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## 6.1 Present Value and Future Value

PLEASE NOTE: This book is currently in draft form; material is not final.

### Learning Objectives

1. Define and explain investments and opportunity cost.
2. Define and explain the concepts of present value and future value.

“Lend me \$10 for the pizza tonight, and I’ll give you \$15 when I get my paycheck next week. Okay?” This was a common way for Mary to hit up John for some cash. Thankfully, Mary didn’t ask too often, and was always good to her word. But John only had \$50 in his wallet, just enough for the new video game that is coming out this weekend….

If John could time-travel (bear with me for a second…), he could loan the money to Mary, travel to next week to collect the \$15, travel back to today, and have enough for the video game all while making \$5. Of course, then Mary probably wouldn’t be asking to borrow John’s money at all, since she could just go forward in time to collect her paycheck. Since, as of this writing, time-travel is still an impossibility, John instead has to make a choice to commit his limited resources faced with a set of potential outcomes; he must choose which investmentA choice to commit limited resources faced with a set potential outcomes. is more desirable. He could deny Mary, but then he wouldn’t gain the extra \$5. But if he lends the money to Mary, he won’t have enough to purchase the video game until Mark pays him back. John therefore faces an opportunity costThe greatest outcome from the set of choices not taken when considering an investment.: the greatest outcome from the set of choices not taken when considering an investment.

### Only the Greatest…

John could also use his \$50 to purchase a math textbook, new socks, or a gift for his professor. But since he values the video game the most, that is the appropriate measure of the opportunity cost.

Mary has already faced a similar decision and made her choice: she is willing to forego \$15 in a week to receive \$10 now (presumably to exchange for pizza for her growling stomach). Specifically, she has determined that the present value (PV)The equivalent value of an outcome in today’s cash. (the equivalent value of an outcome in today’s cash) of \$15 at the time of her paycheck is less than \$10 (otherwise, she wouldn’t be offering those terms).

Figure 6.1 Comparing Outcomes

There is no way to directly compare \$10 today and \$15 in a week, since the \$15 is a future value (FV)A cash flow at any time later than the present.. A future value is a cash flow at any time later than the present, so comparing a PV and an FV would be like comparing apples and oranges. The only proper comparison is to find the present value of each cash flow. Note that the present value of \$10 today must be equal to \$10, by definition.

We have to make similar choices every day (for example, should I put in an hour of work studying now with the hopes of eventually earning a higher grade on the exam, but foregoing that television program). Technically, we could talk about the present value of an hour of labor in the future in terms of today’s hour of labor. For now, we will restrict ourselves to only considering financial investments, where the inputs and the outcomes are all quantifiable in monetary terms.

Which is worth more, a dollar today or a dollar tomorrow? (Feel free to substitute any currency.) Mary is willing to pay \$15 dollars in a week for \$10 today, so today’s dollars must be worth more to her. The reason she is offering the higher future amount is, presumably, because she intends to do something with those dollars in the meantime that is of higher value than paying \$5 of interest. Could the reverse ever be true? No! A dollar today could easily be turned into a dollar tomorrow by just holding on to it and not doing anything, so no one would ever accept fewer dollars tomorrow.

### Picking Nits

If there was some cost to holding dollars, say because you had so many you had to hire a guard to make sure nobody stole your cash, then it might be possible for a dollar tomorrow to be worth more. But that assumes there is nothing else for you to do with your money better than holding it (even spending it!), and that the cost of the guard is not trivial. For all practical purposes, a dollar today is going to be worth more than a dollar tomorrow.

### Key Takeaways

• Everyone’s resources are limited, so committing to an investment has an opportunity cost equal to the best alternative use of the resources.
• Comparing PVs with FVs tell us nothing. Comparisons can only be made with cash flows evaluated at the same point in time.
• A dollar (or euro, or yen, etc.) today is worth more than a dollar tomorrow.

### Exercises

1. What is the relationship between present value and future value?
2. What are the opportunity costs of attending college? In general? For your 9am class tomorrow morning?