Bond Price Yield Duration Pricing and Yield Curve

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

General Bond Characteristics Price q Face or par value q Coupon rate q Compounding and payment frequency q Indenture, i. e. attached options, covenants, etc. q Investments 15 2

Example from April 23, 2015 WSJ q q 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: Ø Ø Maturity Coupon Bid Asked Chg 2/15/2026 6. 000 138. 9141 138. 9766 0. 3047 Asked Yield 1. 978 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 Investments 15 3

Bond Price and Yield (YTM) q Bond Price, P Ø Ø C: Coupon period N: Number of periods F: Face (par) value y: Yield period Investments 15 4

Prices and Yields Price Yield Investments 15 5

Bond Equivalent Yield (BEY) q q q 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 Ø F = $1, 000, C = $40, N = 60, P = $1, 276. 76 Ø Notice the difference among y, y. BEY, and y. EAY BEY is the yield quoted in financial press Ø BEY is just annualized YTM, and we will use them interchangeably Investments 15 6

Term Structure of Interest Rates (Yield Curve) q Is there a single interest rate? Ø US Treasury Yield Curve – Nov 24, 2008 Source: U. S. Treasury at www. ustreas. gov Investments 15 7

Yield Curve and Interest Rate Risk q q 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 Investments 15 8

The Deep End of the Yield Curve q It is typical that the yields on the longest available maturities decrease, since Ø U. S. Treasury bonds do not have close substitutes in longest maturities q q Who can guarantee what happens to any corporate bond in 30 years? Few alternatives in other countries’ bonds Ø Ø 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…) Investments 15 9

Bond Terminology q q q 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 Investments 15 10

Day Count Conventions for Accrued Interest q q q Actual/Actual - Actual number of days between two dates is used. Ø 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. Ø 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. Ø AI = C x days/360 30/360 - All months are assumed to have 30 days. Ø If the first date falls on the 31 st, it is changed to the 30 th. Ø If the second date falls on the 31 th, it is changed to the 30 th, but only if the first date falls on the 30 th or the 31 st. 30 E/360 - All months are assumed to have 30 days. Ø If the first date falls on the 31 st, it is changed to the 30 th. Ø If the second date falls on the 31 th, it is changed to the 30 th Investments 15 11

Example 30 year U. S. Treasury bond q q q 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) Investments 15 12

Example Objectives Find: Accrued Interest Ø Invoice Price Ø Bond Equivalent Yield (BEY) Ø Current Yield Ø Investments 15 13

Example Continued q q q 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 Investments 15 14

Example Continued IRR = BEY 1. 76% Payment Date 1/24/2003 12% Bond Cash Flow -12535. 17 600. 00 Time to Receipt in 6 m Units 0 Discount Factor PV of the Cash Flow 5/15/2004 11/15/2004 5/15/2005 600. 00 10600. 00 0. 62 1. 62 2. 62 3. 62 4. 62 1 0. 9946 0. 9859 0. 9773 0. 9688 0. 9603 -12535. 17 596. 76 591. 55 586. 38 581. 26 10179. 22 Sum of PVs q 5/15/2003 11/15/2003 0. 00 BEY = 1. 76% Investments 15 15

Example Continued q q Ø 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!!! Investments 15 16

Important Takeaways q For premium bonds (like in the Example) Ø Ø q Current Yield > BEY Investors expect capital loss For discount bonds Ø Ø Current Yield < BEY Investors expect capital gain Investments 15 17

Price Sensitivity to Interest Rates q Although 1 -yr and 30 -yr interest rates are closely correlated… Investments 15 18

Price Sensitivity to Interest Rates 1 -yr and 30 -yr bond prices display drastically different interest rate sensitivity! Investments 15 19

Price Sensitivity to Interest Rates q Zero-Coupon Bond Ø Maturity matters!!! Investments 15 20

Price Sensitivity to Interest Rates q 8% Coupon Bond Ø Coupons matter as well!!! Investments 15 21

Duration – Measure of Sensitivity q q 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… Investments 15 22

Duration - Is There a Single Maturity? q 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. Ø Ø Ø q 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 Investments 15 23

Cash Flows and Duration of 8 -yr Bond with 9% annual coupon and 10% YTM 24

Macaulay Duration - Example q q 10% annual coupon 5 years to maturity par bond Par value at the time of issue gives the Yield of 10% Time Cash Flow PV Time*PV 1 10 9. 09 2 10 8. 26 16. 53 3 10 7. 51 22. 54 4 10 6. 83 27. 32 5 110 68. 30 341. 51 100. 00 416. 99 Total Macaulay Duration Investments 15 4. 17 25

Modified Duration q Modified Duration ( D* ) D – Macaulay duration y – YTM k – number of compounding periods per year q Modified duration describes a percentage change in bond price with respect to the yield change Investments 15 26

Using Modified Duration Example q 20 year, 6% coupon (semiannual payments) $100 face value bond Ø Ø Ø q 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% Ø Ø Predicted price change = -10. 5 ×. 001 = -1. 05% Actual price change = -1. 04% Investments 15 27

Using Modified Duration - Continued q Suppose the yield increases from 8% to 10% Ø Ø q 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! Investments 15 28

Duration Takeaways q q Duration provides an answer the question “What happens to the value of my bond portfolio when interest rates change”… Duration Limitations Ø Ø Ø Accurate only for small yield changes Assumes a flat yield curve and parallel shifts Bonds are assumed option-free Investments 15 29

Concepts Check q How does Duration vary with maturity? q How does Duration vary with coupon? q How does Duration vary with yield? q How does Callability affect previous answers? Investments 15 30

Duration – Graphic Interpretation Price Tangent Line Yield-to-Price Curve Current Price Duration Prediction Error New Price Predicted Price Current Yield Investments 15 New Yield 31

Convexity measures the curvature of the bond Yield-to-Price curve q Positive convexity implies that duration underestimates the price increase when yields drop, and overestimates the price decrease when yields increase q Ø q It means that a long position benefits from positive convexity All non-callable bonds have positive convexity Investments 15 32

Immunization q q q 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 Ø Ø q 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 Investments 15 33

Immunization Procedure q q q 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 Investments 15 34

Immunization Rebalancing q q 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 Ø Ø Ø q 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 Investments 15 35

Immunization Limitations q q q Immunization matches duration, which assumes a flat yield curve Immunization only protects against parallel yield curve shifts Immunization is not a risk-free strategy Investments 15 36

Immunization Takeaways q q 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 q Pension Funds q Life Insurance Companies q Banks Investments 15 37

Wrap-up q q q 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 Investments 15 38