Bond Share Valuation

 

Bond Definitions • Bond • Par value (face value) • Coupon rate • Coupon payment • Maturity date • Yield or Yield to maturity

 

Present Value of Cash Flows as Rates Change • Bond Value = PV of coupons + PV of par • Bond Value = PV annuity + PV of lump sum • Rememb Remember, er, as inte interest rest rates rates incr increas easee the the PV’s PV’s decrease • So, as interest ratesversa increase, bond prices decrease and vice

 

Valuing a Discount Bond with Annual Coupons • Consider a bond with a The coupon ofis10% and coupons paid annually. par rate value $1000 and the bond has 5 years to maturity. The yield to maturity is 11%. What is the value of the bond?  – Using  –  Using the formula: • B = PV of annuity + PV of lump sum • B = 100[1 –  100[1 –  1/(1.11)  1/(1.11)5] / .11 + 1000 / (1.11) 5  • B = 369.59 + 593.45 = 963.04

 – Using  –  Using the calculator:

•  N = 5; I/Y = 11; PMT = 100; FV = 1000 • CPT PV = -963.04

 

Valuing a Premium Bond with Annual

Coupons • Suppose you are looking at a bond that has a 10% annual coupon and a face value of $1000. There are 20 years to maturity and the yield to maturity is 8%. What is the price of this bond?  – Using  –  Using the formula: • B = PV of annuity + PV of lump sum • B = 100[1 –  100[1 –  1/(1.08)  1/(1.08)20] / .08 + 1000 / (1.08)20  • B = 981.81 + 214.55 = 1196.36

 – Using  –  Using the calculator: •  N = 20; I/Y = 8; PMT = 100; FV = 1000 • CPT PV = -1196.36

 

Graphical Relationship Between Price and Yield-to-maturity 1500 1400 1300 1200 1100 1000 900 800 700 600 0%

2%

4%

6%

8%

10%

12%

14%

 

Bond Prices: Relationship Between Coupon and Yield • If YTM = coupon rate, then par value = bond  price • If YTM > coupon rate, then par value > bond  price

 – Why?  –  Why?  – Selling  –  Selling at a discount, called a discount bond

• If YTM < coupon rate, then par value < bond  price  – Why?  –  Why?  – Selling  –  Selling at a premium, called a premium bond

 

The Bond-Pricing Equation

1  1 t  Bond Value  C  (1  r) r   

   F t  (1  r) 

 

Example • Find present values based on the payment  period  – How  –  How many coupon payments are there?  –  What is the semiannual coupon payment?  – What  – What  –  What is the semiannual yield?  – B  –  B = 70[1 –  70[1 –  1/(1.08)  1/(1.08)14] / .08 + 1000 / (1.08)14 = 917.56  – Or  –  Or PMT = 70; N = 14; I/Y = 8; FV = 1000; CPT PV = -917.56

 

Interest Rate Risk • Price Risk

 – Change  –  Change in price due to changes in interest rates  – Long-term  –  Long-term bonds have more price risk than shortterm bonds

• Reinvestment Rate Risk  – Uncertainty  –  Uncertainty concerning rates at which cash flows can be reinvested  – Short-term  –  Short-term bonds have more reinvestment rate risk than long-term bonds

 

Figure

 

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 similar to the process for finding r withand an is 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)

 

YTM with Annual Coupons • Consider a bond with a 10% annual coupon rate, 15 years to maturity and a par value of $1000. The current price is $928.09.  –  Will the yield be more or less than 10%?  – Will  –  N  N = 15; PV = -928.09; FV = 1000; PMT = 100  – CPT  –  CPT I/Y = 11%

 

YTM with Semiannual Coupons • Suppose a bond with a 10% coupon rate and semiannual coupons, has a face value of $1000, 20 years to maturity and is selling for $1197.93.

 – Is  –  Is the YTM more or less than 10%?  – What  –  What is the semiannual coupon payment?  –  How many periods are there?  – How  –  N  N = 40; PV = -1197.93; PMT PMT = 50; FV = 1000; CPT I/Y = 4% (Is this the YTM?)

