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

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Investments 152 General Bond Characteristics Price Face or par value Coupon rate Compounding and payment frequency Indenture, i.e. attached options, covenants, etc.

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Investments 153 Example from April 23, 2015 WSJ U.S. Treasury Notes and Bonds are typically sold with face value of $10,000, but quoted in the WSJ as a percentage of face (par) value, and pay semi-annual coupons The following bond quoted in the April 23, 2015 WSJ: Matures on February 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 bond price is $13,897.66 MaturityCouponBidAskedChg Asked Yield 2/15/20266.000 138.9141138.9766 0.3047 1.978

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Investments 154 Bond Price and Yield (YTM) Bond Price, P C: Coupon per period N: Number of periods F: Face (par) value y: Yield per period

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Investments 155 Prices and Yields Price Yield

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Investments 156 Pricing Bond price higher … If coupon rate is higher If interest rate (yield) is lower Premium Bond P > par value Discount Bond P < par value Par Bond P=par value

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Investments 157 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, y BEY, and y EAY BEY is the yield quoted in financial press BEY is just annualized YTM, and we will use them interchangeably

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Investments 158 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

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Investments 159 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)

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Investments 1510 Example Objectives Find: Accrued Interest Invoice Price Current Yield

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Investments 1511 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

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Investments 1512 Example Continued Current yield = $1200 / $12,303.125 = 9.75% At the same time BEY = 1.76% Current yield is high, but BEY is low !!! This is because investors expect capital loss!!!

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Investments 1513 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

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Investments 1514 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

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Investments 1515 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 or 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

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Investments 1516 Price Sensitivity to Interest Rates Although 1-yr and 30-yr interest rates are closely correlated…

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Investments 1517 Price Sensitivity to Interest Rates 1-yr and 30-yr bond prices display drastically different interest rate sensitivity!

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Investments 1518 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…

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Investments 1519 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 PVC i – 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

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Investments 1520 Macaulay Duration - Example 10% annual coupon 5 years to maturity par bond Par value at the time of issue gives the Yield of 10% TimeCash FlowPVTime*PV 1109.09 2108.2616.53 3107.5122.54 4106.8327.32 511068.30341.51 Total100.00416.99 Macaulay Duration4.17

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Investments 1521 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

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Investments 1522 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%

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Investments 1523 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!

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Investments 1524 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

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Investments 1525 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?

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Investments 1526 Recap 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

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Contents Method 1: –Pricing bond from its yield to maturity –Calculating yield from bond price Method 2: –Pricing bond from Duration –Pricing bond from.

Contents Method 1: –Pricing bond from its yield to maturity –Calculating yield from bond price Method 2: –Pricing bond from Duration –Pricing bond from.

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