Introduction to CFA Level II Formula Sheet Equations and Latex Code-QUANTITATIVE and ECONOMICS

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Introduction to CFA Level II Formula Sheet Equations and Latex Code-QUANTITATIVE 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, Mundell-Fleming 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 REGRESSION-Standard Error of the Estimate
• LINEAR REGRESSION-Prediction Interval
• LINEAR REGRESSION-Prediction Interval
• MULTIPLE REGRESSION
• MULTIPLE REGRESSION-Hypothesis Test
• MULTIPLE REGRESSION-p value
• MULTIPLE REGRESSION-F-stat
• MULTIPLE REGRESSION-Coefficient of Determination
• MULTIPLE REGRESSION-Adjusted R squared
• TIME SERIES ANALYSIS-Linear trend model
• TIME SERIES ANALYSIS-Log linear trend model
• TIME SERIES ANALYSIS-Autoregressive (AR)
• TIME SERIES ANALYSIS-t Test
• TIME SERIES ANALYSIS-AR(1) Mean-reverting level
• TIME SERIES ANALYSIS-Random 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
• Mundell-Fleming 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 REGRESSION-Standard Error of the Estimate

  SEE-Standard Error of the Estimate

  Equation

  
  $$SEE = \frac{\sum^{n}_{i=1} (Y_{i}-\hat{b}_{0}-\hat{b}_{1}X_{i})^{2}}{n-2}$$
  

  Latex Code

              SEE = \frac{\sum^{n}_{i=1} (Y_{i}-\hat{b}_{0}-\hat{b}_{1}X_{i})^{2}}{n-2}
          

  Explanation

  Latex code for SEE-Standard 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 REGRESSION-Prediction Interval

  Prediction Interval

  Equation

  
  $$s^{2}_{f} = s^{2}(1 + \frac{1}{n} + \frac{X-\bar{X}^2}{(n-1) s^{2}_{x}})$$
  $$\hat{Y} \pm t_{c}s_{f}$$
  

  Latex Code

              s^{2}_{f} = s^{2}(1 + \frac{1}{n} + \frac{X-\bar{X}^2}{(n-1) 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 t-value
  
  

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• MULTIPLE REGRESSION

  Confidence interval for regression coefficients

  Equation

  
  $$\hat{b}_{j} \pm (t_{c} \times S_{b_{j}})$$
  

  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 t-value
  • $$S_{b_{j}}$$: Coefficient standard error
  
  

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• MULTIPLE REGRESSION-Hypothesis Test

  Hypothesis test on each regression coefficient

  Equation

  
  $$\text{t-stat} = \frac{\text{Estimated regression coefficient−Hypothesized value of regression coefficient}}{\text{Standard error of regression coefficient}}$$
  

  Latex Code

              \text{t-stat} = \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 REGRESSION-p value

  p-value

  Equation

  
  $$p-value$$
  

  Latex Code

              p-value
          

  Explanation

  • $$p-value$$: Lowest level of significance at which we can reject the null hypothesis that the population value of the regression coefficient is zero in a two-tailed test (the smaller the p-value, the weaker the case for the null hypothesis)

• MULTIPLE REGRESSION-F-stat

  F-stat

  Equation

  
  $$\text{F-stat} = \frac{MSR}{MSE} = \frac{RSS/k}{SSE/(n-(k+1))}$$
  

  Latex Code

              \text{F-stat} = \frac{MSR}{MSE} = \frac{RSS/k}{SSE/(n-(k+1))}
          

  Explanation

  • $$\text{F-stat}$$: use a one-tailed F-test and reject null hypothesis if F-statistic 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 REGRESSION-Coefficient of Determination

  F-stat

  Equation

  
  $$R^{2} = \frac{Total variation − Unexplained variation}{Total variation} = \frac{SST-SSE}{SST} = \frac{RSS}{SST}$$
  

  Latex Code

              R^{2} = \frac{Total variation − Unexplained variation}{Total variation} = \frac{SST-SSE}{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 REGRESSION-Coefficient of Determination

  Coefficient of Determination

  Equation

  
  $$R^{2} = \frac{Total variation − Unexplained variation}{Total variation} = \frac{SST-SSE}{SST} = \frac{RSS}{SST}$$
  

