Constructing a portfolio
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Constructing a Portfolio. Novel Mathematical Models for Profit Optimization. Barry Dynkin, Scott Huang, Benjamin Leibowicz, Sam Panzer, David Rosengarten John L. Miller Great Neck North High School. The Problem. Our team has been given $30,000 to create a portfolio.

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Constructing a portfolio

Constructing a Portfolio

Novel Mathematical Models for Profit Optimization

Barry Dynkin, Scott Huang, Benjamin Leibowicz, Sam Panzer, David Rosengarten

John L. Miller Great Neck North High School


The problem

The Problem

  • Our team has been given $30,000 to create a portfolio.

  • We can invest in a maximum of six stocks chosen from a group of eighteen computer software and services companies.

  • Our ultimate goal is to purchase a combination of stocks so as to maximize the net profit earned when we sell the stocks one year after the date of purchase.

  • To accomplish this, we create a basic form for a mathematical model and then continue to improve the model through various revisions.

  • For each version of the model, we use an original JAVA program and a proportional division of our investment to determine a portfolio.

  • We test our methods using data accumulated since the date of purchase.


The indicators

The Indicators

  • I. Free Cash Flow (FCF)

    • The FCF is the ratio of cash flow to the number of shares of the stock. It measures a company’s financial strength.

  • II. Return on Invested Capital (ROIC)

    • The ROIC is the ratio of net income to invested capital. A high ROIC indicates a strong company.

  • III. Price to Earnings Ratio (P/E)

    • The P/E is the ratio of the price per share of stock to the earnings per share of stock. A low P/E makes a stock attractive.

  • IV. Price to Sales Ratio (P/S)

    • The P/S is the ratio of the price per share of stock to the revenue per share of stock. A low P/S indicates a good value investment.

  • V. Beta Coefficient (β)

    • β is a measure of risk or volatility. β = 1 indicates that the stock is moving with the market, β > 1 indicates that the stock is moving more drastically than the market, and β < 1 indicates that the stock is moving less drastically than the market.


Optimal values of the indicators

Optimal Values of the Indicators

There is no true optimal value of β since it is not an indicator of stock quality. However, since extremely risky stocks are unattractive, for the purposes of this study we choose the industry average of β as the optimal β value.


Basic quality assessment qa model

Basic Quality Assessment (QA) Model

  • Our first attempt to select six stocks from a portfolio is based purely on the four stock quality indicators and one stock volatility indicator provided.

  • First, we assume that it is beneficial to invest in the maximum of six different stocks. By producing a larger, more balanced portfolio, we eliminate much of the risk associated with the poor performance of an individual stock.


Basic quality assessment qa model1

Basic Quality Assessment (QA) Model

  • To construct a basic form for a mathematical model, we develop an equation called the Quality Assessment (QA) equation (next slide).

  • One assumption underlying this equation is that the list of eighteen stocks provided is a representative microcosm of the computer industry as a whole.

  • Based on this assumption, the industry average ROIC is equal to the average ROIC of the eighteen stocks, the industry average β is equal to the average β of the eighteen stocks, and so on.

  • A second assumption is that the four stock quality indicators are equally significant predictors of performance.

  • Due to this assumption, the ratios of the indicator values to their averages contained in the QA equation are preceded by coefficients that standardize them to be equally weighted in the formula.

  • For example, the first term is divided by 1.5 because the optimal value baseline for FCF is 50% higher than industry average.

  • In other words, for all four quality indicators, a ratio that matches the optimal value baseline yields a term in the QA equation equal to 1.


Basic quality assessment qa model2

Basic Quality Assessment (QA) Model

The Quality Assessment (QA) equation. Coefficients such as 1.5, 1.3, and 2 ensure that the four stock quality indicators are weighted equally, according to the assumption. Large deviations from the average β are viewed unfavorably.


Calculating the qa score for each stock

Calculating the QA Score for Each Stock

  • We calculate the QA for each of the eighteen stocks using our QA equation.

  • To facilitate this process, we create a computer program in JAVA.

  • The program obtains the values of the indicators from a text file and substitutes them into the QA equation, determining a QA score for each stock.

  • It then prints, in descending order of QA score, the eighteen stocks and their respective scores.

  • The results of the computer program were checked manually and confirmed.

  • The table below shows the six stocks with the highest QA scores.

  • Based on the values of their indicators, it seems reasonable that these six stocks would perform very well.


