis Test
 MULTIPLE REGRESSIONp value
 MULTIPLE REGRESSIONFstat
 MULTIPLE REGRESSIONCoefficient of Determination
 MULTIPLE REGRESSIONAdjusted R squared
 TIME SERIES ANALYSISLinear trend model
 TIME SERIES ANALYSISLog linear trend model
 TIME SERIES ANALYSISAutoregressive (AR)
 TIME SERIES ANALYSISt Test
 TIME SERIES ANALYSISAR(1) Meanreverting level
 TIME SERIES ANALYSISRandom walk
 Machine Learning
 Classification Evaluation
 2.ECONOMICS
 Exchange rate
 Marking to market a position on a currency forward
 Covered interest rate parity
 Uncovered interest rate parity
 Relative purchasing power parity
 Fisher and international Fisher effects
 FX carry trade
 MundellFleming model
 3.ECONOMIC GROWTH
 Growth accounting equation
 Labor productivity growth accounting equation
 Classical growth model (Malthusian model)
 Neoclassical growth model (Solow’s model)
 Endogenous growth model
 Convergence
 4.ECONOMICS OF REGULATION

1.QUANTITATIVE METHODS
LINEAR REGRESSIONStandard Error of the Estimate
SEEStandard Error of the EstimateEquation
$$SEE = \frac{\sum^{n}_{i=1} (Y_{i}\hat{b}_{0}\hat{b}_{1}X_{i})^{2}}{n2}$$
Latex Code
SEE = \frac{\sum^{n}_{i=1} (Y_{i}\hat{b}_{0}\hat{b}_{1}X_{i})^{2}}{n2}
Explanation
Latex code for SEEStandard Error of the Estimate
 $$X_{i}$$: Independent Variable
 $$Y_{i}$$: Target Value to predict
 $$\hat{b}_{0}$$: Bias Term
 $$\hat{b}_{1}$$: Estimated regression coefficient
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LINEAR REGRESSIONPrediction Interval
Prediction IntervalEquation
$$s^{2}_{f} = s^{2}(1 + \frac{1}{n} + \frac{X\bar{X}^2}{(n1) s^{2}_{x}})$$
$$\hat{Y} \pm t_{c}s_{f}$$
Latex Code
s^{2}_{f} = s^{2}(1 + \frac{1}{n} + \frac{X\bar{X}^2}{(n1) s^{2}_{x}}) \\ \hat{Y} \pm t_{c}s_{f}
Explanation
Latex code for Prediction Interval of Linear Regression.
 $$s^{2}_{f}$$: Standard Deviation of Forecast
 $$t_{c}$$: Statistics of critical tvalue
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MULTIPLE REGRESSION
Confidence interval for regression coefficientsEquation
$$\hat{b}_{j} \pm (t_{c} \times S_{b_{j}})$$
Latex Code
\hat{b}_{j} \pm (t_{c} \times S_{b_{j}})
Explanation
Latex code for Confidence interval for regression coefficients
 $$\hat{b}_{j}$$: Estimated regression coefficient
 $$t_{c}$$: Critical tvalue
 $$S_{b_{j}}$$: Coefficient standard error
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MULTIPLE REGRESSIONHypothesis Test
Hypothesis test on each regression coefficientEquation
$$\text{tstat} = \frac{\text{Estimated regression coefficient−Hypothesized value of regression coefficient}}{\text{Standard error of regression coefficient}}$$
Latex Code
\text{tstat} = \frac{\text{Estimated regression coefficient−Hypothesized value of regression coefficient}}{\text{Standard error of regression coefficient}}
Explanation
Latex code for Hypothesis test on each regression coefficient
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MULTIPLE REGRESSIONp value
pvalueEquation
$$pvalue$$
Latex Code
pvalue
Explanation
 $$pvalue$$: Lowest level of significance at which we can reject the null hypothesis that the population value of the regression coefficient is zero in a twotailed test (the smaller the pvalue, the weaker the case for the null hypothesis)

MULTIPLE REGRESSIONFstat
FstatEquation
$$\text{Fstat} = \frac{MSR}{MSE} = \frac{RSS/k}{SSE/(n(k+1))}$$
Latex Code
\text{Fstat} = \frac{MSR}{MSE} = \frac{RSS/k}{SSE/(n(k+1))}
Explanation
 $$\text{Fstat}$$: use a onetailed Ftest and reject null hypothesis if Fstatistic is larger than critical value
 $$\text{RSS}$$: Regression Sum of Squares
 $$\text{SSE}$$: Standard error of the estimate
 $$\text{MSR}$$: Mean Sum of Squares
 $$\text{MSE}$$: Mean Standard error of the estimate
 $$\text{SST}$$: Sum of Squares Total

