TIME VALUE OF MONEY

Present Value and Future Value Suppose a rich relative, Uncle Bill, offers to give you one of two, mutually exclusive gifts just for being his niece:

If you choose Gift A, he would give you $100 a year from now, $100 two years from now, $100 three years from now and $250 six years from now.

If you choose Gift B, he would give you $325 five years from today and another $325 six years from today.

Which gift should you pick?

To answer this question, you need to know a few things: Is this guy really your uncle? Mom says he’s her brother, but are you sure? In other words, is this guy true to his word? Will he keep his promise over the full six years? What’s the expected rate of inflation over the next six years? What rate can you earn investing your money? Are you able to save what Uncle Bill gives you or are you likely to go to the mall with each installment?

We will talk more about all of these factors later, but for now let’s assume you will save the money and you can earn 10% per year by investing it. So, i=10%.

Here are the two gifts: To solve this dilemma, you can’t just say, Oh, with A, I get only $550 but with B, I get $650, so I’ll take B. Remember, you can make 10% on your investments, and with Gift A, you get the money a lot sooner and can earn lots more interest on it.

To answer the question, then, you need to take all the dollar amounts to the same, common point. It could be time 0 (present value), it could be time 6 (future value), it could be time 4. It doesn’t matter. Whichever gift is better at time 0 will also be better at time 6 or time 4. It just has to be a common point for all cash flows in order to be able to compare apples with apples.

Since we’re making the choice today at time 0, let’s start with calculating the present value of both gifts at time 0. For each gift, discount all the payments back to time 0 using an interest rate of 10%.

Let’s start with A: The first $100 is a full year away. It’s worth only 100/(1.10) 1 or 90.91. The second $100 is two years in the future so its value today is only 100/(1.10) 2 82.64. The third $100 is worth only 100/(1.10) 3 or 75.13 today. The 250 payment can be discounted back to the present by 250/(1.10) 6 or 141.12. Add up the four present values: 90.91 + 82.64 + 75.13 + 141.12 = 389.80.

Use your calculator or click the button below to use the Fin 125 Cash Flow Calculator.

Enter these values: discount or interest rate = 10; CF ₀ = 0; CF ₁ = 100; n ₁ = 3; CF ₂ = 0; n ₂ = 2; CF ₃ = 250; n ₃ = 1. Click the NPV button and the present value is 389.80.

You could also try the problem using the Fin 125 PV/FV Calculator. With this approach, you’d need to find the present value of the three 100s as an annuity and add the present value of the 250.

Before we look at Gift B to see how it stacks up, let’s think about what that 389.80 means.

389.80 is the amount you could put away today in a savings account that earns 10% interest and be able to withdraw 100 at the end of each of the next three years and 250 at the end of year 6 and have nothing left over.