Q. 86 - Precalculus Enhanced with Graphing Utilities

Question

A young couple has \(25,000 to invest. As their financial consultant, you recommend that they invest some money in Treasury bills that yield 7%, some money in corporate bonds that yield 9%, and some money in junk bonds that yield 11%. Prepare a table showing the various ways that this couple can achieve the following goals:

(a) \)1500 per year in income

(b) \(2000 per year in income

(c) \)2500 per year in income

(d) What advice would you give this couple regarding the income that they require and the choices available?

Step-by-Step Solution

Answer

Various combinations exist; higher income requires more allocation to higher-yield (riskier) bonds.

Step 1: Set Up the System

Let \(x\) = amount in Treasury bills (7%), \(y\) = amount in corporate bonds (9%), \(z\) = amount in junk bonds (11%).
\(x + y + z = 25000\)
Income equation: \(0.07x + 0.09y + 0.11z = \text{target income}\)

Step 2: Solve for Part (a): $1500 Income

\(0.07x + 0.09y + 0.11z = 1500\)
From the two equations, \(z = 25000 - x - y\), substitute:
\(0.07x + 0.09y + 0.11(25000 - x - y) = 1500\)
\(-0.04x - 0.02y = 1500 - 2750 = -1250\)
\(4x + 2y = 125000\), or \(2x + y = 62500\).
With \(x + y + z = 25000\), we can parameterize: let \(z = t\), then solve for various non-negative values.

Step 3: Solve for Parts (b), (c), (d)

(b) For $2000: \(0.07x + 0.09y + 0.11z = 2000\) leads to \(2x + y = 37500\).
(c) For $2500: \(0.07x + 0.09y + 0.11z = 2500\) leads to \(2x + y = 12500\).
(d) Advice: Higher returns require more investment in riskier junk bonds. The couple should balance risk tolerance with desired income. A diversified approach is recommended.