Math Problem Statement

A stock had returns of 20.47% (1 year ago), -9.32% (2 years ago), X (3 years ago), and 14.45% (4 years ago) in each of the past 4 years. Over the past 4 years, the compound return for the stock was -4.28 percent. What was the arithmetic average annual return for the stock over the past 4 years?

Input instructions: Input your answer as the number that appears before the percentage sign. For example, enter 9.86 for 9.86% (do not enter .0986 or 9.86%). Round your answer to 2 decimal places.

Solution

To solve this problem, we calculate the arithmetic average annual return and validate the conditions provided.

Steps to Calculate the Arithmetic Average Annual Return

  1. Let the returns over the four years be:

    • R1=20.47%R_1 = 20.47\%,
    • R2=−9.32%R_2 = -9.32\%,
    • R3=XR_3 = X,
    • R4=14.45%R_4 = 14.45\%.

    The arithmetic average is given by: Arithmetic Average Return=R1+R2+R3+R44.\text{Arithmetic Average Return} = \frac{R_1 + R_2 + R_3 + R_4}{4}.

  2. The compound annual return (CAGR) formula is: (1+CAGR)=(1+R1)(1+R2)(1+R3)(1+R4)4.(1 + \text{CAGR}) = \sqrt[4]{(1 + R_1)(1 + R_2)(1 + R_3)(1 + R_4)}.

    Substituting the compound return (−4.28%=−0.0428-4.28\% = -0.0428): (1−0.0428)=(1+0.2047)(1−0.0932)(1+R3)(1+0.1445)4.(1 - 0.0428) = \sqrt[4]{(1 + 0.2047)(1 - 0.0932)(1 + R_3)(1 + 0.1445)}.

    Simplify the left side: 0.9572=(1.2047)(0.9068)(1+R3)(1.1445)4.0.9572 = \sqrt[4]{(1.2047)(0.9068)(1 + R_3)(1.1445)}.

  3. Raise both sides to the 4th power: 0.95724=(1.2047)(0.9068)(1+R3)(1.1445).0.9572^4 = (1.2047)(0.9068)(1 + R_3)(1.1445).

    Calculate 0.957240.9572^4: 0.95724≈0.8386.0.9572^4 \approx 0.8386.

  4. Solve for 1+R31 + R_3: 0.8386=(1.2047)(0.9068)(1.1445)(1+R3).0.8386 = (1.2047)(0.9068)(1.1445)(1 + R_3).

    Compute the product of (1.2047)(0.9068)(1.1445)(1.2047)(0.9068)(1.1445): (1.2047)(0.9068)(1.1445)≈1.2484.(1.2047)(0.9068)(1.1445) \approx 1.2484.

    Thus: 0.8386=1.2484(1+R3).0.8386 = 1.2484(1 + R_3).

  5. Divide by 1.2484 to isolate 1+R31 + R_3: 1+R3=0.83861.2484.1 + R_3 = \frac{0.8386}{1.2484}.

    1+R3≈0.6718.1 + R_3 \approx 0.6718.

  6. Solve for R3R_3: R3≈0.6718−1=−0.3282 or −32.82%.R_3 \approx 0.6718 - 1 = -0.3282 \, \text{or} \, -32.82\%.

Find the Arithmetic Average Return

Substitute the values into the arithmetic average formula: Arithmetic Average Return=20.47−9.32−32.82+14.454.\text{Arithmetic Average Return} = \frac{20.47 - 9.32 - 32.82 + 14.45}{4}.

Simplify: Arithmetic Average Return=−7.224≈−1.81.\text{Arithmetic Average Return} = \frac{-7.22}{4} \approx -1.81.

Final Answer:

−1.81\boxed{-1.81}

Let me know if you want more details or have any questions! Here are five related questions for deeper understanding:

  1. What is the difference between compound and arithmetic average returns?
  2. How does one interpret a negative compound return over a multi-year period?
  3. What scenarios cause arithmetic and compound averages to diverge significantly?
  4. Why does a large negative return have a substantial impact on average returns?
  5. How is the geometric mean used in calculating CAGR?

Tip: Always use the CAGR formula for multi-year growth rates to account for compounding effects accurately.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Average
Compound Annual Growth Rate (CAGR)
Exponentiation
Algebraic Manipulation

Formulas

Arithmetic Average Return = (R1 + R2 + R3 + R4) / 4
(1 + CAGR) = (1 + R1)(1 + R2)(1 + R3)(1 + R4)^(1/4)

Theorems

Geometric Mean for CAGR Calculation

Suitable Grade Level

Grades 11-12 and College Level

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