How to Value Bonds and Stocks

What is a Bond? A bond is a legally binding agreement between a borrower and a lender

Bond Terminology Face value (F) or Principal Coupon rate For a corporate bond this is generally $1,000 Coupon rate This is a Stated Annual rate Determines the coupon payment Coupon payment (C ) Zero- coupon bond Yield to maturity Rating

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, which is a function of: Time to maturity Longer term bonds are riskier Risk of default Risk is measured by bond ratings

Pure Discount Bonds Makes no coupon payments 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 face value A pure discount bond cannot sell for more than par. WHY?

Pure Discount Bonds Present value of a pure discount bond at time 0: 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%.

Coupon Bonds Make periodic coupon payments in addition to repaying the principal Coupon payments are the same each period Coupon payments are typically semi-annual.

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

Coupon Bond Pricing Equation An annuity plus a lump sum

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

Valuing a Corporate Bond DuPont issued a 30 year maturity 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=

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?

Coupon Rate and YTM Coupon rate = YTM Coupon rate > YTM

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 6 3/8 800 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%? What is the semi-annual coupon payment? How many periods are there? What is the YTM?

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

The effect of changes in interest rates on 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%, if interest rates rise from 8% to 10%

Interest Rates and Time to Maturity The longer a bond has till maturity, the greater the price impact of a change in interest rates WHY?

Interest Rates and Bond Prices Bond Prices and Interest Rates have an Inverse Relationship

Pricing Stocks Remember: The value of any asset is the present value of its expected future cash flows. Bond Cash flows are: Stock produces cash flows from:

Stock Valuation Terminology Dt or Divt –dividend expected at time t P0 – market price of stock at time 0 Pt – expected mkt price of stock at time t g- expected growth rate of dividends rs or re- required rate of return on equity D1 / P0 – expected one-year dividend yield (P1 - P0)/ P0 – expected one year capital gain The stocks total return = div yield + cap. gain

Valuing Common Stock The price of a share is simply the present value of the expected future cash flows An investor planning on selling his share in a year is willing to pay: The investor buying the share next year plans on selling it a year later so he is only willing to pay:

Keep Going Using summation: P0 = H Dh / (1 + r)h + PH / (1 + r)H This process can be repeated into the future Using summation: P0 = H Dh / (1 + r)h + PH / (1 + r)H What happens to PH as H approaches infinity?

Dividend Valuation Model As H approaches infinity PH goes to zero Because of this we only need to be concerned with the stock’s future dividends The price of a stock is equal to the present value of its expected future dividends

Constant Dividend How do you value a stock that will pay a constant dividend? Hint: what does the cash flow stream look similar to?

Constant Dividend Example What is the value of a stock that is expected to pay a constant dividend of $2 per share? The required rate of return is 10%

Growing Dividends Now we are assuming that the firm’s dividends will grow at a constant rate, g forever This is similar to a: So the price of a share is:

Growing Dividend Example Geneva steel just paid a dividend of $2.10. Dividend payments are expected to grow at a constant rate of 6%. The appropriate discount rate is 12%. What is the price of Geneva stock? Div1 = P0 =

Valuing Stock with Changing g Find the PV of dividends during the period of non-constant growth, PA Find the price of the stock at the end of the non-constant growth period, PN Discount the price found in 2 back to the present, PB Add the two present values (1+3) to find the intrinsic value (price) of the stock P0 = PA + PB

Differential Growth Rates Dividends will grow at g1 for N years and g2 thereafter Step 1: An N-year annuity growing at rate g1 Step 2: A growing perpetuity at rate g2 PN = DivN+1 / (R-g2) Step 3: PB = PN / (1+R)N Step 4: P0 = PA + PB

Non-Constant Growth Example (Given) Websurfers Inc, a new internet firm is expected to do very well during its initial growth period. Investors expect its dividends to grow at 25% for the next 3 years. Obviously one cannot expect such extraordinary growth to continue forever, and it is expected that dividends will grow at 5% after year 3 in perpetuity. Its current dividend is $1/share. Required rate of return on the stock = 10%. Calculate what the current price should be.

Websurfer Inc, Example (Given) 1*1.253 *1.052 = 2.15 PA=[(1*1.25)/(0.10-0.25)]*[1-{1.25/1.10}3] = 3.90 PN ={1*1.253*1.05}/(0.10-0.05) = 41.00 PB =41.00/(1.103) = 30.80 P0 = PA + PB = 3.90+ 30.80 = $34.70 1*1.253*1.05 = 2.05 1*1.25 = 1.25 1*1.252 =1.56 1*1.253 = 1.95 0 1 2 3 4 5

A Differential Growth Example A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. What is the stock worth? The discount rate is 12%. 0 1 2 3 4 5

Solution A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. R=12% PA = PN = PB = P0 = PA + PB =

Important Parameters The value of a firm depends on the discount rate, the growth rate, and the initial dividend.

