Q$4.$ Risk and Return Analysis: Creating an Optimal Portfolio with Monthly Data $(20$ Points$)$ Objective:

The aim is to construct an optimal investment portfolio for $2018$ using historical monthly price data for Apple Inc. $($AAPL$)$ and Alphabet Inc. $($GOOGL$),$ in combination with a risk$-$free asset. Your strategy should be informed by continuously compounded returns from January $2012$ through December $2017.$ The portfolio must reflect your specific risk tolerance, combining both theoretical understanding and practical application to achieve your investment goals.

Instructions:

$1.$ Data:

$$ Use the provided five years of monthly closing stock price data for AAPL and GOOGL. This data has been downloaded from Yahoo Finance $($January $2012-$ December $2017).$

$2.$ Risk$-$Free Rate Adjustment: $(2$ Points$)$

$$ The annual risk$-$free rate for $2018$ is provided as $2\mathrm{.}5\%.$ Convert this to a monthly continuously compounded rate using the formula: rm$=$ln$(1+$rannual$)/12$

$3.$ Risk Tolerance Identification: $(2$ Points$)$

$$ Clearly state your level of risk tolerance before proceeding with portfolio construction. Consider categorizing risk tolerance as low, moderate, or high, and explain what this means in terms of potential loss and return expectations.

$4.$ Analysis and Calculations: $(5$ Points$)$

$$ Calculate the continuously compounded monthly returns for AAPL and GOOGL:

Return$=$ln$($Pricet$/$Pricet$-1)$

$$ Determine the average return, standard deviation, and correlation between AAPL and GOOGL based on these returns.

$5.$ Tangency Portfolio Construction: $(2$ Points$)$

$$ Utilize the calculated statistics to determine the weights of AAPL and GOOGL in the tangency portfolio, $($Note: Tangency portfolio is the combination of AAPL and GOOGL with the highest possible Sharpe ratio$).$

$$ Calculate the expected return and standard deviation of the tangency portfolio.

$6.$ Desired Portfolio Construction: $(3$ Points$)$

$$ With your stated risk tolerance $($from part $3)$ as a guide, mix the tangency portfolio with the risk$-$free asset to form your desired portfolio.

$$ Compute the expected return of this portfolio.

$7.$ Investment Decision: $(6$ Points$)$

$$ Submit a brief video in mp$4$ format $(2-6$ minutes$)$

$$ Reflecting on the principles of diversification and Modern Portfolio theory.

$$ Discuss the alignment of your portfolio choice $($Your desired portfolio$)$ with the principles of modern portfolio theory and how it reflects your risk tolerance.

$$ You may submit a written report instead of a Video. However, in that case the maximum mark possible for this part will be $3/6$