Question 1 Which security is riskier? Why?
For stocks held individually, the best measure of risk would be the coefficient of variation, which is the proportional fraction of standard deviation to the expected/required returns. The coefficient of variation (CV) measures the ratio of risks to a single return.
The coefficient of variation is calculated as follows:
CV = stdev_r / ER
Where:
CV is the coefficient of variation
Stdev_r is the standard deviation of returns, and
ER is the expected rate of return
Security A CV would be 30 / 6 = 5 while security B would be 10 / 11 =0.909.
Therefore, security A is riskier with coefficient of variation (CV) of 5. Meanwhile, security B's CV is only 0.91 which implies it is less risky.
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Question 2 what is Stock A’s beta?
The formula for solving the beta of the stock is ( Oleinikova, Kravets, & Silnov, D. S. 2016) :
B = Covariance / Variance
Where:
B = Stock's beta
Let Cov = Covariance between the stock and the market.
In order to solve for the this covariance (cov), the formula below is used:
Cov = C or ∗ market std ∗ stock std
Where:
Cor = Correlation between the stock and the market
std = Standard deviation
The variance of the market is simply the market's standard deviation multiplied by itself.
Market variance = 20 * 20 = 400
Cov = 0.70 * 20 * 40 = 560
B = 560 / 400
Therefore, beta value for stock A is 1.4
Question 3 what is Crisp’s required return?
The formula for the multi-factor APT model is:
RR = rf + b1 ∗ (r1 − rf) + b2 ∗ (r2 − rf)
Where:
RR = Required return
rf = Risk-free rate
b = Sensitivity or volatility of an asset
r1 or r2 = Expected return on individual assets.
RR = 6 + 0.7 * (12 – 6) + 0.9 * (8 – 6)
The required return is 12%
Question 4 What are the expected return and standard deviation of a portfolio invested 30% in Stock A and 70% in Stock B?
The standard deviation is calculated as the square root of the variance (Tong et al., 2014). So, first I will use the formula given below to calculate the variance of the portfolio. And, thereafter calculate the standard deviation
VarP=Wt_a^2 ∗ stdevn_a^2 + Wt_b^2 ∗ stdevn_b^2 + 2 ∗ Cor ∗ stdevn_a ∗ stdevn_b
Where:
Var = Variance
Wt = Weight of the stock (or percentage of investment)
stdevn = Standard deviation
Cor = Correlation coefficient
VarP = 0.3^2 * 40%^2 + 0.7^2 * 60%^2 + 2*0.2*40%*60% = 0.2868
The square root of 0.2868 = 0.5355 which is equivalent to 53.55 %
Therefore, the portfolio has a standard deviation of 53.55%
The weighted mean of the stock returns that are included in the portfolio will form its the expected return.
The expected return rate is calculated using the following formula:
ER = Wa * Era + Wb * Erb
Where:
W is weight of invested asset
Er is the expected return of an asset
ER = 0.30 * 0.12 + 0.70 * 0.18 = 0.162
Now 0.162 * 100 to convert back into percentage
Therefore, the portfolio has an expected return rate of 16.2%
References
Wan, X., Wang, W., Liu, J., & Tong, T. (2014). Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC medical research methodology , 14 (1), 135.
Oleinikova, S. A., Kravets, O. J., & Silnov, D. S. (2016). Analytical estimates for the expectation of the beta distribution on the basis of the known values of the variance and mode. International Information Institute (Tokyo). Information , 19 (2), 343.
