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## CHAPTER 1:  The Time Value of Money

### 1.1 Introduction

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ould you lend someone \$10,000 today in return for \$10,000 in five years? Probably not. In fact, you should not, because \$10,000 in five years is not worth \$10,000 today. Money received at different times in the future will have different values today. For example, \$10,000 in 20 years is probably worth less to you than \$10,000 in 10 years. The principle behind this is called the “time value of money.”

So what is \$10,000 in the future worth today? The answer depends on the interest rate. The following interactive calculator shows you what happens when the interest rate is 6%.

Time is on the X-axis. The red chart is \$10,000 at times in the futures, the yellow chart shows you the value today. By placing your mouse over the dot on the top of the yellow chart, you can see that \$10,000 in six years is worth \$7,049 today. Similarly, \$10,000 in 10 years is worth just about half that amount today. You can change the numbers and click OK.  For example, you can verify that if the interest rate is 8%, the \$10,000 in five years is worth \$6,805 today. Click the numeric button to see all the values.

Now try increasing the number of periods to 30, and click OK.  This lets you see how the curve changes with time to maturity.  You will see the yellow chart take on a pronounced “convex” shape. This convexity results from discounting because the present value decreases at an increasing rate with the time to maturity. You can also explore how the curve itself changes with interest rates.  Scroll the interest rate to see how the whole curve moves as you increase or decrease interest rates. You can see how interest rates capture the time value of money: the higher the interest rate, the lower the value today of money in the future.

The Time Value of Money is a central concept of finance because it affects everything that involves borrowing or lending. This includes car loans, mortgages, student loans, bonds, savings accounts, and so on.

For example, if you want to borrow money, you can go to a financial institution such as a bank. If you satisfy the bank that you have the ability to repay the loan, plus interest, it will lend you the money. This creates a fixed-income security, which is a contract specifying the timing and amounts of cash flows over time. The "timing" is how often you make payments, and the "amount" is the dollar amount you pay each time. If you take out a car loan, you have the same type of security: a fixed amount that has to be paid over a period of time by you to the bank.

For these fixed-income securities, you are the "seller" and the bank is the "buyer" (it “owns” the loan). That is, you sell a legal obligation to repay the loan plus interest, and in return, you receive the amount of the loan. The interest rate is what the bank charges you for use of the money. It compensates the bank for not being able to lend the money to anyone else for the term of your loan.

Bonds are another large class of fixed-income securities. When an economic unit, such as the U.S. Treasury, borrows cash from the general investing public, it sells bonds. The public is now the buyer that lends money to the Treasury. This fixed-income security specifies the timing and amounts of cash payments by the U.S. Treasury (the seller) to the owner of the fixed-income security.

Fixed-income securities are traded in primary and secondary markets. The primary market is the market in which the security is first issued. For example, the U.S. Treasury initially auctions Treasury bills. Once issued in the primary market, securities are re-traded in the secondary market.  These markets determine a price for each fixed-income security. Since a fixed-income security specifies a series of future payments (or cash flows), the price of the security represents the value today of future cash flows.