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1 Bond Pricing

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9 Problems with YTM YTM is the stated internal rate of return It is not guaranteed to be the realized return due to interest rate risk It may not be a correct description of expected return We want to take into account Re-investment risk Price risk Default risk (for defaultable bonds)

10 Re-investment Risk YTM assumes that coupons are reinvested at the same yield As coupons are paid over time, the interest rate of debt instruments available for reinvestment is actually changing Consider the realized compound return (r) of a T-period investment Initial Investment x (1 + r) T = Total Value at time T Note the total value is affected by re-investment income from interim payoffs and price changes of the security If the re-investment rate differs from YTM, then the realized compound return will be different

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13 Price Risk Price risk: the risk due to change of the sale price of a security If YTM is constant over time, return = YTM But, if YTM changes in the future, return will change as well Example: A 10% annual coupon bond with maturity of 2 years and a face value of \$1,000 is selling at par 1. What is the YTM of this bond? 2. If the investor sells the bond in a year and the YTM remains constant, what is the return? 3. If the investor sells the bond in a year and the YTM changes to 8%, what is the return?

14 YTM vs. Realized Compound Return For coupon bonds, YTM = Realized Compound Return if: Re-investment rate = YTM, and Investor holds bond to maturity Re-investment rate = YTM, and YTM on sale date is the same as YTM on purchase date How about for zero-coupon bonds?

15 Default and Expected YTM: Example Casino Royale Co. issues bonds with 10 years to maturity, par value of \$1,000, and a coupon rate of 9% with semi-annual payments; The bond is currently selling for \$ Calculate the stated YTM (What assumption about the bond s cash flows is implicit in stated YTM?) 2. Suppose you expect Casino Royale will make good on all of the coupon payments but can only pay 70% of the face value at maturity. What is the expected YTM?

16 Yield Curve Term structure of interest rates: Relation between YTM and maturity Yield curve: a graph of the yields on Treasury bonds (y-axis) relative to the number of years to maturity (x-axis) Yield curve gives comparison of expected returns on short-term and long-term debt instruments

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18 US Yield Curve as of March. 23, 2015

19 Use Yield Curve to Price Bonds So far, we ve assumed a constant interest rate Yields on different-maturity bonds are equal Each cash flow is discounted at the same interest rate, r In reality, interest rates are different Yields on different maturity bonds are not all equal Each cash flow must be considered as a stand-alone zero-coupon bond where y(t) is the yield to maturity of a t-year zero-coupon bond The interest rates depend on the time-to-maturity

20 Use Yield Curve to Price Bonds: Example Consider the following prices and yields to maturities on zero-coupon bonds (\$1000 face value) Find the price and YTM of a 3-year, 10% annual payment coupon Bond

21 Term Structure of Interest Rates Theories which explain the shape of the yield curve Pure Expectations Theory Liquidity Premium Theory Segmented Markets Theory

22 Theories of The Term Structure Expectations Hypothesis Yield curve slope reflects market expectations of future interest rates Assumes investor has no maturity preferences and transaction costs are low Implied forward rate is the market s forecast of the future interest rate Upward (downward) yield curve means that the market is expecting higher (lower) future short term rates Flat yield curve means that short rate is expected to stay at the same level

23 Implied Forward Rates By the expectations hypothesis, use the term structure of interest rates below and find the implied forward rates: E.g. 1-year Treasury bill 1% p.a. (per annum) 2-year Treasury note 2% p.a. 3-year Treasury note 3% p.a. What s the one-year forward rate from year 1 to 2 What s the two-year forward rate from year 1 to 3

24 Theories of The Term Structure Liquidity Preference Investors prefer to hold short-term bonds (because the return on longer-term bonds are uncertain) They require a risk premium to hold longer-term bonds This liquidity premium compensates short-term investors for the uncertainty about future prices

25 Interpreting the Term Structure The yield curve reflects expectations of future interest rates The forecasts of future rates are clouded by other factors, such as liquidity premiums An upward sloping curve could indicate: Rates are expected to rise, and/or Investors require large liquidity premiums to hold long term bonds The yield curve is a good predictor of the business cycle Long term rates tend to rise in anticipation of economic expansion Inverted yield curve may indicate that interest rates are expected to fall and signal a recession

26 Predicting recessions Expected higher interest rate levels in the future. Interest rates low now means we are in expansive monetary policy. Low interest rates to stimulate growth. Expanding economy ahead Expected lower interest rate levels in the future Tight monetary policy: High interest rates now to prevent economy from overheating. Investors demand longer term bonds so much that long term yields are low. Willingness to lock in low rates could mean few investment opportunities in economy. Recession coming soon?