  Latex Code

              R^{2} = \frac{Total variation − Unexplained variation}{Total variation} = \frac{SST-SSE}{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 REGRESSION-Adjusted R squared

  Adjusted R Squared

  Equation

  
  $$\bar{R}^{2} = 1 - \frac{n - 1}{n - k - 1}(1 - R^{2})$$
  

  Latex Code

              \bar{R}^{2} = 1 - \frac{n - 1}{n - k - 1}(1 - R^{2})
          

  Explanation

• TIME SERIES ANALYSIS-Linear trend model

  Time Series Analysis Linear Trend

  Equation

  
  $$y_{t} = b_{0} + b_{1}t + \epsilon_{t}, t=1,2,...,T$$
  

  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 ANALYSIS-Log linear trend model

  Time Series Analysis Log Linear Trend

  Equation

  
  $$\ln y_{t} = b_{0} + b_{1}t + \epsilon_{t}, t=1,2,...,T$$
  

  Latex Code

              \ln y_{t} = b_{0} + b_{1}t + \epsilon_{t}, t=1,2,...,T
          

  Explanation

  Log-linear trend model: predicts that the dependent variable exhibits exponential growth.

• TIME SERIES ANALYSIS-Autoregressive (AR)

  Equation

  
  $$x_{t} = b_{0} + b_{1}x_{t-1} + \epsilon_{t}, t=1,2,...,T$$
  

  Latex Code

              x_{t} = b_{0} + b_{1}x_{t-1} + \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 ANALYSIS-t Test

  Equation

  
  $$\text{t-stat} = \frac{Residual autocorrelation for lag}{Standard error of residual autocorrelation}$$
  

  Latex Code

              \text{t-stat} = \frac{Residual autocorrelation for lag}{Standard error of residual autocorrelation}
          

  Explanation

  T-test 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 ANALYSIS-AR(1) Mean-reverting level

  Equation

  
  $$\text{Mean Reverting Level} = x_{t} = \frac{b_{0}}{1-b_{1}}$$
  

  Latex Code

              \text{Mean Reverting Level} = x_{t} = \frac{b_{0}}{1-b_{1}}
          

  Explanation

• TIME SERIES ANALYSIS-Random Walk

  Equation

  
  $$x_{t} = x_{t-1} + \epsilon_{t}, E(\epsilon_{t}) = 0, E(\epsilon_{t}^{2})=\sigma^{2}, E(\epsilon_{t}\epsilon_{s})=0$$
  

  Latex Code

              x_{t} = x_{t-1} + \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
  • K-nearest 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

  • $$TP$$ True Positive Samples
  • $$TN$$ True Negative Samples
  • $$FP$$ False Positive Samples
  • $$FN$$ False Negative Samples

• 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.

  • Cross-rates 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 cross-rate bid price.

    Multiply ask prices to obtain the cross-rate ask price.

    Triangular arbitrage is possible if the dealer's cross-rate bid (ask) price is above (below) the interbank market's implied cross-rate 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 all-in 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

  Covered interest rate parity

  • Currency with the higher risk-free 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 foreign-domestic nominal yield spread will be determined by the foreign-domestic expected inflation rate differential

• FX carry trade

  Equation

  
  

  Latex Code

  Explanation

  FX carry trade

  • taking long positions in high-yield currencies and short positions in low-yield currencies (return distribution is peaked around the mean with negative skew and fat tails)

• Mundell-Fleming model

  Equation

  
  

  Latex Code

  Explanation

  Mundell-Fleming 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.

  Mundell-Fleming 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

  Cobb-Douglas production function

• Labor productivity growth accounting equation

  Equation

  
  $$\text{Growth rate in potential GDP} = \text{Long-term growth rate of labor force} + \text{Long-term growth rate in labor productivity}$$

  Latex Code

              \text{Growth rate in potential GDP} = \text{Long-term growth rate of labor force} + \text{Long-term 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

  Endogenous growth model

  • 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 self-sustaining 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

  Convergence

  • 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

  ECONOMICS OF REGULATION

  • Economic rationale for regulatory intervention: informational frictions (resulting in adverse selection and moral hazard) and externalities (free-rider 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