If A 180-Day T-Note With A Face Value Of 10,000 Is Purchased

If A 180 Day T Note With A Face Value Of 10000 Is Purchased At A

Calculate the price of a 180-day T-note with a face value of $10,000 purchased at a 6% requested yield.

Determine the return on a 90-day note with a face value of $100,000 that is issued at a 7% yield per year, sold with 60 days to maturity at a 7.4% yield per annum.

Find the price of a $100 zero-coupon bond with a six-year maturity, a 5% annual interest rate, and a face value of $100.

Paper For Above instruction

Introduction

Understanding the valuation of debt instruments such as Treasury notes and zero-coupon bonds is essential for investors, policymakers, and financial analysts. These securities are widely used in financial markets due to their relative safety and predictable cash flows. This paper explores the calculations involved in determining the price of a 180-day Treasury note, calculating the return on a short-term bond, and valuing a zero-coupon bond based on given interest rates and maturities. These computations are fundamental in assessing the attractiveness of fixed-income securities and understanding interest rate impacts on bond prices.

Valuation of a 180-Day Treasury Note

The price of a Treasury note is inversely related to its requested yield or yield to maturity (YTM). For the 180-day T-note with a face value of $10,000 and a requested yield of 6%, the valuation involves discounting the face value to present value using the YTM. Since the note is semi-annual, the yield must be converted to the period yield to appropriately discount the cash flows.

The formula for the present value (price) of the note is:

P = FV / (1 + r/2)^n

Where:

  • FV = Face value = $10,000
  • r = Annual yield = 6% or 0.06
  • n = Number of periods = 1 (since it's 180 days, exactly half a year)

Calculating:

P = 10,000 / (1 + 0.06/2)^1 = 10,000 / (1 + 0.03) = 10,000 / 1.03 ≈ $9,708.74

Therefore, the price of the T-note is approximately $9,708.74.

Return Calculation for the Short-Term Note

The second scenario involves a $100,000 note issued at a 7% yield, sold with 60 days remaining at a 7.4% yield. To compute the return, we analyze the purchase price at issuance and the resale price based on the new yield, considering the change in yield and remaining maturity.

First, compute the purchase price at issuance:

P₁ = FV / (1 + r₁ × t)

Where:

  • FV = $100,000
  • r₁ = 7% per annum = 0.07
  • t = 90/360 (assuming 360-day year) = 0.25 years

P₁ = 100,000 / (1 + 0.07 × 0.25) = 100,000 / (1 + 0.0175) ≈ 98,343.10

Next, estimate the resale price considering a 7.4% yield with 60 days to maturity:

P₂ = FV / (1 + r₂ × t)

Where:

  • r₂ = 7.4% per annum = 0.074
  • t = 60/360 = 0.1667

P₂ = 100,000 / (1 + 0.074 × 0.1667) ≈ 100,000 / (1 + 0.0123) ≈ 98,538.52

The return is then calculated as:

Return = (P₂ - P₁ + Interest accrued) / P₁

Interest accrued over 60 days at the original yield:

Interest = FV × r₁ × t = 100,000 × 0.07 × 0.1667 ≈ $1,167

Thus, total proceeds upon resale are approximately:

Proceeds = P₂ + interest accrued ≈ 98,538.52 + 1,167 = $99,705.52

The total return:

Return = (99,705.52 - 98,343.10) / 98,343.10 ≈ 0.01384 or 1.384%

Annualized, considering 60 days, the return approximates around 8.32% (since 1.384% over 60 days scaled to a year).

Valuation of a Zero-Coupon Bond

The price of a zero-coupon bond is computed by discounting its face value using the annual interest rate over its maturity period. The formula is:

P = FV / (1 + r)^t

Where:

  • FV = $100
  • r = 5% or 0.05
  • t = 6 years

Calculating:

P = 100 / (1 + 0.05)^6 = 100 / (1.05)^6 ≈ 100 / 1.3401 ≈ $74.64

Thus, the current price of the zero-coupon bond is approximately $74.64.

Conclusion

This analysis highlights the importance of yield-to-maturity calculations in bond pricing and the impact of interest rate changes on the valuation of debt securities. The calculated prices and returns are vital for investors seeking to optimize their fixed-income portfolios. Understanding these principles allows for better risk management and strategic decision-making in bond investment.

References

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