Q1: Calculating Expected Return, Variance, and Standard Deviation for Stock YSJ Inc.
To calculate the expected return for stock YSJ Inc., we need to multiply the probability of occurrence by the corresponding return and sum them up:
Expected Return = (0.15 * -22%) + (0.25 * 4%) + (0.25 * 17%) + (0.35 * 26%)
Expected Return = -3.3% + 1% + 4.25% + 9.1%
Expected Return = 11.05%
To calculate the variance, we need to calculate the squared deviation from the expected return for each return, multiply it by its probability of occurrence, and sum them up:
Variance = (0.15 * (-22% – 11.05%)^2) + (0.25 * (4% – 11.05%)^2) + (0.25 * (17% – 11.05%)^2) + (0.35 * (26% – 11.05%)^2)
Variance = (0.15 * 121%) + (0.25 * 49%) + (0.25 * 36.2025%) + (0.35 * 225%)
Variance = 18.15% + 12.25% + 9.050625% + 78.75%
Variance = 118.200625%
Finally, we can calculate the standard deviation by taking the square root of the variance:
Standard Deviation = √(118.200625%)
Standard Deviation ≈ 10.87%
The standard deviation measures the dispersion or volatility of the returns of a stock. In this case, the standard deviation for stock YSJ Inc. is approximately 10.87%. This means that the returns of this stock are expected to deviate, on average, by around 10.87% from its expected return of 11.05%.
Q2: Calculating Portfolio’s Standard Deviation and Expected Return
To calculate the portfolio’s standard deviation and expected return, we need to consider the weights assigned to each stock and their correlation.
For a portfolio with 50% allocation to each stock (YSJ Inc. and YFK Inc.), we can calculate the portfolio’s expected return as follows:
E(r) Portfolio = (0.5 * E(r) YSJ Inc.) + (0.5 * E(r) YFK Inc.)
E(r) Portfolio = (0.5 * 11.05%) + (0.5 * 14%)
E(r) Portfolio = 12.525%
To calculate the portfolio’s standard deviation, we can use the following formula:
σ Portfolio = √((w1^2 * σ1^2) + (w2^2 * σ2^2) + (2 * w1 * w2 * ρ1,2 * σ1 * σ2))
Where:
w1 and w2 are the weights assigned to each stock
σ1 and σ2 are the standard deviations of each stock
ρ1,2 is the correlation between the two stocks
Plugging in the values, we get:
σ Portfolio = √((0.5^2 * (10.87%^2)) + (0.5^2 * (19%^2)) + (2 * 0.5 * 0.5 * 0.38 * 10.87% * 19%))
σ Portfolio ≈ √((0.25 * 118.200625%) + (0.25 * 361%) + (0.10327))
σ Portfolio ≈ √(29.55015625% + 90.25% + 0.10327)
σ Portfolio ≈ √(119.90342625%)
σ Portfolio ≈ √(1.1990342625)
σ Portfolio ≈ 10.95%
Therefore, the portfolio’s standard deviation is approximately 10.95%.
Q3: Evaluating Stock Replacement
To evaluate whether replacing stock YFK Inc. with stock YGR Inc. is beneficial, we need to compare their expected returns and consider their covariance.
For a portfolio with equal allocation to each stock:
E(r) Portfolio = (0.5 * E(r) YSJ Inc.) + (0.5 * E(r) YGR Inc.)
E(r) Portfolio = (0.5 * 11.05%) + (0.5 * 14%)
E(r) Portfolio = 12.525%
To calculate the portfolio’s standard deviation, we again use the formula:
σ Portfolio = √((w1^2 * σ1^2) + (w2^2 * σ2^2) + (2 * w1 * w2 * ρ1,2 * σ1 * σ2))
Plugging in the values:
σ Portfolio = √((0.5^2 * (10.87%^2)) + (0.5^2 * (19%^2)) + (2 * 0.5 * 0.5 * -0.01572 * 10.87% * 19%))
σ Portfolio ≈ √((0.25 * 118.200625%) + (0.25 * 361%) – (0.01572))
σ Portfolio ≈ √(29.55015625% + 90.25% – 0.01572)
σ Portfolio ≈ √(119.78443625%)
σ Portfolio ≈ √(1.1978443625)
σ Portfolio ≈ 10.94%
Therefore, with a covariance of -0.01572, replacing stock YFK Inc. with stock YGR Inc., while keeping an equal allocation, results in a slightly lower standard deviation of approximately 10.94%.
Q4: Impact of Updated Covariance on Portfolio’s Standard Deviation
If the covariance between YSJ Inc. and YGR Inc is updated to -0.02218, we need to recalculate the portfolio’s standard deviation using this new value.
Plugging in the updated covariance value into the formula:
σ Portfolio = √((0.5^2 * (10.87%^2)) + (0.5^2 * (19%^2)) + (2 * 0.5 * 0.5 * -0.02218 * 10.87% * 19%))
σ Portfolio ≈ √((0.25 * 118.200625%) + (0.25 * 361%) – (0.02218))
σ Portfolio ≈ √(29.55015625% + 90.25% – 0.02218)
σ Portfolio ≈ √(119.77897625%)
σ Portfolio ≈ √(1.1977897625)
σ Portfolio ≈ 10.94%
As we can see, even with the updated covariance value, the portfolio’s standard deviation remains approximately unchanged at around 10.94%.
Therefore, the updated covariance does not have a significant impact on the portfolio’s standard deviation.
Q5: Computing Sharpe Ratios for Portfolios
To compute the Sharpe Ratios for the portfolios in questions #2 and #4, we need to use the formula:
Sharpe Ratio = (E(r) Portfolio – Risk-Free Rate) / σ Portfolio
Assuming a risk-free rate of:
Treasury bills yielding 3%
Long-term Government of Canada bonds yielding 5.5%
For question #2:
Sharpe Ratio = (12.525% – 3%) / 10.95%
Sharpe Ratio ≈ 8 / 10.95%
Sharpe Ratio ≈ 7.31
For question #4:
Sharpe Ratio = (12.525% – 3%) / 10.94%
Sharpe Ratio ≈ 8 / 10.94%
Sharpe Ratio ≈ 7.31
Both portfolios have a Sharpe Ratio of approximately 7.31.
The Sharpe Ratio measures the excess return per unit of risk and helps assess the risk-adjusted performance of a portfolio or investment strategy.
In conclusion, both portfolios maintain the same Sharpe Ratio despite changes in stock selection and updated covariance values.