 – YTM  –  YTM = 4%*2 = 8%

 

Table

 

Bond Pricing Theorems • Bonds of similar risk (and maturity) will be  priced to yield about the same return, regardless of the coupon rate • If you know the price of one bond, you can estimate its YTM and use that to find the price of the second bond • This is a useful concept that can be transferred to valuing assets other than bonds

 

Differences Between Debt and Equity • Debt

 –  Not an ownership interest  –  Not  –  Creditors do not have voting rights  –  Interest is considered a cost of doing business and is tax deductible  –  Creditors have legal recourse if interest or  principal missed payments are  –  Excess debt can lead to financial distress and  bankruptcy

• Equity

 –  Ownership interest  –  Common stockholders vote for the board of directors and other issues  –  Dividends are not considered a cost of doing  business and are not tax deductible  –  Dividends of the firm are andnot a liability stockholders have no legal recourse if dividends are not paid  –  An all equity firm can not go bankrupt

 

The Bond Indenture • Contract between the company and the  bondholders and includes  – The  –  The basic terms of the bonds  –  The total amount of bonds issued  – The  – A  –  A description of property used as security, if applicable  –  Sinking fund provisions  – Sinking  – Call  –  Call provisions  – Details  –  Details of protective covenants

 

Bond Classifications • Registered vs. Bearer Forms • Security  – Collateral  –  Collateral –   –  secured  secured by financial securities  – Mortgage  –  Mortgage –   –  secured  secured by real property, normally land or buildings  – Debentures  –  Debentures –   –  unsecured  unsecured  –  Notes –   Notes –  unsecured  unsecured debt with original maturity less than 10 years

• Seniority

 

Bond Characteristics and Required Returns • The coupon rate depends on the risk characteristics of the bond when issued • Which bonds will have the higher coupon, all else equal?  – Secured  –  Secured debt versus a debenture  – Subordinated  –  Subordinated debenture versus senior debt  –  A bond with a sinking fund versus one without  – A  – A  –  A callable bond versus a non-callable bond

 

Bond Ratings  – Investment Quality • High Grade

 – Moody’s Aaa and S&P AAA –  capacity  capacity to pay is extremely strong  – Moody’s Aa and S&P AA –  capacity  capacity to pay is very strong

• Medium Grade  – Moody’s A and S&P A –  capacity  capacity to pay is strong, but more susceptible to changes in circumstances circumstances  – Moody’s Baa and S&P B BBB BB –  capacity  capacity to pay is adequate, adverse conditions will have more impact on the firm’s ability to pay  pay 

 

Bond Ratings - Speculative • Low Grade

 – Moody’s Ba, B, Caa and Ca  Ca   – S&P  –  S&P BB, B, CCC, CC  – Considered  –  Considered speculative with respect to capacity to pay. The “B” ratings are the lowest l owest degree of speculation.

• Very Low Grade  – Moody’s C and S&P C –  income  income bonds with no interest being paid  – Moody’s D and S&P D –  in  in default with  principal and interest in arrears

 

Government Bonds • Treasury Securities

 –  Federal government debt  –  T-bills T-bills –   –  pure  pure discount bonds with original maturity of one year or less  –  T-notes T-notes –   –  coupon  coupon debt with original maturity between one and ten years  –  T-bonds coupon debt with original or iginal maturity greater  than  than

ten years

• Municipal Securities  –  Debt of state and local governments  –  Varying degrees of default risk, rated similar to corporate debt  –  Interest received is tax-exempt at the federal level

 

Example • A taxable bond has a yield of 8% and a municipal bond has a yield of 6%  – If  –  If you are in a 40% tax bracket, which bond do you prefer?

