Question

If the risk-free rate of return is 6%, and if a risky asset is available with a return of 9% and a standard deviation of 3%, what is the maximum rate of return you can achieve if you are willing to accept a standard deviation of 2%? What percentage of your wealth would have to be invested in the risky asset?

Short answer

Max return: 8.01%, with 67% in the risky asset.

Step-by-step solution

Understand the Problem

We are given the return and standard deviation of both a risk-free asset and a risky asset. We need to calculate the maximum return for a portfolio with a given standard deviation by mixing these assets.

Identify Parameters

Given:- Risk-free rate, \( R_f = 6\% \)- Return of the risky asset, \( R = 9\% \)- Standard deviation of the risky asset, \( \sigma = 3\% \)- Desired portfolio standard deviation, \( \sigma_P = 2\% \)

Use the Capital Allocation Line (CAL) Formula

The expected return on a portfolio, \( E(R_P) \), with a proportion \( x \) in the risky asset is given by:\[E(R_P) = x \cdot R + (1-x) \cdot R_f\]The portfolio standard deviation is given by:\[\sigma_P = x \cdot \sigma\]

Solve for Proportion `x`

We are given \( \sigma_P = 2\% \) and \( \sigma = 3\% \). We solve in terms of \( x \):\[2\% = x \cdot 3\%\]\[x = \frac{2}{3} \, ext{or} \, 0.67\]This means 67% of the wealth is invested in the risky asset.

Calculate the Expected Portfolio Return

Using the proportion \( x = 0.67 \):\[E(R_P) = 0.67 \cdot 9\% + 0.33 \cdot 6\%\]Calculate:\[E(R_P) = 0.67 \cdot 9\% + 0.33 \cdot 6\% = 6.03\% + 1.98\% = 8.01\%\]The maximum expected rate of return for a standard deviation of 2% is 8.01%.

Key concepts

Risk-Free Rate

In finance, the risk-free rate is a foundational concept used in investment theory and portfolio management. It represents the return on an investment with no risk of financial loss. Typically, this is demonstrated through government bonds, like U.S. Treasury bills, as they are considered virtually free of default risk.

• The risk-free rate provides a baseline for evaluating investment opportunities.
• Investors use it to compare the potential returns of other investments.
• This rate affects decisions on how to allocate resources between risky and safe assets.

In the exercise, the risk-free rate is given as 6%. This means this is the minimum return investors expect if they take no risks. By comparing it with other potential returns, they can decide if taking on more risk is justified.

Capital Allocation Line

The capital allocation line (CAL) is an important tool in portfolio optimization. It graphically represents all possible combinations of risk and reward for different allocations between a risk-free asset and a risky asset.

• The CAL helps investors visualize the risk-return trade-off.
• Its slope is depicted by the market risk premium, which is the extra return for taking on additional risk.
• Investors can choose their preferred risk level by selecting where on the line they want their portfolio to be.

In our example, the CAL assists us in determining the optimal portfolio mix for a specific risk level. By using the CAL formula, investors can identify the expected return and the percentage allocation needed between the risk-free and risky assets.

Risky Asset

A risky asset is an investment that carries a certain level of risk of financial loss, alongside the potential for higher returns. Stocks, corporate bonds, and real estate often qualify as risky assets.

• Risky assets offer higher potential returns compared to risk-free assets.
• The risk is usually quantified by the asset's standard deviation.
• Investors accept that there's a chance of variability in returns.

For this exercise, the risky asset has a return of 9% and a standard deviation of 3%. These parameters provide the basis for calculating potential portfolio returns, emphasizing the relationship between risk and possible rewards.

Standard Deviation

Standard deviation is a key mathematical tool in finance that measures the dispersion or volatility of a set of returns. It indicates how much an investment's returns can deviate from its average over a period.

• A low standard deviation suggests returns will be relatively close to the average.
• A high standard deviation means there is a higher variability in returns.
• Investors use it to gauge the risk level associated with particular investments.

In the context of this exercise, both the risky asset and the portfolio have their standard deviations specified. The given standard deviation of 3% for the risky asset and a desired portfolio standard deviation of 2% drive the portfolio allocation decision. It's this measure that allows investors to customize their portfolios to suit their risk appetite.