Q1: You are presented with the following data for stock YSJ Inc.:
Probability of occurrence(s) Return(s)
0.15 – 22%
0.25 + 4%
0.25 + 17%
0.35 + 26%
Calculate: the expected return, variance and standard deviation; also, explain your results.
Q2: Assume that you combine stock YSJ Inc. (from Q1) with stock YFK Inc., whose E(r) and are 14% and 19%, respectively; if the correlation between each stock = + 0.38, what is the portfolio’s standard deviation and E(r) (*50% of the funds are allocated to each stock)?
Q3: While doing research, you identify a stock opportunity for company YGR Inc., whose E(r) and are the same as YFK Inc., but it has a covariance of – 0.01572 with YSJ Inc.; should you replace stock YFK Inc. (from Q2) with YGR Inc. (assume a 50% distribution to each stock, and show all your work)?
Q4: Your colleague re-computes the statistics for YGR Inc., and the covariance of YSJ Inc., YGR Inc. = -0.02218; how does this impact the portfolio’s standard deviation, and would your answer from Q3 change?
Q5: Compute the Sharpe Ratios (reward-to-variability) for the portfolios in Q #’s 2 & 4 (i.e., YSJ Inc. + YFK Inc., and YSJ Inc. + YGR Inc.)- assume 90-day Treasury bills are yielding 3%, and that 30-year long-term Government of Canada bonds = 5.5%.
Sample solution
Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell.
In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.
God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.
Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.
To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.
References
Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.
Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.
Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.
Sample Answer
Sample Answer
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.