• 8%(1 - .4) = 4.8% • The after-tax return on the corporate bond is 4.8%, compared to a 6% return on the municipal

 – At  –  At what tax rate would you be indifferent between the two bonds? • 8%(1 8%(1 –   –  T)  T) = 6% • T = 25%

 

Zero-Coupon Bonds • Make no periodic interest payments (coupon rate = 0%) • The entire yield-to-maturity comes from the difference  par value between the purchase price and the • Cannot sell for more than par value •  bonds Sometimes called zeroes, or deep discount • Treasury Bills and principal only Treasury strips are good examples of zeroes

 

Floating Rate Bonds • Coupon rate floats depending on some index value • Examples Examples –   –  adjustable  adjustable rate mortgages and inflationlinked Treasuries • There is less price risk with floating rate bonds  –  The coupon floats, so it is less likely to differ substantially from the yield-to-maturity

• Co Coup upon onss ma may yh hav avee a “c “col olla lar” r” –  the  the rate cannot go above a specified “ceiling” or below a specified “floor”   “floor”

 

Other Bond Types • Disaster bonds • Income bonds • Convertible bonds • Put bond • There are many other types of provisions that can be added to a bond and many bonds have several provisions –  provisions –  it  it is important to recognize how these provisions affect required returns

 

Bond Markets • Primarily over-the-counter transactions with dealers connected electronically • Extremely large number of bond issues, but generally low daily volume in single issues • Makes getting up-to-date prices difficult,  particularly on small company o orr municipal issues • Treasury securities are an exception

 

Inflation and Interest Rates • Real rate of interest –  interest –  change  change in purchasing  power •  Nominal rate of interest interest –   –  quoted  quoted rate of interest, change in purchasing power and inflation • The ex ante nominal rate of interest includes our desired real rate of return plus an adjustment for expected inflation

 

The Fisher Effect • The Fisher Effect defines the relationship  between real rates, nominal rates and inflation • (1 + R) = (1 + r)(1 + h), where  –  R = nominal rate  – R  – rr = real rate  –   – h  –  h = expected inflation rate

• Approximation  – R  –  R=r+h

 

Example • If we require a 10% real return and we expect inflation to be 8%, what is the nominal rate? • R = (1.1)(1.08) –  (1.1)(1.08) –  1  1 = .188 = 18.8% • Approximation: Approximation: R = 10% + 8% = 18% • Because the real return and expected inflation are relatively high, there is significant difference between the actual Fisher Effect and the approximation.

 

Term Structure of Interest Rates • Term structure is the relationship between time to maturity and yields, all else equal • It is important to recognize that we pull out the effect of default risk, different coupons, etc. • Yield curve –  curve –  graphical  graphical representation of the term structure  –  Normal –   Normal –  upward-sloping,  upward-sloping, long-term yields are higher than short-term yields  – Inverted  –  Inverted –   –  downward-sloping,  downward-sloping, long-term yields are lower than short-term yields

 

Figure  – Upward-Sloping Yield Curve

 

Figure – Downward-Sloping Yield Curve

 

Factors Affecting Required Return • Default risk premium –  premium –  remember  remember bond ratings • Taxability premium –  premium –  remember  remember municipal versus taxable • Liquidity premium –  premium –  bonds  bonds that have more frequent trading will generally have lower required returns • Anything else that affects the risk of the cash flows to the bondholders, will affect the required returns

 

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 are bond ratings and why are they important? • How does inflation affect interest rates? • What is the term structure of interest rates? • What factors determine the required return on  bonds?

 

Cash Flows for Stockholders • If you buy a share of stock, you can receive cash in two ways  – The  –  The company pays dividends  – You  –  You sell your shares, either to another investor in the market or back to the company

• As with bonds, the price of the stock is the  present value of these expected cash flows

 

One Period Example • Suppose you are thinking of purchasing the stock of Moore Oil, Inc. and you expect it to  pay a $2 dividend in one year and you believe that you can sell the stock for $14 at that time. If you require a return of 20% on investments of this risk, what is the maximum you would would  be willing to pay?

