Bond Cash Flows Prices and Yields Bond Terminology Face Value Notional amount used to compute the interest payments Coupon Rate Determines the amount of each coupon payment expressed as an ID: 446223
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Slide1
Valuing BondsSlide2
Bond Cash Flows, Prices,
and Yields
Bond Terminology
Face ValueNotional amount used to compute the interest paymentsCoupon RateDetermines the amount of each coupon payment, expressed as an annual percentage rateCoupon PaymentSlide3
Zero-Coupon Bonds
Zero-Coupon
Bond (Zero)
Does not make coupon paymentsAlways sells at a discount (a price lower than face value), so they are also called pure discount bondsTreasury Bills are U.S. government zero-coupon bonds with a maturity of up to one year. Slide4
Zero-Coupon Bonds
Suppose that a one-year, risk-free, zero-coupon bond with a $100,000 face value has an initial price of $96,618.36. The cash flows would be:
Although the bond pays no “interest,” your compensation is the difference between the initial price and the face value.Slide5
Zero-Coupon Bonds
Yield to Maturity
The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond.
Price of a Zero-Coupon bondSlide6
Zero-Coupon Bonds
Yield to Maturity
For the one-year zero coupon bond:
Thus, the YTM is 3.5%.Slide7
Example
Suppose the following three zero-coupon bonds are trading at the prices shown. What is the yield to maturity for each bond?
Maturity
1 Year2 Years
3 years
Price
980.39
942.60
901.94Slide8
Example
The general formula is found by re-arranging the present value equation:
The specific solutions are:
1 Year – YTM
$1,000/$980.39
-1
2.0%
2 Years – YTM
($1,000/$942.60)
1/2
- 1
3.0%
3 Years – YTM
($1,000/$901.94)
1/3
- 1
3.5%Slide9
Zero-Coupon Bonds
Risk-Free Interest Rates
A default-free zero-coupon bond that matures on
date n provides a risk-free return over that time period equal to the yield to maturity. Thus, the Law of One Price guarantees that the risk-free interest rate equals the yield to maturity on such a bond.Risk-Free Interest Rate (risk free zero coupon yield or spot rate)
with Maturity
n
The zero coupon yield curve is a plot of the spot rates for different maturities.
Things will not be this simple when we consider coupon bonds.Slide10
Coupon Bonds
Yield to Maturity
The YTM is the
single discount rate that equates the present value of the bond’s remaining cash flows to its current price.Yield to Maturity of a Coupon BondSlide11
YTM – Example
On 9/1/95, PG&E bonds with a maturity date of 3/01/25 and a coupon rate of 7.25% were selling for 92.847% of par, or $928.47 per $1,000 of face value. What is their YTM?
Semiannual coupon payment = 0.0725*1000/2 = $36.25.
Number of semiannual periods to maturity
= 30*2 – 1 = 59.Slide12
YTM - Example
r/2 can only be found by trial and error. However, calculators and spread sheets have algorithms to speed up the search.
Searching reveals that r/2 = 3.939% or a stated annual rate (YTM) of r = 7.878%.
This is an effective annual rate of:Slide13
Example
Suppose that now the yield to maturity for the PG&E bonds has changed to 7% (on a stated annual basis, APR, with semi-annual compounding).
What must be the current price of the bond?
Note that the bond itself has not changed (assume for simplicity that the change occurs instantaneously) only the pricing.Slide14
Example
We find the price using the present value of the bond payments discounted at the YTM.
Does it make sense that the price changed like this?Slide15
Dynamic Behavior of Bond Prices
Discount
A bond is selling at a
discount if the price is less than the face value.ParA bond is selling at par if the price is equal to the face value.Premium
A bond is selling at a
premium
if the price is greater than the face value.Slide16
Discounts and Premiums
If a coupon bond trades at a discount, an investor will earn a return both from receiving the coupons and from receiving a face value that exceeds the price paid for the bond.
If a bond trades at a discount, its yield to maturity will exceed its coupon rate
.If a coupon bond trades at a premium it will earn a return from receiving the coupons but this return will be diminished by receiving a face value less than the price paid for the bond.If a bond trades at a premium, its coupon rate will exceed its yield to maturity.
