# Jacoby, Stangeland and Wajeeh, 20001 Valuation of Bonds and Stock uFirst Principles: u Value of financial securities = PV of expected future cash flows.

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Jacoby, Stangeland and Wajeeh, 20001 Valuation of Bonds and Stock uFirst Principles: u Value of financial securities = PV of expected future cash flows uTo value bonds and stocks we need to: u Estimate future cash flows: size (how much) and timing (when) u Discount future cash flows at an appropriate rate Chapter 5

Jacoby, Stangeland and Wajeeh, 20002 Bond Features lWhat is a bond - l debt issued by a corporation or a governmental body. l A bond represents a loan made by investors to the issuer. l In return for his/her money, the investor receives a legal claim on future cash flows of the borrower. lThe issuer promises to: l make regular coupon payments every period until the bond matures, and l pay the face (par) value of the bond when it matures. lDefault l an issuer who fails to pay is subject to legal action on behalf of the lenders (bondholders).

3 Pure-Discount (Zero-Coupon) Bonds uInformation needed for valuing pure discount bonds: u Time to maturity (T): T = Maturity date - today’s date u Face value (F) u Discount rate (r) 0 12 … T |-------------------|-------------------|------ … ------| F uValue of a pure discount bond: PV = F / (1 + r) T

4 Examples - Pure Discount Bonds Q1.Consider a zero-coupon bond, with a face value of \$1,000, maturing in 5 years. Suppose that the appropriate discount rate is 8%. What is the current value of the bond? A1.This is a simple TVM problem: Use the above PV equation to solve: P V = F / (1 + r) T = 1,000 / (1.08) 5 = \$ Q2.Suppose 6 months have past. What is the bond value now? A1.Again, use the above PV equation to solve: P V = F / (1 + r) T = 1,000 / (1.08) 4.5 = \$ Note:As we get closer to maturity(T), the z.c. bond value increases (PV m ), since we have to wait less time to receive \$1,000 012345 Year: (r = 8%) 1,000PV 0

5 Level-Coupon Bonds uInformation needed to value level-coupon bonds: u Coupon payment dates and Time to maturity (T) u Coupon (C) per payment period and Face value (F) u Discount rate 0 1 2 … T |----------------|------------------|------- … ------| Coupon Coupon Coupon + F uValue of a Level-coupon bond: PV= C/(1+r) + C/(1+r) 2 +.. + C/(1+r) T + F/(1+r) T = C (1/r) { 1 - [1 / ( 1 + r ) T ] } + F/(1 + r) T = PV of coupon payments + PV of face value

6 Example - Coupon Bonds Q1.Consider a coupon bond paying a 4% coupon rate annually, with a face value of \$1,000, maturing in 10 years. Suppose that the appropriate discount rate is 6%. What is the current value of the bond? A1.The time line: Define: c = annual coupon rate (%) C = dollar periodic coupon payment = c % F In the above example: c = %C = c % F = = \$ F = \$T = yearsr = % Use the above PV equation to solve: PV= C (1/r){1 - [1 / (1 + r) T ]} + F/(1 + r) T = 40(1/0.06){1 - [1 / (1.06) 10 ]} + 1,000/(1.06) 10 = \$ 012910 (Years) (r = 6%) …

7 First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above bond example) PV of a Bond in your HP 10B Calculator PMT I/YR N PV Key in coupon payment Key in discount rate Key in number of periods to maturity Compute PV of the bond Display should show: -852.79825897 40 6 10 FV Key in face value (paid at maturity) 1,000 Yellow C C ALL

8 Example - Discount, Premium and Par Bonds Q2.For the above coupon bond: when discount rate is 6% and coupon rate is 4% (c < r), the value of the bond is \$852.80, less than its face value (PV < F). In this case we say that the bond is priced at discount. Recalculate the PV of the above bond with discount rates of 2% and 4%. A2.r = 2% We have : r = 2% < 4% = c. Use the above PV equation to solve: PV= C (1/r){1 - [1 / (1 + r) T ]} + F/(1 + r) T = 40(1/0.02){1 - [1 / (1.02) 10 ]} + 1,000/(1.02) 10 = \$1,179.65 We see that when c > r, the bond is priced at premium (PV > F). r = 4% We have : r = c = 4%. Use the above PV equation to solve: PV= 40(1/0.04){1 - [1 / (1.04) 10 ]} + 1,000/(1.04) 10 = \$1,000 We say that when c = r, the bond is priced at par (PV = F).

