Chapter Six How to Value Bonds and Stocks 6

Chapter Outline 6.1 Definition and Example of a Bond 6.2 How to Value Bonds 6.3 Bond Concepts 6.4 The Present Value of Common Stocks 6.5 Estimates of Parameters in the Dividend-Discount Model 6.6 Growth Opportunities 6.7 The Dividend Growth Model and the NPVGO Model (Advanced) 6.8 Comparables 6.9 Valuing the Entire Firm 6.10 Stock Market Reporting 6.11 Summary and Conclusions

Valuation of Bonds and Stock First Principles: Value of financial securities = PV of expected future cash flows To value bonds and stocks we need to: Estimate future cash flows: Size (how much) and Timing (when) Discount future cash flows at an appropriate rate: The rate should be appropriate to the risk presented by the security. LO 6.1

Definition and Example of a Bond A bond is a legally binding agreement between a borrower (bond issuer) and a lender (bondholder): Specifies the principal amount of the loan. Specifies the size and timing of the cash flows: In dollar terms (fixed-rate borrowing) As a formula (adjustable-rate borrowing) LO 6.1

Definition and Example of a Bond Consider a Government of Canada bond listed as 5.000 December 2018. The Face 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 5.000 the payment is $25.00. On January 1, 2015 the size and timing of cash flows are: LO 6.1

How to Value Bonds Bond value is determined by the present value of the coupon payments and face value. We have to identify the size and timing of cash flows. Discount at the correct discount rate. Discount rates are inversely related to present (i.e., bond) values. If you know the price of a bond and the size and timing of cash flows, the yield to maturity (YTM) is the discount rate. LO 6.2

Pure Discount Bonds Make no periodic interest payments (coupon rate = 0%) The entire yield to maturity comes from the difference between the purchase price and the face value. Cannot sell for more than face value Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs) Treasury Bills and principal-only Treasury strips are good examples of zeroes. LO 6.2

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: LO 6.2

Pure Discount Bonds: Example Find the present value of a 30-year zero-coupon bond with a $1,000 face value and a YTM of 6%. LO 6.2

Pure Discount Bonds: Example Find the present value of a 30-year zero-coupon bond with a $1,000 face value and a YTM of 6%. N 30 I/Y 6 FV 1,000 CPT PV – 174.11 LO 6.2

Level Coupon Bonds Make periodic coupon payments in addition to the maturity value The payments are equal each period. Therefore, the bond is just a combination of an annuity and a terminal (maturity) value. Coupon payments are typically semiannual. LO 6.2

Level-Coupon Bonds Information needed to value level-coupon bonds: Coupon payment dates and time to maturity (T) Coupon payment (C) per period and Face value (F) Discount rate Value of a Level-coupon bond = PV of coupon payment annuity + PV of face value LO 6.2

Level-Coupon Bonds: Example Find the present value (as of January 1, 2015), of a 6-3/8 coupon Government of Canada bond with semi-annual payments, and a maturity date of December 31, 2020 if the YTM is 5-percent. The Face 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, 2015 the size and timing of cash flows are: LO 6.2

Level Coupon Bond: Example On January 1, 2015, the required annual yield is 5%. The present value of the bond is: LO 6.2

Level-Coupon Bonds: Example Find the present value (as of January 1, 2015), of a 6-3/8 coupon Government of Canada bond with semi-annual payments, and a maturity date of December 31, 2020 if the YTM is 5-percent. N 12 I/Y 5 1,000×0.06375 2 PMT 31.875 = FV 1,000 CPT PV – 1,070.52 LO 6.2

Bond Pricing with a Spreadsheet There are specific formulas for finding bond prices and yields on a spreadsheet. PRICE(Settlement,Maturity,Rate,Yld,Redemption, Frequency,Basis) YIELD(Settlement,Maturity,Rate,Pr,Redemption, Frequency,Basis) Settlement and maturity need to be actual dates The redemption and Pr need to be given as % of face value Click on the first Excel icon for pricing the bond in the 6-3/8 December 2020 bond. Click on the second Excel icon for another example. Please note that you need to have the analysis tool pack add-ins installed to access the PRICE and YIELD functions. If you do not have these installed on your computer, you can use the PV and the RATE functions to compute price and yield as well. Click on the TVM tab to find these calculations. LO 6.2

Bond Market Reporting The Government of Canada issued this bond The bond pays an annual coupon rate of 6.375% The bond matures on December 31, 2020 The bond is selling at 107.05% of the face value of $1,000 The bond’s quoted annual yield to maturity is 5% LO 6.2

Consols Not all bonds have a final maturity. British consols pay a set amount (i.e., coupon) every period forever. These are examples of a perpetuity. LO 6.2