MULTIPLE REGRESSIONCoefficient of Determination
FstatEquation
$$R^{2} = \frac{Total variation − Unexplained variation}{Total variation} = \frac{SSTSSE}{SST} = \frac{RSS}{SST}$$
Latex Code
R^{2} = \frac{Total variation − Unexplained variation}{Total variation} = \frac{SSTSSE}{SST} = \frac{RSS}{SST}
Explanation
 $$R^{2}$$: Coefficient of Determination, value ranging from 0 to 1
 $$\text{RSS}$$: Regression Sum of Squares
 $$\text{SSE}$$: Standard error of the estimate
 $$\text{MSR}$$: Mean Sum of Squares
 $$\text{MSE}$$: Mean Standard error of the estimate
 $$\text{SST}$$: Sum of Squares Total

MULTIPLE REGRESSIONCoefficient of Determination
Coefficient of DeterminationEquation
$$R^{2} = \frac{Total variation − Unexplained variation}{Total variation} = \frac{SSTSSE}{SST} = \frac{RSS}{SST}$$
Latex Code
R^{2} = \frac{Total variation − Unexplained variation}{Total variation} = \frac{SSTSSE}{SST} = \frac{RSS}{SST}
Explanation
 $$R^{2}$$: Coefficient of Determination, value ranging from 0 to 1
 $$\text{RSS}$$: Regression Sum of Squares
 $$\text{SSE}$$: Standard error of the estimate
 $$\text{MSR}$$: Mean Sum of Squares
 $$\text{MSE}$$: Mean Standard error of the estimate
 $$\text{SST}$$: Sum of Squares Total

MULTIPLE REGRESSIONAdjusted R squared
Adjusted R SquaredEquation
$$\bar{R}^{2} = 1  \frac{n  1}{n  k  1}(1  R^{2})$$
Latex Code
\bar{R}^{2} = 1  \frac{n  1}{n  k  1}(1  R^{2})
Explanation

TIME SERIES ANALYSISLinear trend model
Time Series Analysis Linear TrendEquation
$$y_{t} = b_{0} + b_{1}t + \epsilon_{t}, t=1,2,...,T$$
Latex Code
y_{t} = b_{0} + b_{1}t + \epsilon_{t}, t=1,2,...,T
Explanation
Linear trend model: predicts that the dependent variable grows by a constant amount in each period.

TIME SERIES ANALYSISLog linear trend model
Time Series Analysis Log Linear TrendEquation
$$\ln y_{t} = b_{0} + b_{1}t + \epsilon_{t}, t=1,2,...,T$$
Latex Code
\ln y_{t} = b_{0} + b_{1}t + \epsilon_{t}, t=1,2,...,T
Explanation
Loglinear trend model: predicts that the dependent variable exhibits exponential growth.

TIME SERIES ANALYSISAutoregressive (AR)
Equation
$$x_{t} = b_{0} + b_{1}x_{t1} + \epsilon_{t}, t=1,2,...,T$$
Latex Code
x_{t} = b_{0} + b_{1}x_{t1} + \epsilon_{t}, t=1,2,...,T
Explanation
Autoregressive (AR) time series model: use past values of the dependent variable to predict its current value. AR model must be covariance stationary and specified such that the error terms do not exhibit serial correlation and heteroskedasticity in order to be used for statistical inference.

TIME SERIES ANALYSISt Test
Equation
$$\text{tstat} = \frac{Residual autocorrelation for lag}{Standard error of residual autocorrelation}$$
Latex Code
\text{tstat} = \frac{Residual autocorrelation for lag}{Standard error of residual autocorrelation}
Explanation
Ttest for serial (auto) correlation of the error terms (model is correctly specified if all the error autocorrelations are not significantly different from 0)

TIME SERIES ANALYSISAR(1) Meanreverting level
Equation
$$\text{Mean Reverting Level} = x_{t} = \frac{b_{0}}{1b_{1}}$$
Latex Code
\text{Mean Reverting Level} = x_{t} = \frac{b_{0}}{1b_{1}}
Explanation

TIME SERIES ANALYSISRandom Walk
Equation
$$x_{t} = x_{t1} + \epsilon_{t}, E(\epsilon_{t}) = 0, E(\epsilon_{t}^{2})=\sigma^{2}, E(\epsilon_{t}\epsilon_{s})=0$$
Latex Code
x_{t} = x_{t1} + \epsilon_{t}, E(\epsilon_{t}) = 0, E(\epsilon_{t}^{2})=\sigma^{2}, E(\epsilon_{t}\epsilon_{s})=0
Explanation
Random Walk is a special AR(1) model that is not covariance stationary (undefined mean reverting level).