Jacoby, Stangeland and Wajeeh, 20009 Some Tips on Bond Pricing uBond prices and market interest rates move in opposite directions. u u When coupon rate = market rate (r) => price = par value. (par bond) u When coupon rate > market rate (r) => price > par value (premium bond) u When coupon rate price < par value (discount bond)

Jacoby, Stangeland and Wajeeh, 200010 PV r (%) 1,000 = F 4 = c26 (\$) Premium Bond (r F) Discount Bond (r > c, and PV { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/14/4253970/slides/slide_10.jpg", "name": "Jacoby, Stangeland and Wajeeh, 200010 PV r (%) 1,000 = F 4 = c26 (\$) Premium Bond (r F) Discount Bond (r > c, and PV c, and PV

11 uThe bond’s indenture provides: F, c, and T uThe bond price (B) is set by the market uGiven F, c, T, and B, what return (y) does the market demand for holding the bond? uTo find y, solve the following equation: uThere is no analytical solution (use calculator) Yield-To-Maturity

Jacoby, Stangeland and Wajeeh, 200012 Example - Bond’s YTM Q.Consider a coupon bond paying a 7% coupon rate annually, with a face value of \$1,000, maturing in 20 years. The current market price of the bond is \$1,072.93. What is the yield to maturity (YTM) of the bond? A.We have: c = %C = c % F = =\$ F = \$T = B = \$ Use your HP 10B Financial Calculator: PMT PV N I/YR 1) Key in coupon payment 3) Key in the bond price (PV) 4) Key in number of periods to maturity 5) Compute YTM Display should show: 6.346178% 70 1,072.93 20 FV 2) Key in face value 1,000 +/- Yellow C C ALL

13 Canadian Bonds Canadian bonds usually pay coupons every six months (semiannually) Q.Consider a GofC bond paying semiannual coupons at an annual rate of 6%, with a face value of \$1,000, maturing in 8 years. The bond’s YTM is 7% per year compounded semiannually. What is the value of the bond? A.The time line: Define: c = (stated) annual coupon rate (%) = 6% C = dollar periodic coupon payment = (c/2) % F= = \$ We also have: F = 1,000N = s.a. periodsy 1/2 = 7% per year comp. s.a. Use the following PV equation to solve: 0121516 (6-month Periods) ( y 1/2 = 7% per year comp. s.a.) …

14 First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above bond example) PV of a S.A. Coupon Bond in your HP 10B Calculator PMT I/YR N PV Key in the s.a. coupon payment Key in the effective s.a. discount rate Key in number of 6-month periods to maturity Compute PV of the bond Display should show: -939.52941596 30 3.5 16 FV Key in face value (paid at maturity) 1,000 Yellow C C ALL

15 Finding the YTM of Canadian Bonds Q.Consider a GofC bond paying semiannual coupons at an annual rate of 12%, with a face value of \$1,000, maturing in 25 years. The bond’s market value is \$1,057.98. What is the yield to maturity (YTM) of the bond per year compounded semiannually? A.We have: c = 12%C = (c/2) % F= = \$ F = 1,000N = s.a. periodsB= 1,057.98 Use your HP 10B Financial Calculator: Since ( y 1/2 /2) = 5.649995%, the YTM of the bond per year compounded semiannually is given by: y 1/2 = 2 % 5.649995% = 11.299990% PMT PV N I/YR 1) Key in s.a. coupon payment 3) Key in the bond price (PV) 4) Key in number of s.a. periods to maturity 5) Compute the effective YTM PER 6 MONTHS Display should show: 5.649995% 60 1,057.98 50 FV 2) Key in face value 1,000 +/- Yellow C C ALL

