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 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, MaturityCouponBidAskedChg Asked Yield 2/15/

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 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 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,  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 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

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

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

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

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

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

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

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

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

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

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…

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

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 Total Macaulay Duration4.17

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

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 = 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 1523 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 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

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?

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