15.3 Relationship Between Risk And Return In Portfolio
Calculate and Interpret the Expected Return of a Portfolio of Securities
The relationship between risk and return is a foundational concept in investment theory and practice. When considering a portfolio, you need to understand both how to calculate the expected return and interpret its implications. Here’s how to do that:
Calculating Expected Return
The expected return of a portfolio is the weighted average of the expected returns of the individual securities in the portfolio. This can be calculated using the formula:
$$
(E(R_p) = \sum_{i=1}^{n} w_i E(R_i))
$$
Where:
 \(E(R_p)\) is the expected return of the portfolio
 \(w_i\) is the weight of security i in the portfolio
 \(E(R_i)\) is the expected return of security i
Example
If you have a portfolio with three securities with the following expected returns and weights:
 Security A: Expected return = 10%, weight = 50%
 Security B: Expected return = 8%, weight = 30%
 Security C: Expected return = 6%, weight = 20%
Then the expected return of the portfolio would be calculated as:
$$
(E(R_p) = (0.50 \times 0.10) + (0.30 \times 0.08) + (0.20 \times 0.06))
$$
$$ E(R_p) = 0.05 + 0.024 + 0.012 = 0.086 = 8.6% $$
Thus, the portfolio’s expected return is 8.6%.
Benefits and Challenges of Combining Securities in a Portfolio
Benefits of Combining Securities (Diversification)

Risk Reduction:
 By investing in a variety of securities, specific risk (also known as unsystematic risk) can be reduced.
 Diversification diminishes the impact of any one security’s performance on the overall portfolio.

Potential for Higher Returns:
 Diversification provides access to different securities with varying performance, some of which may yield higher returns.

Smoothing Out Volatility:
 Grouping a mix of securities helps stabilize returns, as not all investments will react to market changes in the same way or at the same magnitude.
Challenges of Diversification

Diminishing Returns:
 After a certain point, adding more securities provides fewer benefits in risk reduction and may not significantly affect returns.

Higher Transaction Costs:
 More securities mean more transactions, which leads to increased transaction costs and management fees.

Complexity:
 Managing a diversified portfolio can be complex and might require time, resources, and expertise beyond the average investor.
Key Takeaways
 Expected Return Calculation: The expected return of a portfolio is a weighted sum of the expected returns of individual assets.
 Risk and Return TradeOff: Proper diversification can help reduce risk without dramatically impacting potential returns.
 Diversification Benefits and Challenges: While reducing specific risk, diversification comes with certain logistical and cost challenges.
Frequently Asked Questions (FAQs)
What is the significance of the expected return in portfolio management?
The expected return provides an estimate of the potential gain from the portfolio over a certain period, helping investors compare different portfolios and set realistic financial goals.
How does diversification reduce risk?
Diversification reduces risk by spreading investments across diverse assets, which generally lowers unsystematic risk as the poor performance of one security can be offset by better performance of others in the portfolio.
Is there a limit to the benefits of diversification?
Yes, after a certain level, additional securities bring diminishing benefits in risk reduction. This phenomenon is due to various securities’ inherent correlation, where most nonsystematic risks get diversified away with a sufficiently large and varied portfolio.
Glossary
 Expected Return: The average return an investor anticipates receiving from an investment or a portfolio.
 Portfolio: A group of financial securities that an investor holds.
 Diversification: The strategy of reducing risk by investing in a variety of assets.
 Unsystematic Risk: Risk that is unique to a specific company or industry.
Visualizations
Diversification Effect on Risk
pie
title Diversification's Impact on Risk Level
"Unsystematic Risk Reduced" : 70
"Remaining Risk" : 30
Conclusion
Understanding the relationship between risk and return is crucial in creating a wellbalanced investment portfolio. Calculating and interpreting the expected return, while appreciating the pros and cons of diversification, can significantly assist investors in achieving their financial goals.
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## What is the primary focus when calculating the expected return of a portfolio?
 [ ] The return from the highestperforming security
 [x] The weighted average return of all securities in the portfolio
 [ ] The return from the lowestperforming security
 [ ] The median return of the securities in the portfolio
> **Explanation:** The expected return of a portfolio is calculated as the weighted average return of all securities within the portfolio, taking into account the proportion of the total investment in each security.
## What is a significant benefit of combining different securities in a portfolio?
 [ ] Increased risk exposure
 [ ] Singular source of return
 [x] Diversification benefits
 [ ] Decreased liquidity
> **Explanation:** Combining different securities in a portfolio allows for diversification, which can help to spread and potentially reduce the overall risk.
## What does diversification in a portfolio generally aim to achieve?
 [ ] Eliminate all investment risk
 [ ] Increase the portfolio's expected return above market average
 [ ] Guarantee positive returns
 [x] Reduce unsystematic risk
> **Explanation:** Diversification aims to reduce unsystematic risk (also known as specific or idiosyncratic risk), which is associated with individual securities. It does not eliminate systematic risk.
## How can the expected return of a portfolio be described?
 [x] The weighted average of the expected returns of its constituent securities
 [ ] The sum of the returns of its constituent securities
 [ ] The median return of its constituent securities
 [ ] The highest return among its constituent securities
> **Explanation:** The expected return of a portfolio is the weighted average of the expected returns of the individual securities within the portfolio.
## Which type of risk remains even after achieving diversification in a portfolio?
 [ ] Unsystematic risk
 [ ] Companyspecific risk
 [ ] Currency risk
 [x] Systematic risk
> **Explanation:** Systematic risk, which is the risk inherent to the entire market or market segment, remains even after diversifying a portfolio.
## What happens to the overall risk when more uncorrelated securities are added to a portfolio?
 [ ] It remains constant
 [ ] It increases
 [ ] It doubles
 [x] It decreases
> **Explanation:** Adding more uncorrelated securities to a portfolio generally decreases the overall risk due to the diversification effect.
## Why might an investor combine securities with low correlation in their portfolio?
 [x] To minimize total risk through diversification
 [ ] To increase the overall expected return
 [ ] To maximize the volatility of the portfolio
 [ ] To ensure all securities perform similarly
> **Explanation:** Low correlation between securities can help minimize total risk through diversification as the performance of those securities is less likely to move in tandem.
## What is a challenge when combining securities in a portfolio?
 [ ] Increasing total expected return
 [x] Identifying truly uncorrelated assets
 [ ] Reducing portfolio complexity
 [ ] Increasing unsystematic risk
> **Explanation:** One of the challenges when combining securities is identifying assets that are truly uncorrelated, which can help achieve proper diversification.
## How does adding more securities usually affect the portfolio's expected return?
 [ ] It nullifies the return
 [x] It generally maintains the expected return while reducing risk
 [ ] It guarantees higher returns
 [ ] It lowers the overall return
> **Explanation:** Adding more securities can maintain or potentially increase expected return while reducing risk due to diversification benefits.
## When combining securities, what is a potential downside an investor may face?
 [ ] Increased unsystematic risk
 [ ] Decreased investment choices
 [ ] Enhanced volatility
 [x] Higher transaction and management costs
> **Explanation:** One potential downside of combining more securities into a portfolio is the higher transaction and management costs, which may accrue as the portfolio becomes more complex.
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