16 Finding the Maturity of Canadian Bonds Q.Consider a GofC bond paying semiannual coupons at an annual rate of 7%, with a face value of \$1,000. The bond’s market value is \$1,026.82, with a YTM of 6.6% per year compounded semiannually. What is the time to maturity of this bond in years? A.We have: c = 7%C = (c/2) % F= = \$ F = 1,000 ( y 1/2 /2) = = % B=\$ Use your HP 10B Financial Calculator: Thus, T = 0.5 % (# of 6-month periods to maturity) = 0.5 % 17.99806470 = 8.99903235 years to maturity PMT PV I/YR N 1) Key in s.a. coupon payment 3) Key in the bond price (PV) 4) Key in the effective YTM per 6 months 5) Compute the number of 6 MONTHS PERIODS to maturity Display should show: 17.99806470 35 1,026.82 3.3 FV 2) Key in face value 1,000 +/- Yellow C C ALL

17 Finding the Coupon Rate of Canadian Bonds Q.Consider a 30-year GofC bond paying semiannual coupons, with a face value of \$1,000. The bond’s market value is \$912.83, with a YTM of 8.4% per year compounded semiannually. What is the the bond’s annual coupon rate? A.We have: F = 1,000( y 1/2 /2) = = % B= 912.83 N = s.a. periods Use your HP 10B Financial Calculator: Since: C = (c/2) % F, we get: c = (C/F) % 2 = (38.00001999/1,000) % 2 = 7.600004% per annum N PV I/YR PMT 1) Key in number of s.a. periods to maturity 3) Key in the bond price (PV) 4) Key in the effective YTM per 6 months 5) Compute the SEMI ANNUAL DOLLAR coupon payment Display should show: 38.00001999 60 912.83 4.2 FV 2) Key in face value 1,000 +/- Yellow C C ALL

Jacoby, Stangeland and Wajeeh, 200018 The Yield Curve

19 Four theories: I. Expectations theory e.g. if investors expect next years yield to be 12%, then the forward rate will also be 12%: 1 f 2 = 12% Example: two alternative investments l B 1 : a zero coupon bond with: T = 1, YTM: 0 r 1 = 8% l B 2 : a zero coupon bond with: T = 2, YTM: 0 r 2 = 9%, The Term Structure of Interest Rates

Jacoby, Stangeland and Wajeeh, 200020 The following investments of \$1 must be equivalent: (i) investing in B 2 for 2 years. At t = 2, receive: 1.09 2 (ii) investing in B 1 for 1 year. At t = 1, investing in a new 1-year bond at a rate 1 f 2. At t = 2, receive: 1.08(1+ 1 f 2 ) The forward rate ( 1 f 2 ) must take a value such that: The Term structure of Interest Rates

21 This implies the following forward rate for year-2: The general case: Note: This formula can be used only for zero-coupon bonds Example: 3 alternative zero-coupon bonds, with the following spot rates: 0 r 1 = 8% 0 r 2 = 10% 0 r 3 = 12% Calculating 1 f 2 and 2 f 3 :

22 Example - Using the Term Structure Q1.You observe the above spot rates for GofC zero-coupon bonds for different maturities: 0 r 1 = 8%, 0 r 2 = 10%, and 0 r 3 = 12%. A zero- coupon bond has a face value of \$1,000 and maturity of 2 years. What must be its price today? A1.Since: (1+ 0 r 2 ) 2 = (1+ 0 r 1 ) (1+ 1 f 2 ), we can use either spot rates or forward rates (same result) to find B: Q2.Assume that the Pure Expectations Hypothesis (PEH) holds, what do you expect the bond price to be one year from today? A2.One year from today, the bond will have one year remaining to maturity. Based on the PEH: expected spot rate for second year = 1 f 2 Thus, the expected bond price in a year is:

23 II. Liquidity Premium Theory If you invest for (t+1) years, you commit to reinvest in every year after the 1st year, and thereby lose liquidity and ask for a liquidity premium: III. Augmented Expectations Theory Combines the pure expectations theory with the liquidity premium theory: Example - Suppose: 0 r 1 = 8% and 0 r 2 = 9%. By the Expectations Theory: By the Liquidity Premium Theory, when L 2 =1%, we get: 1 f 2 = E[ 1 r 2 ] + L 2. = E[ 1 r 2 ] + 1% Both theories together, give:

