Stock and Bond Valuation

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					Lecture: 3 - Stock and Bond Valuation

How to Get a “k” to Discount Cash Flows - Two Methods

I.      Required Return on a Stock (k) - CAPM (Capital
     Asset Pricing Model) or Other Securities

     k = rf + B(rm - rf)

     where     rf = Risk-Free Rate
              B = Beta (Risk Measure)
              rm = Expected Market Return

Example: k = .05 + 2(.12 - .05) = .19

II. Risk Premium Approach

          kb = Rb = rreal + rrisk + E(I)
             = Productivity Growth + Risk Premium +
                                           Expected Inflation
             = 2 - 4% + 0 - 10% + 2 - 6% (Estimates)
Lecture: 3 - Stock and Bond Valuation

III.   Bond Valuation

       Bond    = Annuity Plus Single Par Payment
               (often Semi-Annual Payments)

       a. Par Value -face value, maturity value -
                 usually $1000

       b. Coupon Interest Rate - stated as a % of Par -
            10% coupon on $1000 par => coupon = $100.

       c. Maturity - length of time until Par value is paid off
Lecture: 3 - Stock and Bond Valuation
Bond Valuation
“A Bond is an Annuity Plus a Single Face Value Payment”
Annual Coupon
Lecture 3 - Stock and Bond Valuation

I.      Bond Valuation
        General Formula

        B0   = I[PVAk,n] + M[PVk,n]

        B0= Bond Price
        I = Interest (coupon) Payment
        M = Par Value

II.     Example:
        Suppose a bond offers a 10% coupon, on $1000
        par, for 3 years, and the expected inflation rate
        is 2%, the real rate is 3% and the bond’s risk is
        1%. What is its price?

        B0   = $100[PVA.06,3] + $1000[PV.06,3]
             = $100(2.673) + $1000(.84)
             = $1107
QUESTION: If the company only agrees to pay $1000 at
       maturity, won’t those who buy this bond lose $107
       at maturity?

QUESTION: Would you buy this bond? Why? - greater
       coupon than par bonds.

A par bond would cost $1000 but only pays a $60 coupon.
The present value of the difference in coupons (100 -
60)(2.673) = 107 which is the difference in price between
this bond and a par bond.

Alternatively, a bond that offered a 2% coupon when rates
are 6% will have a price of B0 = 20[PVA.06, 3] + 1000[PV.06, 3]
= 893 or $107 less than the par bond.
Bond Valuation
“A Bond is an Annuity Plus a Single Face Value Payment”
Semi-Annual Coupon
Lecture 3 - Stock and Bond Valuation

I.     Adjustments For Semi-Annual Coupon Bonds
       a. n = the number of semi-annual payments
                            (maturity x 2)
       b. k = one-half the bond’s annual yield
       c. I = one-half the bond’s annual coupon

II.    Example:
       Suppose a bond pays 10% coupon, semi-annually,
       has 10 years until maturity and has a required
       return (or Yield to Maturity) of 8%. What is its price?

       B0   = $50[PVA.04,20] + $1000[PV.04,20]
            = $50(13.59) + $1000(.456)
            = $1135.5

QUESTION: Consider two identical bonds except that one
     pays an annual coupon and the other a semi-annual
     coupon. Which should have the higher price?

ANSWER: The semi-annual bond.
Yield to Maturity and Realized Yield
Lecture 3 - Stock and Bond Valuation

YIELD TO MATURITY - The return one can expect on an
        investment in a bond if the bond pays all its
        coupons and par and yields do not change after
        you purchase the bond.

PROBLEM: Suppose you observe a bond in the market
       with a price of $803 that pays a coupon of 10%
       till maturity in 5 years. What is its implied yield
       to maturity?

Try 16%
          803       = 100(PVA?,5) + 1000(PV?,5)

                    = 100(3.274) + 1000(.476) = 803

REALIZED YIELD - The actual return one receives on the
       initial investment in a bond.

QUESTION: If you buy a 20% coupon, par bond, with 3
       years maturity and you hold it for three years are
       you sure to earn 20%?