Machine Learning
Equation
Latex Code
Explanation
 Penalized regression
 Support Vector Machine SVM
 Knearest neighbour (KNN) is generally used to classify an new observation into an existing group based on data factors.
 Classification and Regression Tree (CART): a decision tree which can be use to visually classify observations into groups at terminal nodes.
 Ensemble learning : combines the predictions of multiple ML algorithms to classify observations. This may be different types of models run once (voting), or the same type of model run numerous times (bagging).
 Random Forest: Random forest classifier uses the bagging method by using a large number of decision tree

Classification Evaluation
$$\text{Accuracy} = \frac{TP + FN}{TP + FP + TN + FN}$$ $$\text{Precision(P)} = \frac{TP}{TP+FP}$$ $$\text{Recall(R)} = \frac{TP}{TP+TN}$$ $$\text{F1 score} = \frac{2 PR}{P+R}$$
Equation
Latex Code
\text{Accuracy} = \frac{TP + FN}{TP + FP + TN + FN}, \text{Precision(P)} = \frac{TP}{TP+FP}, \text{Recall(R)} = \frac{TP}{TP+TN}, \text{F1 score} = \frac{2 PR}{P+R}
Explanation

2.Economics
Exchange rate
Equation
$$p=\text{price currency}$$ $$b=\text{base currency}$$ $$\text{convertion}=p/b$$ $$\frac{JPY}{EUR}=\frac{JPY}{USD} \times \frac{USD}{EUR}$$Latex Code
p=\text{price currency}, b=\text{base currency}, \text{convertion}=p/b, \frac{JPY}{EUR}=\frac{JPY}{USD} \times \frac{USD}{EUR}
Explanation
 Exchange rates:Exchange rates are expressed using the convention p/b, i.e. number of units of currency p (price currency) required to purchase one unit of currency b (base currency). USD/GBP = 1.5125 means that it will take 1.5125 USD to purchase 1 GBP
 Exchange rates with bid and ask prices
For exchange rate p/b, the bid price is the price at which the client can sell currency b (base currency) to the dealer. The ask price is the price at which the client can buy currency b from the dealer.
The b/p ask price is the reciprocal of the p/b bid price.
The b/p bid price is the reciprocal of the p/b ask price.
 Crossrates with bid and ask prices
Bring the bid‒ask quotes for the exchange rates into a format such that the common (or third) currency cancels out if we multiply the exchange rates
Multiply bid prices to obtain the crossrate bid price.
Multiply ask prices to obtain the crossrate ask price.
Triangular arbitrage is possible if the dealer's crossrate bid (ask) price is above (below) the interbank market's implied crossrate ask (bid) price.

Marking to market a position on a currency forward
Equation
Latex Code
Explanation
Marking to market a position on a currency forward
 Create an equal offsetting forward position to the initial forward position.
 Determine the allin forward rate for the offsetting forward contract.
 Calculate the profit/loss on the net position as of the settlement date.
 Calculate the PV of the profit/loss.

Covered interest rate parity
Equation
$$F_{PC/BC}=S_{PC/BC} \times \frac{1+(i_{PC} \times \frac{Actual}{360})}{1+(i_{BC} \times \frac{Actual}{360})}$$ $$\text{Forward premium (discount) as a %}=\frac{F_{PC/BC}S_{PC/BC}}{S_{PC/BC}}$$Latex Code
F_{PC/BC}=S_{PC/BC} \times \frac{1+(i_{PC} \times \frac{Actual}{360})}{1+(i_{BC} \times \frac{Actual}{360})}, \text{Forward premium (discount) as a %}=\frac{F_{PC/BC}S_{PC/BC}}{S_{PC/BC}}
Explanation
 Currency with the higher riskfree rate will trade at a forward discount

Uncovered interest rate parity
Equation
$$S^{e}_{FC/DC}=S_{FC/DC} \times \frac{1+i_{FC}}{1+i_{DC}}$$Latex Code
S^{e}_{FC/DC}=S_{FC/DC} \times \frac{1+i_{FC}}{1+i_{DC}}
Explanation
Uncovered Covered interest rate parity
 Expected appreciation/ depreciation of the currency offsets the yield differential

Relative purchasing power parity
Equation
$$\text{RelativePPP}: E(S^{T}_{FC/DC})=S^{0}_{FC/DC}(\frac{1+\pi_{FC}}{1+\pi_{DC}})^{T}$$Latex Code
\text{RelativePPP}: E(S^{T}_{FC/DC})=S^{0}_{FC/DC}(\frac{1+\pi_{FC}}{1+\pi_{DC}})^{T}
Explanation
Relative purchasing power parity
 high inflation leads to currency depreciation

Fisher and international Fisher effects
Equation
$$\text{Fisher Effect}:i=r+\pi^{e}, \text{International Fisher effect}:(i_{PC}i_{DC})=(\pi^{e}_{FC}\pi^{e}_{DC})$$Latex Code
\text{Fisher Effect}:i=r+\pi^{e}, \text{International Fisher effect}:(i_{PC}i_{DC})=(\pi^{e}_{FC}\pi^{e}_{DC})
Explanation
Fisher and international Fisher effects
 If there is real interest rate parity, the foreigndomestic nominal yield spread will be determined by the foreigndomestic expected inflation rate differential