Jacoby, Stangeland and Wajeeh, 200024 Shapes of the Term Structure under the Liquidity Premium Theory & the Augmented Expectations Theory The shape of the term structure depends on the magnitude of the premium: l when there exist constant expectations: l when there exist decreasing expectations: % t f `` f ` t % f E[ t r t+1 ] L

25 IV. Market Segmentation Theory l Different segments of investors choose to invest in assets with different investment horizons: e.g. mutual funds that commit to invest in long term bonds. l Thus, demand and supply in each segment could set different rates: t % Short Term Bonds Medium Term Bonds Long Term Bonds

Jacoby, Stangeland and Wajeeh, 200026 Common Stocks lWhat are stocks - l legal representation of of ownership in a corporation (equity) l a stock holder is entitled to receive profit distributions of the corporation (dividends) lDividends: l cash payments made by the corporation to stockholders l since stocks have no expiration date, we assume that dividends will be paid forever lValuation l the value of stocks at any point in time equals the present value of all future dividends

Jacoby, Stangeland and Wajeeh, 200027 Common Stock Valuation uThe value of a stock = PV of all expected future cash flows uThus, the information needed to value common stocks: u Common Stock Dividends (D t ) u Discount rate (r) uPV 0 = D 1 /(1 + r) 1 + D 2 /(1 + r) 2 + D 3 /(1 + r) 3 +... forever.. uWe have to estimate future dividends

Jacoby, Stangeland and Wajeeh, 200028 Case 1: Zero Growth uAssume that dividends will remain at the same level forever, i.e. D 1 = D 2 =…= D t = D uSince future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity: P t = D t+1 / r

Jacoby, Stangeland and Wajeeh, 200029 Example - Valuation of Common Stocks with Zero Growth Q.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 expected price of a share of ABC stock? A.The stock price is given by the the present value of the perpetual stream of dividends: P 0 = D 1 / r = = 01234 \$\$\$\$

Jacoby, Stangeland and Wajeeh, 200030 Case 2: Constant Growth uAssume that dividends will grow at a constant rate, g, forever, i. e., D 1 = D 0 x (1+g) D 2 = D 1 x (1+g) = D 0 x (1+g) 2 ® D t = D 0 x (1+g) t ® uSince future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity: P t = D t+1 / (r - g)

Jacoby, Stangeland and Wajeeh, 200031 Examples - Valuation of Common Stocks with Constant Growth Q1.XYZ Corp. has a common stock that paid its annual dividend this morning. It is expected to pay a \$3.60 dividend one year from now, and following dividends are expected to grow at a rate of 4% per year into the foreseeable future (forever)in perpetuity. If stocks of similar risk earn 16% effective annual return, what is the price of a share of XYZ stock? A1.The stock price is given by the the present value of the perpetual stream of growing dividends: P 0 = D 1 / (r-g) = = 01234 \$3.60\$3.60 % forever...

32 Q2.In the above example, assume that XYZ’s common stock that paid its quarterly dividend two months ago. It is expected to pay a \$0.90 dividend in one month, and following quarterly dividends are expected to grow at a rate of 1% per quarter into the foreseeable future. Recall that the effective annual required rate of return on XYZ stock is 16%. What is the price of a share of XYZ stock now? A2.Time line of the quarterly dividends: We first need to calculate EPR 1/4 and EPR 1/12 : Using: EPR n = (1+EAR) n - 1, we get: EPR 1/4 = 3.780199% and EPR 1/12 = 1.244514% The stock price in one month (after D 1 month is paid): P 1 month = D 4 months / (EPR 1/4 - g) = (0.90 % 1.01)/(0.03780199 - 0.01) = \$32.69550129 The stock price today: P 0 = (D 1 month + P 1 month ) / (1+EPR 1/12 ) = (0.90+ 32.69550129) / 1.01244514 = \$33.18 01 month4 months7 months10 months \$0.90\$0.90 % forever...