Purchasing stocks

Purchasing Stocks

  • To determine the amount of money to invest in each of our top six stocks, we simply divide the $30,000 between the six stocks such that the money invested in each stock is proportional to its QA score.

  • The table below shows the amount of money to invest in each stock and the number of shares of each stock to purchase.

  • According to our Basic QA Model, we purchase 151 shares of CAI, 645 shares of QADI, 129 shares of COGN, 165 shares of BMC, 142 shares of MSFT, and 159 shares of SRX.


Revised qa model

Revised QA Model

  • The information we were provided states that recent results suggest that a relatively high ROIC and relatively low P/E ratio are strong indicators of the value in a stock.

  • To account for these findings, we adjust the coefficients of the QA equation to weight the ROIC and P/E terms more heavily.

  • The computer program was edited to contain variable coefficients that can be easily adjusted to change the weighting factor of the ROIC and P/E terms.

  • The revised QA equation is show below.

  • C1 and C2 are the variable coefficients.

  • We constructed two different portfolios for this version of the model, one for C1=C2=2.0 and another for C1=C2=3.0.


Calculating the qa score for each stock1

Calculating the QA Score for Each Stock

  • The QA scores for each stock are calculated the same way that they were calculated for the basic QA model.

  • The top six stocks and their respective QA scores for both weighting scenarios are displayed in the table below.

  • The results were checked manually and confirmed.

  • For the most part, the order of the stocks is not that different from the Basic QA Model, but there are slight changes. The only new member in the top six stocks for both scenarios is ORCL, which replaces SRX in the sixth spot.


Purchasing stocks1

Purchasing Stocks

  • The money to invest was again divided proportionally to the QA scores.

  • The money to invest in each stock and the shares of each stock to purchase are given in the tables below, representing the two weighting scenarios.

  • For the C1=C2=2.0 weighting scenario, we purchase 124 shares of CAI, 194 shares of MSFT, 640 shares of QADI, 165 shares of BMC, 119 shares of COGN, and 240 shares of ORCL.

  • For the C1=C2=3.0 weighting scenario, we purchase 215 shares of MSFT, 110 shares of CAI, 634 shares of QADI, 164 shares of BMC, 113 shares of COGN, and 263 shares of ORCL.

C1=C2=2.0

C1=C2=3.0


Toward a final qa model

Toward a Final QA Model

Of the five indicators provided, β is the most difficult to interpret and the hardest to incorporate into a mathematical model.

The problem with β is that it indicates nothing about whether the stock is moving upward at a certain pace or downward at that pace.

A far better indicator is the Reward to Risk ratio (Rew/Ris), which unlike β accounts for the direction in which the stock is moving.

The formula for Rew/Ris, which comes from the Capital Asset Pricing Model (CAPM), is shown below.

In the equation above, E(Ri) is the expected return on the capital asset, Rf is the risk-free interest rate, and E(Rm) is the expected return of the market. The expression including E(Ri) refers to the individual capital asset while the expression including E(Rm) applies to the market as a whole.


Toward a final qa model1

Toward a Final QA Model

Rf, estimated by the one-year constant maturity treasury rate, is currently 5.06%.

This means that an investment in a government treasury bond would return 5.06% interest over one year with essentially no risk.

Expected returns are generally calculated using data from the previous year.

To calculate E(Ri), we begin by taking the difference between the March 2, 2006 and March 2, 2007 closing prices for each of the eighteen stocks.

Then, we divide each difference by the March 2, 2006 closing price and multiply by 100% to obtain E(Ri) for that stock.


Toward a final qa model2

Toward a Final QA Model

To calculate E(Rm), we begin by taking the difference between the March 2, 2006 and March 2, 2007 closing prices for each of the eighteen stocks.

Then, we divide each difference by the March 2, 2006 closing price, multiply the result by the average stock volume traded daily, and divide by the sum of the average stock volumes.

Multiplying by 100%, we obtain E(Rm).

In this manner, we calculate a weighted average of the returns of each of the eighteen stocks in the market.

This accounts for the fact that not all stocks are traded in equal quantities.


Final qa model security market line

Final QA Model: Security Market Line

A common technique for representing the rewards of specific assets with respect to their volatilities is the Security Market Line (SML) based on the CAPM.

It helps to indicate whether a stock’s returns are worth the risk associated with the stock.

The x-axis of the SML is β and the y-axis is the expected return of the assets.