ANSWER: No because the calculation of YTM assumes
      that the coupons are reinvested at 20%, if rates
      change your realized yield will change because
      you'll earn more or less than 20% on their
      reinvested coupons.
Example: When you bought the bond, YTM was 20%. But
  suppose rates fell to 5% the day after you bought and
  stay there for three years. Your realized yield will be:

use PV = FV[PVk,n]

1000      = (200(1+.05)2 + 200(1+.05) + 1200)[PVk,3]

          = 1630.5[PVk,3]

          => 1000/1630.5 = [PVk,3] = .6133

=> k = 17% realized yield falls when reinvestment rate falls

QUESTION: Then how can you truly lock-in a rate?

ANSWER: Buy a bond with no coupons - called zero coupon

QUESTION: How would you price a zero coupon bond?

ANSWER: Use the second term in the bond pricing formula.

QUESTION: Some find this attractive but is there a problem
       with being locked-in?

ANSWER: Yes. How about if rates rise. You lose out on
      earning extra interest on reinvested coupons.
Stock Valuation
“Valuation is Based Upon Expected Dividend Flow and the
Future Expected Market Value of the Stock”
Lecture 3 - Stock and Bond Valuation

I.         Common Stock Valuation
           General Formula
           P0   = E(D1)/(1+ke) + E(D2)/(1+ke)2 + ... +
                   E(Dn)/(1+ke)n + E(Pn)/(1+ke)n

      where E means expectation, Dt means dividend at time t,
      P means stock price, and ke is the cost of equity.

II.        Example:
           Suppose a firm pays $4 in Dividends, which will
           increase by $1 in each of the next 3 years and we
           expect the price to be $30 at the end of the 3rd year.
           Assume stock beta = 1, the risk free rate is 10%, and
           the expected market return is 15%. What is the stock

           Ke = .10 + 1(.15 - .10)
           P0 = $4/(1.15) + $5/(1.15)2 + $6/(1.15)3 +$30/(1.15)3
           = $4(.87) + $5(.756) + $6(.658) + $30(.658)
           = $30.94
  Stock Valuation - Constant Growth
  “Valuation is Based Upon Expected Dividends That Grow at
  a Constant Rate For Ever”
  Lecture 3 - Stock and Bond Valuation

 Constant Growth Discounted Cash Flow Model, or DCF.
                   D0 (1  g )        D1
           P0 =                  =
                    kg              kg

           where      D0 = present dividend paid at time 0
                      D1 = dividend expected at time 1
                      g = constant future growth in dividends
                      k = required return (discount rate)

 Note: Works only for k>g and dividend paying firm

PROBLEM: Suppose a company will pay a dividend of $5 in
    one year, has a required return of 10% and dividends
    grow 5% per year. What is the stock price?
                       5      5
          P =                    100
                   (.10.05) .05

Mixture of Dividends

PROBLEM: Suppose a firm will pay a dividend of $5 per
       year for 5 years and then increases the dividend
       by 10% per year thereafter. The firm has a required
       return of 15%. What is its price now?
          P   = 5(PVA.15, 5) +   (.15.10) [PV.15, 5]

              = 5(3.352) + 110(.497) = 71.43
Stock Valuation - Implied Required Returns and
                Growth Rates
“Just Manipulate the Constant Growth Formula”
Lecture 3 - Stock and Bond Valuation

PROBLEM: Suppose we know the price of the stock in the
       market is $80, and it pays a dividend of $3 that
       will grow by 10% per year. What is the return the
       market requires on the stock?
                 D0 (1  g )
          k =                g

             =         .10
             = .14

PROBLEM: If the market price of a stock is $50, its required
       return is 15 percent and next year’s dividend is
       expected to be $5, by what percent must the
       market expect the company’s dividends to grow.

          g = k - D1

          so g = .15 - 5          = .05
“PE’s Are Commonly Used to Compare Stocks”

Lecture 3 - Stock and Bond Valuation

PE Ratio - This is the number of dollars investors are willing
            to pay for each dollar of a company’s earnings.

You can use the growth model to see why stocks have
different price-earnings ratios

                    P =
                          (k  g)
                          (k  g)

                P          d
          =>        =
                E       (k  g)

where     E         = earnings per share      D
          d         = dividend payout ratio =

Clearly, a larger growth rate and payout ratio and a smaller
discount rate (k) makes for a larger price earnings ratio.

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