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## Bond and Stock Valuation

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**Bond and Stock Valuation**The application of the present value concept Fin 351: lecture 3**Today’s plan or learning goal**• Review of the present value concept • Bond valuation • Interest rates and compounding • Some terminology about bonds • Value bonds • The yield curve • Default risk • Stock valuation • Some terminology about a stock • Value a stock • Simple dividend discount model • Dividend growth model Fin 351: lecture 3**Lottery example**• Paper reports: Today’s JACKPOT = $20mm !! • paid in 20 annual equal installments. • payment are tax-free. • odds of winning the lottery is 13mm:1 • Should you invest $1 for a ticket? • assume the risk-adjusted discount rate is 8% Fin 351: lecture 3**My solution**• Should you invest ? • Step1: calculate the PV • Step 2: get the expectation of the PV • Pass up this this wonderful opportunity Fin 351: lecture 3**Mortgage-style loans**• Suppose you take a $20,000 3-yr car loan with “mortgage style payments” • annual payments • interest rate is 7.5% • “Mortgage style” loans have two main features: • They require the borrower to make the same payment every period (in this case, every year) • The are fully amortizing (the loan is completely paid off by the end of the last period) Fin 351: lecture 3**Mortgage-style loans**• The best way to deal with mortgage-style loans is to make a “loan amortization schedule” • The schedule tells both the borrower and lender exactly: • what the loan balance is each period (in this case - year) • how much interest is due each year ? ( 7.5% ) • what the total payment is each period (year) • Can you use what you have learned to figure out this schedule? Fin 351: lecture 3**My solution**Ending balance Total payment Interest payment Principle payment year Beginning balance 0 $20,000 $1,500 $6,191 $7,691 $13,809 1 7,154 13,809 1,036 6,655 7,691 2 7,154 7,691 0 7,154 537 3 Fin 351: lecture 3**Interest**• Simple interest - Interest earned only on the original investment. • Compound interest - Interest earned on interest. • In Finance, we have only compound interest rates Fin 351: lecture 3**Simple interest**Example Simple interest is earned at a rate of 6% for five years on a principal balance of $100. Fin 351: lecture 3**Simple interest**Today Future Years 12345 Interest Earned 6 6 6 6 6 Value 100 106 112 118 124 130 Value at the end of Year 5 = $130 Fin 351: lecture 3**Compound interest**Example Compound interest is earned at a rate of 6% for five years on $100. Today Future Years 1 2 34 5 Interest Earned 6.00 6.36 6.74 7.15 7.57 Value 100 106.00 112.36 119.10 126.25 133.82 Value at the end of Year 5 = $133.82 Fin 351: lecture 3**Interest compounding**• The interest rate is often quoted as APR, the annual percentage rate. • If the interest rate is compounded m times in each year and the APR is r, the effective annual interest rate is Fin 351: lecture 3**Compound Interest**i ii iii iv v Periods Interest Value Annually per per APR after compounded year period(i x ii) one year interest rate 1 6% 6% 1.06 6.000% 2 3 6 1.032 = 1.0609 6.090 4 1.5 6 1.0154 = 1.06136 6.136 12 .5 6 1.00512 = 1.06168 6.168 52 .1154 6 1.00115452 = 1.06180 6.180 365 .0164 6 1.000164365 = 1.06183 6.183 Fin 351: lecture 3**Compound Interest**Fin 351: lecture 3**Interest Rates**Example Given a monthly rate of 1% (interest is compounded monthly), what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? Fin 351: lecture 3**Solution**Fin 351: lecture 3**Interest Rates**Example If the interest rate 12% annually and interest is compounded semi-annually, what is the Effective Annual Rate (EAR)? What is the Annual Percentage Rate (APR)? Fin 351: lecture 3**Solution**• APR=12% • EAR=(1+0.06)2-1=12.36% Fin 351: lecture 3**Nominal and real interest rates**• Nominal interest rate • What is it? • Real interest rate • What is it? • Inflation • What is it? • Their relationship • 1+real rate =(1+nominal rate)/(1+inflation) Fin 351: lecture 3**Bonds**• Bond – a security or a financial instrument that obligates the issuer (borrower) to make specified payments to the bondholder during some time horizon. • Coupon - The interest payments made to the bondholder. • Face Value (Par Value, Face Value, Principal or Maturity Value) - Payment at the maturity of the bond. • Coupon Rate - Annual interest payment, as a percentage of face value. Fin 351: lecture 3**Bonds**• A bond also has (legal) rights attached to it: • if the borrower doesn’t make the required payments, bondholders can force bankruptcy proceedings • in the event of bankruptcy, bond holders get paid before equity holders Fin 351: lecture 3**An example of a bond**• A coupon bond that pays coupon of 10% annually, with a face value of $1000, has a discount rate of 8% and matures in three years. • The coupon payment is $100 annually • The discount rate is different from the coupon rate. • In the third year, the bondholder is supposed to get $100 coupon payment plus the face value of $1000. • Can you visualize the cash flows pattern? Fin 351: lecture 3**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. Fin 351: lecture 3**Bond Valuation**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. Fin 