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28 Bond market 2 -Markets and Institutions Ruichang LU ( 卢瑞昌 ) Department of Finance Guanghua School of Management Peking University

29 Outline Treasury notes and bonds Municipal bonds Corporate bonds International bond market Advanced concept in bond market Duration Convexity

30 Bond Market Instruments Outstanding

31 US debt ceiling

32 Debt/GDP

33 Size of the market Global Bond volume totaled \$6.29tr in According to the Federal Reserve Bank of New York, the average daily trading volume in T-note and T-bond issues for the week ended July 7, 2010 was \$ billion. Source: Dealogic.com

34 T-notes and T-bonds T-notes and T-bonds issued by the U.S. Treasury to finance the national debt and other federal government expenditures Backed by the full faith and credit of the government and are default risk free Pay relatively low rates of interest (yields to maturity) Given their longer maturity, not entirely risk free due to interest rate fluctuations Pay coupon interest (semiannually): notes have maturities from 1-10 years; bonds years

35 Types of issue The Treasury issues two types of notes and bonds: fixed principal and inflation-indexed. While both types pay interest twice a year, the principal value used to determine the percentage interest payment (coupon) on inflation-indexed bonds is adjusted to reflect inflation (measured by the consumer price index). Thus, the semiannual coupon payments and the final principal payment are based on the inflation-adjusted principal value of the security.

36 Types of issue For example, a two-year, 10 percent coupon (annual) bond issued with a principal value (face value) of \$1,000 will pay a total of \$10 and \$10 in the first and second years. An indexed (annual) bond when inflation is 10 percent in the first year and the second year will pay a 10 percent coupon based on principal values of \$1,000 X(1.1) =\$1,100 and \$1,000X(1.1)X1.1= \$1,210, respectively. That is, the first year coupon will be 10% \$1,100 \$11 and the second year coupon will be 10% \$1,210 \$12.10.

37 Separate Trading of Registered Interest and Principal Securities (STRIPS). A treasury security in which the individual interest payments are separated from the principal payment Effectively creates sets of securities--one for each semiannual interest payment one one for the final principal payment Often referred to as Treasury zero-coupon bonds Created by U.S. Treasury in response to separate trading of treasury security principal and interest developed by securities firms; only available through FIs and government securities brokers

38 Creation of STRIP

39 Usage of STRIP Using a STRIP to Immunize against Interest Rate Risk Investors needing a lump sum payment in the distant future (e.g., life insurers) would prefer to hold the principal portion of the STRIP. Investors wanting nearer-term cash flows (e.g., commercial banks) would prefer the interest portions of the STRIP. Also, some state lotteries invest the present value of large lottery prizes in STRIPs to be sure that funds are available to meet required annual payments to lottery winners. Pension funds purchase STRIPs to match payment cash flows received on their assets (STRIPs) with those required on their liabilities (pension contract payments).

40 Auction in T-bond market Similar to the primary market T-bill sales, the Treasury sells T-notes and bonds through competitive and noncompetitive auctions Auction Pattern for Treasury Notes and bonds Security Purchase Minimum General Auction Schedule 2-year note \$1,000 Monthly 5-year note \$1,000 Feb, May-Aug, Nov 10-year note \$1,000 Feb, May-Aug, Nov

41 Most government bond auctions are conducted as uniform price auctions. Investors submit bids expressed in terms of % p.a. yield. Uniform payment at the price determined at the cut-off bid. For competitive auctions, securities are allotted from the lowest to the highest bidder. At the highest cutoff bid, the allotment is on a pro-rated basis _5.pdf

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43 Municipal bonds Securities issued by state and local governments Tax receipts or revenues generated are the source of repayment Attractive to household investors because interest (but not capital gains) are tax exempt (in contrast, interest payments on Treasury securities are exempt only from state and local income taxes) As a result, the interest borrowing cost to the state or local government is lower, because investors are willing to accept lower interest rates on municipal bonds relative to comparable taxable bonds such as corporate bonds.

