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Bond Price, Yield, Duration

Bond Price, Yield, Duration. Pricing and Yield Yield Curve Duration Immunization. General Bond Characteristics. Price Face or par value Coupon rate Compounding and payment frequency Indenture, i.e. attached options, covenants, etc. Example from July 1, 2004 WSJ.

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Bond Price, Yield, Duration

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  1. Bond Price, Yield, Duration Pricing and Yield Yield Curve Duration Immunization

  2. General Bond Characteristics • Price • Face or par value • Coupon rate • Compounding and payment frequency • Indenture, i.e. attached options, covenants, etc.

  3. Example from July 1, 2004 WSJ • U.S. Treasury Notes and Bonds are typically issued with face value of $10,000, and pay semi-annual coupons • The following bond quoted in the July 1, 2004 WSJ: • Matures in February 2026 (2/15/2026) • Coupon rate is 6%. Semi-annual coupon payments are made on 2/15 and 8/15 of each year in the amount of (0.06 x $10,000)/2 = $300 • At maturity (2/15/2026) the payment is the coupon of $300 plus the principal of $10,000 • Quoted decimal price per $100 par is $108 8/32 = $108.25 • Quoted bond price is $10,825.00

  4. Bond Price and Yield (YTM) • Bond Price, P • C: Coupon per period • N: Number of periods • F: Face (par) value • y: Yield per period

  5. Prices and Yields Price Yield

  6. Bond Equivalent Yield (BEY) • Bond Equivalent Yield (BEY) is the interest rate that makes the present value of a bond’s payments equal to its price assuming semi-annual compounding convention • Example: What’s YTM of the following bond • F = $1,000, C = $40, N = 60, P = $1,276.76 • Notice the difference among y, yBEY, and yEAY • BEY is the yield quoted in financial press • BEY is just annualized YTM, and we will use them interchangeably

  7. Term Structure of Interest Rates (Yield Curve) • Is there a single interest rate? • US Treasury Yield Curve – Nov 24, 2008 Source: U.S. Treasury at www.ustreas.gov

  8. Yield Curve and Interest Rate Risk • On one hand, yield curve rates reflect today’s expectations of interest rates in the future and inflation in coming years • If either inflation of the real interest rate are expected to change in the future, then long term rates will differ from short term rates • On the other hand, yield curve rates also reflect the risk premium over longer maturities, since holding long-term bonds could be risky • Typically, forward rates are higher than expected actual rates, reflecting the risk premium

  9. The Deep End of the Yield Curve • It is typical that the yields on the longest available maturities decrease, since • U.S. Treasury bonds do not have close substitutes in longest maturities • Who can guarantee what happens to any corporate bond in 30 years? • Few alternatives in other countries’ bonds • e.g. no big Latin American government has ever fully repaid a 30-year bond • It is impossible to immunize a 30 year U.S. Treasury bond (will see later…)

  10. Bond Terminology • Flat Price is quoted in financial press • Accrued Interest is not accounted for in the Flat Price • Invoice Price is the actual price a buyer pays for the bond Invoice Price = Flat Price + Accrued Interest • Current Yield = Annual Coupon / Bond Price • Discount Bond sells below par value • Premium Bond sells above par value

  11. Day Count Conventions for Accrued Interest • Actual/Actual - Actual number of days between two dates is used. • AI = C x days/actual days in the year • Actual/365 - Actual number of days between two dates is used as the numerator. All years are assumed to have 365 days. • AI = C x days/365 • Actual/360 – Actual number of days between two dates is used as the numerator. All years are assumed to have 360 days. • AI = C x days/360 • 30/360 - All months are assumed to have 30 days. • If the first date falls on the 31st, it is changed to the 30th. • If the second date falls on the 31th, it is changed to the 30th, but only if the first date falls on the 30th or the 31st. • 30E/360 - All months are assumed to have 30 days. • If the first date falls on the 31st, it is changed to the 30th. • If the second date falls on the 31th, it is changed to the 30th

  12. Example 30 year U.S. Treasury bond • Issued on 5/15/75 • Coupon rate = 12% • Semi-annual coupon payments on 5/15 and 11/15 • Par value = $10,000 • Flat (Quoted) Price on January 23, 2003 = $12303.125 • Next day settlement (January 24, 2003)

  13. Example Objectives Find: • Accrued Interest • Invoice Price • Bond Equivalent Yield (BEY) • Current Yield

  14. Example Continued • Semi-annual coupon = (0.12 x $10,000)/2 = $600 • Days between coupon payments on 11/15/2002 and 5/15/2003 = 181 • Days past since last coupon payment on 11/15/2002 until the settlement date on 1/24/2003 = 70 • Accrued interest (January 23, 2003) = (70/181)*$600 = $232.044 • Invoice price = $12303.125 + $232.044 = $12,535.17

  15. Example Continued • BEY = 1.76%

  16. Example Continued • Current yield = $1200 / $12,303.125 = 9.75% • Recall BEY = 1.76% • Current yield is high, but BEY is low !!! This is because investors expect capital loss!!!

