Chapter 2 Bond Prices and Yields FIXED-INCOME SECURITIES

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 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

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

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. (Individual need vs. Generalized preference) 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

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

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 C 1 = C 0 (1+ r ) –After n period, capital is C n = C 0 (1+ r ) n –Interests : I = C n - C 0 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

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 = $ after 1 year Equivalent to % annual rate with annual compounding

The effective equivalent annual (i.e., compounded once a year) rate r a is defined as the solution to 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 Frequency or

The equivalent annual rate of a 6% continuously compounded interest rate is e 6% –1 = % Very convenient: present value of X is Xe -rT One may of course easily obtain the effective equivalent annual r a Continuous Compounding What happens if we compound infinitely often, i.e. we use continuous compounding? The amount of money obtained per dollar invested after T years is

Bond Prices Bond price Shortcut when cash-flows are all identical (Exercise: can you prove it?) Coupon bond

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

Perpetuity When the bond has infinite maturity (consol bond) 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 –Perpetuities are issued by the British government (consol bonds)

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

Yield to Maturity (YTM) Definition: It is the interest rate that makes the present value of the bond’s payments equal to its price. (Memorize it!) 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)

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

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 on 01/07/2002 –What is the YTM and EAY? Solution –There are 128 calendar days between 01/07/2002 and the next coupon date (05/15/2002) EAY is

Quoted Bond Prices - Screen

Quoted Bond Prices - Paper

Quoted Bond Prices Bonds are –Sold in denominations of $1,000 par value –Quoted as a percentage of par value Prices –Integer number + n/32 ths (Treasury bonds) or + n/8 ths (corporate bonds) –Example: 112:06 = 112 6/32 = % –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)

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 (23/32)%=103.72% –And obtain the following equation –With solution y/2 = 3% or y = 6%

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 –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) = $ –The investor will pay = for this bond

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

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

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

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

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?

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%

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