The Time Value of Money

You may have heard the old adage a dollar today is worth more than a dollar tomorrow. But how much more? Would you rather have $100 today or $120 in a month? What about $100 today or $120 in ten years? The study of interest rates (or more precisely risk-free interest rates) tries to answer the question: what is the monetary value of time? How much should you be compensated simply for waiting?

Most discussions of interest rates begin with the assumption that there exists a fictitious bank from which anyone can borrow or deposit any amount of money at a fixed rate r_f

Let’s go back to the hypotheticals posed earlier and assume that r_f is 2% (continuously compounded). Should we choose $100 now or $120 in a month? If we really needed money today (to pay bills, buy something immediately) we might be tempted to choose $100 today. However, using the bank we can simply borrow $100 today and pay back the debt with the $120 we get a month from now. Let’s compare the two options:

Receive $100 Today Borrow $100 Today
Today \$100 \$100
Tomorrow 0 \$120-\$100e^{(.02)(\frac{1}{12})}=\$19.83

Clearly, choosing to receive the $120 in a month is a better option. Now lets consider $100 today or $120 in ten years.

Receive $100 Today Borrow $100 Today
Today \$100 \$100
Tomorrow 0 \$120-\$100e^{(.02)(10)}=-\$2.14

In this scenario the $120 isn’t enough to pay back our debt, so we should choose $100 today. If we really didn’t mind waiting the 10 years and would rather have more money in the future, we could simply take the $100 today and put it in the bank. In 10 years, your $100 would turn into \$100e^{(.01)(10)}=\$122.14, which is more than the $120 we would have received.

The central idea here is that we can convert money promised in the future into money today, and vice-versa, by using the bank. $100 today is exactly the same as \$100e^{(.02)(1)}=\$102.02 in a year, \$100e^{(.02)(5)}=\$110.52 in five years, and \$100e^{(.01)(10)}=\$122.14 in ten years: We can put $100 in the bank today and withdraw any of those amounts in the future. Similarly, $120 in 10 years is exactly the same as \$120e^{(.02)(-10)} = 98.25 today, \$120e^{(.02)(-9)}=\$100.23 in one year, and \$120e^{(.02)(-5)}=\$108.58: We can borrow these amounts now, in a year, or in five years and pay back our debt with the $120 in ten years.

They are totally and unambiguously interchangeable.

Let’s consider another example. Suppose you want to open a restaurant but are deciding between opening a cheap restaurant and a fancy restauran. For each option you know exactly how much money it will cost to open the restaurant, how much money you’ll make in every year for the next ten years.

Year Cheap Fancy
0 -280,000 -720,000
1 120,000 120,000
2 110,400 120,000
3 101,568 120,000
4 93,442 120,000
5 85,967 120,000
6 79,089 120,000
7 72,762 120,000
8 66,941 120,000
9 61,586 120,000
10 56,659 120,000

When comparing these two options, its convenient to convert future cash flows into today-money. This is called discounting future cash flows to their present value or PV. For the Cheap restaurant, this comes out to be:

\displaystyle -\$280{,}000 + \$120{,}000e^{(.02)(-1)} + \$110{,}400e^{(.02)(-2)}+ \cdots + \$56{,}659e^{(.02)(-10)}=\$655{,}771.12

Thus, running the cheap restaurant for ten years is exactly the same as producing $655,771 today. The PV of the cashflows for the fancy restaurant is:

\displaystyle -\$720{,}000 + \$120{,}000e^{(.02)(-1)} + \$120{,}000e^{(.02)(-2)}+ \cdots + \$120{,}000e^{(.02)(-10)}=\$621{,}744.99

Therefore opening the cheap restaurant is a better investment than opening the fancy restaurant.

Unfortunately, in the real world this fictitious bank doesn’t exist. As an alternative we can deposit our money in U.S. government bonds, or if we’re a creditworthy bank we can deposit money with or borrow money from other creditworthy banks. These deposit and borrow rates typically depend on the length of time you’re depositing or borrowing for, and fluctuate daily. We’ll discuss all of these complexities of real-world risk-free interest rates in a future article.

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