Chap 5. Applications of time value: Bond valuation

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Chap 5. Applications of time value Bond valuation

• Bond - fixed income asset
• Contracts that promise to pay a cash flow stream
• Debt contracts specify
• Bond covenants Lay out requirements/restrictions
  that the borrower must meet
• Asset covenant governs the firms acquisition
  and disposition of assets
• Dividend covenants Limits payment of dividends
• Financing covenant Restrictions on the issuance
  of additional debt and their claims
• Bonding covenant Mechanism for enforcing
  covenants - auditing, trustee for bondholders

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• Options Allows borrowers and lenders to
  terminate the contract under certain conditions
• Callability Retire bonds prior to maturity by
  paying a prespecified price
• Convertibility Allows lender to convert the bond
  to another security, typically stock. Conversion
  price or the number of shares that the bondholder
  can get in return are specified
• Exchangeability Allows lenders to convert from
  one type of bond (fixed rate) to another
  (variable rate)
• Putability Allows the bondholder to sell the
  bond back to the firm

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Bond ratings

• Ranking of quality for a debt issue
• Agencies Moodys, SP, Duff Phelps and Fitch
• Agencies review the quality of the issuer and
  continue to monitor the quality
• Debt can be up/downgraded
• SP Ratings
• AAA through BBB- Gilt edge to lower medium
• BB through B- Low grade to speculative
• CCC through C Poor quality to default
• Spread between AAA and BBB- bonds averaged about
  70 basis points in 1993

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Ratings versus yields

• DATE "AAA" "BAA Difference
• 12/28/94 8.43 9.08 0.65
• 12/27/95 6.76 7.43 1.27
• 12/25/96 7.22 7.91 0.69
• 12/31/97 6.7 6.91 0.21
• 2/25/98 6.71 7.28 0.57

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Valuing bonds

• Simplest type of bond to value is a zero coupon
  bond
• Zero coupon bonds
• Only one payment Face value of the bond
• Paid at maturity
• Ex treasury bills
• Price PV(Future cash flows)

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Example pricing a zero-coupon bond

• Pays 100,000 in exactly two years.
• The interest rate is 5.6
• Price 100,000/(1.056)2 (face value)/(1 r)n
  
• What would price be if the interest rate on
  similar bonds was 10 or 6?
• Inverse relation between price and interest rate

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Coupon bonds

• Pay periodic payments called coupons
• Also a balloon payment equal to the face value of
  the bond
• Quoted with a percentage equal to the yearly
  coupon
• For example, a 10 coupon means that 10 of the
  face value is paid each year as a coupon. 
• Coupons are paid semiannually, which means that
  the 10 coupon bond pays 5 of the face value
  every six months. 

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Coupon bonds

• Ex (pg 100) WSJ - November 1998 - Default free
  government bond- 13s of November 2002 I.e. coupon
  is 13. The face value is 1,000 so the annual
  coupon payment is 130 0.13 x 1000. Interest
  is paid each May and November so the semi-annual
  coupon is 65 130/2. The face value will be
  paid out in November 2002 I.e. 4 years from now.
  If the market interest rate is 10, the present
  value of the bond is
• Price of bond PV(Annuity) PV(Face)
• Since coupons are semi-annual, there is
  semi-annual compounding (discounting) at the rate
  of 10/2 5
• Hence PV(Annuity) 65 x 6.463 420.09
• PV(Face) 1,000 x 0.677 677
• Hence the price of the bond 1,097.09

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• What if the interest rate on similar bonds is
• 8
• 4

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• 5.7-Effect of time on bond pricing
• Two bonds A and B coupon rates 10, par
  1,000
• A has 20 years to maturity and B has 10
• If the relevant market interest rate is 10, what
  are the prices of A and B?
• If the market rate rises to 12 what happens?

