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Calculate Effective Annual Rate of Return for 2-Year Stock Investment: $26,300 to $34,700
Mathematics (Finance)
High School Grade 12 / College Introductory Finance
Question Content
Steven purchased 1000 shares of a certain stock for $26,300 (including commissions). He sold the shares 2 years later and received $34,700 after deducting commissions. Find the effective annual rate of return on his investment over the 2-year period. (Round your answer to two decimal places.)
Correct Answer
14.77%
Detailed Solution Steps
1
Step 1: Identify the variables for effective annual rate (EAR) calculation: Initial Investment (P) = $26,300, Future Value (A) = $34,700, Time (t) = 2 years.
2
Step 2: Use the compound growth formula: A = P(1 + r)^t, where r is the effective annual rate.
3
Step 3: Rearrange to solve for r: (1 + r) = (A/P)^(1/t).
4
Step 4: Substitute values: (1 + r) = ($34,700 / $26,300)^(1/2) ≈ (1.31939)^0.5 ≈ 1.1477.
5
Step 5: Calculate r: r = 1.1477 - 1 = 0.1477, or 14.77% when converted to a percentage and rounded to two decimal places.
Knowledge Points Involved
1
Effective Annual Rate (EAR)
EAR is the actual annual rate of return on an investment, accounting for compound growth over multiple periods. For multi-year investments with a single initial and final value, it is calculated using the formula r = (Future Value/Initial Value)^(1/t) - 1, where t is the number of years.
2
Compound Growth Formula
The formula A = P(1 + r)^t describes how an initial amount (P) grows to a future amount (A) at an annual compound rate (r) over t years. It is used for investments with annual compounding of returns.
3
Stock Investment Rate of Return
For stock investments, the effective annual rate accounts for the total return (including all costs like commissions) over the holding period, expressed as an annualized percentage to compare with other investments.
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