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Lecture 3 How to value bonds and common stocks Corporate Finance Lecturer: Quan, Qi Fall 2012 Exercises for lecture 2 1. As winner of a breakfast cereal competition, you can choose one of the following prizes: p a. $100,000 now. p b. $180,000 at the end of five years. p c. $11,400 a year forever. p d. $19,000 for each of 10 years. p e. $6,500 next year and increasing thereafter by 5 percent a year forever. If the interest rate is 12 percent, which is the most valuable prize? Solution a. PV = $100,000 b. PV = $180,000/(1+0.12)5 = $102,137 c. A perpetuity: PV = $11,400/0.12 = $95,000 (a perpetuity) d. An annuity: PV = $19,000 ´ [Annuity factor, 12%, t = 10] = $19,000 ´ 5.650 = $107,350 e. A growing perpetuity: PV = $6,500/(0.12 - 0.05) = $92,857 Prize (d) is the winner with the highest present value Question 2 You are considering the purchase of an apartment complex that will generate a net cash flow of $400,000 per year. You normally demand a 10 percent rate of return on such investments. Future cash flows are expected to grow with inflation at 4 percent per year. How much would you be willing to pay for the complex if it: p a. Will produce cash flows forever? p b. Will have to be torn down in 20 years? Assume that the site will be worth $5 million at that time net of demolition costs. (The $5 million includes 20 years’ inflation.) Now calculate the real discount rate corresponding to the 10 percent nominal rate. Redo the calculations for parts (a) and (b) using real cash flows. (Your answers should not change.) Solution to question 2 With the nominal cash flows: a. Given the inflation rate of 4%, the cash flow in 1 year will be $416,000 and: PV = $416,000/(0.10 - 0.04) = $6,933,333 b. The nominal cash flows are growing annuity for 20 years, with an additional payment of $5 million at year 20: Second, with real cash flows: The real rate (r) is (1 + 0.10) = (1 + r) ´ (1.04) Þ r = 0.0577 = 5.77% a. PV = $400,000/(0.0577) = $6,932,409 b. Now, the real cash flows are $400,000 per year for 20 years and $5 million (nominal) in 20 years. In real terms, the $5 million dollar payment is: $5,000,000/(1.04)20 = $2,281,935 Thus, the present value of the project is: Topics covered p How to value bonds p How to value common stocks What is a bond? p What is a bond n A certificate showing that a borrower owes a specified sum; the borrower agrees to make interest and principal payments on designated dates n Components of bonds: face value, coupon rate, maturity, how often the coupons are paid (annually or semiannually?) Bond terminology p Bond certificate n States the terms of the bond p Maturity date n Final repayment date p Term n The time remaining until the repayment date p Coupon n Promised interest payments Bond terminology p Face value n Notional amount used to compute the interest payments p Coupon rate n Determines the amount of each coupon payment, expressed as an APR p Coupon payment Zero-coupon bond 1) p Does not make coupon payments n Always sells at a discount (a price lower than face value), so they are also called pure discount bonds n Treasury Bills are U.S. government zero-coupon bonds with a maturity of up to one year. Zero-coupon bond 2) p Suppose that a one-year, risk-free, zero-coupon bond with a $100,000 face value has an initial price of $96,618.36. The cash flows would be: n The bond pays no “interest” and your compensation is the difference between the initial price and the face value. Zero-coupon bond 3) p Yield to maturity n The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond p Price of a Zero-Coupon bond Zero-coupon bond 4) p Yield to maturity n For the one-year zero coupon bond: p Thus, the YTM is 3.5%. Zero-coupon bond 5) p Yield to maturity n Yield to maturity of an n-Year Zero-Coupon Bond Coupon bonds p Coupon bonds n Pay face value at maturity n Pay regular coupon interest payments p Treasury notes n U.S. Treasury coupon security with original maturities of 1 –10 years p Treasury bonds n U.S. Treasury coupon security with original maturities over 10 years The cash flows of a coupon bond The U.S. Treasury has just issued a five-year,$1000 bond with a 5% coupon rate and semiannual coupons. What cash flows will you receive if you hold this bond until maturity Coupon bonds p Yield to maturity n The YTM is the single discount rate that equates the present value of the bond’s remaining cash flows to its current price. n Yield to maturity of a coupon Bond Compute the Yield to Maturity Dynamic behavior of bond prices p Discount n A bond is selling at a discount if the price is less than the face value. p Par n A bond is selling at par if the price is equal to the face value. p Premium n A bond is selling at a premium if the price is greater than the face value. Discounts and premiums Determine the discount or premium of a coupon bond The effect of time on bond prices Interest rate changes and bond prices p There is an inverse relationship between interest rates and bond prices. n As interest rates and bond yields rise, bond prices fall. n As interest rates and bond yields fall, bond prices rise p The sensitivity of a bond’s price to changes in interest rates is measured by the bond’s duration. n Bonds with high durations are highly sensitive to interest rate changes. n Bonds with low durations are less sensitive to interest rate changes. Corporate bonds p Corporate Bonds n Issued by corporations p Credit Risk n Risk of default Bond ratings p Investment Grade Bonds p Speculative Bonds n Also known as Junk Bonds or High-Yield Bonds Corporate yield curves p Default Spread n Also known as Credit Spread n The difference between the yield on corporate bonds and Treasury yields Corporate yield curves for various ratings, February 2009 Source: Reuters Yield spreads and the financial crisis Source: Bloomberg.com How to value common stocks p The discounted-cash-flow (DCF) formula for the present value of a stock is the same as that of any other asset p What do shareholders receive from the company? n A stream of dividends n PV(stock) = PV(expected future dividends) Common stocks terminology p Common stock n Ownership shares in a publicly held corporation p Secondary market n Market in which already issued securities are traded by investors p Dividend n Periodic