44 Conversion of a Municipal Bond Rate to a Tax Equivalent Rate i a = i b (1 - t) Where: i a = After-tax (equivalent tax exempt) rate of return on a taxable bond i b = Before-tax rate of return on a taxable bond t = Income tax rate of the marginal bond holder Example: You can invest in taxable corporate bonds that are paying 10% annually on munis. Your marginal tax rate is 28%. The after- tax rate of return on the taxable bond is: 10%(1-.28) = 7.2%

45 Primary Market Placement Choices for Munis General Public Offering underwriter is selected either by negotiation or by competitive bidding the underwriter offers the bonds to the general public Rule 144A Placement bonds are sold on a semi-private basis to qualified investors (generally Fis, Qualified Institutional Buyers) Secondary market trading is thin. Why? 6-45 McGraw-Hill/Irwin

46 Corporate Bonds All long-term bonds issued by corporations Minimum denominations publicly traded corporate bonds is \$1,000 Generally pay interest semiannually Bond indenture 6-46 McGraw-Hill/Irwin

47 Indenture The bond indenture is the legal contract that specifies the rights and obligations of the bond issuer and the bond holders. The bond indenture contains a number of covenants associated with a bond issue. These bond covenants describe rules and restrictions placed on the bond issuer and bond holders. the ability to call the bond issue restrictions as to limits on the ability of the issuer to increase dividends paid to equity holders. the bond indenture helps lower the risk (and therefore the interest cost) of the bond issue.

48 Convertible bonds Convertible bonds are bonds that may be exchanged for another security of the issuing firm (e.g., common stock) at the discretion of the bond holder. If the market value of the securities the bond holder receives with conversion exceeds the market value of the bond, the bond holder can return the bonds to the issuer in exchange for the new securities and make a profit. An example of a convertible bond issue is the bond issue by General Motors Corporation in These bonds pay an annual coupon of 6.25 percent. Furthermore, on their maturity date, the holder of the bond has the option to choose between receiving the nominal value in cash or converting the bond into 21 shares of the General Motors Corporation stock.

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52 Junk bond market history In 1980s, Junk bond issuance 170 Billion, Drexel Burnham Lambert underwrote 80 Billion.

53 International Aspects of Bond Markets International bond market trades bonds that are underwritten by an international syndicate offer bonds simultaneously to investors in several countries issue bonds outside the jurisdiction of any single country offer bonds in unregistered form 6-53 McGraw-Hill/Irwin

54 Eurobonds, Foreign Bonds, Brady Bonds and Sovereign Bonds Eurobonds Foreign Bonds Sovereign Bonds 6-54 McGraw-Hill/Irwin

55 Eurobonds Eurobonds are long-term bonds issued and sold outside the country of the currency in which they are denominated (e.g., dollardenominated bonds issued in Europe or Asia). Perhaps confusingly, the term Euro simply implies the bond is issued outside the country in whose currency the bond is denominated. Thus, Euro -bonds are issued in countries outside of Europe and in currencies other than the euro. Indeed, the majority of issues are still in U.S. dollars and can be issued in virtually any region of the world.

56 Eurobonds Eurobonds were first sold in 1963 as a way to avoid taxes and regulation. U.S. corporations were limited by regulations on the amount of funds they could borrow domestically (in the United States) to finance overseas operations, while foreign issues in the United States were subject to a special 30 percent tax on their coupon interest. In 1963, these corpora- tions created the Eurobond, by which bonds were denominated in various currencies and were not directly subject to U.S. regulation. Even when these regulations were abandoned, access to a new and less-regulated market by investors and corporations created sufficient demand and supply for the market to continue to grow.

57 Foreign Bonds Foreign Bonds. Foreign bonds are long-term bonds issued by firms and governments outside of the issuer s home country and are usually denominated in the currency of the country in which they are issued rather than in their own domestic currency for example, a Japanese company issuing a dollar-denominated public bond rather than a yen-denominated bond in the United States.

58 Sovereign bonds Sovereign bonds are government-issued debt. Sovereign bonds have historically been issued in foreign currencies, in either U.S. dollars or euros.

59 Institutions

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62 Dirty price/clean price At settlement, the buyer must pay the seller the purchase price of the T-note or T-bond plus accrued interest. The sum of these two is often called the full price or dirty price of the security. The price without the accrued interest added on is called the clean price.