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Conventions regarding bond price quotations

• When calculating price, it is important to
  discount future cash flows with respect to the
  date on which the bond trade will be settled
• Settlement date - varies between one and three
  trading days after the trade is executed
• Actual price paid is its quoted price plus
  accrued interest
• Accrued interest - is the interest owed on the
  bond when the bond is purchased between coupon
  payment dates
• Quoted price is called the flat price
• Actual price paid is the full price flat
  price accrued interest

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• Ex Feb.97 WSJ T bonds 8 1/8s of May 21 are
  priced at 117.04 (actual price)
• Coupon is 8 1/8 of the face value
• Bond matures in 2021
• Since bond prices are quoted in 32nds the price
  is 117.125
• This bond pays semi-annual coupons on May 15 and
  November 15 each year
• Suppose you purchased the bond such that the
  settlement date is March 13,1998. How much do
  you pay for the bond?
• You pay the quoted price plus accrued interest
• Accrued interest interest accrued for the time
  since the last coupon paid on Nov.15,97

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• Accrued interest
• (Coupon amount) x ( of days since last coupon /
  of days in current coupon period)
• Number of days since last coupon (Nov.15,97)
  118 days
• Number of days in current coupon period 181
  days
• Coupon amount 1/2 x (0.081250 x 100) 8.125/2
  4.0625
• Hence accrued interest 4.0625 x 118/181
  2.6485
• Hence the full price 117.125 2.6485
  119.7735

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Yield to Maturity (YTM)

• YTM is the single discount rate that equates the
  discounted cash flows to the price
• It is the geometric average annual rate of return
  earned each year if the bond is held to maturity
• Ex
• 20 year zero-coupon bond
• Face value of 1,000,000
• Current price 195,616.39.
• Solve for 195,616.39 1,000,000 / (1r)20
• r 8.5

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• Treasury Note And Bond Quotes
• (From Cantor Fitzgerald Securities)
• As of Midafternoon Thursday, February 5, 1998
• Issue Bid Ask Chg Yld
• 7 7/8
• 04/15/1998-N 100.13 100.15 -1 5.21
• 5 1/8
• 04/30/1998-N 99.29 99.31 -1 5.19
• 5 5/8
• 02/28/2001-N 100.18 100.20 -3 5.40
• 6 3/8
• 03/31/2001-N 102.21 102.23 -4 5.42

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Term structure of interest rates

• Yields to maturity could differ at different
  maturities
• Plot of YTM against the time to maturity is the
  yield curve that reveals the term structure of
  interest rates
• Yield curve could be rising (previous numbers) or
  falling or both
• Can use information in yield curves to determine
  the direction of future interest rates

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Term structure of interest rates

• Suppose that we have two bonds
• A one year zero-coupon bond with 100 face value
  and price of 94.34.
• A two year zero coupon bond with 100 face value,
  whose current price is 86.55.
• What does the yield curve look like?
• The 1 year yield to maturity using the one year
  bond
• Ytm1 (100/94.34) - 1 6.00
• The two year yield to maturity using the two year
  bond
• Price 86.55 100/(1ytm2)2 ? ytm2 7.49
• Upward sloping yield curve

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Term structure of interest rates

• YTM on the 1 year bond one year interest rate
• YTM on the 2 year bond average of the annual
  interest rate earned over two years
• (1ytm2)2 (1 ytm1) x (1 f2 )
• (1.0749)2 1.06 x (1 f2 ) ? f2 9.
• f is the forward rate or the expected interest
  rate
• Hence according to these numbers, the current
  interest rate is around 6
• Furthermore, assets are being priced to imply
  that the market expects the interest rate next
  year to be 9

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Example - Using the term structure to estimate
forward rates

• Appendix 5A - page 131 - Example Assume the
  following set of interest rates (Spot rates) at
  different years to maturity (Year)
• Year Spot rates
• 1 5
• 2 6
• 3 7
• 4 6
• What are the forward rates for each of the 4
  years?

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• The forward rate one year from now or the
  expected spot rate one year from now
• f2 1.062 / 1.05 - 1 7.01
• Similarly the one year rates two and three years
  from now
• f3 1.073 / 1.062 - 1 9.03
• f4 1.064 / 1.073 - 1 3.06

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Stock valuation - What determines a stock's value?

• Book value
• Records what a company has paid for its assets
  with a simple deduction for depreciation and no
  allowance for inflation.
• Liquidation value
• What the company could realize by selling its
  assets and repaying debts.
• It does not measure the value of a going
  concern. It ignores intangible assets, (patents,
  brand name)
• It ignores that firms may be able to make
  profitable investments in the future because of
  their position in the industry.

