## Expected Return for a Two Asset Portfolio

The expected return of a portfolio is equal to the weighted average of the returns on individual assets in the portfolio.

R_{p} = w_{1}R_{1} + w_{2}R_{2}

- R
_{p}= expected return for the portfolio - w
_{1}= proportion of the portfolio invested in asset 1 - R
_{1}= expected return of asset 1

## Expected Variance for a Two Asset Portfolio

The variance of the portfolio is calculated as follows:

σ_{p}^{2} = w_{1}^{2}σ_{1}^{2} + w_{2}^{2}σ_{2}^{2} + 2w_{1}w_{2}Cov_{1,2}

- Cov
_{1,2}= covariance between assets 1 and 2 - Cov
_{1,2 }= ρ_{1,2}* σ_{1}* σ_{2}; where ρ = correlation between assets 1 and 2

The above equation can be rewritten as:

_{p}

^{2}= w

_{1}

^{2}σ

_{1}

^{2}+ w

_{2}

^{2}σ

_{2}

^{2}+ 2w

_{1}w

_{2}ρ

_{1,2}σ

_{1}σ

_{2}

Keep in mind that this is the calculation for portfolio variance. If a test question asks for the ** standard deviation** then you will need to take the square root of the variance calculation. Percentage values can be used in this formula for the variances, instead of decimals.

**Example**** **

The following information about a two stock portfolio is available:

| Stock A | Stock B |
---|---|---|

Amount | 20,000 | 30,000 |

Expected Returns | 12% | 20% |

Standard Deviation | 20% | 30% |

Correlation | 0.25 |

The weights for the two assets are:

w_{A}= 20,000/50,000 = 40%

w_{B}= 30,000/50,000 = 60%

**Expected Returns** = 0.40*0.12 + 0.60*0.20 = 16.8%

**Variance** = (0.40)^{2}(0.20)^{ 2} + (0.60)^{ 2} (0.30)^{ 2} + 2(0.40)(0.60)(0.25)(0.20)(0.30)

= 0.046

**Standard deviation = **Sqrt(0.046) = 0.2145 or 21.45%

## Expected Variance for a Three Asset Portfolio

σ_{p}^{2} = w_{1}^{2}σ_{1}^{2} + w_{2}^{2}σ_{2}^{2} + w_{3}^{2}σ_{3}^{2} + 2w_{1}w_{2}Cov_{1,2} + 2w_{1}w_{3}Cov_{1,3 }+ 2w_{2}w_{3}Cov_{2,3}