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

rockingdingo 2024-01-30 #CFA #CFA II #AI Courses

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

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

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

• #### 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.

• #### 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

• #### 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

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)

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))}


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}


• #### 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}


#### 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})


• #### 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.

#### 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.

#### 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)

#### 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}}


#### 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).

#### Latex Code



• #### Classification Evaluation

$$\text{Accuracy} = \frac{TP + FN}{TP + FP + TN + FN}$$ $$\text{Precision(P)} = \frac{TP}{TP+FP}$$ $$\text{Recall(R)} = \frac{TP}{TP+TN}$$ $$\text{F1 score} = \frac{2 PR}{P+R}$$

#### 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}


• ### 2.Economics

#### 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.

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.

#### Latex Code



#### Explanation

• 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.

#### 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 risk-free rate will trade at a forward discount

#### 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

• Expected appreciation/ depreciation of the currency offsets the yield differential

#### 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

• high inflation leads to currency depreciation

#### 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

• If there is real interest rate parity, the foreign-domestic nominal yield spread will be determined by the foreign-domestic expected inflation rate differential

#### Latex Code



#### Explanation

• 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)

#### Latex Code



#### Explanation

• 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.

• 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 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.

#### 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


#### 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}


#### Latex Code



#### Explanation

• 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.

#### Latex Code



#### Explanation

• 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.

#### 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 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.

#### 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.

#### Latex Code



#### Explanation

• 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