# Coupon Bonds and Zeroes

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1 Coupon Bonds and Zeroes Concepts and Buzzwords Coupon bonds Zero-coupon bonds Bond replication No-arbitrage price relationships Zero rates Zeroes STRIPS Dedication Implied zeroes Semi-annual compounding Reading Veronesi, Chapters 1 and 2 Tuckman, Chapters 1 and 2 Coupon Bonds and Zeroes 1

2 Coupon Bonds In practice, the most common form of debt instrument is a coupon bond. In the U.S and in many other countries, coupon bonds pay coupons every six months and par value at maturity. The quoted coupon rate is annualized. That is, if the quoted coupon rate is c, and bond maturity is time T, then for each \$1 of par value, the bond cash flows are: c/2 c/2 c/2 1 + c/2 0.5 years 1 year 1.5 years T years If the par value is N, then the bond cash flows are: Nc/2 Nc/2 Nc/2 N(1 + c/2) 0.5 years 1 year 1.5 years T years U.S. Treasury Notes and Bonds Institutionally speaking, U.S. Treasury notes and bonds form a basis for the bond markets. The Treasury auctions new 2-, 3-, 5-, 7-year notes monthly, and 10-year notes and 30-year bonds quarterly, as needed. See for a schedule. Non-competitive bidders just submit par amounts, maximum \$5 million, and are filled first. Competitive bidders submit yields and par amounts, and are filled from lowest yield to the stop yield. The coupon on the bond, an even eighth of a percent, is set to make the bond price close to par value at the stop yield. All bidders pay this price. See, for example, page=fisearchtreasury for a listing of outstanding Treasuries. Coupon Bonds and Zeroes 2

3 Class Problem The current long bond, the newly issued 30-year Treasury bond, is the 3 7/8 s (3.875%) of August 15, What are the cash flows of \$1,000,000 par this bond? (Dates and amounts.) Bond Replication and No Arbitrage Pricing It turns out that it is possible to construct, and thus price, all securities with fixed cash flows from coupon bonds. But the easiest way to see the replication and no-arbitrage price relationships is to view all securities as portfolios of zero-coupon bonds, securities with just a single cash flow at maturity. We can observe the prices of zeroes in the form of Treasury STRIPS, but more typically people infer them from a set of coupon bond prices, because those markets are more active and complete. Then we use the prices of these zero-coupon building blocks to price everything else. Coupon Bonds and Zeroes 3

4 Zeroes Conceptually, the most basic debt instrument is a zerocoupon bond--a security with a single cash flow equal to face value at maturity. Cash flow of \$1 par of t-year zero: \$1 Time t It is easy to see that any security with fixed cash flows can be constructed, and thus priced, as a portfolio of these zeroes. Zero Prices Let d t denote the price today of the t-year zero, the asset that pays off \$1 in t years. I.e., d t is the price of a t-year zero as a fraction of par value. This is also sometimes called the t-year discount factor. Because of the time value of money, a dollar today is worth more than a dollar to be received in the future, so the price of a zero must always less than its face value: d t < 1 Similarly, because of the time value of money, longer zeroes must have lower prices. Coupon Bonds and Zeroes 4

5 A Coupon Bond as a Portfolio of Zeroes Consider: \$10,000 par of a one and a half year, 8.5% Treasury bond makes the following payments: \$425 \$425 \$ years 1 year 1.5 years Note that this is the same as a portfolio of three different zeroes: \$425 par of a 6-month zero \$425 par of a 1-year zero \$10425 par of a 1 1/2-year zero No Arbitrage and The Law of One Price Throughout the course we will assume: The Law of One Price Two assets which offer exactly the same cash flows must sell for the same price. Why? If not, then one could buy the cheaper asset and sell the more expensive, making a profit today with no cost in the future. This would be an arbitrage opportunity, which could not persist in equilibrium (in the absence of market frictions such as transaction costs and capital constraints). Coupon Bonds and Zeroes 5

6 Valuing a Coupon Bond Using Zero Prices Let s value \$10,000 par of a 1.5-year 8.5% coupon bond based on the zero prices (discount factors) in the table below. These discount factors come from historical STRIPS prices (from an old WSJ). We will use these discount factors for most examples throughout the course. Maturity Discount Factor Bond Cash Flow Value \$425 \$ \$425 \$ \$10425 \$9614 Total \$10430 On the same day, the WSJ priced a 1.5-year 8.5%-coupon bond at /32 (= ). An Arbitrage Opportunity What if the 1.5-year 8.5% coupon bond were worth only 104% of par value? You could buy, say, \$1 million par of the bond for \$1,040,000 and sell the cash flows off individually as zeroes for total proceeds of \$1,043,000, making \$3000 of riskless profit. Similarly, if the bond were worth 105% of par, you could buy the portfolio of zeroes, reconstitute them, and sell the bond for riskless profit. Coupon Bonds and Zeroes 6

7 Class Problems In today s market, the discount factors are: d 0.5 =0.9991, d 1 =0.9974, and d 1.5 = ) What would be the price of an 8.5%-coupon, 1.5-year bond today? (Say for \$100 par.) 2) What would be the price of \$100 par of a 2%-coupon, 1-year bond today? Securities with Fixed Cash Flows as Portfolios of Zeroes More generally, if an asset pays cash flows K 1, K 2,, K n, at times t 1, t 2,, t n, then it is the same as: K 1 t 1 -year zeroes + K 2 t 2 -year zeroes + + K n t n -year zeroes Therefore no arbitrage requires that the asset s value V is V = K 1 d t1 + K 2 d t K n d tn or V = n j=1 K j d t j Coupon Bonds and Zeroes 7

