How much would you pay for a $100 bill? If I offered to sell you hundred dollar bills at any price below $100 each, you'd probably jump at the chance, buying as many as you could. Now suppose that I offer to sell you a $100 bill for less than $100, but the catch is that you won't get your $100 bill until next week. How much would you be willing to pay for it now?

There are at least two important factors to consider. First, you need to trust that I will actually deliver you the $100 the following week and not simply abscond with your money. Second, you won't have access to the amount you pay for the bill during the week prior to its delivery, so if any better opportunity comes along, you won't be able to take advantage of it.

Both of these factors lower the price you would be willing to pay for the bill. This situation may seem far-fetched, but if you replace me by the federal government, and change the length of time to 30 years, you are talking about Treasury bonds or T-Bonds.

The U.S. government is constantly in need of money, so it sells 30-year T-Bonds and other
*treasuries* to raise capital. In return for lending it money, the government pays
you interest just as a bank does when you deposit money into your account. The difference
with Treasury bonds is that they have a *face value* which is typically $1,000.
You purchase the bond for less than the face value, and at *maturity* 30 years later,
the government pays you the face value. The difference between what you paid for the bond
and the face value is the interest earned.

So how much should you pay for a $1,000 T-Bond? The answer to that question depends on the interest rate you want to earn over the 30 year lifetime of the investment. Recall the equation we came up with for compounding interest:

B = P × (1 + r)^{t}

where B is the ending value of your investment of P dollars earning an interest rate of r for t years. Plugging in what we know about our $1,000 T-Bond gives us:

$1000 = Price × (1 + r)^{30}

We want to know the price, so we need to divide both sides of the equation by
(1 + r)^{30}. During the discussion of logarithms, we saw that division
by a number raised to an exponent was the same as multiplying by the same number raised to
the negative of the exponent. So we can achieve the effect we want by multiplying both
sides of the equation by (1 + r)^{−30}:

Price = $1000 × (1 + r)^{−30}

Now we have an equation that determines the fair price of a 30-year T-Bond dependent only on the interest rate we want to earn. In April 2009, the interest rate you could expect to earn on a 30-year treasury was about 3.7 percent. Let's use a calculator to determine the price of the bond:

So you see that you just about triple your investment, but remember that it takes 30 years to realize those gains. Here is a table of prices for 30-year T-Bonds for various interest rates:

Interest Rate | T-Bond Price | |
---|---|---|

2.0% | $552.07 | |

2.5% | $476.74 | |

3.0% | $411.99 | |

3.5% | $356.28 | |

4.0% | $308.32 | |

4.5% | $267.00 | |

5.0% | $231.38 | |

5.5% | $200.64 | |

6.0% | $174.11 |

Notice that as the interest rate rises, the price you pay for the bond falls and vice-versa, which can be confusing. Just remember that you want to pay as little as possible for the bond and to earn as much interest as possible. The lower the price, the better the return.

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