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65 Fundamental Bond relations 1. Bond yields vary inversely with changes in bond prices. 2. Bond price volatility increases as maturity increases. 3. Bond price volatility decreases as coupon rates increase.

66 e.g. Yield decrease with bond price

67 E.g. Volatility increases with maturity

68 E.g. Volatility decreases with coupon rate

69 Macaulay s Duration Duration (or Macaulay s Duration) is the weighted average of the time to receipt of the cash flows with the weights proportional to the present value of the payment A measure of the effective maturity of a bond. Formula for Macaulay s Duration: where the weights (w t ) are determined by

70 Fulcrum Analogy Think of the following bars as weights on a board If you try to balance the board, where do you put the fulcrum?

71 Properties of Duration The greater the duration, the greater is price volatility of the bond for a given change in interest rates. All else equal: Bonds with higher coupon rates have shorter durations, all else equal. Bonds with longer maturities have longer durations. The higher the yield to maturity, the shorter is duration.

72 Duration: An Example Find Macaulay s duration (assume a semi-annual YTM of 5% for both bonds): 8% coupon, semi-annual payment bond with 2-year maturity, face value of \$1,000 Zero-coupon bond, face value of \$1,000

73 Aside: Deriving Macaulay Duration

74 An Example Consider an 8% coupon, annual payment bond with 3-year maturity, face value of \$1000, YTM=10% It is selling at \$ and its Macaulay duration is Suppose yield rises by 50 basis points (0.5%). How much does the bond price change?

75 Risk Management: Immunization Immunization: a strategy that matches the durations of assets and liabilities to shield net worth from interest rate movements Check what will happen when yield goes up/down by 1%.

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79 Building an Immunized Portfolio Steps Toward Immunization Step 1: Find the duration of the liability A one-payment obligation with 7-year duration Step 2: Find the duration of the asset portfolio The duration of the 3-year zero-coupon bond is 3 years The duration of the 11-year zero-coupon bond is 11 years Asset duration = w 3 years + (1 - w) 11 years Note: Duration is additive (very useful!)

80 Bond price curve

81 Convexity The duration approach for estimating the % change in bond price works well for small changes in interest rates, but not for large changes in interest rates. The formula can be modified to work well for large interest changes and the modification is an adjustment for convexity. Convexity incorporates the curvature of the price-yield curve into the estimated percentage price change of a bond given an interest rate change:

82 Convexity effect on price changes

83 Exercises

84 Exercise 1. A bond has a market price that exceeds its face value. Which of the following features currently apply to this bond? I. discounted price II. premium price III. yield-to-maturity that exceeds the coupon rate IV. yield-to-maturity that is less than the coupon rate A. III only B. I and III only C. I and IV only D. II and III only E. II and IV only

85 E. II and IV only

86 2. Which one of the following relationships is stated correctly? A. The coupon rate exceeds the current yield when a bond sells at a discount. B. The call price must equal the par value. C. An increase in market rates increases the market price of a bond. D. Decreasing the time to maturity increases the price of a discount bond, all else constant. E. Increasing the coupon rate decreases the current yield, all else constant.

87 D. Decreasing the time to maturity increases the price of a discount bond, all else constant.

88 3. The bonds issued by Stainless Tubs bear a 6 percent coupon, payable semiannually. The bonds mature in 11 years and have a \$1,000 face value. Currently, the bonds sell for \$989. What is the yield to maturity? A percent B percent C percent D percent E percent

89 This cannot be solved directly, so it's easiest to just use the calculator method to get an answer. You can then use the calculator answer as the rate in the formula just to verify that your answer is correct.

90 4. The Corner Grocer has a 7-year, 6 percent annual coupon bond outstanding with a \$1,000 par value. The bond has a yield to maturity of 5.5 percent. Which one of the following statements is correct if the market yield suddenly increases to 6.5 percent? A. The bond price will increase by \$ B. The bond price will increase by 5.29 percent. C. The bond price will decrease by \$ D. The bond price will decrease by 5.43 percent. E. The bond price will decrease by 5.36 percent.

91 Difference in prices = \$ \$1, = -\$55.83 Percentage difference in prices =

92 Next Class Reading Chapter 5 for Money Market Reading Chapter 9 for FX Market

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