Most coupon bonds have a coupon rate so
that the bonds will
initially
trade at, or very
close to, par.Slide17
Time and Bond Prices
Holding all other things constant, a bond’s yield to maturity will not change over time
.
Over time the price of the bond will change given a constant yield to maturity.Holding all other things constant, the price of a discount or a premium bond will move towards par value over time.Slide18
The Effect of Time on Bond PricesSlide19
Interest Rate Changes and
Bond
Prices
The sensitivity of a bond’s price to changes in interest rates is measured by the bond’s duration.Bonds with high durations are highly sensitive to interest rate changes.Bonds with low durations are less sensitive to interest rate changes.A bond’s term or time to maturity is a major determinant of duration. All else equal, the longer the term the more sensitive to changes in the interest rates is bond’s price.
However, that is not the whole story:Slide20
Problem
Consider a 15-year zero-coupon bond and a 30-year coupon bond with 10%
annual
coupons. By what percentage will the price of each bond change if its yield to maturity increases from 5% to 6%?Slide21
Solution
First consider the price of each bond at the two yields.
Now note that the price of the 15-year zero changes by -13.2% and the price of the 30-year coupon bond changes by only – 12.3%.
Even though the coupon bond has a longer maturity it is less sensitive to rate changes because of its large coupons.There is an inverse relation between interest rates and bond prices
Yield to Maturity
15-Year Zero
30-Year Coupon
5%
6%Slide22
The Yield Curve and Bond Arbitrage
Using the Law of One Price and the yields of default-free zero-coupon bonds, one can determine the price and yield of any other
default-free
bond.The yield curve provides sufficient information to evaluate all such bonds.Slide23
Replicating a Coupon Bond
Replicating a three-year $1000 bond that
pays 10% annual coupon using three zero-
coupon bonds:Slide24
Replicating a Coupon Bond
Yields and Prices (per $100
Face)
for Zero Coupon BondsMaturity
1 year
2 years
3 years
4 years
YTM
3.50%
4.00%
4.50%
4.75%
Price
$96.62
$92.45
$87.63
$83.06Slide25
Replicating a Coupon Bond
By the Law of One Price, the three-year coupon bond must trade for a price of $1153.Slide26
Value
a Coupon Bond
Using Zero-Coupon Yields
The price of a coupon bond must equal the present value of its coupon payments and face value.Price of a Coupon BondSlide27
Coupon Bond Yields
Knowing the price we can of course find the yield to maturity of the coupon bond.
Compare this to the zero-coupon yields.Slide28
Corporate Bonds
Corporate Bonds
Issued by corporations
Credit RiskRisk of defaultInvestors pay less for bonds with credit risk than they would for an otherwise identical default-free bond. The yield of bonds with credit risk will
therefore be
higher than that of otherwise identical default-free bonds.Slide29
Corporate Bond Yields
No Default
Consider a 1-year, zero coupon Treasury Bill with a YTM of 4%.
What is the price?Slide30
Corporate Bond Yields
Certain Default
Suppose
that bond issuer will pay 90% of the obligation. What is the price?Slide31
Corporate Bond Yields
Certain Default
When computing the yield to maturity for a bond with certain default, the promised rather than the actual cash flows are used
.The yield to maturity of a certain default bond is not equal to the expected return of investing in the bond. The yield to maturity will always be higher than the expected return of investing in the bond.Slide32
Corporate Bond Yields
Risk of Default
Consider a one-year, $1000, zero-coupon bond issued.
Assume that the bond payoffs are uncertain. There is a 50% chance that the bond will repay its face value in full and a 50% chance that the bond will default and you will receive $900. Thus, you would expect to receive $950.Because of the uncertainty, the discount rate is 5.1%.Slide33
Corporate Bond Yields
Risk of Default
The price of the bond will be
The yield to maturity will beSlide34
Corporate Bond Yields
Risk of Default
A bond’s expected return will be less than the yield to maturity if there is a risk of default.
A higher yield to maturity does not necessarily imply that a bond’s expected return is higher.