Question

Suppose you buy a 91 -day \(\$ 500,000\) Treasury bill at the price of \(\$ 485,000\) and you hold the bill until it matures. What is the interest rate you earn?

Short answer

The interest rate earned on the Treasury bill when held until maturity is approximately \(12.03\%\).

Step-by-step solution

Write down the given values

Purchase price: \(\$485,000\) Mature price: \(\$500,000\) Number of days until maturity: \(91\)

Calculate the discount rate

To calculate the interest rate, we first need to determine the rate at which the price of the Treasury bill was discounted. The discount rate is calculated as: \(D=\frac{F-P}{F}\) Where: \(D\) = discount rate \(F\) = mature price \(P\) = purchase price \[ D=\frac{500,000-485,000}{500,000} \] Calculating this, we get: \[ D=\frac{15,000}{500,000} = 0.03 \] The discount rate is \(0.03\), or \(3\%\).

Calculate the annual discount rate

The current discount rate is for the 91-day period. To calculate the annual discount rate, we need to multiply this by the number of 91-day periods in a year: \(Annualized\:Discount\:Rate = D \times \frac{365}{n}\) Where: \(n\) = number of days until maturity \[ Annualized\:Discount\:Rate = 0.03 \times \frac{365}{91} \] Calculating this, we get: \[ Annualized\:Discount\:Rate = 0.03 \times 4.0110 \approx 0.1203 \] The annual discount rate is approximately \(0.1203\), or \(12.03\%\).

Determine the interest rate earned

The interest rate earned is equal to the annual discount rate: \(Interest\:Rate = Annualized\:Discount\:Rate\) Thus, the interest rate earned on the Treasury bill is approximately: \(Interest\:Rate = 12.03\%\) So, by holding the Treasury bill until it matures, you earn an interest rate of approximately \(12.03\%\).

Key concepts

Discount Rate

When you purchase a Treasury bill for less than its face value, you do so at a discount. The discount rate is simply a measure of how much less you paid compared to what you will receive at maturity. It tells us what percentage of the face value you have saved through the discount. To find it, subtract the purchase price from the mature (or face) value, and then divide by the mature value. Using the exercise example:

• Face Value (\( F \)) = \(\(500,000\)
• Purchase Price (\( P \)) = \(\)485,000\)

The equation becomes:\[D = \frac{500,000 - 485,000}{500,000} = 0.03\]This gives us a discount rate of \(3\%\). This calculation helps investors gauge the return just based on the discount before considering the effect of holding the bill over time.

Annualized Discount Rate

The discount rate by itself only covers the specific period from purchase to maturity. To better understand how this discount would compare over a full year, we calculate the annualized discount rate. Converting a 91-day interest to an annual rate allows for an easier comparison with other investment opportunities.To annualize the rate, we multiply the discount rate by the ratio of days in a year to the maturity period (91 days in this scenario):\[Annualized\ Discount\ Rate = D \times \frac{365}{91}\]Thus:\[Annualized\ Discount\ Rate = 0.03 \times 4.0110 \approx 0.1203\]This results in an annualized discount rate of approximately \(12.03\%\). This rate shows the effective annual yield, giving investors a clearer picture of their potential earnings if the investment were projected across a full year.

91-Day Treasury Bill

Treasury bills (T-bills) are short-term debt securities issued by the government. The 91-day Treasury Bill is named for its 91-day maturity period, making it a relatively short-term investment. The short duration minimizes exposure to interest rate fluctuations but also means that it typically offers lower returns compared to longer-term securities.Key aspects of the 91-day T-bill include:

• It is sold at a discount and matures at full face value, meaning you earn the difference as interest.
• These are considered low-risk, highly liquid investments, favored by conservative investors looking for safety and steady returns rather than high yields.

In our exercise, buying the T-bill at \(\(485,000\) and holding to maturity nets you the face value of \(\)500,000\), illustrating how this investment type functions. Their short maturities and predictable returns make them an integral part of investment portfolios for managing risk.