The task of choosing the optimal portfolio is a lengthy one but entails some very well defined steps and formulas. The first part of the report deals with formulation of the optimal portfolio consisting of stocks listed with the Dow 30 index, while the last part would deal with utility scores and the clients risk appetite. I would systematically present formulas and explain every part of portfolio construction as the report proceeds.The first in the series of steps is annualizing monthly stock returns by multiplying each value by 12. Excess returns are then calculated by subtracting the T-bill rate (Exhibit 1) from these annualized stock returns.

Exhibit 2a!B5 is the monthly stock returnExhibit 1!C68 is the T-bill rate for year 1990.Averages of these annualized excess returns are then calculated as a nave forecast for expected returns. Bordered Covariance Matrix is next in this series of steps. The covariance matrix is made using the covariance option in Excels Data Analysis tool by selecting the entire excess return data sheet. The actual portfolio construction process starts with the determination of risk return opportunities available to the investor. All such opportunities are summarized and graphically represented by the minimum-variance frontier of risky assets. Formulas used to find portfolio expected return, portfolio standard deviation and the Sharpe ratio are listed below. In addition, Solver was used to determine weights and the portfolio combination with the least amount of variance for a given portfolio expected return. Portfolio expected return= sumproduct(all weights, expected returns)

Since portfolio standard deviation, like the expected return, is the weighted average of individual standard deviations I first calculated standard deviations for each stock using the formula Stock risk = stocks weight * sumproduct( all weights, stocks covariance with all other stocks) as shown in the picture below

Portfolio standard deviation = sum (all stdev of individual stocks) ^ 0.5 was then used to calculate the portfolios variance.The slope or the Sharpe ratio was calculated by simply dividing the portfolio expected return by portfolio standard deviation.To find the minimum variance portfolio, standard deviation was set as the objective function to be minimized by solver. The resulting mean, standard deviation, slope and weights were copied and pasted on a table to chart out the efficient frontier. Next we used the solver ad on to locate the optimum risky point on this frontier i.e. the portfolio with the highest reward to volatility ratio. For this we selected the slope as our objective function and requested the solver to maximize it. The mean, SD, slope and weights that ensured gave me a fair idea as to help me plot other points. I then changed expected return values within the range and asked solver to maximize the slope. All values and their weights were copied and pasted in the table made earlier for construction of the efficient frontier. I in fact mad e to tables, one for short sales and one in which short sales was not allowed.

WeightsThe CAL or the Capital Allocation Line is the one that is tangent to the optimum risky portfolio, this line has the highest Sharpe ratio or in other words the greatest reward to variability ratio. The determination of CAL effectively marks the end of work for the investment manager since allocation of the complete portfolio of T-bills versus the risky portfolio depends on the degree of risk aversion of the client. As can be seen in the picture above, the CAL line was constructed by multiplying each securitys standard deviation with the slope of the optimum risky portfolio. Two different CAL lines were plotted each for short sales and with no short sales. Below is the graph for short sales.

Picture of the table used to plot the CAL line under no short sales criteria is also given below.

Weights The CAL line under this situation is graphed below

ANALYSISThe following table supplements our financial belief that the higher the risk the higher the return. The other insight that we get is that short selling is a much better strategy than going without short selling. T he optimum portfolio gives a much better incremental return to incremental risk than when the manager decides not to sell borrowed security.What is amazing though is the higher Sharpe ratio of the Index Return but with lower volatility and lower expected return than our selected portfolio. One reason for this could be that Index returns are better diversified and have a broader selection of asset classes giving them the cushion and the advantage over actively managed funds.