A US Government Bond With A Face Value Of 1000 Has A Coupon

A US Government Bond With A Face Value Of 1000 Has A Coupon Rate Of

A US government bond with a face value of $1,000 has a coupon rate of 3% with semiannual coupons, and matures in four years and six months from now. The first coupon is due in six months from now. You have obtained the following rates from the web site of the US Treasury. The rates are expressed as nominal annual rates with semiannual compounding as usual. c) What is the present value of the principal amount you receive at maturity? (2 points) d) What should be the current market price of the bond? (2 points) e) Using the above information calculate the yield-to-maturity of this bond? (4 points) f) Calculate the duration of the bond? (5 points) g) How much is the value of your bond holdings going to deteriorate if the interest rate curve shifts up by 1 percentage point? (2 points)

Paper For Above instruction

The valuation and analysis of government bonds are central to understanding fixed-income securities and their sensitivities to interest rate changes. This paper discusses the key concepts involved in valuing a US government bond with a given coupon rate, maturity, and prevailing market rates. It explores calculations for the present value of principal at maturity, determines the current market price of the bond, computes the yield-to-maturity (YTM), measures the bond’s duration, and assesses its price sensitivity to interest rate shifts. Each section integrates theoretical foundations with practical calculation methodologies, demonstrating their application within the context of the specified bond.

Introduction

Government bonds are considered safe-haven assets due to the backing of the US government and are widely analyzed for their sensitivities to interest rate changes. The particular bond under consideration is a four-year and six-month maturity security with a face value of $1,000, a coupon rate of 3%, and semiannual coupon payments. The challenge involves aligning the bond's cash flows with market rates to determine its valuation and risk measures accurately. This analysis highlights the importance of understanding present values, yield calculations, and duration as tools for investors and policymakers to evaluate bond investments.

Present Value of Principal at Maturity

The principal amount of $1,000 received at maturity is straightforward; however, its current value depends on the prevailing market interest rates. Since the question asks specifically for the present value (PV) of the principal, assuming market rates are used to discount this amount at the appropriate rate, the fundamental calculation involves discounting the $1,000 to today’s value using the yield for a similar maturity bond. Given that the rates are semiannual and nominal, the discount rate per period is derived from the nominal annual rate divided by two, and the number of periods is twice the number of years plus additional periods for the months remaining.

Assuming the applicable discount rate is given or inferred from the US Treasury rates, the present value PV of the principal received at maturity is calculated using the discount factor:

PV of principal = $1,000 / (1 + r/2)^{N}

where r is the relevant semiannual market interest rate, and N is the total number of semiannual periods until maturity. For a maturity of 4.5 years, N = 9 semiannual periods.

Current Market Price of the Bond

The current market price is the present value of all future cash flows, which include semiannual coupon payments and the principal repayment. The sequence involves summing the present value of the coupons and the discounted face value:

P = \(\sum_{i=1}^{N} \frac{C}{(1 + r/2)^i} + \frac{F}{(1 + r/2)^N}\)

where:

- C = semiannual coupon payment = (3% of $1,000) / 2 = $15,

- F = face value = $1,000,

- r = market nominal annual interest rate (semiannual compounded),

- N = total number of semiannual periods (9).

The summation considers each coupon discounted back to today, with the final term including the principal repayment. Precise calculation depends on the actual market rate, which needs to be provided or estimated from the US Treasury yield curve.

Yield-to-Maturity Calculation

YTM is the internal rate of return (IRR) that equates the present value of cash flows to the current market price. For semiannual coupons, the YTM is found by solving:

P = \(\sum_{i=1}^{N} \frac{C}{(1 + y/2)^i} + \frac{F}{(1 + y/2)^N}\),

where y is the YTM annualized rate. The iterative process or financial calculator is used to find y such that the right side equals the current market price. Given the complexity, approximation methods like trial-and-error or numerical solvers are typically employed.

Bond Duration

Duration measures the sensitivity of a bond's price to interest rate changes and is calculated as the weighted average time until cash flows are received, weighted by their present value. The Macaulay duration formula is:

D = \(\frac{\sum_{i=1}^{N} t_i \times PV(CF_i)}{\text{Price}}\),

where t_i is the time period, and PV(CF)_i is the present value of each cash flow. Semiannual periods are used to maintain consistency with coupon payments.

Price Change Due to Interest Rate Shift

A 1 percentage point increase in interest rates causes the bond's value to deteriorate. The price change can be approximated by multiplying the duration by the change in yield:

\(\Delta P \approx - D \times \Delta y \times P\),

where D is the bond duration, \(\Delta y\) is the interest rate increase (0.01), and P is the current bond price. This approximation highlights the inverse relationship between interest rates and bond prices.

Conclusion

Understanding the valuation and interest rate sensitivities of government bonds requires a comprehensive grasp of present value concepts, yield calculations, and duration measures. The example detailed here exemplifies how market rates influence bond pricing and risk assessment. These analyses are invaluable for investors seeking to optimize portfolio returns and for policymakers monitoring monetary policy impacts.

References

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