We generate a SML for our market of eighteen stocks by using E(Rm) as the slope of the line and Rf as its y-intercept.

Each individual stock is plotted on the axes with the SML.

The vertical distance from the SML to a stock point is equal to the E(Ri) – Baseline, where Baseline is the y-value of the SML at that β.

This is better than using only β because the more risky a stock gets, the more it must return for the Rew/Ris ratio to yield the same value.

The SML with the eighteen stocks plotted is shown on the next slide.


Final qa model security market line1

Final QA Model: Security Market Line

The SML with individual stocks plotted.


Toward a final qa model3

Toward a Final QA Model

We used the formula shown earlier to calculate the Rew/Ris ratio for each of the eighteen stocks.

These calculations employed the part of the equation that applies to individual capital assets.

We used the SML plot to calculate E(Ri) – Baseline for each stock.

The Rew/Ris ratio and E(Ri) – Baseline for each stock are displayed in the table on the next slide.


Toward a final qa model4

Toward a Final QA Model


Final qa model

Final QA Model

The β term from the previous two QA models is replaced by a term based on the E(Ri) – Baseline.

This is chosen instead of the Rew/Ris ratio because the E(Ri) – Baseline is relative to our market while the Rew/Ris ratio is not.

Since a positive value of this term indicates an attractive investment, the term is added rather than subtracted.

Therefore, the term is simply the ratio of the stock’s E(Ri) – Baseline to the average E(Ri) – Baseline of the eighteen stocks.

The final QA equation is shown below.

The last term is now based on E(Ri) – Baseline rather than β.


Calculating the qa for each stock

Calculating the QA for Each Stock

The change in the final term of the QA equation was incorporated into the computer program and the program was used to calculate the QA score for each stock using the final QA model.

The top six stocks and their respective QA scores are displayed in the table below.

This addendum to the model did not significantly affect the top six stocks. The six stocks are the same as the ones identified by the previous model. The only change is the order in which they appear.


Purchasing stocks2

Purchasing Stocks

We determine the amount of money to invest in each stock and number of shares of each stock to purchase using the same method that was described previously.

These numbers appear in the table below.

According to the Final QA Model, we purchase 208 shares of MSFT, 679 shares of QADI, 111 shares of CAI, 159 shares of BMC, 111 shares of COGN, and 264 shares of ORCL.


Testing the model

Testing the Model

Since the Final QA Model is the most advanced model we produced, we take the results of that model and purchase those stocks in the calculated amounts.

We trace the performance of the six stocks from the purchase date on March 2, 2007 until closing on April 13, 2007.

Although this is a short period of time to assess stock performance, it is of interest to test the predictions of the model.

We recreate the SML using the data from this time period and observe where our stocks fall relative to the line.

In addition, we calculate the return on our stocks if we sold on April 13, 2007.


The security market line

The Security Market Line

The SML based on data from March 2, 2007 to April 13, 2007.


Returns on our investment

Returns on Our Investment

If we purchased our stocks at closing on March 2, 2007 and sold them at closing on April 13, 2007, we would have made $2,504.

This gain is approximately an 8.35% return on our investments.

This return is impressive because, if the $30,000 were invested in each of the eighteen stocks equally, the return would be only 3.75%.

Our stocks outperformed the market by 4.43%.

Although it is impossible to extrapolate to this extent, if this rate of return continues for a period of 365 days following the date of purchase and we sell our stocks at that point, we would earn $30,209.

This gain is approximately an unbelievable 100.7% return on our investment (wouldn’t that be great!).


References

References

"Capital Asset Pricing Model." Wikipedia. 4 March 2007. http://en.wikipedia.org/wiki/capital_asset_pricing_model

Federal Reserve Board. "Interest Rates." MoneyCafe. 2007. 4 March 2007. http://moneycafe.com/library/cmt.htm

Goetzmann, William. "The Arbitrage Pricing Theory." An Introduction to Investment Theory. 1996. 4 March 2007. http://Viking.som.yale.edu/will/finman540/classnotes/class6.html

Intuit. "About Stock Evaluation." Quicken. 2006. 4 March 2007.

https://help.quicken.com/support/investments/stkeval/help/pfhelp.shtml

Investopedia. 4 March 2007. http://investopedia.com

Smart Money. 4 March 2007. http://smartmoney.com

Stock Charts. 4 March 2007. http://stockcharts.com

Yahoo Finance. 4 March 2007. http://finance.yahoo.com.


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