Question

Treasury bills have a fixed face value (say, \(\$ 1,000\) ) and pay interest by selling at a discount. For example, if a one-year bill with a \(\$ 1,000\) face value sells today for \(\$ 950\), it will pay \(\$ 1,000-\$ 950=\$ 50\) in interest over its life. The interest rate on the bill is, therefore, \(\$ 50 / \$ 950=0.0526\), or 5.26 percent. a. Suppose the price of the Treasury bill falls to \(\$ 925\). What happens to the interest rate? b. Suppose, instead, that the price rises to \(\$ 975\). What is the interest rate now? c. (More difficult) Now generalize this example. Let \(P\) be the price of the bill and \(r\) be the interest rate. Develop an algebraic formula expressing \(r\) in terms of \(P\). (Hint: The interest earned is \(\$ 1,000-P\). What is the percentage interest rate?) Show that this formula illustrates the point made in the text: Higher bond prices mean lower interest rates.

Short answer

Answer: The algebraic formula to express the interest rate (r) in terms of the price (P) for a Treasury bill with a fixed face value of $1,000 is: \(r = \dfrac{\$1,000 - P}{P}\).

Step-by-step solution

a. Calculate the interest rate when the price falls to \(\$925\)

To find the interest rate, we will calculate the difference between the face value and the selling price, and then divide that difference by the selling price. In this case: Interest rate = \((\$1,000 - \$925) / \$925\) Interest rate = \(0.0811\) or \(8.11\%\)

b. Calculate the interest rate when the price rises to \(\$975\)

To find the interest rate, we will calculate the difference between the face value and the selling price, and then divide that difference by the selling price. In this case: Interest rate = \((\$1,000 - \$975) / \$975\) Interest rate = \(0.0256\) or \(2.56\%\)

c. Develop the algebraic formula to express the interest rate \(r\) in terms of the price \(P\)

To generalize the example and find the algebraic formula for the interest rate, we can follow the same method we used in parts a and b. Let \(F\) be the fixed face value of the Treasury bill, \(P\) be the selling price, and \(r\) be the interest rate. The formula is as follows: \(r = \dfrac{F - P}{P}\) For this Treasury bill with a face value of \(\$1,000\), we can replace \(F\) with \(\$1,000\): \(r = \dfrac{\$1,000 - P}{P}\) This formula shows the inverse relationship between bond prices and interest rates. As the price of the bond \(P\) increases, the difference \((\$1,000 - P)\) decreases, thereby resulting in a lower interest rate \(r\). Conversely, as the price of the bond \(P\) decreases, the difference \((\$1,000 - P)\) increases, resulting in a higher interest rate \(r\).

Key concepts

Bond Prices

Bond prices refer to the current cost of purchasing a bond. In the context of Treasury bills, these prices are usually lower than the bond's face value because they are sold at a discount.

Understanding bond prices is crucial because:

• They determine the interest earnings for investors, with the difference between the selling price and the face value representing the interest earned.
• Price fluctuations can impact the yield or return on investment for the bondholder.

For example, a Treasury bill with a face value of \(\\(1,000\) sells at \(\\)950\). Here, \(\\(950\) is the bond price. The difference of \(\\)50\) corresponds to the interest earned, calculated as \((\\(1,000 - \\)950)\).

These price shifts occur frequently, affecting the potential returns that an investor might receive. An increase in bond prices typically leads to a decrease in the interest rates and vice versa. Therefore, closely monitoring bond prices is essential for anyone dealing in bonds.

Interest Rates

Interest rates are the percentages that represent the cost of borrowing money or the reward for saving. In Treasury bills, the interest rate is essentially the yield that investors receive from holding the bill until maturity.

Here's how they're calculated:

• The interest earned is the difference between the face value and the purchase price of the bill.
• The interest rate is the interest earned divided by the price paid for the bond.

For example, if a Treasury bill costs \(\\(950\) and matures to \(\\)1,000\), the interest rate is calculated as \(\frac{\\(1,000 - \\)950}{\$950}\), yielding around 5.26%.

Interest rates directly influence how attractive an investment is. Higher rates provide better yields, making the securities more appealing. These rates are significant for investors, as they dictate the profitability of the investment.

Inverse Relationship

The inverse relationship between bond prices and interest rates is a fundamental principle in finance. Understanding this relationship helps investors make informed decisions.

Here's how the inverse relationship works:

• When bond prices rise, the yield or interest rate falls. This is because the fixed payment (face value) results in a smaller difference relative to the increased price.
• Conversely, when bond prices decrease, the yield or interest rate increases, as the fixed payment becomes a larger portion of the lower price.

Mathematically, the relationship is expressed through \[ r = \frac{F - P}{P} \] Where \(r\) is the interest rate, \(F\) is the face value, and \(P\) is the purchase price. This formula clearly shows how an increase in \(P\) (price) leads to a decrease in \(r\) (rate), and vice versa.

In practical terms, this relationship means that market conditions affecting bond prices will inversely affect how much return investors can expect. It's a key concept for those involved in bond markets, as it underscores the delicate balance between risk and return.