Bond Concepts Bond prices and market interest rates move in opposite directions. 2. When coupon rate = YTM, price = face value. When coupon rate > YTM, price > face value (premium bond) When coupon rate < YTM, price < face value (discount bond) A bond with longer maturity has higher relative (%) price change than one with shorter maturity when interest rate (YTM) changes. All other features are identical. 4. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes. All other features are identical. LO 6.3

YTM and Bond Value $1400 When the YTM < coupon rate, the bond trades at a premium. 1300 Bond Value 1200 When the YTM = coupon rate, 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 When the YTM > coupon rate, the bond trades at a discount. LO 6.3

Bond Example Revisited Using our previous example, now assume that the required yield is 11%. How does this change the bond’s price? The bond trades at a discount. LO 6.3

Maturity and Bond Price Volatility Long Maturity Bond Maturity and Bond Price Volatility Bond Value Consider two otherwise identical bonds. The long-maturity bond will have much more volatility with respect to changes in the discount rate Short Maturity Bond Face C Discount Rate LO 6.3

Coupon Rate and Bond Price Volatility Low Coupon Bond Coupon Rate and Bond Price Volatility Bond Value Consider two otherwise identical bonds. The low-coupon bond will have much more volatility with respect to changes in the discount rate High Coupon Bond Discount Rate LO 6.3

Holding Period Return Suppose that on January 1, 2015, you purchased 6.375 coupon Government of Canada bond with semi-annual payments, and a maturity date of December 31, 2022. At that time the YTM was 5-percent, and you paid $1,089.75 (the PV of the bond). Six months later (July 1, 2015), you sold the bond when the YTM was 4-percent. The size and timing of cash flows (as of July 1, 2015 ) were: LO 6.3

Holding Period Return (cont.) Your holding period return was: Given that the YTM at that time was 4-percent, you sold the bond for: This annualizes to an effective rate of: LO 6.3

Computing Yield to Maturity Yield to maturity is the rate implied by the current bond price. Finding the YTM requires trial and error if you do not have a financial calculator and is similar to the process for finding R with an annuity. 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). LO 6.3

YTM with Annual Coupons Find YTM for a bond with a 10% annual coupon rate, 15 years to maturity, and a face value of $1,000. The current price is $928.09. N 15 PV – 928.09 The students should be able to recognize that the YTM is more than the coupon since the price is less than par. PMT 100 FV 1,000 CPT I/Y 11.00 LO 6.3

YTM with Semiannual Coupons Find YTM for a bond with a 10% coupon rate paid semi-annually and a face value of $1,000, 20 years to maturity, and is selling for $1,197.93. What is the semiannual coupon payment? How many periods are there? LO 6.3

YTM with Semiannual Coupons (cont.) Find YTM for a bond with a 10% coupon rate paid semi-annually, a face value of $1,000, 20 years to maturity, and is selling for $1,197.93. N 40 PV – 1,197.93 PMT 50 FV 1,000 CPT I/Y 4.00 => YTM = 4%*2 = 8% LO 6.3

The Present Value of Common Shares The value of any asset is the present value of its expected future cash flows. Share ownership produces cash flows from: Dividends Capital Gains Dividends versus Capital Gains Valuation of Different Types of Shares Zero Growth Constant Growth Differential Growth LO 6.4

Case 1: Zero Growth Assume that dividends will remain at the same level forever Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity: LO 6.4

A Zero Growth Example … P0 = Div1 / r $0.75 1 2 3 4 ABC Corp. is expected to pay $0.75 dividend per annum, starting a year from now, in perpetuity. If stocks of similar risk earn 12% annual return, what is the price of a share of ABC? The share price is given by the present value of the perpetual stream of dividends: $0.75 … 1 2 3 4 P0 = Div1 / r = 0.75/0.12 = $6.25 LO 6.4

Case 2: Constant Growth Assume that dividends will grow at a constant rate, g, forever. i.e. . . . Since future cash flows grow at a constant rate forever, the value of a constant growth share is the present value of a growing perpetuity: LO 6.4

A Constant Growth Example XYZ Corp. has a common share that paid its annual dividend this morning. It is expected to pay a $3.60 dividend one year from now, and future dividends thereafter are expected to grow at a rate of 4% per year forever. If shares of similar risk earn a 16% effective annual return, what is the price of a share of XYZ? LO 6.4

A Constant Growth Example (cont.) The share price is given by the present value of the perpetual stream of growing dividends: $3.60 $3.60´1.04 $3.60´1.042 $3.60´1.043 1 2 3 4 … P0 = Div1 / (r-g) = 3.60/(0.16-0.04) = $30.00 LO 6.4

