Formulas: Portfolio Mathematics

Module: M05

Portfolio Expected Return

$E (R_{p}) = \sum_{i = 1}^{n} w_{i} \cdot E (R_{i})$

$C o v (R_{i}, R_{j}) = E [(R_{i} - E R_{i}) (R_{j} - E R_{j})]$

From joint probability model: $C o v (R_{A}, R_{B}) = \sum_{i} \sum_{j} P (R_{A, i}, R_{B, j}) (R_{A, i} - E R_{A}) (R_{B, j} - E R_{B})$

$ρ (R_{i}, R_{j}) = \frac{C o v ( R _{i} , R _{j} )}{σ _{i} \cdot σ _{j}}$

Also: $C o v (R_{i}, R_{j}) = ρ_{ij} \cdot σ_{i} \cdot σ_{j}$

$σ_{p}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} \cdot C o v (R_{A}, R_{B})$

$= w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} \cdot ρ_{A B} \cdot σ_{A} \cdot σ_{B}$

$σ_{p}^{2} = \sum_{i = 1}^{N} \sum_{j = 1}^{N} w_{i} w_{j} \cdot C o v (R_{i}, R_{j})$

$Z = \frac{X - μ}{σ}$

$S F R a t i o = \frac{E ( R _{p} ) - R _{L}}{σ _{p}}$

where $R_{L}$ = minimum acceptable return (threshold)

Choose portfolio with highest SFRatio.

$P (R_{p} < R_{L}) = N (- S F R a t i o)$

where $N (\cdot)$ is the standard normal CDF