33 Q3.Manitoba Network Operators (MNO) is expected to pay a dividend next year of \$8.06 per share. Both sales and profits for Pale Hose are expected to grow at a rate of 2% per year indefinitely. Its dividend is expected to grow by the same amount. If an investor is currently willing to pay \$62.00 per one MNO share, what is her required return for this investment? A3.We have: P 0 =\$62.00, D 1 =\$8.06, and g=0.02. We are looking for r. The stock price is given by: P 0 = D 1 /(r-g) = Rearranging, we get: r = = In general: r = (D 1 /P 0 ) + g

34 Q4.Vandalay Industries Corp. (VIC) is expected to pay a dividend next year of \$4.32 per share. Its current stock price is \$36. If the required return for this stock is 15%, what is the constant dividend growth rate expected for VIC’s stock starting from the second year forever? A4.We have: P 0 =\$36.00, D 1 =\$4.32, and r=0.15. We are looking for g. The stock price is given by: P 0 = D 1 /(r-g) = Rearranging, we get: g = = In general: g = r - (D 1 /P 0 )

Jacoby, Stangeland and Wajeeh, 200035 Q5.MT&T Inc. has a common stock that paid its annual dividend this morning. You expect future annual dividends to grow at a rate of 2% per year into the foreseeable future (forever). The required return for this stock is 20%, and its current price is \$25.50. What is the dividend that was paid this morning? A5.We have: P 0 =\$25.50, r=0.20, and g=0.02. We are looking for D 0. The stock price is given by: P 0 = D 1 /(r-g) 25.5 = D 1 /(0.20-0.02) Rearranging, we get: D 1 = 25.5(0.20-0.02) = \$4.59 We are looking for D 0. Since D 1 =D 0 (1+g), D 0 is given by: D 0 = D 1 /(1+g) = = \$

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

37 Examples - Differential Growth Q1. Whizzkids Inc. is experiencing a period of rapid growth. Earnings and dividends are expected to grow at a rate of 8 percent during the next three years, and then at a constant rate of 4% thereafter. Whizzkids’ last dividend, which has just been paid, was \$2 per share. If the required rate of return on the stock is 12 percent, what is the price of the stock today? A1.It is given that: r = 12%, D 0 = \$2, g 1 = g 2 = g 3 = 8%, and g 4 = g * = 4% (forever) We calculate: D 1 =\$2 % =\$, D 2 ==\$, D 3 = =\$ With g 4 =g * =4%, we have: D 4 = =\$ Since constant growth rate applies to D 4, we use Case 2 (constant growth) to compute P 3 : P 3 = = \$

Jacoby, Stangeland and Wajeeh, 200038 Expected future cash flows of this stock: 0 1 2 3 |----------|---------|---------|(r = 12%) D 1 D 2 D 3 + P 3 2.16 2.33 2.52 + 32.75 The current (time 0) value of the stock: P 0 = D 1 /(1+r) + D 2 /(1+r) 2 + (D 3 +P 3 )/(1+r) 3 = + + = \$

39 Q2. An investor has just paid \$141.75 for the purchase of one share of UMB Corp. stock. UMB just paid a \$9 dividend per share. Annual dividends paid at the end of the first, second and third years will grow at a rate of 10% per annum, and then grow at a constant annual rate of g * forever. Given the risk inherent in UMB Corp., the investor requires an effective annual rate of 10% on his/her investment. What is the value of g * ? A2.We calculate: D 1 =\$9 % =\$, D 2 = =\$, D 3 = =\$ With g 4 =g *, we have: D 4 = UMB’s current stock price is given by: P 0 = D 1 /(1+r) + D 2 /(1+r) 2 + (D 3 +P 3 )/(1+r) 3 Where: P 3 = D 4 /(r-g * )

Jacoby, Stangeland and Wajeeh, 200040 With the above data: 141.75 = 9.9/1.1 + 10.89/(11) 2 + 11.979/(11) 3 + P 3 /(1.1) 3 Thus, the expected stock price in three years is P 3 = 152.73225 Since, this price is given by: P 3 = D 4 /(r-g * ) = [D 3 (1+g * )]/(r-g * ) We have 152.73225 = Rearranging, we get: g * = %

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