Introduction to CFA Level III Formula Sheet Equations and Latex Code-Portfolio Management

rockingdingo 2024-01-30 #Portfolio Management #CFA III #AI course

Introduction to CFA Level III Formula Sheet Equations and Latex Code-Portfolio Management

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In this blog, we will summarize the latex code for equations and the formula sheet of CFA Level III exam- Section Portfolio Management. Topics include important concepts, formulas of portfolio management, including Total Wealth, Weight of risky asset in the portfolio, Effective Annual Rate EAR, Marginal contribution to total risk MCTR, Actual contribution to total risk ACTR, Percentage of risk contributed by position, Ratio of excess return to MCTR, The Fama-French model, Objective function to maximize utility, Expected after tax return of a bond, Expected after tax return of equities, Expected after tax standard deviation, and After tax rebalancing range. Tag: Portfolio Management,CFA III,AI course

• 1.Portfolio Management
• Total Wealth
• Weight of risky asset in the portfolio
• Effective Annual Rate EAR
• Marginal contribution to total risk MCTR
• Actual contribution to total risk ACTR
• Percentage of risk contributed by position
• Ratio of excess return to MCTR
• The Fama-French model
• Objective function to maximize utility
• Expected after tax return of a bond
• Expected after tax return of equities
• Expected after tax standard deviation
• After tax rebalancing range

1.Portfolio Management

• Total Wealth

  Equation

  
  $$\text{Total Wealth} = \text{Financial Capital} + \text{Extended Capital}$$
  

  Latex Code

              \text{Total Wealth} = \text{Financial Capital} + \text{Extended Capital}
          

  Explanation

  Latex code for Total Wealth

  • $$\text{Total Wealth}$$: Total Wealth
  • $$\text{Extended Capital}$$: Extended Capital
  
  

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• Weight of risky asset in the portfolio

  Equation

  
  $$w^{*}=\frac{1}{\lambda}\frac{\mu-r_{f}}{\sigma^{2}}$$
  

  Latex Code

              w^{*}=\frac{1}{\lambda}\frac{\mu-r_{f}}{\sigma^{2}}
          

  Explanation

  Latex code for Weight of risky asset in the portfolio.

  • $$w^{*}$$: Weight of risky asset in the portfolio
  • $$\lambda$$: Investor’s degree of risk aversion
  • $$r_{f}$$: Risk-free rate of return
  • $$\mu$$: Mean expected return of a risky asset
  • $$\sigma^{2}$$: The variance of the return
  • $$Weight of the risk-free asset$$: Weight of the risk-free asset = 1 – (Weight of risky asset)
  
  

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• Effective Annual Rate EAR

  Equation

  
  $$U_{m}=E(r_{m})-0.005\lambda\sigma_{m}^{2}$$
  

  Latex Code

              U_{m}=E(r_{m})-0.005\lambda\sigma_{m}^{2}
          

  Explanation

  Latex code for Effective Annual Rate EAR.

  • $$U_{m}$$: The investors' utility for a given asset mix
  • $$E(r_{m})$$: The expected return for the given asset mix(%)
  • $$\lambda$$: The investors' degree of risk-aversion
  • $$\sigma_{m}^{2}$$: The expected variance of the given asset mix (%)
  
  

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• Marginal contribution to total risk MCTR

  Equation

  
  $$ \text{MCTR} = \text{(Beta of asset class)} \times \text{(Portfolio standard deviation)} $$
  

  Latex Code

              \text{MCTR} = \text{(Beta of asset class)} \times \text{(Portfolio standard deviation)}
          

  Explanation

  Latex code for Marginal contribution to total risk MCTR.

  • $$\text{(Beta of asset class)}$$: Beta of asset class
  • $$\text{(Portfolio standard deviation)}$$: Portfolio standard deviation
  
  

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• Actual contribution to total risk ACTR

  Equation

  
  $$ \text{ACTR} = \text{(Asset weight)} \times \text{(MCTR)} $$
  

  Latex Code

              \text{ACTR} = \text{(Asset weight)} \times \text{(MCTR)}
          

  Explanation

  Latex code for Actual contribution to total risk ACTR.

  • $$\text{(Asset weight)}$$: Asset weight
  • $$\text{(MCTR)}$$: Marginal contribution to total risk
  
  

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• Percentage of risk contributed by position

  Equation

  
  $$ \text{% of risk contributed by position} = \frac{ACTR}{Total portfolio risk} $$
  

  Latex Code

              \text{% of risk contributed by position} = \frac{ACTR}{Total portfolio risk}
          

  Explanation

  Latex code for Percentage of risk contributed by position.