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What determines a stock's value?

• Market value
• The amount that investors are willing to pay for
  the firm.
• It depends on the earning power of existing
  assets and the profitability of future
  investments
• Use the present value model to price stocks
• Price PV(Cash flows from owning stock)
• Cash flows
• Dividends
• Capital gains

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What determines a stock's value?

• P0 div1 div2 div3 div4 .
  divn Pn
• (1r) (1r)2 (1r)3 (1r)4
  (1r)n (1r)n
• Need to estimate the future dividends and capital
  gains (selling price)
• Also need the discount rate to price these cash
  flows
• What happens to the stock market as a whole when
  there is low inflation and interest rates are
  low?

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What determines a stock's value?

• Special cases
• Where we can make some simplifying assumptions
  about the growth pattern of future dividends
• Case 1. No growth
• Constant dividends div1div2.......Div
• Ex utility stocks, preferred stocks
• P0 Div
• r

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What determines a stock's value?

• Case 2. Dividends expected to grow at constant
  rate (often stated as a company goal)
• We know this wont happen exactly
• Reasonable approximation within the accuracy of
  our estimate
• Growing perpetuity
• P0 DIV1
• (r-g)
• r DIV1 g
• P0
• Market capitalization rate Dividend yield
  (DIV1/P0) expected rate of growth in dividends,
  g

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• 5.11 World Wide Inc. is expected to pay a
  per-share dividend of 3 next year. It also
  expects that this dividend will grow at a rate of
  8 in perpetuity. What price would you expect to
  see for World Wide stock if the appropriate
  discount rate is 12?
• What if the rate of growth of 8 is expected to
  last for 2 years and thereafter dividends (based
  on EPS) are expected to grow at a more modest
  rate of 2 indefinitely?

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• 5.15 Brown Inc. has just paid a 3 dividend per
  share of common stock. The stock is currently
  being sold at 40. Investors expect that Browns
  dividend will grow at a constant rate
  indefinitely. What growth rate is expected by
  investors if they require 8 return on the stock?
• What if they require 10 return on the stock?
• What if the required return is 15?

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• 5.17 Consider the stock of Davidson Company that
  will pay an annual dividend of 2 in the coming
  year. This dividend is expected to grow at a
  constant rate of 5 permanently. The market
  requires a 12 return on the company.
• What is the current price of the stock?
• What will the stock price be 10 years from now?

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• 5.19 Whizkids Inc. is experiencing a period of
  rapid growth. Earnings and dividends are
  expected to grow at 18 during the next two
  years, 15 in the third year and a constant rate
  of 6 thereafter. Their last dividend which has
  already been paid was 1.15. If the required
  rate of return on the stock is 12, what should
  the price of the stock be today?

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Stock price and earnings per share (EPS)

• Investors often distinguish between
• Growth stocks
• Expectation of capital gains, based on future
  growth in earnings
• Income stocks
• Cash dividends

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No Growth stocks

• No Growth stocks
• P0 DIV1 EPS1
• r r
• Growth stocks
• P0 EPS PVGO
• r
• Two components of price
• The capitalized value of earnings under a
  no-growth policy
• Present value of growth opportunities

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Notes to calculations of PVGO

• Prices as of February 1998
• EPS numbers are from http//quote.yahoo.com/l
• Market capitalization rates were calculated using
  the capital asset pricing model

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Rate of growth, g

• g refers to the growth rate of equity - why?
• Price PV(Cash flows paid to shareholders)
• Depends on the rate of growth of equity
• Sustainable growth rate
• Consider a firm wishing to increase sales
• Must be supported by increases in assets such as
  inventory, receivables and productive capacity
• Must be supported by increases in the right hand
  side of the balance sheet I.e. increases in
  liabilities and /or shareholders equity
• Limits to the use of debt
• Hence equity must grow in order to sustain
  expansion in sales

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Estimates of the growth rate, g

• The growth rate g R x ROE
• R Retention ratio 1 - (Dividends/EPS)
• r
• ROE
• Net income
• Shareholders equity
• Net income x Sales x Assets
• Sales Assets Shareholders equity
• Profit margin x Asset Turnover x Financial
  leverage
• P x A x L
• Managerial input into all the components of
  growth, P,A,L and the retention ratio, R

