Question

part e: portfolio allocation – risk-free and risky asset mix
learning outcome: explain the capital allocation line and the investment opportunity set with risky and risk - free assets
instructions: assume:

• risk - free rate = 3%
• expected return of risky portfolio = 9%
• standard deviation of risky portfolio = 18%

calculate the expected return and standard deviation for the following portfolio combinations:

1. 100% risk - free asset
2. 50% risk - free, 50% risky
3. 120% risky, -20% risk - free (borrowed funds used for leverage)

then answer:

• what do these combinations reveal about risk, return, and diversification?

show all formulas and calculations clearly.

Explanation:

Step1: Define portfolio return formula

The expected return of a portfolio combining a risk-free asset ($R_f$) and a risky portfolio ($R_p$) is:
$$R_{portfolio} = w_f R_f + w_p R_p$$
where $w_f$ = weight of risk-free asset, $w_p$ = weight of risky portfolio, and $w_f + w_p = 1$

Step2: Define portfolio standard deviation formula

The standard deviation of the portfolio (since risk-free asset has $\sigma_f=0$) is:
$$\sigma_{portfolio} = w_p \sigma_p$$
where $\sigma_p$ = standard deviation of risky portfolio

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For Portfolio 1: 100% risk-free asset

Step1: Assign weights

$w_f=1$, $w_p=0$, $R_f=0.03$, $\sigma_p=0.18$

Step2: Calculate return

$$R_{portfolio} = 1*0.03 + 0*0.09 = 0.03$$

Step3: Calculate standard deviation

$$\sigma_{portfolio} = 0*0.18 = 0$$

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For Portfolio 2: 50% risk-free, 50% risky

Step1: Assign weights

$w_f=0.5$, $w_p=0.5$, $R_f=0.03$, $R_p=0.09$, $\sigma_p=0.18$

Step2: Calculate return

$$R_{portfolio} = 0.5*0.03 + 0.5*0.09 = 0.015 + 0.045 = 0.06$$

Step3: Calculate standard deviation

$$\sigma_{portfolio} = 0.5*0.18 = 0.09$$

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For Portfolio 3: 120% risky, -20% risk-free

Step1: Assign weights

$w_f=-0.2$, $w_p=1.2$, $R_f=0.03$, $R_p=0.09$, $\sigma_p=0.18$

Step2: Calculate return

$$R_{portfolio} = (-0.2)*0.03 + 1.2*0.09 = -0.006 + 0.108 = 0.102$$

Step3: Calculate standard deviation

$$\sigma_{portfolio} = 1.2*0.18 = 0.216$$

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Interpretation of combinations

Step1: Risk-return relationship

As the weight of the risky portfolio increases (including leveraging by borrowing at the risk-free rate), portfolio expected return rises, and portfolio standard deviation (risk) also rises linearly.

Step2: Diversification note

This combination only uses a single risky portfolio and a risk-free asset, so it does not show diversification across multiple risky assets. Instead, it demonstrates the Capital Allocation Line (CAL) tradeoff: investors can adjust risk-return by mixing a risky portfolio with risk-free assets, or leverage to amplify both risk and return.

Answer:

Portfolio Calculations:

1. 100% risk-free asset:

• Expected return: 3%
• Standard deviation: 0%

1. 50% risk-free, 50% risky:

• Expected return: 6%
• Standard deviation: 9%

1. 120% risky, -20% risk-free:

• Expected return: 10.2%
• Standard deviation: 21.6%

Interpretation:

These combinations reveal a direct, linear tradeoff between risk (standard deviation) and expected return: increasing exposure to the risky portfolio (including using leverage by borrowing at the risk-free rate) increases both the portfolio's expected return and its volatility. This is the core of the Capital Allocation Line: investors can choose their preferred risk-return point by mixing a risky portfolio with risk-free assets. No diversification benefit is shown here, as we are not combining multiple risky assets, only a single risky portfolio with a risk-free asset.