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Chapter 2 Bond Prices and Yields

FIXED-INCOME SECURITIES. Chapter 2 Bond Prices and Yields. Outline. Bond Pricing Time-Value of Money Present Value Formula Interest Rates Frequency Continuous Compounding Coupon Rate Current Yield Yield-to-Maturity Bank Discount Rate Forward Rates. Bond Pricing.

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Chapter 2 Bond Prices and Yields

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  1. FIXED-INCOME SECURITIES Chapter 2 Bond Prices and Yields

  2. Outline • Bond Pricing • Time-Value of Money • Present Value Formula • Interest Rates • Frequency • Continuous Compounding • Coupon Rate • Current Yield • Yield-to-Maturity • Bank Discount Rate • Forward Rates

  3. Bond Pricing • Bond pricing is a 2 steps process • Step 1: find the cash-flows the bondholder is entitled to • Step 2: find the bond price as the discounted value of the cash-flows • Step 1 - Example • Government of Canada bond issued in the domestic market pays one-half of its coupon rate times its principal value every six months up to and including the maturity date • Thus, a bond with an 8% coupon and $5,000 face value maturing on December 1, 2005 will make future coupon payments of 4% of principal value every 6 months • That is $200 on each June 1 and December 1 between the purchase date and the maturity date

  4. Bond Pricing • Step 2 is discounting • Does it make sense to discount all cash-flows with same discount rate? • Notion of the term structure of interest rates – see next chapter • Rationale behind discounting: time value of money

  5. Time-Value of Money • Would you prefer to receive $1 now or $1 in a year from now? • Chances are that you would go for money now • First, you might have a consumption need sooner rather than later • That shouldn’t matter: that’s what fixed-income markets are for • You may as well borrow today against this future income, and consume now • In the presence of money market, the only reason why one would prefer receiving $1 as opposed to $1 in a year from now is because of time-value of money

  6. Present Value Formula • If you receive $1 today • Invest it in the money market (say buy a one-year T-Bill) • Obtain some interest r on it • Better off as long as r strictly positive: 1+r>1 iff r>0 • How much is worth a piece of paper (contract, bond) promising $1 in 1 year? • Since you are not willing to exchange $1 now for $1 in a year from now, it must be that the present value of $1 in a year from now is less than $1 • Now, how much exactly is worth this $1 received in a year from now? • Would you be willing to pay 90, 80, 20, 10 cents to acquire this dollar paid in a year from now? • Answer is 1/(1+r) : the exact amount of money that allows you to get $1 in 1 year

  7. Interest Rates • Specifying the rate is not enough • One should also specify • Maturity • Frequency of interest payments • Date of interest rates payment (beginning or end of periods) • Basic formula • After 1 period, capital is C1= C0 (1+ r ) • After n period, capital is Cn = C0(1+ r )n • Interests : I = Cn - C0 • Example • Invest $10,000 for 3 years at 6% with annual compounding • Obtain $11,910 = 10,000 x(1+ .06)3 at the end of the 3 years • Interests: $1,910

  8. Frequency • Watch out for • Time-basis (rates are usually expressed on an annual basis) • Compounding frequency • Examples • Invest $100 at a 6% two-year annual rate with semi-annual compounding • 100 x(1+ 3%)after 6 months • 100 x(1+ 3%)2 after 1 year • 100 x(1+ 3%)3 after 1.5 year • 100 x(1+ 3%)4 after 2 years • Invest $100 at a 6% one-year annual rate with monthly compounding • 100 x(1+ 6/12%)after 1 month • 100 x(1+ 6/12%)2 after 2 months • …. 100 x(1+ 6/12%)12 = $106.1678 after 1 year • Equivalent to 6.1678% annual rate with annual compounding

  9. Frequency • More generally • Amount x invested at the interest rate r • Expressed in an annual basis • Compounded n times per year • For T years • Grows to the amount • The effective equivalent annual (i.e., compounded once a year) rate ra is defined as the solution to • or

  10. Continuous Compounding • What happens if we get continuous compounding • The amount of money obtained per dollar invested after T years is • Very convenient: present value of X is Xe-rT • One may of course easily obtain the effective equivalent annual ra • The equivalent annual rate of a 6% continuously compounded interest rate is e6% –1 = 6.1837%

  11. Bond Prices • Bond price • Coupon bond • Note that when r=c, P=N (see next example) • Shortcut when cash-flows are all identical (can you prove it?)

