Portfolio Expected Return: Calculation With Two Assets

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Alright guys, let's dive into calculating the expected return of a portfolio containing two assets. This is super important for understanding how your investments might perform. We'll break down the formula and apply it using the data you've provided. So, grab your calculators, and let's get started!

Understanding Expected Return

First, let's quickly recap what expected return actually means. Expected return is the anticipated amount of profit or loss an investor can expect on an investment. It's calculated by considering various possible scenarios and weighting them by their probabilities. Basically, it's a weighted average of possible returns.

In the context of a portfolio, the expected return is not just the sum of the individual asset's expected returns. Instead, it takes into account the proportion of each asset in the portfolio. This means a larger allocation to a higher-returning asset will boost the overall portfolio expected return, and vice versa. Understanding this concept is foundational for any investor looking to optimize their portfolio's performance.

For a single asset, the expected return E(R) can be represented as:

E(R) = Σ [Pi * Ri]

Where:

  • Pi is the probability of scenario i
  • Ri is the return in scenario i

However, when dealing with a portfolio, we need to consider the weight (or proportion) of each asset within that portfolio. That’s where the portfolio expected return formula comes in handy.

Formula for Portfolio Expected Return

The formula to calculate the expected return of a portfolio is relatively straightforward, especially when dealing with only two assets. If you've got more assets, don't worry, the concept extends logically. Here’s the formula:

E(Rp) = w1 * E(R1) + w2 * E(R2) + ... + wn * E(Rn)

Where:

  • E(Rp) is the expected return of the portfolio
  • wi is the weight (or proportion) of asset i in the portfolio
  • E(Ri) is the expected return of asset i

In simpler terms, you multiply the weight of each asset in the portfolio by its expected return and then add up all those values. This gives you the overall expected return of your portfolio. The weights (w1, w2, ..., wn) should always add up to 1 (or 100%) because they represent the entirety of your portfolio allocation. This formula is a cornerstone in modern portfolio theory, allowing investors to estimate potential returns based on asset allocation strategies. Remember, diversification and strategic weighting are key to optimizing your portfolio's risk-return profile!

For a two-asset portfolio, this simplifies to:

E(Rp) = w1 * E(R1) + w2 * E(R2)

Applying the Formula to Your Data

Now, let's use the data you've provided to calculate the expected return of your portfolio. You have two assets with the following characteristics:

  • Asset 1: E(R1) = 12% = 0.12, σ1 = 10%
  • Asset 2: E(R2) = 8% = 0.08, σ2 = 6%
  • Correlation: ρ12 = 0.3
  • Portfolio composition: 60% in Asset 1 and 40% in Asset 2

This means:

  • w1 = 0.60 (weight of Asset 1)
  • w2 = 0.40 (weight of Asset 2)

Plug these values into our formula:

E(Rp) = (0.60 * 0.12) + (0.40 * 0.08)

E(Rp) = 0.072 + 0.032

E(Rp) = 0.104

So, the expected return of your portfolio is 0.104, or 10.4%.

Interpretation and Considerations

The expected return of 10.4% represents the return you can anticipate, on average, from this specific asset allocation. However, it's crucial to remember that this is just an expectation, not a guarantee. The actual return can be higher or lower, depending on various market conditions and the actual performance of the assets.

Also, while we calculated the expected return, we haven't yet considered the portfolio's risk. The correlation between the assets (ρ12 = 0.3) will play a role in determining the overall portfolio risk, which is usually measured by the standard deviation. A lower correlation generally helps in reducing portfolio risk through diversification. In the next sections, we’ll delve into calculating the portfolio's standard deviation to give you a complete picture of your portfolio's risk-return profile.

Calculating Portfolio Standard Deviation (Risk)

Understanding the risk associated with your portfolio is just as crucial as knowing its expected return. The most common measure of risk is the standard deviation (σ), which quantifies the dispersion of possible returns around the expected return. A higher standard deviation means greater volatility and, therefore, higher risk. To calculate the portfolio standard deviation with two assets, we'll use the following formula:

σp = √[w1^2 * σ1^2 + w2^2 * σ2^2 + 2 * w1 * w2 * ρ12 * σ1 * σ2]

Where:

  • σp is the standard deviation of the portfolio
  • w1 and w2 are the weights of Asset 1 and Asset 2, respectively
  • σ1 and σ2 are the standard deviations of Asset 1 and Asset 2, respectively
  • ρ12 is the correlation between Asset 1 and Asset 2

Let's plug in the values you provided:

  • w1 = 0.60
  • w2 = 0.40
  • σ1 = 0.10
  • σ2 = 0.06
  • ρ12 = 0.3

σp = √[(0.60)^2 * (0.10)^2 + (0.40)^2 * (0.06)^2 + 2 * 0.60 * 0.40 * 0.3 * 0.10 * 0.06]

σp = √[0.36 * 0.01 + 0.16 * 0.0036 + 2 * 0.60 * 0.40 * 0.3 * 0.006]

σp = √[0.0036 + 0.000576 + 0.000864]

σp = √[0.00504]

σp ≈ 0.071

So, the standard deviation of your portfolio is approximately 0.071, or 7.1%.

Risk-Return Profile Analysis

Now that we have both the expected return (10.4%) and the standard deviation (7.1%) of your portfolio, let's analyze what this means in terms of the risk-return profile. The expected return gives you an idea of the potential profit, while the standard deviation tells you about the volatility or risk involved in achieving that return.

A portfolio with an expected return of 10.4% and a standard deviation of 7.1% suggests that, on average, you can expect a decent return with a relatively moderate level of risk. To put this into perspective, you can compare this risk-return profile with other investment options. For instance, a risk-free asset (like a government bond) might have a lower return but also a much lower standard deviation.

The Sharpe Ratio is a useful tool for comparing the risk-adjusted return of different portfolios. It’s calculated as:

Sharpe Ratio = (E(Rp) - Rf) / σp

Where Rf is the risk-free rate. A higher Sharpe Ratio indicates a better risk-adjusted return.

Keep in mind that this analysis is based on historical data and assumptions, and future performance can deviate. Regular monitoring and adjustments to your portfolio based on your risk tolerance and investment goals are essential for long-term success.

Diversification and Correlation Benefits

The correlation between assets plays a crucial role in portfolio risk. In your case, the correlation between Asset 1 and Asset 2 is 0.3, which is a positive correlation but not perfectly correlated. This means that the assets tend to move in the same direction, but not always in a perfectly synchronized manner.

Diversification benefits arise when assets are not perfectly correlated. By combining assets with low or negative correlations, you can reduce the overall portfolio risk without sacrificing returns. This happens because when one asset performs poorly, the other asset might perform well, offsetting the losses. A correlation of 0.3 provides some diversification benefits, but greater diversification could be achieved with assets having lower or negative correlations.

To further enhance diversification, you might consider adding more assets to your portfolio that have different risk and return characteristics and low correlations with your existing assets. Regularly reviewing and rebalancing your portfolio can help maintain your desired asset allocation and risk profile.

Conclusion

Calculating the expected return and standard deviation of a portfolio is essential for making informed investment decisions. By understanding these concepts and using the formulas we've discussed, you can assess the potential returns and risks associated with different asset allocations. Remember that the expected return is an estimate, and actual returns may vary.

Diversification is key to managing risk, and considering the correlation between assets is crucial for achieving diversification benefits. Regular monitoring, rebalancing, and adjustments to your portfolio based on your risk tolerance and investment goals will help you stay on track toward achieving your financial objectives. Happy investing, guys!