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• ROE P x A x L
• ()
• DDI Inc. 18.2 12.7 0.94 1.53
• BankAmerica Corp. 13.2 13.1 0.09 11.49
• Duke Power 14.9 15.3 0.35 2.79
• Exxon Corp. 16.0 5.3 1.33 2.26
• Food Lion Inc. 15.7 2.1 3.10 2.40
• HP 20.6 7.7 1.29 2.06
• Nike 20.4 8.4 1.51 2.60
• Nordstrom 11.6 4.0 1.51 1.92
• Southwest Airlines 12.8 6.4 0.88 2.28
• (Reprinted from Higgins, Analysis for Financial
  Mgt.)

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Profit margin and Asset turnover

• Profit margin - fraction of each dollar from
  sales that finally trickles down to profits
• Asset turnover - Sales generated per unit of
  assets
• Generally the two are inversely related
• High profit margins - Value added to the product
  (Duke Power)
• Addition of value requires large assets
• Food Lion - Low profit margin and high asset
  turnover
• Combination of these two is important
• Competitive forces generally prevent both these
  from being high at the same time

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• Profit margin
• Reflects pricing strategy
• Reflects cost controls
• Important to operating managers
• Asset turnover
• Depends on the industry and product
• Management control
• Control of current assets
• Working capital management
• Profit margin x Asset turnover Return on Assets
  (ROA)

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Financial Leverage

• Increases when the proportion of debt increases
• Maximizing this ratio is not necessarily a goal
  (unlike in the previous two cases)
• Although this increases ROE
• Other constraints
• Generally companies with stable and predictable
  cash flows are in a better position to increase
  debt
• Also has to do with the type of assets - tangible
  versus intangibles , liquid versus illiquid

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Phases of growth

• Firms go through various phases of growth
• Initial expansion
• Limited by ability to raise capital
• Too much growth is as bad as too little
• Company expanding too rapidly
• Depleting financial resources
• Financial resources cannot keep up
• Companies can literally grow broke
• Sustainable growth rate versus actual growth rate

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How to finance growth?

• Depends
• Temporary - increase borrowing
• Permanent
• Sell new equity
• Increased equity makes possible increased
  borrowings
• Unfortunately this option is not attractive for a
  lot of smaller companies
• Historically in the U.S. other sources have been
  more important
• On average only about 5 of publicly traded
  companies in the U.S. sell additional common
  stock

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• Increase leverage
• Limits to doing this
• Costs of capital increase
• Increase the retention ratio
• This works only if the company can put the
  retained profits to better use than the investor
  can
• Ex Suppose that you are given the following
  ratios for a firm
• ROE 8 expected to continue indefinitely
• EPS 4
• Market capitalization rate of 10.
• Pay-out ratio of 100.

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• What is the price per share?
• P0 DIV1/(r - g)
• g 0
• P0 DIV/r 4/.10 40
• Suppose the firm announces a change in dividend
  policy such that dividends will now account for
  40 of earnings. What will happen to price?
• Next years dividend .40 ? 4.00 1.60
• g Plow-back Ratio ? ROE .60 ? .08 4.8
• P0 DIV1/(r - g) 1.6/(.10 - .048) 30.77
• Why?
• What if ROE 12?

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• What if ROE 12?
• If all dividends are paid out, P0 4/0.1 40
• With the change in dividend policy, g 0.6 x
  0.12 0.072
• P0 1.6 / (0.1-0.072) 57.14

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• 5.26 California Electronics Inc. expects to
  earn 100 million per year in perpetuity if it
  does not undertake any new projects henceforth.
  The firm however, has an opportunity that
  requires an investment of 15 million today and
  5 million in one year. The new investment will
  begin to generate additional annual earnings of
  10 million two years from today in perpetuity.
  The firm currently has 20 million shares
  outstanding and the required rate of return on
  the stock is 15.
• A. What is the price of the stock if the firm
  does not undertake the new project?
• B. What is the value (present) of growth
  opportunities resulting from the new project?
• C. What will the stock price be if the firm
  decides to take on the new project?

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Problems from Chapter 5

• 5.1-5.6
• 5.14-5.19
• 5.23 - 5.25