The Discount Rate The market consensus of the firm’s required rate This is the Market Capitalization Rate Return that an investor expects to make This is similar to what for a bond?

Where does “r” come from? We generally estimate r from one of the dividend valuation models Using constant dividend growth model: In practice, estimates of r have a lot of estimation error Rearrange and solve for R:

Where does “R” come from? What is D1/P0? What is g?

Classifying Stocks Firms are often classified based on where investors expect to earn their return from “Income/Value stocks”: have a higher dividend yield “Growth stocks”: have a higher growth component As long as both are equally risky, the return should be the same

Where does “g” come from? From analysts' estimates I/B/E/S, Google, Yahoo, or WSJ From earnings re-investment g = plowback ratio * ROE How much does the firm reinvest, and what is the return on the investment Look Familiar?

Link between stock prices and earnings A “new valuation model” : Consider a firm with a 100% payout ratio, so Div = EPS and earnings remain flat.

Present Value of all Future Growth Opportunities (PVGO) The price is composed of the value of the firm’s current assets (100% payout firm) and the firm’s growth opportunities Growth opportunities are opportunities to invest in positive NPV projects. P0 =

Who cares about PVGO? For what type of stock is the PVGO more important? Growth or Value stocks

PVGO Example Assume that a firm has 2 potential projects. Project A & B with NPV’s of $2m, and $3m, respectively. The firm pays out all its earnings as dividends, and paid a dividend of $1/share last year. It has 200,000 shares outstanding. Assume the discount rate is 10%. What is the share price, if the cash flow from the firm's existing assets are expected to remain the same in perpetuity, and the firm takes on Project A, and B?  

Stock Value Represents: Present value of expected future dividends, Present value of free cash flow, Present value of average future earnings under a no-growth policy plus the present value of growth opportunities

Price-Earnings Ratio The price-earnings ratio is calculated as the current stock price divided by annual EPS. The Wall Street Journal uses last 4 quarter’s earnings Many analysts use this to determine how the market feels about a company

Price/Earnings Ratio Is selling at a high P/E good? Why might the P/E be high:

Problem 1 (Given) A firm is expected to grow at 25% for the next 3 years. Its growth is expected to decline to 15% for the following 4 years. It is then expected to grow at 5% in perpetuity. Find the current share price if the current dividend is $1 and the discount rate is 10%. 

Problem 1 (Given) PA=[(1*1.25)/(0.10-0.25)]*[1-{1.25/1.10}3]=3.89 A firm is expected to grow at 25% for the next 3 years. Its growth is expected to decline to 15% for the following 4 years. It is then expected to grow at 5% in perpetuity. Find the current share price if the current dividend is $1 and the discount rate is 10%. PA=[(1*1.25)/(0.10-0.25)]*[1-{1.25/1.10}3]=3.89 PN1 ={1*1.253*1.15}/(0.10-0.15)]* [1-{1.15/1.10}4]=8.76 PB =8.76/(1.103) = 6.58 PN2 ={1*1.253*1.154*1.05}/(0.10-0.05)] = 71.80 PC =71.80/(1.107) = 36.83 P0 = PA + PB + Pc = 3.89+6.58+36.83 = $47.30

Problem 2 (Given) Consider a firm whose dividend growth is expected to decline gradually. For the next two years, the growth is expected to be 20%. In the following years, it is expected to grow at 18%, 13% and 10%. From year 6 onwards, dividends are expected to grow at 5% for perpetuity. Assume the current dividend is $1 and the required rate of return is 10%. What is the current price?

Problem 2: Phase 1 – Years 1-5 (Given) DIV1 = 1.00 * 1.20 = 1.20 DIV2 = 1.20 * 1.20 = 1.44 DIV3 = 1.44 * 1.18 = 1.70 DIV4 = 1.70 * 1.13 = 1.92 DIV5 = 1.92 * 1.10 = 2.11 PA = 1.2/1.1 + 1.44/1.12 + 1.70/1.13 + 1.92/1.14 + 2.11/1.15 = 6.18

Problem 2 Phase 2 – Years 6- (Given) DIV6 = 2.11 * 1.05 = 2.22 PN = DIV6 / (r-g) = 2.22/(0.10-0.05) = 44.4 PB = 44.4 * 1/1.15 = 27.57 Current Price = 6.18 + 27.57 = $33.75

Quick Quiz How do you find the value of a bond, and why do bond prices change? What is a bond indenture, and what are some of the important features? What determines the price of a share of stock? What determines g and R in the DGM? Decompose a stock’s price into constant growth and NPVGO values. Discuss the importance of the PE ratio.

Why We Care Basic real world application of the time value of money Foundation of Investment/ Financial Analysis