  12. Bond Prices - Example • Example • Consider a bond with 5% coupon rate • 10 year maturity • $1,000 face value • All discount rates equal to 6% • Present value • We could have guessed that price was below par • You do not want to pay the full price for a bond paying 5% when interest rates are at 6% • What happens if rates decrease to 5%? • Price = $1,000

  13. Perpetuity • Example • How much money should you be willing to pay to buy a contract offering $100 per year for perpetuity? • Assume the discount rate is 5% • The answer is • When the bond has infinite maturity (consol bond) • Perpetuities are issued by the British government (consol bonds)

  14. Coupon Rate and Current Yield • Coupon rate is the stated interest rate on a security • It is referred to as an annual percentage of face value • It is usually paid twice a year • It is called the coupon rate because bearer bonds carry coupons for interest payments • It is only used to obtain the cash-flows • Current yield gives you a first idea of the return on a bond • Example • A $1,000 bond has a coupon rate of 7 percent • If you buy the bond for $900, your actual current yield is

  15. Yield to Maturity (YTM) • It is the interest rate that makes the present value of the bond’s payments equal to its price • It is the solution to (T is number of periods) • YTM is the IRR of cash-flows delivered by bonds • YTM may easily be computed by trial-and-error • YTM is typically a semi-annual rate because coupons usually paid semi-annually • Each cash-flow is discounted using the same rate • Implicitly assume that the yield curve is flat at a point in time • It is a complex average of pure discount rates (see below)

  16. BEY versus EAY • Bond equivalent yield (BEY): obtained using simple interest to annualize the semi-annual YTM (street convention): y = 2  YTM • One can always turn a bond yield into an effective annual yield (EAY), i.e., an interest rate expressed on a yearly basis with annual compounding • Example • What is the effective annual yield of a bond with a 5.5% annual YTM • Answer is

  17. One Last Complication • What happens if we don’t have integer # of periods? • Example • Consider the US T-Bond with coupon 4.625% and maturity date 05/15/2006, quoted price is 101.739641 on 01/07/2002 • What is the YTM and EAY? • Solution (street convention) • There are 128 calendar days between 01/07/2002 and the next coupon date (05/15/2002) • Fed convention: =1+YTM/2*128/181 • EAY is

  18. Quoted Bond Prices - Screen

  19. Quoted Bond Prices - Paper

  20. Quoted Bond Prices • Bonds are • Sold in denominations of $1,000 par value • Quoted as a percentage of par value • Prices • Integer number + n/32ths (Treasury bonds) or + n/8ths (corporate bonds) • Example: 112:06 = 112 6/32 = 112.1875% • Change -5: closing bid price went down by 5/32% • Ask yield • YTM based on ask price (APR basis:1/2 year x 2) • Not compounded (Bond Equivalent Yield as opposed to Effective Annual Yield)

  21. Examples • Example • Consider a $1,000 face value 2-year bond with 8% coupon • Current price is 103:23 • What is the yield to maturity of this bond? • To answer that question • First note that 103:23 means 103 + (23/32)%=103.72% • And obtain the following equation • With solution y/2 = 3% or y = 6%

  22. Accrued Interest • The quoted price (or market price) of a bond is usually its clean price, that is its gross price (or dirty or full price) minus the accrued interest • Example • An investor buys on 12/10/01 a given amount of the US Treasury bond with coupon 3.5% and maturity 11/15/2006 • The current market price is 96.15625 • The accrued interest period is equal to 26 days; this is the number of calendar days between the settlement date (12/11/2001) and the last coupon payment date (11/15/2001) • Hence the accrued interest is equal to the coupon payment (1.75) times 26 divided by the number of calendar days between the next coupon payment date (05/15/2002) and the last coupon payment date (11/15/2001) • In this case, the accrued interest is equal to $1.75x(26/181) = $0.25138 • The investor will pay 96.40763 = 96.15625 + 0.25138 for this bond

  23. Bank Discount Rate (T-Bills) • Bank discount rate is the quoted rate on T-Bills • where P is price of T-Bill • n is # of days until maturity • Example: 90 days T-Bill, P = $9,800 • Can’t compare T-bill directly to bond • 360 vs 365 days • Return is figured on par vs. price paid

  24. Bond Equivalent Yield • Adjust the bank discounted rate to make it comparable • Example: same as before • BDR versus BEY • (exercise: Show it!)

  25. Spot Zero-Coupon (or Discount) Rate • Spot Zero-Coupon (or Discount) Rate is the annualized rate on a pure discount bond • where B(0,t) is the market price at date 0 of a bond paying off $1 at date t • See Chapter 4 for how to extract implicit spot rates from bond prices • General pricing formula

  26. Bond Par Yield • Recall that a par bond is a bond with a coupon identical to its yield to maturity • The bond's price is therefore equal to its principal • Then we define the par yield c(n) so that a n-year maturity fixed bond paying annually a coupon rate of c(n) with a $100 face value quotes at par • Typically, the par yield curve is used to determine the coupon level of a bond issued at par

  27. Forward Rates • One may represent the term structure of interest rates as set of implicit forward rates • Consider two choices for a 2-year horizon: • Choice A: Buy 2-year zero • Choice B: Buy 1-year zero and rollover for 1 year • What yield from year 1 to year 2 will make you indifferent between the two choices?

  28. Forward Rates (continued) • They are ‘implicit’ in the term structure • Rates that explain the relationship between spot rates of different maturity • Example: • Suppose the one year spot rate is 4% and the eighteen month spot rate is 4.5%

  29. Recap: Taxonomy of Rates • Coupon Rate • Current Yield • Yield to Maturity • Zero-Coupon Rate • Bond Par Yield • Forward Rate

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