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Week 9 Exercise: Risk and return – Portfolio optimization Finance is the study of the timing and riskiness of cash flows. Because most people are generally risk averse (the second derivative of utility with respect to consumption is negative) and because people always want to consume more (the first derivative of utility with respect to consumption is monotonically positive), most people will prefer higher returns for the same risk exposure, or lower risk exposure for the same level of returns. This exercise in portfolio theory depends on those assumptions of risk aversion and consumption. For those who derive utility from accepting unnecessary risk (gamblers) or those who are happier having fewer things in life, this analysis will not hold true. We can generalize these assumptions mathematically by saying that investors always want to maximize the Sharpe Ratio of their investments. The Sharpe Ratio is simply the ratio created by the return on an asset (above and beyond the risk-free rate) divided by the standard deviation of returns for the same asset. Since higher returns are better (numerator), and lower risk is better (denominator), we know that a higher Sharpe Ratio will be associated with greater success for the general investor. When assets are combined in portfolio the assets share some characteristics with the resultant portfolio, but more importantly, the portfolio will take on some unique characteristics depending on the relationship between the component assets. Often (whenever the correlation between the asset returns is less than perfect), the idiosyncratic risk associated with the combination of 2 assets in portfolio is less than the average of the idiosyncratic risks of the individual assets. However, the systematic risk is always a function of the weighted average of the betas of the individual components. Similarly, the returns generated by the portfolio will simply be the weighted average of the returns of the components. The combination of these facts means that portfolios can be created or managed in such a way that all of the return associated with the components remains, while only a fraction of the risk remains (the systematic risk) as long as the proper assets are selected and their weights are carefully determined. Here are the formulas we will use for this exercise, they are all based on a two asset portfolio, here referred to as asset A and asset B, the formulas would need to be modified for adding additional assets into the portfolio: Sharpe Ratio: ????−???? ???? Return on a Portfolio: ???? = ???????? + ???????? Beta of a Portfolio: ???? = ???????? + ???????? Standard Deviation of a Portfolio: ???? = �????2????2 + ????2????2 + 2????????????,?????????? Where (in order of appearance in the above formulas): ???? = the return on the portfolio ???? = the risk free rate ???? = the standard deviation of the portfolio’s returns ???? = the weight of asset A in the portfolio calculated as the value of asset A divided by the value of the portfolio ???? = the return on asset A ???? = the weight of asset B in the portfolio calculated as the value of asset A divided by the value of the portfolio ???? = the return on asset B ???? = the beta of the portfolio ???? = the beta (equity beta) of asset A ???? = the beta (equity beta) of asset B ???? = the standard deviation of asset A’s returns ???? = the standard deviation of asset B’s returns ????,?? = the correlation coefficient of asset a’s returns and asset b’s returns In this exercise you have 2 assets available that you can invest in: small cap stocks (asset A) and large cap stocks (asset B). Your assignment is to determine the ratio (use the weights) of each of those assets in portfolio that will result in you achieving the highest Sharpe ratio for your portfolio. You do not need to be exact, being roughly correct, but understanding why is much more important than being exactly correct. You may use the attached Excel file that contains the formulas and problem data already created, or you may solve the problem any way you wish. The best alternative methods for approaching this problem are deriving an equation from the individual formulas, trial and error and using the formulas to see the impact of changing the weights, or making an educated guess and hoping for the best. For this exercise assume the following inputs (which happen to be historic averages): ????,?? = .30 ???? = 3.46% ???? = 16.47% ???? = 11.95% ???? = 1.53 ???? = 1.0 ???? = .32 ???? = .2 Assignment: What is the optimal ratio of large cap stocks to small cap stocks to hold in your portfolio? Enter your answer as a percentage of small cap stocks, for example, a 50/50 mix would be entered as 1. To get this ratio divide the optimal percentage allocation in large caps by the optimal percentage allocation in small caps. *Note, no short selling is allowed in this exercise. Sheet1 Sharpe Ratio:ERROR:#DIV/0! Return on Portfolio:0 Risk Free Rate: St. Dev of Portfolio:0 Weight A Weight B St. Dev A St. Dev B Correlation Coefficient Return on A Return on B