Fixed Income Module 3 Group Homework 1 6pts Given A Five Yea

Fixed Incomemodule 3 Group Homework1 6pts Given A Five Year 8 Cou

Given a five-year, 8% coupon bond with a face value of $1,000 and coupon payments made annually: a. What is the bond value if it is trading at the yield of 6%? b. What is the bond value if it is trading at the yield of 8%? c. What is the bond value if it is trading at the yield of 10%? d. Comment on the price and yield relation you observe. What are the percentage changes in value when the yield goes from 6% to 8% and when it goes from 8% to 10%?

Suppose an investor bought a 10-year, 10% annual coupon bond at par (face value of $1,000 and paying coupons annually) and then sold it 3.5 years later at a yield of 8%. a. What is the full price? b. What is the accrued interest the investor would receive when he sold the bond? (Use a 30/360-day count convention) c. What is the clean price?

A zero-coupon Treasury bill maturing in 150 days is trading at $98 per $100 face value. Determine the following rates for the T-bill: a. Dealer’s annual discount yield? (use 360-day count convention) b. Yield to maturity? (Use an actual 365-day count convention) c. Logarithmic return (use an actual 365-day count convention)

Calculate both Macaulay and modified durations of the eight-year, 8.5% coupon bond given a flat yield curve at 10%.

Paper For Above instruction

Fixed income securities are essential components of global financial markets, offering both investment opportunities and risk management tools for investors. This paper aims to analyze various aspects of fixed-income instruments, including bond valuation at different yields, bond pricing after purchase and sale, yield calculations on treasury bills, and duration measures critical for risk assessment. The analyses employ fundamental financial formulas and concepts rooted in fixed income theory to elucidate the relationships between yields, prices, and durations.

Bond Valuation at Different Yields

The primary concept in bond valuation is that a bond's price is the present value of its future cash flows, discounted at the prevailing market yield. For a five-year, 8% coupon bond with a face value of $1,000, coupons are paid annually, resulting in fixed annual payments of $80 ($1,000 × 8%). The valuation formulas for different yields involve discounting these cash flows accordingly.

When the yield is 6%, the bond’s price, P, can be calculated as follows:

P = \(\sum_{t=1}^{5} \frac{C}{(1 + y)^t} + \frac{F}{(1 + y)^5}\)

where C = $80, F = $1,000, y = 0.06.

Similarly, for 8% and 10%, the calculations adjust with corresponding yield rates. As yields increase, the present value of future cash flows decreases, leading to a decline in bond prices.

This inverse relationship demonstrates a fundamental principle: bond prices and yields move inversely. When yields go from 6% to 8%, the bond price drops more sharply, reflecting increased discounting. Moving from 8% to 10%, this decline accelerates, indicating heightened interest rate risk. Quantitatively, the percentage change in bond price can be calculated by comparing the prices at different yields.

Bond Price After Purchase and Sale

In the scenario involving a 10-year, 10% coupon bond purchased at par and subsequently sold after 3.5 years at an 8% yield, the full price of the bond must be determined using the standard bond valuation formula, discounting future cash flows at the new market yield of 8%. Given that the bond was originally bought at face value, understanding its price 3.5 years later involves recalculating the present value of remaining coupons and principal at the new yield.

The accrued interest at sale is computed according to the 30/360-day count convention, which assumes 30 days per month and 360 days per year, simplifying interest calculations over partial periods. The accrued interest represents the interest accumulated from the last coupon date to the sale date, and it must be added to the clean price to derive the full (dirty) price.

The clean price is obtained by subtracting accrued interest from the full price. This distinction is critical in fixed income trading because the clean price excludes accrued interest, which is settled separately upon trade settlement.

Rates for a Treasury Bill

The Treasury bill trading at $98 per $100 face value for 150 days involves calculating the annual discount yield, yield to maturity, and logarithmic return. The discount yield, based on a 360-day convention, measures the annualized discount relative to face value, providing a measure of the bill’s cost to the investor.

The yield to maturity employs a 365-day convention, reflecting actual calendar days, and accounts for the total return if held to maturity. The logarithmic return assesses the continuous growth rate over the period, offering a different perspective on investment performance.

Duration Calculations

Duration metrics like Macaulay and modified durations quantify the interest rate sensitivity of bonds. The eight-year, 8.5% coupon bond's durations are calculated using the present value of cash flows discounted at a 10% flat yield curve. Macaulay duration measures the weighted average time to receive cash flows, while modified duration estimates the percentage change in price for a 1% change in yield.

These measures are vital for managing interest rate risk, with higher durations indicating greater sensitivity to interest rate fluctuations. The calculations involve summing the present value-weighted time periods and adjusting for the yield curve to compute precise duration figures.

Conclusion

Understanding the relationships among bond prices, yields, and durations is crucial for investors and financial professionals. As yields increase, bond prices decline, with the extent of this decline captured by duration. Accurate valuation and risk assessment enable better investment decisions, especially in volatile interest rate environments. Mastery of these concepts using precise calculations supports effective portfolio management, risk mitigation, and strategic investment planning in the fixed income market.

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