UF-70569-M1-S1-FINBAN: Lecture 4: Calculating Return and Risk for a Portfolio Composed of Two Financial Assets:

Lecture 4: Calculating Return and Risk for a Portfolio of Two Financial Assets

1. Calculating the Expected Return of a Portfolio with Two Assets

The expected return of a portfolio is the weighted average of the expected returns of the individual assets in the portfolio. The weights represent the proportion of each asset in the portfolio.

Formula : $E (R_{p}) = w_{1} \cdot E (R_{1}) + w_{2} \cdot E (R_{2})$

Where:
$E (R_{p})$ : Expected return of the portfolio
$w_{1}, w_{2}$ : Weights of Asset 1 and Asset 2 in the portfolio ( $w_{1} + w_{2} = 1$ )
$E (R_{1}), E (R_{2})$ : Expected returns of Asset 1 and Asset 2

2. Calculating the Variance and Standard Deviation of a Portfolio with Two Assets

The variance and standard deviation of a portfolio measure the risk associated with the portfolio. Unlike individual assets, the risk of a portfolio also depends on the correlation between the two assets.

Portfolio Variance Formula :

$Var (R_{p}) = w_{12} \cdot σ_{12} + w_{22} \cdot σ_{22} + 2 \cdot w_{1} \cdot w_{2} \cdot ρ_{1, 2} \cdot σ_{1} \cdot σ_{2}$

Where:
$Var (R_{p})$ : Variance of the portfolio
$σ_{1}, σ_{2}$ : Standard deviations (risks) of Asset 1 and Asset 2
$ρ_{1, 2}$ : Correlation coefficient between Asset 1 and Asset 2
$w_{1}, w_{2}$ : Weights of Asset 1 and Asset 2
Portfolio Standard Deviation Formula :

3. Concept of Correlation Coefficient and Its Impact on Portfolio Risk

The correlation coefficient ( $ρ_{1, 2}$ ) measures the degree to which the returns of two assets move together. It ranges from -1 to +1:

Perfect Positive Correlation ( $ρ = + 1$ ) : The assets move in the same direction, and diversification does not reduce risk.
Perfect Negative Correlation ( $ρ = - 1$ ) : The assets move in opposite directions, allowing for maximum risk reduction through diversification.
Zero Correlation ( $ρ = 0$ ) : The assets are unrelated, and combining them reduces risk but not as effectively as with negative correlation.

Impact on Portfolio Risk :

A lower correlation between assets reduces the overall portfolio risk because losses in one asset can be offset by gains in another.
Diversification benefits are maximized when assets have low or negative correlations.

4. Practical Examples and Calculations

Example 1: Calculating Expected Return

Suppose you have a portfolio with two assets:

Asset 1: Expected return = 8%, Weight = 60%
Asset 2: Expected return = 12%, Weight = 40%

$E (R_{p}) = (0.6 \cdot 8%) + (0.4 \cdot 12%) = 4.8% + 4.8% = 9.6%$

The expected return of the portfolio is 9.6% .

Example 2: Calculating Portfolio Variance and Standard Deviation

Given the following data:

Asset 1: Standard deviation ( $σ_{1}$ ) = 10%, Weight ( $w_{1}$ ) = 60%
Asset 2: Standard deviation ( $σ_{2}$ ) = 15%, Weight ( $w_{2}$ ) = 40%
Correlation coefficient ( $ρ_{1, 2}$ ) = 0.5

Step 1: Calculate Portfolio Variance

$Var (R_{p}) = (0. 6^{2} \cdot 1 0^{2}) + (0. 4^{2} \cdot 1 5^{2}) + (2 \cdot 0.6 \cdot 0.4 \cdot 0.5 \cdot 10 \cdot 15)$

$Var (R_{p}) = (0.36 \cdot 100) + (0.16 \cdot 225) + (2 \cdot 0.6 \cdot 0.4 \cdot 0.5 \cdot 150)$

$Var (R_{p}) = 36 + 36 + 36 = 108$

Step 2: Calculate Portfolio Standard Deviation

The portfolio's standard deviation is approximately 10.39%.

Example 3: Impact of Correlation on Portfolio Risk

Using the same data as Example 2 but changing the correlation coefficient:

Case 1: $ρ_{1, 2} = 0$ (No correlation)
$Var (R_{p}) = (0.36 \cdot 100) + (0.16 \cdot 225) + (2 \cdot 0.6 \cdot 0.4 \cdot 0 \cdot 10 \cdot 15) = 36 + 36 + 0 = 72$ $σ_{p} = 72 \approx 8.49%$
Case 2: $ρ_{1, 2} = - 0.5$ (Negative correlation)
$Var (R_{p}) = (0.36 \cdot 100) + (0.16 \cdot 225) + (2 \cdot 0.6 \cdot 0.4 \cdot - 0.5 \cdot 10 \cdot 15)$ $Var (R_{p}) = 36 + 36 - 36 = 36$ $σ_{p} = 36 = 6%$

Conclusion : Lower or negative correlation significantly reduces portfolio risk.

Key Takeaways

Expected Return : The portfolio's expected return is the weighted average of the individual assets' expected returns.
Risk Measurement : Portfolio variance and standard deviation depend on the assets' risks, weights, and their correlation.
Correlation Impact : Lower or negative correlation between assets reduces portfolio risk due to diversification benefits.
Practical Application : Calculating portfolio metrics helps investors optimize their portfolios for better risk-return trade-offs.

By understanding these calculations, investors can make informed decisions about asset allocation and portfolio management.

Modifié le: vendredi 11 juillet 2025, 00:32

إدارة المحافظ المالية/ Portfolio Management
Lecture 4: Calculating Return and Risk for a Portfolio Composed of Two Financial Assets:

Lecture 4: Calculating Return and Risk for a Portfolio of Two Financial Assets

1. Calculating the Expected Return of a Portfolio with Two Assets