Valuing Bonds Bond Cash Flows Prices and Yields

Valuing Bonds

Bond Cash Flows, Prices, and Yields � Bond Terminology � Face Value � Notional � Coupon amount used to compute the interest payments Rate � Determines the amount of each coupon payment, expressed as an annual percentage rate � Coupon Payment

Zero-Coupon Bonds � Zero-Coupon � Does Bond (Zero) not make coupon payments � Always sells at a discount (a price lower than face value), so they are also called pure discount bonds � Treasury Bills are U. S. government zero-coupon bonds with a maturity of up to one year.

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.

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 bond

Zero-Coupon Bonds � Yield � For to Maturity the one-year zero coupon bond: � Thus, the YTM is 3. 5%.

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 Year 2 Years 3 years Price 980. 39 942. 60 901. 94

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%

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 bonds. will not be this simple when we consider coupon

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 Bond

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.

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:

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.

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?

Dynamic Behavior of Bond Prices � Discount �A bond is selling at a discount if the price is less than the face value. � Par �A 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.

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.

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.

The Effect of Time on Bond Prices

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:

Problem � Consider a 15 -year zero-coupon bond 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%?

Solution � First consider the price of each bond at the two yields. Yield to Maturity 15 -Year Zero 30 -Year Coupon 5% 6% � 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

The Yield Curve and Bond Arbitrage � Using the Law of One Price and the yields of defaultfree 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.

Replicating a Coupon Bond � Replicating a three-year $1000 bond that pays 10% annual coupon using three zerocoupon bonds:

Replicating a Coupon Bond � Yields Bonds and Prices (per $100 Face) for Zero Coupon Maturity 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. 06

Replicating a Coupon Bond � By the Law of One Price, the three-year coupon bond must trade for a price of $1153.

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 Bond

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.

Corporate Bonds � Corporate � Issued � Credit � Risk Bonds by corporations Risk of default � Investors 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.

Corporate Bond Yields � No Default � Consider a 1 -year, zero coupon Treasury Bill with a YTM of 4%. � What is the price?

Corporate Bond Yields � Certain Default � Suppose � What that bond issuer will pay 90% of the obligation. is the price?

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.

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%.

Corporate Bond Yields � Risk of Default � The price of the bond will be � The yield to maturity will be

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.