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2. Consider the following expected returns, volatilities, and correlations: \be...
May 7, 2024
a Solution
a
To ensure a risk-free investment with stocks Vital and Mital, we need to find the weights of the portfolio that eliminate risk. Since the correlation between Vital and Mital is -1, a perfectly negative correlation, we can create a risk-free portfolio by allocating the investments in a way that the weighted standard deviations cancel each other out
a
The formula for the weight of Vital (wVitalw_{Vital}) in the risk-free portfolio is given by the ratio of the standard deviation of Mital (σMital\sigma_{Mital}) to the sum of the standard deviations of Vital and Mital (σVital+σMital\sigma_{Vital} + \sigma_{Mital}), since the correlation is -1
a
The weight of Vital is calculated as wVital=σMitalσVital+σMitalw_{Vital} = \frac{\sigma_{Mital}}{\sigma_{Vital} + \sigma_{Mital}}. Plugging in the values, we get wVital=24%6%+24%=2430=0.8w_{Vital} = \frac{24\%}{6\% + 24\%} = \frac{24}{30} = 0.8 or 80%
a Answer
80% of the investment should be allocated to Vital stock to ensure a risk-free investment.
Key Concept
Risk elimination through diversification
Explanation
By investing in negatively correlated assets, it is possible to create a risk-free portfolio where the assets' price movements offset each other.
b Solution
b
To calculate the portfolio's volatility with a long position in Pital and a short position in Mital, we need to use the formula for the standard deviation of a two-asset portfolio:
b
The formula is σp=wPital2⋅σPital2+wMital2⋅σMital2+2⋅wPital⋅wMital⋅σPital⋅σMital⋅ρPital,Mital\sigma_p = \sqrt{w_{Pital}^2 \cdot \sigma_{Pital}^2 + w_{Mital}^2 \cdot \sigma_{Mital}^2 + 2 \cdot w_{Pital} \cdot w_{Mital} \cdot \sigma_{Pital} \cdot \sigma_{Mital} \cdot \rho_{Pital,Mital}}, where ww represents the weights and ρ\rho represents the correlation coefficient between the two stocks
b
Given the long position in Pital is 10,000andtheshortpositioninMitalis10,000 and the short position in Mital is 2,000, the weights are wPital=10,00010,000−2,000=1.25w_{Pital} = \frac{10,000}{10,000 - 2,000} = 1.25 and wMital=−2,00010,000−2,000=−0.25w_{Mital} = \frac{-2,000}{10,000 - 2,000} = -0.25
b
Plugging in the values, we get σp=1.252⋅0.142+(−0.25)2⋅0.242+2⋅1.25⋅(−0.25)⋅0.14⋅0.24⋅0.7\sigma_p = \sqrt{1.25^2 \cdot 0.14^2 + (-0.25)^2 \cdot 0.24^2 + 2 \cdot 1.25 \cdot (-0.25) \cdot 0.14 \cdot 0.24 \cdot 0.7}
b
Calculating the above expression gives us the portfolio's volatility
b Answer
The portfolio's volatility is calculated using the given formula and weights.
Key Concept
Portfolio Volatility Calculation
Explanation
The volatility of a portfolio comprising multiple assets is determined by the individual volatilities, the weights of the assets in the portfolio, and the correlation between the assets.
c Solution
c
To decide which security an investor primarily concerned about systematic risk would favor, we need to consider the beta of each security, which measures its sensitivity to market movements
c
The beta of a security is calculated by multiplying the correlation of the security with the market by the ratio of the security's standard deviation to the market's standard deviation
c
For Artis, the beta is βArtis=ρArtis,Market⋅σArtisσMarket\beta_{Artis} = \rho_{Artis,Market} \cdot \frac{\sigma_{Artis}}{\sigma_{Market}}. However, we do not have the standard deviation of Artis, so we cannot calculate its beta directly
c
For Brotis, the beta is βBrotis=ρBrotis,Market⋅σBrotisσMarket=0.2⋅0.120.10=0.24\beta_{Brotis} = \rho_{Brotis,Market} \cdot \frac{\sigma_{Brotis}}{\sigma_{Market}} = 0.2 \cdot \frac{0.12}{0.10} = 0.24
c
Since we cannot calculate the beta for Artis directly, we must rely on qualitative analysis. Given the high correlation with the market, Artis likely has a higher beta than Brotis, indicating higher systematic risk
c Answer
An investor primarily concerned about systematic risk would favor Brotis due to its lower beta and therefore lower systematic risk.
Key Concept
Systematic Risk and Beta
Explanation
Beta measures a security's systematic risk relative to the market. A lower beta indicates lower sensitivity to market movements and thus lower systematic risk.
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