Introduction to CFA Level III Formula Sheet Equations and Latex CodePortfolio 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 FamaFrench 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 FamaFrench 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

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
$$w^{*}=\frac{1}{\lambda}\frac{\mur_{f}}{\sigma^{2}}$$
Latex Code
w^{*}=\frac{1}{\lambda}\frac{\mur_{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
 $$r_{f}$$: Riskfree rate of return
 $$\mu$$: Mean expected return of a risky asset
 $$\sigma^{2}$$: The variance of the return
 $$Weight of the riskfree asset$$: Weight of the riskfree 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 riskaversion
 $$\sigma_{m}^{2}$$: 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 FamaFrench 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 FamaFrench 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 riskaversion parameter
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Expected after tax return of a bond
Equation
$$ r_{at} = r_{pt} (1t) $$
Latex Code
r_{at} = r_{pt} (1t)
Explanation
Latex code for Expected after tax return of a bond.
 $$ r_{at} $$: The expected aftertax return
 $$ r_{pt} $$: The expected pretax (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 aftertax return
 $$ p_{d} $$: The proportion of $$r_{pt}$$ attributed to dividend income
 $$ r_{pt} $$: The expected pretax (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
$$ \sigma_{at} = \sigma_{pt} (1t) $$
Latex Code
\sigma_{at} = \sigma_{pt} (1t)
Explanation
Latex code for Expected after tax standard deviation.
 $$ \sigma_{at} $$: The expected aftertax standard deviation
 $$ \sigma_{pt} $$: The expected pretax standard deviation
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After tax rebalancing range
Equation
$$ \text{After tax rebalancing range} = \frac{ \text{Pretax rebalancing range}}{ \text{1 – Tax rate}} $$
Latex Code
\text{After tax rebalancing range} = \frac{ \text{Pretax rebalancing range}}{ \text{1 – Tax rate}}
Explanation
Latex code for After tax rebalancing range.
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