8 Coupon Bond Prices in Terms of Zero Prices For example, if a bond has coupon c and maturity T, then in terms of the zero prices d t, its price per \$1 par must be P(c,T) = (c /2) (d d 1 + d d T ) + d T 2T or P(c,T) = (c /2) d s/ 2 + d T s=1 Constructing Zeroes from Coupon Bonds Often people would rather work with Treasury coupon bonds than with STRIPS, because the market is more active. They can imply zero prices from Treasury bond prices instead of STRIPs and use these to value more complex securities. In other words, not only can we construct bonds from zeroes, we can also go the other way. Example: Constructing a 1-year zero from 6-month and 1- year coupon bonds. Coupon Bonds: Maturity Coupon Price in 32nds Price in Decimal % % Coupon Bonds and Zeroes 8

9 Constructing the One-Year Zero Find portfolio of bonds 1 and 2 that replicates 1-year zero. Let N 0.5 be the par amount of the 0.5-year bond and N 1 be the par amount of the 1-year bond in the portfolio. At time 0.5, the portfolio will have a cash flow of N 0.5 x ( /2) + N 1 x /2 At time 1, the portfolio will have a cash flow of N 0.5 x 0 + N 1 x ( /2) We need N 0.5 and N 1 to solve (1) N 0.5 x ( /2) + N 1 x /2 = 0 (2) N 0.5 x 0 + N 1 x ( /2) =100 => N 1 = and N 0.5 = Implied Zero Price So the replicating portfolio consists of long par value of the 1-year bond short 2.10 par value of the 0.5-year bond. Class Problem: Given the prices of these bonds below, Maturity Coupon Price in 32nds Price in Decimal % % what is the no-arbitrage price of \$100 par of the 1-year zero? Coupon Bonds and Zeroes 9

10 Inferring Zero Prices from Bond Prices: Short Cut The last example showed how to construct a portfolio of bonds that synthesized (had the same cash flows as) a zero. We concluded that the zero price had to be the same as the price of the replicating portfolio (no arbitrage). If we don't need to know the replicating portfolio, we can solve for the implied zero prices more directly: Price of bond 1 = ( /2) d 0.5 = Price of bond 2 = (4.375/2) d ( /2) d 1 = d 0.5 = 0.973, d 1 = Class Problems 1) Suppose the price of the 4.25%-coupon, 0.5-year bond is What is the implied price of a 0.5-year zero per \$1 par? 2) Suppose the price of the 4.375%-coupon, 1-year bond is 99. What is the implied price of a 1-year zero per \$1 par? Coupon Bonds and Zeroes 10

11 Replication Possibilities Since we can construct zeroes from coupon bonds, we can construct any stream of cash flows from coupon bonds. Uses: Bond portfolio dedication--creating a bond portfolio that has a desired stream of cash flows funding a liability defeasing an existing bond issue Taking advantage of arbitrage opportunities Market Frictions In practice, prices of Treasury STRIPS and Treasury bonds don't fit the pricing relationship exactly transaction costs and search costs in stripping and reconstituting bid/ask spreads Note: The terms bid and ask are from the viewpoint of the dealer. The dealer buys at the bid and sells at the ask, so the bid price is always less than the ask. The customer sells at the bid and buys at the ask. Coupon Bonds and Zeroes 11

12 Interest Rates People try to summarize information about bond prices and cash flows by quoting interest rates. Buying a zero is lending money--you pay money now and get money later Selling a zero is borrowing money--you get money now and pay later A bond transaction can be described as buying or selling at a given price, or lending or borrowing at a given rate. The convention in U.S. bond markets is to use semi-annually compounded interest rates. Annual vs. Semi-Annual Compounding At 10% per year, annually compounded, \$100 grows to \$110 after 1 year, and \$121 after 2 years: 10% per year semi-annually compounded really means 5% every 6 months. At 10% per year, semi-annually compounded, \$100 grows to \$ after 1 year, and \$ after 2 years: Coupon Bonds and Zeroes 12

13 Annual vs. Semi-Annual Compounding... After T years, at annually compounded rate r A, P grows to Present value of F to be received in T years with annually compounded rate r A is In terms of the semi-annually compounded rate r, the formulas become The key: An (annualized) semi-annually compounded rate of r per year really means r/2 every six months. Zero Rates If you buy a t-year zero and hold it to maturity, you lend at rate r t where r t is defined by Call r t the t-year zero rate or t-year discount rate. Coupon Bonds and Zeroes 13

14 Class Problems: Rate to Price According to market convention, zero prices are quoted using rates. Sample STRIPS rates from our historic WSJ: 0.5-year rate: 5.54% 1-year rate: 5.45% 1) What is the 0.5-year zero price? 2) What is the 1-year zero price? Class Problems: Price to Rate 1) The 1-year zero price implied from coupon bond prices was What was the implied zero rate? 2) In today s market, the 5-year zero price is What is the 5-year zero rate? Coupon Bonds and Zeroes 14

15 Value of a Stream of Cash Flows in Terms of Zero Rates Recall that any asset with fixed cash flows can be viewed as a portfolio of zeroes. So its price must be the sum of its cash flows multiplied by the relevant zero prices: Equivalently, the price is the sum of the present values of the cash flows, discounted at the zero rates for the cash flow dates: Example \$10,000 par of a one and a half year, 8.5% Treasury bond makes the following payments: \$425 \$425 \$ years 1 year 1.5 years Using STRIPS rates from the WSJ to value these cash flows: Coupon Bonds and Zeroes 15

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