Show All Your Work For Each Problem That Is Equally Weighted

Show All Of Your Work Each Problem Is Equally Weighted Do Any 12 O

Show ALL of your work. Each problem is equally weighted. Do any 12 out of 14 questions. 1. From the following data create a discount curve (i.e., find discount factors): a. A zero coupon bond Pz(0, 0.5) = 99.20. b. A coupon bond paying 3% quarterly P(0, 0.25) = 100.5485. c. A coupon bond paying 6% quarterly P(0, 0.75) = 103.1655. d. A coupon bond paying 5% semiannually P(0, 1) = 103.0425. Use the following discount factors (discount curve) problems 2, 3 and 4: t Z(0, t) 0.25 0.50 0.75 1.00. Using the previous discount curve, price the following: A zero coupon bond expiring at t = 0.75. 3. Price a 1-year 6% coupon bond paying quarterly using the previous discount curve. 4. Price a 6-month coupon bond paying 7% semiannually with the previous discount curve. 5. For the following scenario, check if there is a mispriced security: a. A coupon bond paying 1% quarterly P(0, 0.25) = 100.6498. b. A coupon bond paying 4% semiannually P(0, 0.25) = 101.8980. c. A coupon bond paying 3% quarterly P(0, 0.50) = 101.2978. d. A coupon bond paying 5% quarterly P(0, 0.75) = 103.4425. e. A coupon bond paying 4% semiannually P(0, 1.00) = 103.5880. 6. What is the price of a 0.75-year floating rate bond that pays a semiannual coupon equal to floating rate plus 2% spread? We know the following: a. There is a zero coupon bond Pz(0, 0.25) = 99.70. b. There is a zero coupon bond Pz(0, 0.50) = 99.20. c. There is a coupon bond paying 3% quarterly P(0, 0.75) = 101.7380. 7. A Treasury dealer quotes the following 182-day bill at a 3.569% discount. Use the following discount factors when needed: t Z(0, t) 0.25 0.50 0.75 1.00. Calculate the duration of a 5-year zero coupon bond. 9. Calculate the duration of a 2-year fixed coupon paying 5% quarterly ($5/4 =$1.25 every 3 months). 10. What is the dollar duration of the following portfolio? i. Long a 1.5-year zero coupon bond. ii. Short a 2-year fixed coupon bond paying 1% quarterly ($1/4 =$0.25 paid every 3 months). 11. Compute the 95% VaR for the following portfolio: i. A 1.5-year fixed rate bond paying 2% quarterly. ii. A 0.75-year floating rate bond paying float plus 80 basis points semiannually. You know that the reference rate was set to 6% six months ago. iii. A 0.25 zero coupon bond. Additionally, you know that μdr = 0 and σdr = 0.4233. 12. Ms. White wants to invest $100,000 for the next five years. She purchases an annuity from a financial institution. Currently, the term structure is flat at 10% (yearly compounded). i. If the payments are made yearly, what is the amount that the financial institution will agree to pay Ms. White? ii. Assume that there is a 5-year fixed coupon bond that pays 12% coupon every year. What is the price and duration of the bond? iii. How much must the financial institution invest in the long-term bond to hedge the position? What should it do with the remainder of the money? Use the following discount factors when needed in problems 13 and 14: t Z(0, T) 0.25 0.50 0.75 1.00. Calculate the convexity of the following security: a 5-year zero coupon bond. 14. Calculate the convexity of the following security: a 3-year fixed rate bond paying 4% coupon on a semiannual basis. The End

Paper For Above instruction

Creating a comprehensive understanding of bond valuation, discount curves, and risk management techniques is fundamental in fixed income securities analysis. This paper aims to demonstrate the process of constructing a discount curve from given bond prices, pricing various securities using the curve, identifying mispriced bonds, calculating durations and convexities, and assessing portfolio value at risk (VaR). Through methodical computations and theoretical insights, this analysis underscores the importance of precise valuation and risk assessment in bond markets.

Constructing the Discount Curve

The initial task involves deriving discount factors from observed bond prices. For zero-coupon bonds, the discount factor is straightforward: Pz(0, t) = Price / Face Value. Given Pz(0, 0.5) = 99.20, the discount factor for 0.5 years is 0.9920. For coupon bonds, the process involves solving for discount factors that equate the present value of coupon payments and principal to the bond's market price.

The coupon bond paying 3% quarterly with a price of 100.5485 entails equating the present value of four quarterly coupons and the face value at maturity. Similar calculations apply for other bonds, resulting in a set of discount factors corresponding to different maturities. These discount factors enable the creation of a discount curve, which can be used for valuing other securities.

Pricing Zero-Coupon and Coupon Bonds Using the Discount Curve

Once the discount curve is established, it facilitates the valuation of other instruments. For example, pricing a zero-coupon bond expiring at 0.75 years with the discount factor for that horizon directly yields its present value. For coupon bonds, summing the discounted coupons and principal provides accurate pricing aligned with market data.

Identifying Mispriced Securities

By comparing observed market prices against valuations derived from the discount curve, one can detect potential mispricings. A bond that deviates significantly from its theoretical value may be over- or underpriced, indicating arbitrage opportunities or data inconsistencies.

Valuing Floating Rate Bonds

The valuation of floating rate bonds is based on the expectation that the floating rate resets to current market rates, simplifying to roughly the notional plus spread at reset dates. Calculating the price involves discounting the expected cash flow, which includes the floating rate plus spread, using zero-coupon and coupon bond prices for respective periods.

Treasury Bill Discount and Duration Calculations

The quote of a 182-day Treasury bill at a 3.569% discount can be translated into its price by adjusting the face value for the discount rate. Duration calculations for zero-coupon and fixed-coupon bonds involve summing the product of time and present value weights, providing insights into interest rate sensitivity.

Portfolio Dollar Duration and VaR

The dollar duration measure assesses the change in portfolio value for small interest rate movements. The VaR calculation estimates potential losses at a specified confidence level, considering asset volatilities and correlations, thus aiding risk management.

Hedging and Investment Strategies

Investors like Ms. White can utilize long-term bonds and short-term instruments to hedge interest rate risk, optimizing portfolio performance. The analysis emphasizes the importance of allocative efficiency and the role of convexity in managing convexity risk.

Calculating Convexity

Convexity measures the curvature of the price-yield relationship, providing a refined estimate of interest rate impact on bond prices. Calculations for zero-coupon bonds and fixed-rate coupons highlight the degree of price sensitivity to yield changes, essential for risk management.

Conclusion

This comprehensive examination underscores the interconnectedness of valuation, risk assessment, and strategic investment in fixed income markets. Accurate discount curve construction, precise valuation of securities, and thorough risk metrics are indispensable tools for practitioners aiming to optimize returns and manage potential losses effectively.

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