 – Compute  –  Compute the PV of the expected cash flows  – Price  –  Price = (14 + 2) / (1.2) = $13.33  – Or  –  Or FV = 16; I/Y = 20; N = 1; CPT PV = -13.33

 

Two Period Example •  Now what if you decide decide to hold the stock stock for two years? In addition to the dividend in one year, you expect a dividend of $2.10 in and a stock price of $14.70 at the end of year 2. Now how much would you be willing to pay?  – PV  –  PV = 2 / (1.2) + (2.10 + 14.70) / (1.2) 2 = 13.33  – Or  –  Or CF0 = 0; C01 = 2; F01 = 1; C02 = 16.80; F02 = 1; NPV; I = 20; CPT NPV = 13.33

 

Three Period Example • Finally, what if you decide to hold the stock for three periods? In addition to the dividends at the end of years 1 and 2, you expect to receive a dividend of $2.205 at the end of year 3 and a stock price of $15.435. Now how much would you be willing to pay?  – PV  –  PV = 2 / 1.2 + 2.10 / (1.2)2 + (2.205 + 15.435) / (1.2)3 = 13.33  – Or  –  Or CF0 = 0; C01 = 2; F01 = 1; C02 = 2.10; F02 = 1; C03 = 17.64; F03 = 1; NPV; I = 20; CPT  NPV = 13.33

 

Developing The Model • You could continue to push back when you would sell the stock • You would find that the price of the stock is really just the present value of all expected  future dividends  • So, how can we estimate all future dividend  payments?

 

Estimating Dividends: Special Cases • Constant dividend  –  The firm will pay a constant dividend forever  –  This is like preferred preferred stock  –  The price is computed using the perpetuity formula

• Constant dividend growth

 –  The firm will increase the dividend by a constant  percent every every period

• Supernormal growth  –  Dividend growth is not consistent initially, but settles down to constant growth eventually

 

Zero Growth • If dividends are expected at regular intervals forever, then this is like preferred stock and is valued as a perpetuity • P0 = D / R • Suppose stock is expected to pay a $0.50 dividend every quarter and the required return is 10% with quarterly compounding. What is the price?  – P  –  P0 = .50 / (.1 / 4) = $20

 

Dividend Growth Model • Dividends are expected to grow at a constant c onstant  percent per period.  – P  –  P0 = D1 /(1+R) + D2 /(1+R)2 + D3 /(1+R)3 + …  …   – P  –  P0 = D0(1+g)/(1+R) + D0(1+g)2/(1+R)2 + D0(1+g)3/(1+R)3 + …  … 

• With a little algebra, this reduces to:

P0 

D 0 (1  g)

 

R - g



D1 R - g

 

DGM  – Example 1 • Suppose Big D, Inc. just paid a dividend of $.50. It is expected to increase its dividend by 2% per year. If the market requires a return of 15% on assets of this risk, how much should the stock be selling for? • P0 = .50(1+.02) / (.15 - .02) = $3.92

 

DGM  – Example 2 • Suppose TB Pirates, Inc. is expected to pay a $2 dividend in one year. If the dividend is expected to grow at 5% per year and the required return is 20%, what is the price?  – P  –  P0 = 2 / (.2 - .05) = $13.33

 – Why isn’t the $2 in the the numerator multipl multiplied ied by (1.05) in this example?

 

Stock Price Sensitivity to Dividend Growth, g 250

D1 = $2; R = 20% 200   e    i   c 150   r    P    k   c   o    t 100    S

50

0 0

0.05

0.1 Growth Rate

0.15

0.2

 

Stock Price Sensitivity to Required Return, R 250

D1 = $2; g = 5%

200   e    i   c 150   r    P    k   c   o    t 100    S

50

0 0

0.05

0.1

0.15 Growth Rate

0.2

0.25

0.3

 

Example -Gordon Growth Company - I • Gordon Growth Company is expected to pay a dividend of $4 next period and dividends are expected to grow at 6% per year. The required return is 16%. • What is the current price?  – P  –  P0 = 4 / (.16 - .06) = $40  – Remember  –  Remember that we already have the dividend expected next year, so we don’t multiply the dividend by 1+g

 