FX carry trade
Equation
Latex Code
Explanation
 taking long positions in highyield currencies and short positions in lowyield currencies (return distribution is peaked around the mean with negative skew and fat tails)

MundellFleming model
Equation
Latex Code
Explanation
MundellFleming model with high capital mobility
 A restrictive (expansionary) monetary policy under floating exchange rates will result in appreciation(depreciation) of the domestic currency.
 A restrictive (expansionary) fiscal policy under floating exchange rates will result in depreciation (appreciation) of the domestic currency.
 If monetary and fiscal policies are both restrictive or both expansionary, the overall impact on the exchange rate will be unclear.
MundellFleming model with low capital mobility (trade flows dominate)
 A restrictive (expansionary) monetary policy will lower (increase) aggregate demand, resulting in an increase (decrease) in net exports. This will cause the domestic currency to appreciate (depreciate).
 A restrictive (expansionary) fiscal policy will lower (increase) aggregate demand, resulting in an increase (decrease) in net exports. This will cause the domestic currency to appreciate (depreciate).
 If monetary and fiscal stances are not the same, the overall impact on the exchange rate will be unclear.
Monetary models of exchange rate determination (assumes output is fixed)
 Monetary approach: higher inflation due to a relative increase in domestic money supply will lead to depreciation of the domestic currency.
 Dornbusch overshooting model: in the short run, an increase in domestic money supply will lead to higher inflation and the domestic currency will decline to a level lower than its PPP value; in the long run, as domestic interest rates rise, the nominal exchange rate will recover and approach its PPP value.

3.ECONOMIC GROWTH
Growth accounting equation
Equation
$$\Delta Y/Y = \Delta A/A + \alpha \Delta K/K + (1\alpha) \Delta L/L$$Latex Code
\Delta Y/Y = \Delta A/A + \alpha \Delta K/K + (1\alpha) \Delta L/L
Explanation
CobbDouglas production function

Labor productivity growth accounting equation
Equation
$$\text{Growth rate in potential GDP} = \text{Longterm growth rate of labor force} + \text{Longterm growth rate in labor productivity}$$Latex Code
\text{Growth rate in potential GDP} = \text{Longterm growth rate of labor force} + \text{Longterm growth rate in labor productivity}
Explanation
Labor productivity growth accounting equation

Classical growth model (Malthusian model)
Equation
Latex Code
Explanation
Labor productivity growth accounting equation
 Growth in real GDP per capita is temporary: once it rises above the subsistence level, it falls due to a population explosion.
 In the long run, new technologies result in a larger (but not richer) population.

Neoclassical growth model (Solow’s model)
Equation
Latex Code
Explanation
Neoclassical growth model (Solow’s model)
 Both labor and capital are variable factors of production and suffer from diminishing marginal productivity.
 In the steady state, both capital per worker and output per worker are growing at the same rate, $$\theta/(1\alpha)$$, where $$\theta$$ is the growth rate of total factor productivity and $$\alpha$$ is the elasticity of output with respect to capital.
 Marginal product of capital is constant and equal to the real interest rate.
 Capital deepening has no effect on the growth rate of output in the steady state, which is growing at a rate of $$\theta/(1\alpha) + n$$, where n is the labor supply growth rate.

Endogenous growth model
Equation
Latex Code
Explanation
 Capital is broadened to include human and knowledge capital and R\&D.
 R\&D results in increasing returns to scale across the entire economy.
 Saving and investment can generate selfsustaining growth at a permanently higher rate as the positive externalities associated with R&D prevent diminishing marginal returns to capital.
 Capital deepening has no effect on the growth rate of output in the steady state, which is growing at a rate of $$\theta/(1\alpha) + n$$, where n is the labor supply growth rate.

Convergence
Equation
Latex Code
Explanation
 Absolute: regardless of their particular characteristics, output per capita in developing countries will eventually converge to the level of developed countries.
 Conditional: convergence in output per capita is dependent upon countries having the same savings rates, population growth rates and production functions.
 Convergence should occur more quickly for an open economy.

4. ECONOMICS OF REGULATION
Equation
Latex Code
Explanation
 Economic rationale for regulatory intervention: informational frictions (resulting in adverse selection and moral hazard) and externalities (freerider problem)
 Regulatory interdependencies: regulatory capture, regulatory competition, regulatory arbitrage
 Regulatory tools: price mechanisms (taxes and subsidies), regulatory mandates/restrictions on behaviors, provision of public goods/financing for private projects.
 Costs of regulation: regulatory burden and net regulatory burden (private costs – private benefits)
 Sunset provisions: regulators must conduct a new cost benefit analysis before regulation is renewed