Chapter 5 Valuing Bonds

Chapter 5 Topic Overview uBond Characteristics uReading Bond Quotes uAnnual and Semi-Annual Bond Valuation uFinding Returns on Bonds uBond Risk and Other Important Bond Valuation Relationships

Bond Characteristics uFace (or Par) Value = stated face value that is the amount the issuer must repay, usually $1,000 uCoupon Interest Rate uCoupon (cpn) = Coupon Rate x Face Value uMaturity Date = when the face value is repaid. uThis makes a bond’s cash flows look like this:

Characteristics of Bonds  Bonds pay fixed coupon (interest) payments at fixed intervals (usually every 6 months) and pay the face value at maturity n $I $I $I $I $I $I+$M

The Financial Pages: Treasury Bonds Maturity Ask Rate Mo/Yr Bid Asked Chg Yld 6.5 Oct 06n112:17 112: Most values expressed as a %age of par ($1000). xxx:## = xxx and ##/32 nd % of par Asked = investor purchase price = /32% of $1000 = $1, Bid = investor selling price = $1, Rate = Annual coupon rate = 6.5% of par $65/year: $32.50 semiannually Chg = change in price from previous day in 32nds of % of par Ask Yld = 2.23% annual rate of return if purchased and held until maturity in Oct 2006

Bonds WARNING The coupon rate IS NOT the discount rate used in the Present Value calculations. The coupon rate merely tells us what cash flow the bond will produce. Since the coupon rate is listed as a %, this misconception is quite common.

Bond Pricing The price of a bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.

Bond Valuation Discount the bond’s cash flows at the investor’s required rate of return. –the coupon payment stream (an annuity). –the face (par) value payment (a single sum). –PV = cpn (PVAF r, t) + par /(1+r)t n cpn cpncpn+par cpn cpn cpn+par

Bond Valuation Example #1 uDuff’s Beer has $1,000 par value bonds outstanding that make annual coupon payments. These bonds have an 8% annual coupon rate and 12 years left to maturity. Bonds with similar risk have a required return of 10%, and Moe Szyslak thinks this required return is reasonable. uWhat’s the most that Moe is willing to pay for a Duff’s Beer bond?

P/Y = 1 12 = N 10 = I/Y 1,000 = FV 80 = PMT CPT PV = -$ Note: If the coupon rate < discount rate, the bond will sell for less than the par value: a discount.

Let’s Play with Example #1 uHomer Simpson is interested in buying a Duff Beer bond but demands an 8 percent required return. uWhat is the most Homer would pay for this bond?

P/Y = 1 12 = N 8 = I/Y 1,000 = FV 80 = PMT CPT PV = -$1,000 Note: If the coupon rate = discount rate, the bond will sell for its par value.

Let’s Play with Example #1 some more. uBarney (belch!) Barstool is interested in buying a Duff Beer bond and demands on a 6 percent required return. uWhat is the most Barney (belch!) would pay for this bond?

P/Y = 1 12 = N 6 = I/Y 1,000 = FV 80 = PMT CPT PV = -$1, Note: If the coupon rate > discount rate, the bond will sell for more than the par value: a premium.

Bond Prices and Interest Rates have an inverse relationship!

Bonds with Semiannual Coupons uDouble the number of years, and divide required return and annual coupon by 2. V B = annual cpn /2(PVAF r/2,2t ) + par V B = annual cpn /2(PVAF r/2,2t ) + par /(1+r/2) 2t

Semiannual Example uA $1000 par value bond with an annual coupon rate of 9% pays coupons semiannually with 15 years left to maturity. What is the most you would be willing to pay for this bond if your required return is 8% APR? uSemiannual coupon = 9%/2($1000) = $45 u15x2 = 30 remaining coupons

P/Y = 1 15x2 =30 = N 8/2 = 4 = I/Y 1,000 = FV 90/2 = 45 = PMT CPT PV = -$1,086.46

Bond Yields Current Yield - Annual coupon payments divided by bond price. Yield To Maturity - Interest rate for which the present value of the bond’s payments equal the price.

Bond Yields Calculating Yield to Maturity (YTM=r) If you are given the price of a bond (PV) and the coupon rate, the yield to maturity can be found by solving for r.

Yield to Maturity Example u$1000 face value bond with a 10% coupon rate paid annually with 20 years left to maturity sells for $ uWhat is this bond’s yield to maturity?

P/Y = = PV 20 = N 1,000 = FV 100 = PMT CPT I/Y = 9% = YTM

Bond Yields Rate of Return - Earnings per period per dollar invested.

Let’s try this together. Imagine a year later, the discount (required) rate for the bond from the YTM example fell to 8%. What is the bond’s expected price? What is the rate of return, if we sell the bond at this time assuming we bought the bond a year earlier at ? PMT =100, FV = 1000

YTM for semiannual coupon bonds: back to our T-bond Maturity Ask Rate Mo/Yr Bid Asked Chg Yld 6.5 Oct 06n112:17 112: $1000 par value, today’s price = $ = PV Semiannual coupon = $1000(6.5%/2) = $32.50 Assume = 3 years to maturity x 2 = 6 semiannual payments left. -1, = PV, = PMT, $1000 = FV, 6 = N, CPT I/Y = 1.1% semiannually Annual YTM = 2(1.1%) = 2.2% APR

Bond Value Changes Over Time uReturning to the original example #1, where k = 10%, N = 12, cpn (PMT) = $80, par (FV) = $1000, & PV = $ uWhat is bond value one year later when N = 11 and r is still = 10%? u80 = PMT, 1000 = FV, 11 = N, 10 = I/Y, CPT PV = PV = $80(PVAF 10%,11 ) + $1000/(1.10) 11 = $870.10

What is the bond’s return over this year? uRate of Return = (Annual Coupon + Price Change)/Beg. Price uAnnual Coupon = $80 uBeg. Price = $863.73, End Price = $ uPrice Change = $ $ = $6.37 uRate of Return = ($80 + $6.37)/$ = 10%

Bond Prices over time approach par value as maturity date approaches assuming same YTM

Interest Rate Risk Measures Bond Price Sensitivity to changes in interest rates. In general, long-term bonds have more interest rate risk than short- term bonds.

Interest Rate Risk Example Recall from our earlier example (#1), the 12-year, 8% annual coupon bond has the following values at k d = 6%, 8%, & 10%. Let’s compare with a 2-yr, 8% annual coupon bond. 12-year bond2-year bond r=6%: PV = $1,167.68PV = $1, r=8%: PV = $1,000PV = $1,000 r=10%: PV = $863.73PV = $965.29

Bond Price Sensitivity Graph

Default Risk Credit risk Default premium Investment grade Junk bonds

Default Risk

Other Types of Bonds uZero Coupon Bonds: no coupon payments, just par value. uConvertible Bonds: can be converted into (fixed # of) shares of stock. uFloating Rate (Indexed) Bonds: coupon payments and/or par value indexed to inflation. uTIPs: Indexed US Treasury coupon bond, fixed coupon rate, face value indexed. uCallable Bonds: Company can buy back the bonds before maturity for a call price. More likely as interest rates fall. uYield to Call: calculate like yield to maturity but use time to earliest call date as N, and call price as FV.