An investor will use $2{,}000 to purchase 10 shares of a certain stock at $200 per share. At the same...

Let's visualize this problem to make it crystal clear...

Phase 1: Owning the Dataset

Visual Representation

Let's create a simple investment flow diagram:

Initial Investment → After 1 Year → Earnings
Stock: $2,000     → Stock Value    → Stock Gain/Loss
CD: X amount      → X + (X × 0.02) → CD Interest

Target: Total Earnings = $360

Given Information

• Stock purchase: \(\mathrm{10\,shares \times \$200/share = \$2,000}\)
• CD interest rate: 2% simple interest annually
• Goal: Total net earnings = $360 after one year
• Two scenarios to consider:
• Stock increases by 10%
• Stock decreases by 5%

Phase 2: Understanding the Question

We need to find how much to invest in the CD for each scenario such that:

\(\mathrm{Stock\,Earnings + CD\,Interest = \$360}\)

Where:

• Stock Earnings = \(\mathrm{Final\,Stock\,Value - \$2,000}\)
• CD Interest = \(\mathrm{CD\,Investment \times 0.02}\)

Phase 3: Finding the Answer

Scenario 1: Stock Increases by 10%

Step 1: Calculate stock performance

• Initial stock value: $2,000
• After 10% increase: \(\mathrm{\$2,000 \times 1.10 = \$2,200}\)
• Stock earnings: \(\mathrm{\$2,200 - \$2,000 = \$200}\)

Step 2: Calculate required CD earnings

• Total earnings needed: $360
• Earnings from stock: $200
• Required from CD: \(\mathrm{\$360 - \$200 = \$160}\)

Step 3: Calculate CD investment amount

• CD Interest = \(\mathrm{CD\,Investment \times 0.02}\)
• \(\mathrm{\$160 = CD\,Investment \times 0.02}\)
• CD Investment = \(\mathrm{\$160 ÷ 0.02 = \$8,000}\)

Scenario 2: Stock Decreases by 5%

Step 1: Calculate stock performance

• Initial stock value: $2,000
• After 5% decrease: \(\mathrm{\$2,000 \times 0.95 = \$1,900}\)
• Stock earnings: \(\mathrm{\$1,900 - \$2,000 = -\$100}\) (loss)

Step 2: Calculate required CD earnings

• Total earnings needed: $360
• Earnings from stock: -$100
• Required from CD: \(\mathrm{\$360 - (-\$100) = \$460}\)

Step 3: Calculate CD investment amount

• CD Interest = \(\mathrm{CD\,Investment \times 0.02}\)
• \(\mathrm{\$460 = CD\,Investment \times 0.02}\)
• CD Investment = \(\mathrm{\$460 ÷ 0.02 = \$23,000}\)

Phase 4: Solution

Final Answer:

• CD assuming 10% gain: $8,000
• CD assuming 5% loss: $23,000