cash distribution from the firm to the shareholders p P/E Ratio n Price per share divided by earnings per share Common stocks terminology p Book value n Net worth of the firm according to the balance sheet p Liquidation value n Net proceeds that would be realized by selling the firm’s assets and paying off its creditors How to value common stocks p What determines today’s price? n The cash payoff to owners of common stocks comes in two forms p 1) cash dividends 2) capital gains n The current price of a share is P0;the expected price at the end of a year is P1; the expected dividend per share is DIV1 Expected return= r = This expected return is often called the market capitalization rate, which is percentage yield that an investor forecasts from a specific investment over a set period of time If the P0=100, Div1=5 and P1=110, we have r=15% How to value common stocks p You can also predict today’s price like this Price=P0= p How do we decide on r? n All securities in an equivalent risk class are priced to offer the same expected return n P0=(5+100)/(1+0.15)=100 What if today’s price is higher (lower) than 100? How to value common stocks p What determines next year’s price? n P1= How to value common stocks p How far can we look if we continue working like this? p As time goes by, the dividend stream accounts for an increasing proportion of present value. If the horizon is infinite, we have Constant growth model (Gordon growth model) p Suppose the expected dividends grow at a constant rate p What does that look like? n Like a growing perpetuity n Constant-growth DCF formula P0 = p Cautions about using this formula: g must be less than r; g cannot be abnormally high forever Example Edison Inc plans to pay $2.36 per share in dividends in the coming year. If the cost of capital is 7.5% and dividends are expected to grow by 1.5% per year in the future, what is the value of its stock? Constant dividend growth p Can we use this growing perpetuity formula to derive r, the discount rate (market capitalization rate)? r= DIV1/P0+g There are two parts in the formula: Dividend yield (DIV1/P0) and the expected rate of growth in dividends (g) Dividends versus investment and growth 1) p A simple model of growth n Dividend Payout Ratio p The fraction of earnings paid as dividends each year Dividends versus investment and growth 2) p A Simple Model of Growth n Assuming the number of shares outstanding is constant, the firm can do two things to increase its dividend: p Increase its earnings (net income) p Increase its dividend payout rate n A firm can do one of two things with its earnings: p It can pay them out to investors. p It can retain and reinvest them. Dividends versus investment and growth 3) p A Simple model of growth n Plowback ratio (retention rate) p Fraction of current earnings that the firm plows back Dividends versus investment and growth 4) p A simple model of growth n If the firm keeps its retention rate constant, then the growth rate in dividends will equal the growth rate of earnings. Dividends versus investment and growth 5) p Profitable Growth n If a firm wants to increase its share price, should it cut its dividend and invest more, or should it cut investment and increase its dividend? p The answer will depend on the profitability of the firm’s investments. § Cutting the firm’s dividend to increase investment will raise the stock price if, and only if, the new investments have a positive NPV. Example-profitable growth Example-profitable growth Example-unprofitable growth Changing growth rates p Dividend-discount model with constant long-term growth Example-valuing a firm with two different growth rates Calculate the present value of growth opportunities 1) p In general, we can think of stock price as the capitalized value of average earnings under a no-growth policy, plus PVGO, the net present value of growth opportunities p Example: how to value a growth stock using this model? Calculate the present value of growth opportunities 2) p Suppose A company’s market capitalization rate, r, is 15% and it is expected to pay a dividend of $5 in the first year. Also, the constant growth rate of the dividend is 10% Calculate the present value of growth opportunities 3) Our company forecasts to pay a $8.33 dividend next year, which represents 100% of its earnings. This will provide investors with a 15% expected return. Instead, we decide to plow back 40% of the earnings at the firm’s current return on equity of 25%. What is the value of the stock before and after the plowback decision? With Growth No Growth Calculate the present value of growth opportunities 4) If the company did not plowback some earnings, the stock price would remain at $55.56. With the plowback, the price rose to $100.00. The difference between these two numbers) is called the Present Value of Growth Opportunities (PVGO). Exercise Pigeon express currently plows back 40 percent of its earnings and earns a return of 20 percent on this investment. The dividend yield on the stock is 4 percent. a. Assuming that Pigeon express can continue to plow back this proportion of earnings and earn a 20 percent return on the investment, how rapidly will earnings and dividends grow? What is the expected return on Pigeon express stock? b. Suppose that management suddenly announces that future investment opportunities have dried up. Now Pigeon express intends to pay out all its earnings. How will the stock price change? c. Suppose that management simply announces that the expected return on new investment would in the future be the same as the market capitalization rate. Now what is Pigeon express’ stock price? a. g = plowback ratio ´ ROE=0.40 ´ 0.20=8% Using r = dividend yield + growth rate = 0.04 + 0.08 = 0.12 = 12.0% b. Since the dividend yield = 4%, we have DIV1/P0 = 0.04 and DIV1=0.04*P0 A plowback ratio of 0.4 implies a payout ratio of 0.6, and we have DIV1/EPS1 = 0.6 DIV1 = 0.6 ´ EPS1 Equating these two expressions for DIV1 gives a relationship between price and earnings per share: 0.04 ´ P0 = 0.6 ´ EPS1 EPS1/P0 = 0.066 Also, we know that: With (EPS1/P0) = 0.066 and r = 0.12, the ratio of the present value of growth opportunities to price is 34.4 percent. Thus, if there are suddenly no future investment opportunities, the stock price will decrease by 44.4 percent.