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## How to Value Bonds and Stocks

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**What is a Bond?**• A bond is a legally binding agreement between a borrower and a lender • IOU**Bond Terminology**• Face value (F) or Principal • For a corporate bond this is generally $1,000 • Zero- coupon bond • Coupon Rate • This is a Stated Annual Rate • Determines the coupon payment • Coupon payment (C ) • Rating**Bond Pricing Terminology**• Par • The price of the bond equals its face value • Premium • The price of the bond is greater than its face value • Discount • The price of the bond is less than its face value • Yield to Maturity**Yield to Maturity**• YTM is the return that the bond is offering if you bought it today and held it till maturity • The YTM is determined by the riskiness of the bond • Risk comes from: • Risk of default • Risk is often measured with bond ratings • Investment Grade / Junk • Time to maturity • Longer term bonds are riskier**Pure Discount Bonds**• Have no coupon • Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs) • Example: T-Bill • Yield to Maturity comes only from the difference between the purchase price and principal repayment • A pure discount bond cannot sell at a premium WHY?**Pure Discount Bonds**Information needed for valuing pure discount bonds: • Time to maturity (T) = Maturity date - today’s date • Face value (F) • Discount rate (r) Present value of a pure discount bond at time 0:**Pure Discount Bond: Example**Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%. 1,000/(1.0630) = 174.11**Coupon Bonds**• Make periodic coupon payments in addition to repaying the principal • Coupon payments are the same each period • Typically occur semi-annual • An investor’s return is comprised of: • Difference between the purchase price & face value • Coupon payments**Valuing a Coupon Bond**• The value of a bond is simply the present value of it’s future cash flows • We value a bond is a package of two investments: • Present value of the coupon payments • Present value of the principal repayment**Determining Coupon Payments**Coupon ($)= (Principal * Coupon Rate) / Frequency • Ex: • 8% semi-annual • (1,000 * 0.08) / 2 = 40 • 12% monthly • (1,000 * 0.12) / 12 = 10 • 20% annual • (1,000 * 0.20) / 1 = 200**Coupon Bond Pricing Equation**Annuity Coupon Payments Lump Sum Principal Repayment**Coupon Bond Pricing: BA II plus**• N = The number of coupon payments • I/Y = The rate corresponding to the coupon frequency • PV = The price of the bond today • PMT= The amount of the coupon payment • FV = The principal that will be repaid**Coupon Example 2**30 = 15 * 2 3.42% ?? -1,200 45 = (1000 * 0.08)/2 1,000 The 3.42% is a 6 month rate, the YTM = 6.84% What is the yield to maturity of a 9% 15 year, bond that sells for $1,200%? N = I/Y = PV = PMT = FV =**Coupon Example 1**20 = 10 * 2 12 402.44 ?? 40 = (1000 * 0.08)/2 1,000 What is the present value of a 8% 10 year, bond with the yield to maturity is 12%? N = I/Y = PV = PMT = FV =**Valuing a Corporate Bond**• DuPont issued a 30 year bonds with a coupon rate of 7.95%. • Interest is paid semi-annually • These bonds currently have 28 years remaining to maturity and are rated AA. • The bonds have a par value of $1,000 • Newly issued AA bonds with maturities greater than 10 years are currently yielding 7.73% • What is the value of DuPont bond today?**DuPont example (continued)**• Annual interest ($) = • Semiannual coupon payment = • Semiannual discount rate = • Number of semiannual periods= • PV=**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = • Semiannual discount rate = • Number of semiannual periods= • PV=**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = 79.5/2= 39.75 • Semiannual discount rate = • Number of semiannual periods= • PV=**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = 79.5/2= 39.75 • Semiannual discount rate = 0.0773/2 =0.03865 • Number of semiannual periods= • PV=**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = 79.5/2= 39.75 • Semiannual discount rate = 0.0773/2 =0.03865 • Number of semiannual periods= 28*2 = 56 • PV= N = ??, I/Y = ??, PV= ????, PMT =??, FV=??**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = 79.5/2= 39.75 • Semiannual discount rate = 0.0773/2 =0.03865 • Number of semiannual periods= 28*2 = 56 • PV= • N= 56, I/Y = ??, PV= ????, PMT =??, FV=??**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = 79.5/2= 39.75 • Semiannual discount rate = 0.0773/2 =0.03865 • Number of semiannual periods= 28*2 = 56 • PV= N = 56, I/Y = 3.865, PV= ????, PMT = ??, FV= ??**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = 79.5/2= 39.75 • Semiannual discount rate = 0.0773/2 =0.03865 • Number of semiannual periods= 28*2 = 56 • PV= N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV= ??**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = 79.5/2= 39.75 • Semiannual discount rate = 0.0773/2 =0.03865 • Number of semiannual periods= 28*2 = 56 • PV= N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV = 1,000**DuPont example (continued)**• Annual interest ($) = 0.0795*1000 =79.50 • Semiannual coupon payment = 79.5/2= 39.75 • Semiannual discount rate = 0.0773/2 =0.03865 • Number of semiannual periods= 28*2 = 56 • PV= 1,025.06 • The bond is currently selling for 1,025.06 N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV= 1,000**Level Coupon Bond: Example (Given)**• Consider a U.S. government bond with a 6 3/8% coupon that expires in December 2010. • The Par Value of the bond is $1,000. • Coupon payments are made semi-annually (June 30 and December 31 for this particular bond). • Since the coupon rate is 6 3/8%, the payment is $31.875. • On January 1, 2006 the size and timing of cash flows are: • The require annual rate is 5%**Level Coupon Bond: Example (Given)**• Coupon Rate 6 3/8%, pay semi-annually • 10 Semi-Annual Payments of $31.875. • Maturity December 2010, Start Jan. 2006 • The Par Value of the bond is $1,000. • The require annual rate is 5% • N = 10, I/Y = 2.5, PV=???, PMT = 31.875, FV=1,000::: PV = $1,060.17**Valuing a Corporate Bond (Given)**• Value a bond with the following characteristics (calculator): • Face value: $1,000 • Coupon rate (C ): 8% • Time to maturity: 4 years • Discount rate: 9% • Present Value: $967.02 • You should know how to get any one of these numbers given the other 4.