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Expected Return and Variance for a Two Asset Portfolio | Finance Train

Lesson 3 of 29

Expected Return and Variance for a Two Asset Portfolio

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_1R_1 + w_2R_2

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Mean-Variance Analysis Assumptions

The Minimum Variance Frontier & Efficient Frontier

Portfolio Management

29 lessons

Lessons

CFA Level 2: Portfolio Management – Introduction

Mean-Variance Analysis Assumptions

Expected Return and Variance for a Two Asset Portfolio

The Minimum Variance Frontier & Efficient Frontier

Diversification Benefits

The Capital Allocation Line – Introducing the Risk-free Asset

The Capital Market Line

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Adding an Asset to a Portfolio – Improving the Minimum Variance Frontier

The Market Model for a Security’s Returns

Adjusted and Unadjusted Beta

Multifactor Models

Arbitrage Portfolio Theory (APT) – A Multifactor Macroeconomic Model

Risk Factors and Tracking Portfolios

Markowitz, MPT, and Market Efficiency

International Capital Market Integration

Domestic CAPM and Extended CAPM

Changes in Real Exchange Rates

International CAPM (ICAPM) - Beyond Extended CAPM

Measuring Currency Exposure

Company Stock Value Responses to Changes in Real Exchange Rates

ICAPM vs. Domestic CAPM

The J-Curve – Impact of Exchange Rate Changes on National Economies

Moving Exchange Rates and Equity Markets

Impacts of Market Segmentation on ICAPM

Justifying Active Portfolio Management

The Treynor-Black Model

Portfolio Management Process

The Investor Policy Statement

w

2

R2

$R_p$ = 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:

\sigma_{p}^2 = w_{1}^2\sigma_{1}^2 + w_{2}^2\sigma_{2}^2 + 2w_{1}w_{2}Cov_{1,2}

$Cov_{1,2}$ = covariance between assets 1 and 2
$\text{Cov}_{1,2} = \rho_{1,2} \cdot \sigma_{1} \cdot \sigma_{2}$ ; where ρ = correlation between assets 1 and 2

The above equation can be rewritten as:

\sigma_{p}^2 = w_{1}^2\sigma_{1}^2 + w_{2}^2\sigma_{2}^2 + 2w_{1}w_{2} \rho_{1,2} \sigma_{1} \sigma_{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:

\begin{align*} w_A &= \frac{20,000}{50,000} = 40\% \\\ w_B &= \frac{30,000}{50,000} = 60\% \end{align*}

\textbf{Expected Returns} = 0.40 \times 0.12 + 0.60 \times 0.20 = 16.8\%

\textbf{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

\textbf{Standard deviation} = \sqrt{0.046} = 0.2145 \text{ or } 21.45\%

Expected Variance for a Three Asset Portfolio

\sigma_p^2 = w_1^2 \sigma_{1}^2 + w_2^2 \sigma_{2}^2 + w_3^2 \sigma_{3}^2 + 2w_1w_2 \text{Cov}_{1,2} + 2w_1w_3 \text{Cov}_{1,3} + 2w_2w_3 \text{Cov}_{2,3}