Question

It is a fact that \((1+0.12)^{3}=1.40 .\) Knowing that to be true, what is the present value of \(\$ 140\) received in three years if the annual interest rate is 12 percent? a. \(\$ 1.40\) b. \(\$ 12\) c. \$100. d. \(\$ 112\)

Short answer

The present value is \( \$100 \), which corresponds to option c.

Step-by-step solution

Identify Given Information

We know that the future value is \( \$140 \) and it will be received in 3 years. The annual interest rate is 12%.

Understand Present Value Formula

The present value \( PV \) can be calculated using the formula:\[PV = \frac{FV}{(1 + r)^n}\]where \( FV \) is the future value, \( r \) is the annual interest rate, and \( n \) is the number of years.

Substitute Known Values into Formula

Substitute \( FV = 140 \), \( r = 0.12 \), and \( n = 3 \) into the present value formula.\[PV = \frac{140}{(1 + 0.12)^3}\]

Simplify the Denominator

We know from the problem that \((1 + 0.12)^3 = 1.40\). Substitute this into the equation:\[PV = \frac{140}{1.40}\]

Calculate the Present Value

Divide 140 by 1.40:\[PV = 100\]Therefore, the present value of \( \\(140 \) received in three years at an interest rate of 12% is \( \\)100 \).

Key concepts

Future Value

The future value represents the amount of money you receive at a future date. This amount includes all the interest accrued over a period of time.
For example, if someone offers you $140 three years from now, that $140 is the future value.

• The future value is important because it tells you how much your money grows.
• It helps in making decisions about where and when to invest.

In our context, understanding the future value is crucial for calculating how much $140 is worth today.

Interest Rate

The interest rate is a percentage that tells you how much your money will grow per year. It acts like a growth rate for your investments.
A 12% annual interest rate means your money gets 12% bigger each year.

• This percentage compounding adds up quickly over time.
• High interest rates can significantly increase your future value.

In our exercise, the 12% rate helps us understand how the $140 grows over three years.

Calculate Present Value Formula

To figure out what future money is worth today, we use the present value formula. This formula helps take future cash and find its equivalent value now.
Here's the formula again for clarity: \[PV = \frac{FV}{(1 + r)^n}\]
This formula means:

• PV = Present Value (the amount worth today)
• FV = Future Value (the amount worth in the future)
• \(r\) = interest rate
• \(n\) = number of years

By using the present value formula, you can decide if an investment will be worth it.

Time Value of Money

The time value of money is a principle that money now is worth more than the same sum in the future due to its potential earning capacity. It's a core concept in finance because it affects all decisions about money.
Here's why money now is valued more:

• It can be invested immediately to earn returns.
• It may lose value in the future due to inflation.

Understanding this principle allows people to evaluate investments and make informed financial choices.
In our exercise, knowing the time value of money helps us see why receiving $140 in the future is different from having a smaller amount today that's equally valuable.