To model a simple stock portfolio with two stocks, we’ll write down a system of two equations in two variables. We hope to find a unique solution to this system, so let’s make sure we understand two key ideas.

- We need two variables.
- We need two equations.

Why are these important?

##### Two Variables?

The variables represent the two unknown quantities we are looking for. Since we want to know how much two invest in each stock in a tow stock portfolio, the two variables will represent the amounts of money invested in each stock. If we had more stocks in the portfolio, we would need more variables to correspond to.

##### Two Equations?

If we hope to solve our system of linear equations for a unique solution, the number of equations must match the number of variables. This assumes that one of these equations is not redundant. For this model, we’ll get our equations from two pieces of information, the total amount invested in the portfolio and the total return desired.

For a larger portfolio, we would need more equations to specify a unique solution. In that case we would need more information such as an average beta for the portfolio.

##### Write Out the System

For this portfolio, we will model two securities:

Based on data from the end of January 2016, we know the following information.

Security | Annual Dividend Yield | Beta |

Tootsie | 1.16% | 0.7 |

Diebold | 4.45% | 1.51 |

Our goal for this example is to invest a total of $50,000 with a total dividend return of 3%. This is attainable since one security in the portfolio has a higher yield and the other a lower yield. It would be impossible to combine the stocks in a portfolio to get a total yield higher that the highest yielding stock of lower than the lowest yielding stock.

Start with your variables. I’ll call mine *x*_{1} and *x*_{2} and describe them as

*x*_{1}: amount invested in Tootsie

*x*_{2}: amount invested in Diebold

Once you understand what these are, it is easy to use the information in the problem to write out the two equations.

##### Total Amount Invested Is $50,000

We can start to get mathematical by writing

Total Amount Invested = 50,000

To finish the equation, we need to write the left side of the equation in terms of the variables. A “total” indicates addition so write

*x*_{1} + *x*_{2} = 50,000

##### Total Dividend Return is 3%

If the total dividend return needs to be 3% of $50,000, we need a total of

3% of $50,000 = (0.03)(50,000) = 1500

This total dividend will come from the dividend on the Tootsie stock,

1.16% of the amount invested *x*_{1} = 0.0116 *x*_{1}

and the dividend on the Diebold stock,

4.45% of the amount invested *x*_{2} = 0.0445 *x*_{2}

So if the total dividend from the portfolio is $1500 and this is the sum of the dividend from each stock in the portfolio,

0.0116 *x*_{1} + 0.0445 *x*_{2} = 1500

##### Model for a Two Stock Portfolio

Combining the two equations together gives a system of two linear equations in two variable,

*x*_{1} + *x*_{2} = 50,000

0.0116 *x*_{1} + 0.0445 *x*_{2} = 1500

We can solve these graphically or algebraically. If we use the substitution method and solve them graphically, solve for *x*_{1} in the first equation to give

*x*_{1} = 50,000 – *x*_{2}

Putting this into the second equation leads to

0.0116 (50,000 – *x*_{2})+ 0.0445 *x*_{2} = 1500

580 – .0116 *x*_{2} + 0.0445 *x*_{2} = 1500

0.0329 *x*_{2} = 920

* x*_{2} ≈ 27,963.53

If $27,963.53 is invested in Diebold, then *x*_{1} ≈ 50,000 – 27,963.53 or $22,036.47 must be invested in Tootsie.

The sum of these amounts is $50,000 as desired and the total dividend is

0.0116(22036.47) + 0.0445(27963.53) ≈ 1500

as it should be.