Future Value

Question

For the 2009 fiscal year, Lowe's Companies, Inc. reported an annual net income of $48,230,000. Assume the income can be reinvested continuously at an annual rate of return of 5.1% compounded continuously and that Lowe's will maintain this annual net income for the next 5 years.
(a) What is the future value of its 5-year net income?

Answer 1

Hi Logan!

We can solve this problem using the future value formula for a continuous income stream with continuous compounding.

The important detail is that Lowe's earns income continuously over the next five years, rather than receiving the entire amount as one lump sum.

Step 1: Identify the given information

We are given:

Annual net income = $48,230,000

Annual interest rate = 5.1% = 0.051

Investment period = 5 years

Compounding = Continuous

We want to calculate the future value of the income accumulated over five years.

Step 2: Write the future value formula

For a continuous income stream, the future value is:

FV = R(e^(rt) - 1)/r

Where:

FV = Future value

R = Annual income rate

r = Annual interest rate

t = Number of years

e = Euler's number, approximately 2.71828

This formula accounts for the fact that income earned earlier has more time to accumulate interest than income earned later.

Step 3: Substitute the given values

FV = 48,230,000(e^(0.051 × 5) - 1)/0.051

Multiply the exponent:

0.051 × 5 = 0.255

Therefore:

FV = 48,230,000(e^0.255 - 1)/0.051

Step 4: Evaluate the exponential expression

Using a calculator:

e^0.255 ≈ 1.290462

Subtract 1:

1.290462 - 1 = 0.290462

Now substitute:

FV ≈ 48,230,000(0.290462)/0.051

Evaluating with full precision:

FV ≈ $274,685,568.13

Step 5: Understand what the answer means

Without earning any interest, Lowe's would accumulate:

$48,230,000 × 5 = $241,150,000

However, because the company continuously reinvests its income at 5.1%, the money earns additional returns.

Total future value:

$274,685,568.13

Original income:

$241,150,000

Additional investment earnings:

$274,685,568.13 - $241,150,000

Investment earnings = $33,535,568.13

Final Answer

(a) The future value of Lowe's five-year net income is approximately $274,685,568.13.

This is approximately $274.69 million.

Key Takeaway

When money is earned continuously and reinvested with continuous compounding, we use:

FV = R(e^(rt) - 1)/r

This differs from the standard lump-sum compound interest formula because the company receives income throughout the five-year period rather than investing all the money at the beginning.

I hope this helps!