Example  – Gordon Growth Company - II • What is the price expected to be in year 4?  –  P4 = D4(1 + g) / (R –   – P (R –  g)  g) = D5 / (R –  (R –  g)  g)  – P  –  P4 = 4(1+.06)4 / (.16 - .06) = 50.50

• What is the implied return given the change in  price during the the four year period?  – 50.50  –  50.50 = 40(1+return)4; return = 6%  – PV  –  PV = -40; FV = 50.50; N = 4; CPT I/Y = 6%

• The price grows at the same rate as the dividends

 

Nonconstant Growth Problem Statement • Suppose a firm is expected to increase dividends by 20% in one year and by 15% in two years. After that dividends will increase at a rate of 5% per year indefinitely. If the last dividend was $1 and the required return is 20%, what is the price of the stock? • Remember that we have to find the PV of all   expected future dividends.

 

Nonconstant Growth  – Example Solution • Compute the dividends until growth levels off  –  D1 = 1(1.2) = $1.20  – D  – D  –  D2 = 1.20(1.15) = $1.38

 – D  –  D3 = 1.38(1.05) = $1.449

• Find the expected future price  – P  –  P2 = D3 / (R –  (R –  g)  g) = 1.449 / (.2 - .05) = 9.66

• Find the present value of the expected future cash flows  – P  –  P0 = 1.20 / (1.2) + (1.38 + 9.66) / (1.2) 2 = 8.67

 

Quick Quiz  – Part I • What is the value of a stock that is expected to  pay a constant dividend of $2 per year if the the required return is 15%? • What if the company starts increasing dividends by 3% per year, beginning with the next dividend? The required return stays at 15%.

 

Using the DGM to Find R • Start with the DGM:

P0



D 0 (1  g) R - g



D1 R - g

rearrangeand solve for R  D 0 (1  g) R  

P0

D1 

g



  P0



g

 

Finding the Required Return - Example • Suppos Supposee a firm’s firm’s stock stock is sell selling ing for $10.50 $10.50.. They just paid a $1 dividend and dividends are expected to grow at 5% per year. What is the required return?  – R  –  R = [1(1.05)/10.50] + .05 = 15% • What is the dividend yield?  – 1(1.05)  –  1(1.05) / 10.50 = 10%

• What is the capital gains yield?  – g  –  g =5%

 

Table - Summary of Stock Valuation

 

Feature of Common Stock • Voting Rights • Proxy voting • Classes of stock • Other Rights  –  Share proportionally in declared dividends  – Share  – Share  –  Share proportionally in remaining assets during liquidation  – Preemptive  –  Preemptive right –  right –  first  first shot at new stock issue to maintain proportional ownership if desired

 

Dividend Characteristics • Dividends are not a liability of the firm until a dividend has been declared by the Board • Consequently, a firm cannot go bankrupt for not declaring dividends • Dividends and Taxes  –  Dividend payments are not considered a business expense, therefore, therefore, they are not tax deductible  –  Dividends received by individuals are taxed as ordinary income  –  Dividends received by corporations have a minimum 70% exclusion from taxable income

 

Features of Preferred Stock • Dividends  – Stated  –  Stated dividend that must be paid before dividends can be paid to common stockholders  –  Dividends are not a liability of the firm and  – Dividends  preferred dividends can be deferred deferred indefinitely  – Most  –  Most preferred dividends are cumulative cumulative –   –  any  any missed preferred dividends have to be paid before common dividends can be paid

• Preferred stock generally does not carry voting rights

 

Stock Market • Dealers vs. Brokers •  New York Stock Exchange Exchange (NYSE)  – Largest  –  Largest stock market in the world  – Members  –  Members • • • • •

Own seats on the exchange Commission brokers Specialists Floor brokers Floor traders

 – Operations  –  Operations  – Floor  –  Floor activity

 

Quick Quiz  – Part II • You observe a stock price of $18.75. You expect a dividend growth rate of 5% and the most recent dividend was $1.50. What is the required return? • What are some of the major characteristics chara cteristics of common stock? • What are some of the major characteristics chara cteristics of  preferred stock?