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
Novel Mathematical Models for Profit Optimization
Barry Dynkin, Scott Huang, Benjamin Leibowicz, Sam Panzer, David Rosengarten
John L. Miller Great Neck North High School
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.
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.
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.
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.
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.
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.
The SML with individual stocks plotted.
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.
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.
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.
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.
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 SML based on data from March 2, 2007 to April 13, 2007.
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!).
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