1. Chapter 2 Bond Prices and Yields
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 - Example
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 Examples
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

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

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 Present value
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 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
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 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 Prices Ask yield
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 = %
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 To answer that question
With solution y/2 = 3% or y = 6%
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

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
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

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

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