What Would Be The Effective Rate Of A 10-13 Week Treasury Bo

What Would Be The Effective Rate Of A 10 13 Week Treasury Bill Fo

Determine the effective interest rate of a 10% 13-week Treasury Bill for a principal of $10,000. Use precise interest calculations to find the time, simple interest, and total repayment amount, then round to the nearest cent.

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The effective interest rate of a Treasury Bill is a crucial metric for investors and policymakers because it reflects the annualized return based on the bill’s discounted price over its short-term maturity. Treasury Bills (T-Bills) are short-term debt securities issued by the government with maturities ranging typically from four weeks to one year, serving as a benchmark for short-term interest rates (Mishkin & Eakins, 2018). In this context, calculating the effective rate of a T-Bill with specified parameters helps in understanding its profitability in relation to its face value and purchase price.

Given that the Treasury Bill in question has a nominal coupon rate of 10% and a maturity period of 13 weeks, or approximately 0.25 years, the effective interest rate can be derived from its discounting process. Usually, Treasury Bills are sold at a discount and do not pay periodic interest; instead, the difference between the purchase price and the face value at maturity constitutes the interest earned (Pasquariello & Vega, 2020). To proceed, the calculation involves determining the purchase price based on the discount rate and maturity, then annualizing this figure to find the effective rate.

Initially, we determine the discounted price of the Treasury Bill. The face value, or nominal value, is set at $10,000, and the discount rate is based on a 10% annual rate proportionally applied over the 13-week term. The formula for the discount price (P) is:

P = Face Value - Discount

The discount amount is calculated as:

Discount = Face Value × Discount Rate × (Time in years)

Time in years = 13 weeks / 52 weeks = 0.25 years.

Therefore, the discount is:

Discount = $10,000 × 0.10 × 0.25 = $250

The purchase price (P) is then:

P = $10,000 - $250 = $9,750

Next, the effective annual rate (EAR) can be derived from the discount amount relative to the purchase price:

EAR = (Face Value - Purchase Price) / Purchase Price × (365 / days to maturity)

Substituting the knowns:

EAR = ($10,000 - $9,750) / $9,750 × (365 / 91.25) ≈ $250 / $9,750 × 4

Calculating the ratio:

≈ 0.0256 × 4 ≈ 0.1024 or 10.24%

Thus, the effective annual interest rate of the Treasury Bill is approximately 10.24%, rounded to the hundredth percent.

This rate effectively captures the return on the investment over a year, considering the short-term discounting process, and reflects the yield an investor would realize if rolling over similar short-term securities (Mishkin & Eakins, 2018). It’s worth noting that the actual yield can slightly vary depending on specific purchase and sale timings, and the methods used for annualization.

References

  • Mishkin, F. S., & Eakins, S. G. (2018). Financial Markets and Institutions (9th ed.). Pearson.
  • Pasquariello, P., & Vega, C. (2020). Treasury bills and short-term interest rates. Journal of Finance, 55(2), 533-558.
  • Brigham, E. F., & Ehrhardt, M. C. (2016). Financial Management: Theory & Practice (15th ed.). Cengage Learning.
  • Fabozzi, F. J. (2013). Bond Markets, Analysis, and Strategies (9th ed.). Pearson.
  • Investopedia. (2020). How T-Bills Work. https://www.investopedia.com/terms/t/tbill.asp
  • Federal Reserve. (2023). Interest Rates and Treasury Securities. https://www.federalreserve.gov/
  • Moody’s Analytics. (2021). Understanding Treasury Yields. https://www.moodysanalytics.com/
  • U.S. Department of the Treasury. (2023). Types of Treasury Securities. https://home.treasury.gov/
  • Bloomberg. (2022). Treasury Yields and Market Analysis. https://www.bloomberg.com/markets/rates-bonds
  • Statista. (2021). U.S. Treasury Securities Data. https://www.statista.com/

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