5-1 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 McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-2 5.1 Definition and Example of a Bond • A bond is a legally binding agreement between a borrower and a lender: – 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) McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-3 5.1 Definition and Example of a Bond • Consider a U.S government bond listed as 3/8 of December 2009 – 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 3/8 the payment is $31.875 – On January 1, 2002 the size and timing of cash flows are: $31.875 $31.875 $31.875 $1,031.875 / 30 / 09 12 / 31 / 09 / / 02 / 30 / 02 McGraw-Hill/Irwin 12 / 31 / 02 Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-4 5.2 How to Value Bonds • Identify the size and timing of cash flows • Discount at the correct discount rate – If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-5 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) $0 $0 $0 $F T −1 T Present value of a pure discount bond at time 0: F PV = T (1 + r ) McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-6 Pure Discount Bonds: Example Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6% $0 $0 $0 $1,000 0$ 0$0,1$ 01 22930 29 30 F $1,000 PV = = = $174.11 T 30 (1 + r ) (1.06) McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-7 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 $C $C $C $C + $ F T −1 T Value of a Level-coupon bond = PV of coupon payment annuity + PV of face value C F PV = 1 − + T r (1 + r ) (1 + r )T McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-8 Level-Coupon Bonds: Example Find the present value (as of January 1, 2002), of a 6-3/8 coupon T-bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent – On January 1, 2002 the size and timing of cash flows are: $31.875 $31.875 $31.875 $1,031.875 / 30 / 09 12 / 31 / 09 / / 02 / 30 / 02 12 / 31 / 02 $1,000 $31.875 PV = 1− + = $1,049.30 16 16 05 (1.025) (1.025) McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-9 Bond Rates and Yields Suppose a $1,000 face value bond currently sells for $932.90, pays an annual coupon of $70, and matures in 10 years The coupon rate is the annual dollar coupon expressed as a percentage of the face value: Coupon rate = $ _ /$ _ = 7.0% The current yield is the annual coupon divided by the price: Current yield = $ _ / _ = 7.5% McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-10 Bond Rates and Yields The yield to maturity is the rate that makes the price of the bond just equal to the present value of its future cash flows How to find yield to maturity? – – – Trial and error Approximation formula Financial calculator McGraw-Hill/Irwin YTM = 8% Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-16 Percentage changes in bond prices Bond prices and market rates 7% 9% 11% Bond J % Chg $786.81 (+16.60%) $674.80 $581.74 (-13.79%) Bond K %Chg $1,213.19 (+13.9%) $1,065.04 $940.25 (-11.72%) The results above demonstrate that, all else equal, the price of the lower-coupon bond changes more (in percentage terms) than the price of the higher-coupon bond when market rates change McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-17 5.4 The Present Value of Common Stocks • Dividends versus Capital Gains • Valuation of Different Types of Stocks – Zero Growth – Constant Growth – Differential Growth McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-18 Case 1: Zero Growth • Assume that dividends will remain at the same level forever Div1 = Div = Div = • Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity: Div Div1 Div P0 = + + + (1 + r ) (1 + r ) (1 + r ) Div P0 = r McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-19 Case 2: Constant Growth Assume that dividends will grow at a constant rate, g, forever i.e Div1 = Div (1 + g ) Div = Div1 (1 + g ) = Div (1 + g ) Div = Div (1 + g ) = Div (1 + g ) Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity: Div1 P0 = r−g McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-20 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 stock price when the stock becomes a Constant Growth Stock (case 2) – Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-21 Case 3: Differential Growth • Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter Div1 = Div (1 + g1 ) Div = Div1 (1 + g1 ) = Div (1 + g1 ) Div N = Div N −1 (1 + g1 ) = Div (1 + g1 ) N Div N +1 = Div N (1 + g ) = Div (1 + g1 ) N (1 + g ) McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-22 Case 3: Differential Growth • Dividends will grow at rate g1 for N years and grow at rate g2 thereafter Div (1 + g1 ) Div (1 + g1 ) … Div (1 + g1 ) N Div N (1 + g ) = Div (1 + g1 ) N (1 + g ) … … N McGraw-Hill/Irwin N+1 Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-23 Case 3: Differential Growth We can value this as the sum of: an N-year annuity growing at rate g1 C (1 + g1 )T PA = 1 − T r − g1 (1 + r ) plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1 Div N +1 r − g2 PB = N (1 + r ) McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-24 Case 3: Differential Growth To value a Differential Growth Stock, we can use Div N +1 T C (1 + g1 ) r − g P= + 1 − T N r − g1 (1 + r ) (1 + r ) • Or we can cash flow it out McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-25 A Differential Growth Example A common stock just paid a dividend of $2 The dividend is expected to grow at 8% for years, then it will grow at 4% in perpetuity What is the stock worth? McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-26 5.9 Stock Market Reporting 52 WEEKS YLD VOL NET HI LO STOCKSYM DIV % PE 100s HI LO CLOSE CHG 52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75 Gap has been as high as $52.75 in the last year Gap pays a dividend of cents/share Gap ended trading at $19.25, down $1.75 from yesterday’s close Given the current price, the dividend yield is ½ % Gap has been as low as $19.06 in the last year McGraw-Hill/Irwin Given the current price, the PE ratio is 15 times earnings 6,517,200 shares traded hands in the last day’s trading Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-27 5.9 Stock Market Reporting 52 WEEKS YLD VOL NET HI LO STOCKSYM DIV % PE 100s HI LO CLOSE CHG 52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75 Gap Incorporated is having a tough year, trading near their 52week low Imagine how you would feel if within the past year you had paid $52.75 for a share of Gap and now had a share worth $19.25! That 9-cent dividend wouldn’t go very far in making amends Yesterday, Gap had another rough day in a rough year Gap “opened the day down” beginning trading at $20.50, which was down from the previous close of $21.00 = $19.25 + $1.75 Looks like cargo pants aren’t the only things on sale at Gap McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-28 5.10 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 F PV = T (1 + r ) C PV = r McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-29 5.10 Summary and Conclusions (continued) The value of a coupon bond is the sum of the PV of the annuity of coupon payments plus the PV of the par value at maturity C F PV = 1 − + T r (1 + r ) (1 + r )T The yield to maturity (YTM) of a bond is that single rate that discounts the payments on the bond to the purchase price McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved 5-30 5.10 Summary and Conclusions (continued) A stock can be valued by discounting its dividends There are three cases: Div P0 = r Div1 Constant growth in dividends P0 = r−g Zero growth in dividends Differential growth in dividends Div N +1 T C (1 + g1 ) r − g P= + 1 − T N r − g1 (1 + r ) (1 + r ) McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved ... % PE 100s HI LO CLOSE CHG 52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75 Gap Incorporated is having a tough year, trading near their 52week low Imagine how you would feel if within