Case 3: Differential Growth Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter. To value a Differential Growth Stock, we need to: Estimate future dividends in the foreseeable future. Estimate the future share price when the share becomes a Constant Growth Stock (case 2). Compute the total present value of the estimated future dividends and future share price at the appropriate discount rate. LO 6.4

Case 3: Differential Growth Assume that dividends will grow at rate g1 for N years and then grow at rate g2 thereafter . . . . . . LO 6.4

Case 3: Differential Growth (cont.) Dividends will grow at rate g1 for N years and then grow at rate g2 thereafter … 0 1 2 … … N N+1 LO 6.4

Case 3: Differential Growth We can value this as the sum of: an N-year annuity growing at rate g1 plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1 LO 6.4

Case 3: Differential Growth To value a Differential Growth Stock, we can use Or we can cash flow it out. LO 6.4

A Differential Growth Example A common share just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then grow at 4% in perpetuity. If shares of similar risk earn a12% effective annual return, what is the share worth? LO 6.4

With the Formula LO 6.4

A Differential Growth Example (cont.) … 0 1 2 3 4 The constant growth phase beginning in year 4 can be valued as a growing perpetuity at time 3. 0 1 2 3 LO 6.4

A Differential Growth Example (cont.) A common share 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? $28.89 = 5.58 + 23.31 First find the PV of the supernormal dividend stream then find the PV of the steady-state dividend stream. N 3 N 3 1.12 1.08 –1 ×100 I/Y 3.70 = I/Y 12 2×(1.08)3 ×(1.04) .08 2×1.08 1.08 FV 32.75 = PMT $2 = CPT PV – 23.31 CPT PV – 5.58 LO 6.4

Estimates of Parameters in the Dividend-Discount Model The value of a firm depends upon its growth rate, g, and its discount rate, r. Where does g come from? Where does r come from? LO 6.5

Where does g come from? Formula for Firm’s Growth Rate (g): The firm will experience earnings growth if its net capital investment (total investment-depreciation) is positive. To grow, the firm must retain some of its earnings. This leads to: Earnings next Year Earnings this Year Retained earnings this Year Return on retained earnings = + ´ LO 6.5

Where does g come from? (con.) Dividing both sides by this year’s earnings, we get: This leads to the formula for the firm’s growth rate: g = Retention ratio × Return on retained earnings The return on retained earnings can be estimated using the firm’s historical return on equity (ROE) Earnings this Year Earnings next Year 1 Retained earnings this Year Return on retained earnings = + ´ 1 + g Retention ratio LO 6.5

Where does g come from? An Example Ontario Book Publishers (OBP) just reported earnings of $1.6 million, and it plans to retain 28% of its earnings. If OBP’s historical ROE was 12%, what is the expected growth rate for OBP’s earnings? With the above formula: g = 0.28 × 0.12 = 0.0336 = 3.36% Or: Total earnings Change in earnings = $1.6 million (0.28×$1.6 million) ×0.12 0.0336 LO 6.5

Where does r come from? From the constant growth case, we can write: The discount rate can be broken into two parts. The dividend yield The growth rate (in dividends) In practice, there is a great deal of estimation error involved in estimating r. LO 6.5

Where does r come from? An Example Manitoba Shipping Co. (MSC) is expected to pay a dividend next year of $8.06 per share. Future dividends for MSC are expected to grow at a rate of 2% per year indefinitely. If an investor is currently willing to pay $62.00 per one MSC share, what is her required return for this investment? With the above formula: r = (8.06/62) + 0.02 = 0.15 = 15% LO 6.5

Some more on “r” and “g” The previous approaches only estimate r and g based on numerous assumptions. subject to error r is dependent on g Some analysts argue that an industry average of r should be used instead What happens when the company is not currently paying any dividends? Remember that g is in perpetuity and cannot equal r in the long term. LO 6.5

Growth Opportunities Growth opportunities are opportunities to invest in positive NPV projects. The value of a firm can be conceptualized as the sum of the value of a firm that pays out 100-percent of its earnings as dividends (is a cash cow) and the net present value of the growth opportunities. LO 6.6

Growth Opportunities - Example Zybrid currently has earnings per share of $1.50, with 1 million shares outstanding. The annual dividend is equal to the EPS. This is expected to occur in perpetuity if the company makes no new investments. The company has decided to spend $1.5 million on a new project at Date 1, that will increase the earnings by $250,000. This represents a 16.7% return on the project. The firm’s discount rate is 12%. LO 6.6