351: lecture 3**Zero coupon bonds**• Zero coupon bonds are the simplest type of bond (also called stripped bonds, discount bonds) • You buy a zero coupon bond today (cash outflow) and you get paid back the bond’s face value at some point in the future (called the bond’s maturity ) • How much is a 10-yr zero coupon bond worth today if the face value is $1,000 and the effective annual rate is 8% ? Face value PV Time=0 Time=t Fin 351: lecture 3**Zero coupon bonds (continue)**• P0=1000/1.0810=$463.2 • So for the zero-coupon bond, the price is just the present value of the face value paid at the maturity of the bond • Do you know why it is also called a discount bond? Fin 351: lecture 3**Coupon bond**The price of a coupon 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. Fin 351: lecture 3**Bond Pricing**Example What is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%. Fin 351: lecture 3**Bond Pricing**Example What is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%. Fin 351: lecture 3**Bond Pricing**Example (continued) What is the price of the bond if the required rate of return is 6 %? Fin 351: lecture 3**Bond Pricing**Example (continued) What is the price of the bond if the required rate of return is 15 %? Fin 351: lecture 3**Bond Pricing**Example (continued) What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually? Fin 351: lecture 3**Bond Pricing**Example (continued) What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually? Fin 351: lecture 3**Bond Pricing**Example (continued) Q: How did the calculation change, given semi-annual coupons versus annual coupon payments? Fin 351: lecture 3**Bond Pricing**Example (continued) Q: How did the calculation change, given semi-annual coupons versus annual coupon payments? Time Periods Paying coupons twice a year, instead of once doubles the total number of cash flows to be discounted in the PV formula. Fin 351: lecture 3**Bond Pricing**Example (continued) Q: How did the calculation change, given semi-annual coupons versus annual coupon payments? Time Periods Paying coupons twice a year, instead of once doubles the total number of cash flows to be discounted in the PV formula. Discount Rate Since the time periods are now half years, the discount rate is also changed from the annual rate to the half year rate. Fin 351: lecture 3**Bond Yields**• Current Yield - Annual coupon payments divided by bond price. • Yield To Maturity (YTM)- Interest rate for which the present value of the bond’s payments equal the market price of the bond. Fin 351: lecture 3**An example of a bond**• A coupon bond that pays coupon of 10% annually, with a face value of $1000, has a discount rate of 8% and matures in three years. It is assumed that the market price of the bond is the fundamental value of the bond. • What is the current yield? • What is the yield to maturity. Fin 351: lecture 3**My solution**• First, calculate the bond price • P=100/1.08+100/1.082+1100/1.083 • =$1,051.54 • Current yield=100/1051.54=9.5% • YTM=8% Fin 351: lecture 3**Bond Yields**Calculating Yield to Maturity (YTM=r) If you are given the market price of a bond (P) and the coupon rate, the yield to maturity can be found by solving for r. Fin 351: lecture 3**Bond Yields**Example What is the YTM of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? The market price of the bond is $1,010.77 Fin 351: lecture 3**Bond Yields**• In general, there is no simple formula that can be used to calculate YTM unless for zero coupon bonds • Calculating YTM by hand can be very tedious. We don’t have this kind of problems in the quiz or exam • You may use the trial by errors approach get it. Fin 351: lecture 3**Bond Yields (3)**• Can you guess which one is the solution? • 6.6% • 7.1% • 6.0% • 5.6% • My solution is (d). Fin 351: lecture 3**The bond price, coupon rates and discount rates**• If the coupon rate is larger than the discount rate, the bond price is larger than the face value. • If the coupon rate is smaller than the discount rate, the bond price is smaller than the face value. Fin 351: lecture 3**The rate of return on a bond**Example: An 8 percent coupon bond has a price of $110 dollars with maturity of 5 years and a face value of $100. Next year, the expected bond price will be $105. If you hold this bond this year, what is the rate of return? Fin 351: lecture 3**My solution**• The expected rate of return for holing the bond this year is (8-5)/110=2.73% • Price change =105-110=-$5 • Coupon payment=100*8%=$8 • The investment or the initial price=$110 Fin 351: lecture 3**The Yield Curve**Term Structure of Interest Rates - A listing of bond maturity dates and the interest rates that correspond with each date. Yield Curve - Graph of the term structure. Fin 351: lecture 3**The term structure of interest rates (Yield curve)**Fin 351: lecture 3**YTM for corporate and government bonds**• The YTM of corporate bonds is larger than the YTM of government bonds • Why does this occur? Fin 351: lecture 3**Default Risk**• Default risk • The risk associated with the failure of the borrower to make the promised payments • Default premium • The amount of the increase of your discount rate • Investment grade bonds • Junk bonds Fin 351: lecture 3