  17. Important Takeaways • For premium bonds (like in the Example) • Current Yield > BEY • Investors expect capital loss • For discount bonds • Current Yield < BEY • Investors expect capital gain

  18. Price Sensitivity to Interest Rates • Although 1-yr and 30-yr interest rates are closely correlated…

  19. Price Sensitivity to Interest Rates 1-yr and 30-yr bond prices display drastically different interest rate sensitivity!

  20. Pricing • Bond price higher … • If coupon rate is higher • If interest rate (yield) is lower Premium Bond P > par value Par Bond P=par value Discount Bond P < par value

  21. Duration – Measure of Sensitivity • Duration is a measure of bond price sensitivity to interest rate changes • It is a characteristic of a security or a portfolio at a particular point in time, which changes over time along with changes in maturity, yield, and coupon payments • It provides a quantitative measure that can be used in risk management, hedging, immunization... • There are more than one duration measure, i.e. Macaulay, Modified, Dollar, etc…

  22. Duration - Is There a Single Maturity? • Macaulay Duration ( D ) is the weighted average of the times to each coupon or principal payment made by the bond. The weights are given by discounted values of coupon or principal payments. • D – Macaulay duration • PVCi – present value of cash flow at time i • P – current bond price • Macaulay duration is the most intuitive duration measure, and gives explanation as to why the name Duration came into being

  23. Macaulay Duration - Example • 10% annual coupon 5 years to maturity par bond • Par value at the time of issue gives the Yield of 10%

  24. Modified Duration • Modified Duration ( D* ) D – Macaulay duration y – YTM k – number of compounding periods per year • Modified duration describes a percentage change in bond price with respect to the yield change

  25. Using Modified Duration Example • 20 year, 6% coupon (semiannual payments) $100 face value bond • Currently yields 8%, and is priced at $80.21 • Macaulay Duration D = 10.92 years • Modified Duration D* = 10.92/(1.04) = 10.5 • Suppose the yield increases from 8% to 8.1% • Predicted price change = -10.5 × .001 = -1.05% • Actual price change = -1.04%

  26. Using Modified Duration - Continued • Suppose the yield increases from 8% to 10% • Predicted price change = -10.5 × .02 = -21% • Actual price change = -18.11% • Duration approach to estimating price changes is only accurate for small yield changes!

  27. Duration Takeaways • Duration provides an answer the question “What happens to the value of my bond portfolio when interest rates change”… • Duration Limitations • Accurate only for small yield changes • Assumes a flat yield curve and parallel shifts • Bonds are assumed option-free

  28. Concepts Check • How does Duration vary with maturity? • How does Duration vary with coupon? • How does Duration vary with yield? • How does Callability affect previous answers?

  29. Duration – Graphic Interpretation Price Tangent Line Yield-to-Price Curve Current Price Duration Prediction Error New Price Predicted Price Current Yield New Yield Yield

  30. Convexity • Convexity measures the curvature of the bond Yield-to-Price curve • Positive convexity implies that duration underestimates the price increase when yields drop, and overestimates the price decrease when yields increase • It means that a long position benefits from positive convexity • All non-callable bonds have positive convexity

  31. Immunization • Suppose you need some pattern of cash flows in the future • To meet these cash needs requires holding a suitable portfolio of bonds • Ideally one would like to hold a portfolio of zero coupon bonds, or Strips • Such approach is known as “cash flow matching” • Zero coupon bonds may not be the best because of possible unattractive relative pricing • It may be necessary to use a portfolio of coupon bonds

  32. Immunization Procedure • Choose an initial immunization portfolio with the modified duration that equals the modified duration of a set of liabilities • Fund the immunization portfolio so that its present value matches the present value of the set of liabilities, discounting at the rate given by the yield of the immunization portfolio • Rebalance the investment portfolio to adjust for interest rate changes and liabilities payments

  33. Immunization Rebalancing • How often do you need to rebalance the immunization portfolio? • You need to rebalance as soon as a significant discrepancy in durations between liabilities and the immunization portfolio occurs due to • changes in interest rates • payments made by immunization securities • liabilities been paid off • There is no one-fits-all answer to determine the size of a significant discrepancy – it depends on your objectives and risk tolerance

  34. Immunization Limitations • Immunization matches duration, which assumes a flat yield curve • Immunization only protects against parallel yield curve shifts • Immunization is not a risk-free strategy

  35. Immunization Takeaways • Immunization is a dynamic portfolio managing strategy that allows to meet a set of liabilities out of proceeds from a self-financing bond portfolio • Immunization allows to meet future liabilities without having to use a zero coupon bond portfolio Major Users of Immunization Policies • Pension Funds • Life Insurance Companies • Banks

  36. Wrap-up • How to evaluate a bond? • What’s the meaning of yield? • Yield Curve concept • Interest rate risk measures the bond price reaction to the change in interest rate • Duration is a simple measure for interest rate risk • Immunization is a passive but dynamic strategy to limit interest rate risk

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