Growth Opportunities – Example (cont.) The NPV of the new project at Date 1 is: -$1,500,000 + $250,000/.12 = $583,333 Discounted back for one period: the value of the project at Date 0: $583,333/1.12 = $520,833 NPVGO per share = $520,833/1,000,000 = $0.52 Price of share = LO 6.6

Growth Opportunities Two conditions must be met in order to increase value: Earnings must be retained so that projects can be funded. The projects must have positive NPV – that is the return on the project must be greater than the firm’s discount rate A firm’s value increases when it invests in growth opportunities with positive NPVs. LO 6.6

The No-Dividend Firm Firms that pay no dividends sell at positive prices (e.g. Amazon.com) These firms have many growth opportunities and investors believe these firms will pay dividends at some point or they have invested for capital gains returns. The model for constant growth of dividends does not apply. LO 6.6

The Dividend Growth Model and the NPVGO Model (Advanced) We have two ways to value a share: The dividend discount model. The price of a share can be calculated as the sum of its price as a cash cow plus the per-share value of its growth opportunities. LO 6.7

The Dividend Growth Model and the NPVGO Model Consider a firm that has EPS of $5 at the end of the first year, a dividend-payout ratio of 30-percent, a discount rate of 16-percent, and a return on retained earnings of 20-percent. The dividend at year one will be $5 × .30 = $1.50 per share. The retention ratio is .70 ( = 1 -.30) implying a growth rate in dividends of 14% = .70 × 20% From the dividend growth model, the price of a share is: LO 6.7

Price using the NPVGO Model First, we must calculate the value of the firm as a cash cow. Second, we must calculate the value of the growth opportunities. Finally, LO 6.7

Comparables – Price-Earnings Ratio Many analysts frequently relate earnings per share to price. The price earnings ratio is a.k.a the multiple Calculated as current share price divided by annual EPS The National Post uses last 4 quarters’ earnings to estimate EPS Firms whose shares are “in fashion” sell at high multiples. Growth stocks for example. Firms whose shares are out of favour sell at low multiples. Value stocks for example. LO 6.8

Comparables – Price-Earnings Ratio Relating the PE ratio to the NPVGO approach, we see: PE multiple is negatively related to r. As r increases, the PE multiple declines. PE multiple is positively related to g. As g increases, PE multiple increases. Accounting methods to determine EPS, will impact the PE ratio for that company’s share. Generally accepted that companies with conservative accounting practices will have higher PE multiples LO 6.8

Other Price Ratio Analysis Many analysts frequently relate price to variables other than EPS, e.g.: Price/Cash Flow per share Ratio cash flow = net income + depreciation = cash flow from operations or operating cash flow Price/Sales current share price divided by annual sales per share Price/Book value per share (a.k.a. Market to Book Ratio) price divided by book value of equity, which is measured as assets - liabilities LO 6.8

Valuing the Entire Firm - Example The value of a firm can be calculated to be the PV of all future cash flows that the firm will generate. Assume non-constant growth for years 1-5, 3% terminal growth rate, 15% discount rate and 10 million shares o/s. LO 6.9

Stock Market Reporting Teck has been as high as $34.47 in the last year. Teck has been as low as $21.11 in the last year. Teck pays a dividend of 0.9 dollars/share Given the current price, the dividend yield is 3.2 % Given the current price, the P/E ratio is 18.5 times earnings 4,138,000 shares traded hands in the last day’s trading Teck ended trading at $27.86, down $1.62 from yesterday’s close LO 6.10

Stock Market Reporting Teck Resources is having a volatile year, and now trading in between its 52-week high and low. Imagine how you would feel if within the past year you had paid $34.47 for a share of Teck and now had a share worth $27.86! That $0.9 dividend wouldn’t go very far in making amends. LO 6.10

Summary and Conclusions In this chapter, we used the time value of money formulae from previous chapters to value bonds and stocks. The value of a zero-coupon bond is The value of a perpetuity is LO 6.11

Summary and Conclusions (cont.) The value of a coupon bond is the sum of the PV of the annuity of coupon payments plus the PV of the face value at maturity. The yield to maturity (YTM) of a bond is that single rate that discounts the payments on the bond to the purchase price. LO 6.11

Summary and Conclusions (cont.) A stock can be valued by discounting its dividends. There are three cases: Zero growth in dividends Constant growth in dividends Differential growth in dividends LO 6.11

Summary and Conclusions (cont.) The growth rate can be estimated as: g = Retention ratio × Return on retained earnings An alternative method of valuing a stock was presented. The NPVGO values a stock as the sum of its “cash cow” value plus the present value of growth opportunities. LO 6.11

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? What determines g and r in the Dividend Growth Model ? Decompose a stock’s price into constant growth and NPVGO values. Discuss the importance of the P/E ratio.