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Chapter 5: Problem 74

If an investment appreciates by \(10 \%\) per year for 5 years (compounded annually) and then depreciates by \(10 \%\) per year (compounded annually) for 5 more years, will it have the same value as it had originally? Explain your answer.

Short Answer

Expert verified
No, the investment will not have the same value as it had originally. After appreciation and depreciation, the final value of the investment is given by \(A_2 = P(1.10)^5 (0.90)^5\), which is not equal to the initial value P, since \((1.10)^5 (0.90)^5 \ne 1\).

Step by step solution

01

Understand the compound interest formula

The compound interest formula is given by: \(A = P(1 + r/n)^{nt}\) Where: A = Final amount P = Principal amount (Initial investment) r = Annual interest rate (as a decimal) n = Number of compounding periods per year t = Number of years Since the investment appreciates and depreciates annually (compounded annually), we can simplify the formula as follows: \(A = P(1 + r)^t\)
02

Calculate the value of the investment after 5 years of appreciation

Let's denote the initial investment as P. After 5 years of appreciation at 10% per year, the value of the investment will be: \(A_1 = P(1 + 0.10)^5\)
03

Calculate the value of the investment after 5 years of depreciation

Now, let's calculate the value of the investment after it depreciates by 10% per year for 5 more years. We can denote the initial value for this phase as \(A_1\) from step 2: \(A_2 = A_1(1 - 0.10)^5\)
04

Substitute the value of A_1 from step 2 into step 3

Now, we'll substitute the value of \(A_1\) from step 2 into the formula in step 3: \(A_2 = [P(1 + 0.10)^5](1 - 0.10)^5\)
05

Simplify the expression

Now, we should simplify the expression for A_2: \(A_2 = P(1.10)^5 (0.90)^5\)
06

Compare the final value to the initial value

Now, we need to compare the final value \(A_2\) to the initial value P. If they are equal, then the investment has the same value as it had originally. Since we're looking for an equality, then: \(A_2 = P\) So we're comparing: \(P(1.10)^5 (0.90)^5 = P\) After simplification, we get: \((1.10)^5 (0.90)^5 \ne 1\) So, the final value of the investment is not equal to the initial value, meaning the investment will not have the same value as it had originally.

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