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

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

Investments 143 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 = $  Quoted bond price is $10, Rate Maturity BidAskedChg Asked Mo/YrYield 6 Feb :07 108:

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

Investments 155 Prices and Yields Price Yield

Investments 156 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,  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

Investments 157 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

Investments 158 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

Investments 159 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…)

Investments 1510 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

Investments 1511 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

Investments 1512 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 = $  Next day settlement (January 24, 2003)

Investments 1513 Example Objectives Find:  Accrued Interest  Invoice Price  Bond Equivalent Yield (BEY)  Current Yield

Investments 1514 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 = $  Invoice price = $ $ = $12,535.17

Investments 1515 Example Continued  BEY = 1.76% IRR = BEY1.76% Payment Date1/24/20035/15/200311/15/20035/15/200411/15/20045/15/ % Bond Cash Flow Time to Receipt in 6m Units Discount Factor PV of the Cash Flow Sum of PVs0.00

Investments 1516 Example Continued  Current yield = $1200 / $12, = 9.75%  Recall BEY = 1.76%  Current yield is high, but BEY is low !!! This is because investors expect capital loss!!!

Investments 1517 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

Investments 1518 Price Sensitivity to Interest Rates  Although 1-yr and 30-yr interest rates are closely correlated…

Investments 1519 Price Sensitivity to Interest Rates 1-yr and 30-yr bond prices display drastically different interest rate sensitivity!

Investments 1520 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

Investments 1521 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…

Investments 1522 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

Investments 1523 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 Total Macaulay Duration4.17

Investments 1524 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

Investments 1525 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 = years  Modified Duration D* = 10.92/(1.04) = 10.5  Suppose the yield increases from 8% to 8.1%  Predicted price change = ×.001 = -1.05%  Actual price change = -1.04%

Investments 1526 Using Modified Duration - Continued  Suppose the yield increases from 8% to 10%  Predicted price change = ×.02 = -21%  Actual price change = %  Duration approach to estimating price changes is only accurate for small yield changes!

Investments 1527 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

Investments 1528 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?

Investments 1529 Duration – Graphic Interpretation Yield Price Yield-to-Price Curve Current Price Current Yield Tangent Line New Yield Duration Prediction Error New Price Predicted Price

Investments 1530 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

Investments 1531 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

Investments 1532 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

Investments 1533 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

Investments 1534 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

Investments 1535 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

Investments 1536 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