Corporate Finance Optimal Portfolio Choice

Project I – Optimal Portfolio Choice

Jenny Duan 3035507679

Sining Gu 3035507370

Lu Wang 3035507447

Wonwoo Jang 303550715

Zhao Liu 3035507409

Q1, (a) Results of parameter estimation

First, based on the raw historical data of Hang Seng and the nine selected stocks’ (hereinafter referred to as stock 1,2,3,4,5,6,7,8 and 9) return index, we use the formula  to calculate the average monthly return for each stock and obtain 60 observations. The reason why we use return index to calculate the average monthly return instead of stock price is that return index includes dividends, interest, rights offerings and other distributions realized over a given period of time while price index does not.[pic 1]

We assume that the probability distribution for returns in the future is the same as (or similar to) what we observe in the past. Therefore, we use the historical estimation method to do the parameter estimation:

• For the mean returns of the nine stocks, we use the formula  / “=AVERAGE ()”[pic 2][pic 3]
• For the standard deviation of the nine stocks, we use the sample standard deviation formula

  / [pic 4]

“=STDEV ()”[pic 6][pic 5]

• For the covariance of the nine stocks, we use the formula  / “=COVAR(,)*60/(60-1)”[pic 7][pic 8][pic 9]

• For the variance of the portfolio, we use the formula [pic 11][pic 10]

Note: In the screenshot, we use weights from the MVF with 0.2% expected return as  and .[pic 12][pic 13]

• For the correlation of the nine stocks, we use the formula  / [pic 15][pic 14]

“=CORREL (,)”[pic 16][pic 17]

• For the portfolio return, we use the formula [pic 18]

Q1, (b) Weights of individual stocks for those points for constructing Minimum Variance Frontier

The minimum variance frontier is the set of optimal portfolios that are at the lowest risk for a given level of expected return. Given that the expected returns are from 0.2% to 1.2%, by an increment of 0.1%, we could use solver tool in excel to calculate the corresponding risk level (Standard Deviation) as well as the weightings of the 9 stocks of the 11 portfolios and plot the MVF along with the historical means and variance-covariance matrix as estimates.

Below is how we set up the solver when the expected return is 0.2%.

Objective:

Minimize the total risk; [pic 19]

Constrains:

(1) Invest all the money; ([pic 20]

(2) Achieve pre-specified expected return 0.2%; [pic 21]

(3) No short selling is allowed; [pic 22]

[pic 23]

The result of the portfolio weightings, variance and standard deviation are:[pic 24]

[pic 25]

We then run solver for another 10 times by increasing expected return by 0.1% each time (hereinafter referred to as portfolio A, B, C…K), while objective and other constrains maintain the same.

The weights, variance and standard deviation for the 11 portfolios on MVF are as follows.[pic 26]

Q1, (c) Graph of Minimum Variance Frontier

We set the 11 given expected returns as Y values and 11 corresponding minimum standard deviations from Q1, (b) as X values to plot the MVF.

[pic 27]

Q1, (d) Weights of individual stocks in the tangency portfolio

The tangency portfolio is the optimal portfolio of risky assets, known as the market portfolio. In order to get this tangency portfolio, we need to introduce Sharpe ratio which is the average return earned in excess of the risk-free rate per unit of volatility or total risk. The tangency portfolio is a

portfolio that generates the highest “reward to risk ratio” (Sharpe ratio).

For Sharpe ratio of the portfolio, we use the formula , where  ,  is the return of risk-free bank deposit which is 0.1% monthly.[pic 28][pic 29][pic 30]

Then we use solver tool in excel to calculate weightings for T with parameters as follows.

Objective:

Maximize the Sharpe ratio; [pic 31]

Constrains:

(1) Invest all the money; ([pic 32]

(2) No short selling is allowed; [pic 33]

[pic 34]

The calculated weightings, Sharpe value, expected return and standard deviation are:[pic 35]

[pic 36]

Q1, (e) Weights of individual stocks and risk-free asset in the recommended portfolio

The half-line of CAL shown in the figure is the new efficient frontier when risk-free asset is introduced. It is tangent to MVF at the pure risky portfolio with the highest Sharpe ratio. Its vertical intercept represents a portfolio with 100% of holdings in the risk-free asset; the tangency with the hyperbola represents a portfolio with no risk-free holdings and 100% of assets held in the portfolio occurring at the tangency point; points between those points are portfolios containing positive amounts of both the risky tangency portfolio and the risk-free asset.

Portfolios on the Capital Allocation Line (CAL) formed by Tangency Portfolio (T) and Risk-free asset (F) dominate all other portfolios because they offer higher returns for any given level of risk.

We use the risk-free bank deposit return as vertical interception (0,0.1%) and the Tangency point (3.81%,0.85%), which we calculated in the Q1, (d), to construct the CAL.

Therefore, the CAL equation is Y = 0.1967*X+0.1%

[pic 37]

In our case, the client expects a monthly return of 0.8%. We need to allocate his money between T and F according to his risk-return preference.