Chapter 7.3, Problem 92E

To determine

To Find: How much is invested in each type of investment?

Answer to Problem 92E

  $-125000+2a,500000-2a,125000-a,a$ amount invested in certificates of deposit, municipal bonds, blue-chip stocks and growth stocks

Explanation of Solution

Given information: The certificates of deposit pay $2\%$ annually, municipal bonds pay $4\%$ annually. Over a five-year period, the investor expects the blue-chip stocks to return $10\%$ annually and growth stocks to return $14\%$ annually and one-fourth of the portfolio invested in stocks

Concept used:Gaussian Elimination is the process of using valid row operations on a matrix until it is in reduced row echelon form

Calculation:

Let $c,m,b$ and g represent the amount invested in certificates of deposit, municipal bonds, blue-chip stocks and growth stocks

From the given information

Total amount of investment

  $c+m+b+g=500,000$

The investor wants a combined annual return of $6\%$

  $0.02c+0.04m+0.1b+0.14g=\frac{500,000\times 6}{100}=300,00$

The investor wants to have only one-fourth of the portfolio invested in stocks

  $b+g=\frac{500,000}{4}=125,000$

There are $4$ variables but only $3$ equations, then there are infinitely many solutions

So, let $g=a$

  $\begin{array}{l}c+m+b+g=500,000\text{ Equation }1\\ 0.02c+0.04m+0.1b+0.14g=300,00\text{ Equation }2\\ b+g=125,000\text{ Equation }3\\ g=a\text{ Equation }4\end{array}$

Adding $-0.02$ times the first equation to the second equation

  $\begin{array}{l}-0.02c-0.02m-0.02b-0.02g=-100,00\\ 0.02c+0.04m+0.1b+0.14g=300,00\end{array}$ ______________________________________________________________.

  $0.02m+0.08b+0.12g=200,00$

Adding -0.03 times the first equation to the second equation produces a new second equation

  $\begin{array}{l}c+m+b+g=500,000\text{ Equation }1\\ 0.02m+0.08b+0.12g=200,00\text{ Equation }2\\ b+g=125,000\text{ Equation }3\\ g=a\text{ Equation }4\end{array}$

From equation $4$ the value of $g\text{ is }a$ . To solve for b , substitute $g=a$ into equation 3 to obtain

  $b=125,000-a$

  $g=a$ , $b=125000-a$ substitute into equation $2$ to obtain

  $m=500,000-2a$

  $g=a$ , $b=125000-a$ , $m=500000-2a$ substitute into equation $1$ to obtain

  $c=-125,000+2a$

  $-125000+2a,500000-2a,125000-a,a$ amount invested in certificates of deposit, municipal bonds, blue-chip stocks and growth stocks