**YTM and Bond Prices**• How are prices and YTM related? • Inversely, as one goes up the other goes down • As you pay more for the bond you earn a lower return**Coupon Rate and YTM**• Coupon rate = YTM • Price = Face, Bond is selling at Par • Coupons provide all the required return • Coupon rate > YTM • Price > Face, Bond is selling at a Premium • Coupons provide more than the required return • Coupon rate < YTM • Price < Face, Bond is selling at a Discount • Coupons do not provide the required return need to increased the return by paying less**6 3/8**YTM and Bond Value When the YTM < coupon, the bond trades at a premium. 1300 1200 Bond Value When the YTM = coupon, the bond trades at par. 1100 1000 800 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Discount Rate Coupon Rate When the YTM > coupon, the bond trades at a discount.**Computing Yield to Maturity**• Finding the YTM requires trial and error if you do not have a financial calculator • If you have a financial calculator, enter N, PV, PMT, and FV, • Remembering the sign convention • PMT and FV need to have the same sign, PV the opposite sign**YTM with Semiannual Coupons**• A bond has a 10% coupon rate, 20yrs to maturity, makes coupon payments semi-annually, a $1,000 face, and is selling at $1,197.93 • Is the YTM more or less than 10%? • LESS • What is the semi-annual coupon payment? • (1,000 * 0.10) / 2 = $50 • How many periods are there? • 20 * 2 = 40 • What is the YTM? N= 40,I/Y = ?, PV= -1197.93, PMT = 50, FV= 1,000→ 3.99% • YTM = 7.99998011%**YTM with Annual Coupons (Given)**• Consider a bond with a 10% annual coupon rate, 15 years to maturity, and a par value of $1,000. The current price is $928.09. • Will the YTM be more or less than 10%? • MORE • What is the YTM? • N = 15 • I/Y = ???? = 11% • PV = 928.09 • PMT = 100 • FV = 1000**Rate Changes and Bond Prices**• Known as interest rate risk • Consider two identical 8% coupon bonds except that one matures in 4 years, the other matures in 10 years • Calculate the change in the price of each bond if: • Interest rates fall from 8% to 6% • Interest rates rise from 8% to 10%**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _ • 4 years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 years @ 6%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = _, PV=_,PMT = _, FV = _ • 4 years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 years @ 6%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=_, PMT = _, FV = _ • 4 years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 years @ 6%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=?, PMT = _, FV = _ • 4 years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 years @ 6%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=?, PMT = 40, FV = _ • 4 years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 years @ 6%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=?, PMT = 40, FV = 1,000 • 4 years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 years @ 6%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_ PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV =$1,070.20 • 4 years @ 10%, 8% Coupon • N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000 • PV = $935.37 • 10 years @ 6%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV =$1,070.20 • 4 years @ 10%, 8% Coupon • N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000 • PV = $935.37 • 10 years @ 6%, 8% Coupon • N=20, I/Y = _, PV=_ PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV =$1,070.20 • 4 years @ 10%, 8% Coupon • N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000 • PV = $935.37 • 10 years @ 6%, 8% Coupon • N=20, I/Y = 3, PV=_ PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV =$1,070.20 • 4 years @ 10%, 8% Coupon • N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000 • PV = $935.37 • 10 years @ 6%, 8% Coupon • N=20, I/Y = 3, PV=? PMT = _, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV =$1,070.20 • 4 years @ 10%, 8% Coupon • N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000 • PV = $935.37 • 10 years @ 6%, 8% Coupon • N=20, I/Y = 3, PV=? PMT = 40, FV = _ • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV =$1,070.20 • 4 years @ 10%, 8% Coupon • N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000 • PV = $935.37 • 10 years @ 6%, 8% Coupon • N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000 • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV =$1,070.20 • 4 years @ 10%, 8% Coupon • N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000 • PV = $935.37 • 10 years @ 6%, 8% Coupon • N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV = $1,148.77 • 10 Years @ 10%, 8% Coupon • N=_, I/Y = _, PV=_, PMT = _, FV = _**Rate Change and Bond Pricing**• 4 years @ 6%, 8% Coupon • N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV =$1,070.20 • 4 years @ 10%, 8% Coupon • N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000 • PV = $935.37 • 10 years @ 6%, 8% Coupon • N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000 • PV = $1,148.77 • 10 Years @ 10%, 8% Coupon • N=20, I/Y = _, PV=_ PMT = _, FV = _