  
  

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• Ratio of excess return to MCTR

  Equation

  
  $$ \text{Ratio of excess return to MCTR} = \frac{\text{Expected return − Risk free rate}}{\text{MCTR}} $$
  

  Latex Code

              \text{Ratio of excess return to MCTR} = \frac{\text{Expected return − Risk free rate}}{\text{MCTR}}
          

  Explanation

  Latex code for Ratio of excess return to MCTR.

  
  

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• The Fama-French model

  Equation

  
  $$ R_{i} - R_{f} = B_{1} (R_{m} - R_{f}) + B_{2} (SMB) + B_{3}(HML) + E_{i} $$
  

  Latex Code

              R_{i} - R_{f} = \beta_{1} (R_{m} - R_{f}) + \beta_{2} (SMB) + \beta_{3}(HML) + E_{i}
          

  Explanation

  Latex code for the Fama-French model.

  • $$R_{i}$$: Total return of a stock or portfolio i at time
  • $$R_{f}$$: Risk free rate of return at time t
  • $$R_{m}$$: Surplus return
  • $$R_{i} - R_{f}$$: Expected excess return
  • $$R_{m} - R_{f}$$: Excess return on the market portfolio
  • $$SMB$$: Size premium (small minus big)
  • $$HML$$: Value premium (high minus low)
  • $$\beta_{1},\beta_{2},\beta_{3}$$: Factor coefficients
  • $$E_{I}$$: Error term
  
  

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• Objective function to maximize utility

  Equation

  
  $$ \text{Utility} (U_{m}) = E(R_{m}) − 0.005 \times \lambda \times Var_{m} $$
  

  Latex Code

              \text{Utility} (U_{m}) = E(R_{m}) − 0.005 \times \lambda \times Var_{m}
          

  Explanation

  Latex code for Objective function to maximize utility.

  • $$ E(R_{m}) $$: Expected surplus return
  • $$ Var_{m} $$: Variance of the surplus return
  • $$ \lambda $$: Portfolio risk-aversion parameter
  
  

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• Expected after tax return of a bond

  Equation

  
  $$ r_{at} = r_{pt} (1-t) $$
  

  Latex Code

              r_{at} = r_{pt} (1-t)
          

  Explanation

  Latex code for Expected after tax return of a bond.

  • $$ r_{at} $$: The expected after-tax return
  • $$ r_{pt} $$: The expected pre-tax (gross) return
  • $$ t $$: The expected tax rate
  
  

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• Expected after tax return of equities

  Equation

  
  $$ r_{at} = p_{d} r_{pt} (1 - t_{d}) + p_{a} r_{pt} ( 1 - t_{cg}) $$
  

  Latex Code

              r_{at} = p_{d} r_{pt} (1 - t_{d}) + p_{a} r_{pt} ( 1 - t_{cg})
          

  Explanation

  Latex code for Expected after tax return of equities.

  • $$ r_{at} $$: The expected after-tax return
  • $$ p_{d} $$: The proportion of $$r_{pt}$$ attributed to dividend income
  • $$ r_{pt} $$: The expected pre-tax (gross) return
  • $$ p_{a} $$: Proportion of r_{pt} attributed to price appreciation
  • $$ t_{d} $$: The dividend tax rate
  • $$ t_{cg} $$: The capital gains tax rate
  
  

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• Expected after tax standard deviation

  Equation

  
  $$ \sigma_{at} = \sigma_{pt} (1-t) $$
  

  Latex Code

              \sigma_{at} = \sigma_{pt} (1-t)
          

  Explanation

  Latex code for Expected after tax standard deviation.

  • $$ \sigma_{at} $$: The expected after-tax standard deviation
  • $$ \sigma_{pt} $$: The expected pre-tax standard deviation
  
  

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• After tax rebalancing range

  Equation

  
  $$ \text{After tax rebalancing range} = \frac{ \text{Pre-tax rebalancing range}}{ \text{1 – Tax rate}} $$
  

  Latex Code

              \text{After tax rebalancing range} = \frac{ \text{Pre-tax rebalancing range}}{ \text{1 – Tax rate}}
          

  Explanation

  Latex code for After tax rebalancing range.

  
  

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