R tutorial - The Portfolio Return

### Understanding Portfolio Returns: A Comprehensive Guide

When analyzing the composition of an investment portfolio, it becomes evident that the weight of each asset plays a crucial role in determining the overall value of the portfolio. The larger the weight assigned to an asset, the more significant its influence on the future value of the entire portfolio. This concept is fundamental for investors who seek to evaluate the performance and potential outcomes of their investments.

Investors typically focus on analyzing changes in investment values not in absolute terms but rather in relative terms. This approach leads them to compute **simple returns**, which are defined as the change in value over a specific period divided by the initial value. The formula for calculating simple return is:

\[

\text{Simple Return} = \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}}

\]

For instance, if an investment has an initial value of $100 and a final value of $120, the simple return would be calculated as follows:

\[

\text{Return} = \frac{120 - 100}{100} = 0.20 \quad \text{(or 20%)}

\]

In the slide provided, we can see how this definition is applied to compute portfolio returns. The process involves three key steps:

1. **Initial Date Calculation**: Compute the total value invested by summing the values of all individual investments.

2. **Final Date Calculation**: Sum the final values of each investment to determine the total portfolio value.

3. **Portfolio Return Calculation**: Determine the percentage change in the portfolio's final value compared to its initial value.

### Example: Two-Asset Portfolio

Consider a two-asset portfolio where $200 is invested in Asset 1 and $300 is invested in Asset 2. The initial values of these assets are $180 and $330, respectively.

1. **Initial Total Value**:

\[

\$200 + \$300 = \$500

\]

2. **Final Total Value**:

\[

\$180 + \$330 = \$510

\]

3. **Portfolio Return**:

\[

\frac{\$510 - \$500}{\$500} = 0.02 \quad \text{(or 2%)}

\]

This method provides a straightforward way to calculate portfolio returns, but it does not account for how the portfolio weights influence the overall return. To address this limitation, we can use an alternative formula that computes the portfolio return as the weighted average of the returns of its underlying assets.

### Weighted Average Return Calculation

The calculation of the weighted average return involves three steps:

1. **Determine Initial Weights**: Calculate the proportion of each asset's initial value relative to the total portfolio value.

2. **Compute Individual Returns**: Determine the return for each asset individually.

3. **Calculate Portfolio Return**: Sum the products of the initial weights and their respective returns.

**Example: Two-Asset Portfolio**

1. **Initial Weights Calculation**:

- Asset 1:

\[

\frac{\$200}{\$500} = 0.40 \quad \text{(or 40%)}

\]

- Asset 2:

\[

\frac{\$300}{\$500} = 0.60 \quad \text{(or 60%)}

\]

2. **Individual Returns**:

- Asset 1: Return = -10%

- Asset 2: Return = +10%

3. **Portfolio Return Calculation**:

- Weighted Return for Asset 1:

\[

0.40 \times (-0.10) = -0.04

\]

- Weighted Return for Asset 2:

\[

0.60 \times (+0.10) = +0.06

\]

- Total Portfolio Return:

\[

-0.04 + 0.06 = +0.02 \quad \text{(or 2%)}

\]

As demonstrated, the portfolio return calculated using the weighted average method yields the same result as the simple return calculation (+2%). This consistency reinforces the validity of both approaches, but the weighted average method provides additional insight into how individual asset weights and their respective returns contribute to the overall portfolio performance.

### Conclusion and Next Steps

Understanding the relationship between portfolio weights and returns is essential for investors seeking to make informed decisions about their investments. By analyzing these relationships, investors can better assess the risks and potential rewards associated with their portfolios.

The next interactive exercises will allow you to apply this theory in practice, further solidifying your understanding of how portfolio weights influence overall returns.

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60 % times at which its return of 10% which gives us plus 6% adding - 4 % + + 6 % gives us the portfolio return of plus 2% which is exactly the same number as obtained before the next interactive exercises put this theory in practiceanalyzing the portfolio weights reveals the investment bets the larger the weight of an asset in the portfolio the more influential it will be in determining the future value of the portfolio when studying this impact investors typically do not analyze the change and the investment value in absolute terms but in the relative terms this leads them to compute simple returns defined as a change in value over the period relatively to the initial value the simple return is thus the final value minus the initial value divided by the initial value as an example suppose the initial value is $100 the final value is $120 then the return on that investment equals 20% obtained by taking the difference between 120 and 180 it by 100 in the slide I show you how we can apply this definition for computing portfolio returns this involves three steps first for the initial date we need to compute the total value invested as the sum of the values of a different investments second for the final date we have to sum the final values of the individual investments to obtain the final portfolio value then we can compute the portfolio return as a percentage change of the final value compared to the initial value as an example let us consider a two asset portfolio that invests $200 to nest one and $300 in asset to the ant value is $180 and $330 if we sum the values we find that the total initial value of the portfolio is $500 while the total final value is five hundred and ten dollars it follows that the simple return on the portfolio is a $10 change in Val you divide by the initial five hundos invested which gives us a return of two percent this advantage of this calculation method is that it does not show how the portfolio weights determine the portfolio return let us therefore consider a different formula in which the portfolio return is computed as the weighted average of the returns of the underlying assets its calculation proceeds also in three steps first the initial weights of the positions are computed secondly the return on each of the individual positions is determined then in the third step the portfolio return is computed as a sum over the products between the initial weights and the corresponding returns in the slide you see how to apply this formula to compute the return for our example portfolio with two assets we first compute the initial portfolio weights since the initial value of the first asset is $200 and the total value invested is $500 the initial weight of asset 1 is 40% the remainder of 60% is a weight of acid tomb then in a second step we need to compute the returns for each of the assets for s1 we obtain that the individual return is minus 10% while for acid due the return is plus 10% finally we can combine those results and compute the portfolio return by summing over the weights multiplied by their respective returns the first term is a weight of 40 percent times the return of minus 10% which gives us minus 4% the second term is a weight of asset - 60 % times at which its return of 10% which gives us plus 6% adding - 4 % + + 6 % gives us the portfolio return of plus 2% which is exactly the same number as obtained before the next interactive exercises put this theory in practice\n"