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					Introduction to Valuation
     Stock Valuation


   Financial Management

       P.V. Viswanath

 For a First course in Finance
           Absolute and Relative Pricing
 In economics, we tend to price goods and assets by
  considering the factors affecting the supply and demand for
  them.
 The number of goods and assets are very many. Each of
  them is different in some way or another from the other.
 Computing the price of one good does not allow us to price
  another good, except to the extent that other goods are
  substitutes or complements for the first good.
 In finance, the number of assets can be reasonably
  characterized in terms of a smaller number of basic
  characteristics.
 Hence most assets can, to a first approximation be priced by
  considering them as combinations of more fundamental
  assets.

                         P.V. Viswanath                      2
         Relative pricing of financial assets

 Consider first riskless financial assets, i.e, assets that are claims
  on riskless cashflows over time.
 Consider a fundamental asset, i, defined by a claim to $1 at time t
  = i.
 There can be T such fundamental assets, corresponding to the t =
  1,..,T time units.
 Then, any arbitrary riskless financial asset that is a claim to $ci at
  time i, i = 1,..,T can be considered a portfolio of these T
  fundamental assets.
 Hence, the price, P* of any such asset is related to the prices of
  these first T fundamental assets.                              T
 In fact, the price of this asset would simply be           
                                                        P *  ci Pi
                                                              i 1


                              P.V. Viswanath                               3
     Relative pricing of risky financial assets

 What about risky financial assets?
 We can equivalently imagine, for every level of risk, a set of T
  fundamental risky assets. Then, for any arbitrary risky asset of
  this level of risk, we can equivalently write:        T
                                                P   ci Pi
                                                  *

                                                        i 1


 Of course, this is not entirely satisfactory, because we’d have TxM
  fundamental assets corresponding to each of M levels of risk. We
  will come back to this when we talk about the CAPM.
 In any case, we need to examine how this pricing is established in
  the market-place.


                             P.V. Viswanath                          4
       Arbitrage and the Law of One Price

 Law of One Price: In a competitive market, if two assets
  generate the same cash (utility) flows, they will be priced the
  same.
 How is this enforced?
 If the law is violated – if asset 1 sells for more than asset 2,
  then investors can make a riskless profit by buying asset 2
  and selling it as asset 1!
 In practice – we need to take transactions costs into account.
 Also, it may be difficult to execute the two transactions at
  the same time – prices might change in that interval – this
  introduces some risk.


                           P.V. Viswanath                        5
       Exchange Rates and Triangular Arbitrage

 Consider the exchange rates reigning at closing on
  January 30.
      The yen/euro rate was 157.87 yen per euro
      The euro/$ rate was $1.4835 per euro.
      The yen/$ rate was 106.4 yen per dollar.
 If we start with a dollar, we can buy 106.4 yen;
  these can then be used to buy 106.4/157.87 or 0.674
  euros, which can, in turn, be used to acquire
  $0.9998, which is very close to a dollar.


                          P.V. Viswanath               6
           Triangular Currency Arbitrage
 Suppose the euro/$ rate had been $1.50 per euro.
 Then, it would have been possible to start with one dollar,
  acquire 0.674 euros, as above, and then get (0.674)(1.5) or
  $1.011, or a gain of 1.1% on the initial investment of a
  dollar.
 This would imply that the dollar was too cheap, relative to
  the euro and the yen.
 Many traders would attempt to perform the arbitrage
  discussed above, leading to excess supply of dollars and
  excess demand for the other currencies.
 The net result would be a drop a rise in the price of the
  dollar vis-à-vis the other currencies, so that the arbitrage
  trades would no longer be profitable.


                          P.V. Viswanath                         7
                           Risk Arbitrage
  In this case, trading will continue until there are no more
   riskfree profit opportunities.
  Thus, arbitrage can ensure that the sorts of pricing relationships
   referred to above can be supported in the marketplace, viz:
                                   T
                            P *   ci Pi
                                    i 1
 What if there are still opportunities that will, on average, lead
  to profit, but the investors intending to benefit from this profit
  will have to take on some risk?
 Presumably investors will trade off the risk against the
  expected profit so that there will be few of these expected
  profit opportunities, as well; this brings us to the notion of the
  informational efficiency of financial markets.
                              P.V. Viswanath                           8
     Efficient Markets Hypothesis – EMH

 An asset’s current price reflects all available
  information– this is the EMH.
 If it didn’t, there would be an incentive for
  investors to act on that information.
 Suppose, for example, that investors noticed that
  good news led to stock prices rising slowly over
  two consecutive days.
 This would mean that at the end of the first day, the
  good news was not all incorporated in the stock
  price.

                      P.V. Viswanath                  9
              Efficient Markets Hypothesis

 In this situation, it would be optimal for traders to buy even
  more of a stock that was noted to be rising on a given day,
  since the stock would rise more the next day, giving the trader
  an unusually good chance of making money on the trade.
 But if many traders pursue this strategy, the stock price would
  rise on the first day, itself, and the informational inefficiency
  would be eliminated.
 Empirically, financial markets seem to be reasonably close to
  being efficient.
 This allows us to price financial assets with respect to
  fundamentals without worrying about deviations from these
  fundamental prices.

                             P.V. Viswanath                       10
            Stock Price Fundamentals

 What determines the price of a stock? Or, in other
  words, why would an investor hold stocks?
 The answer is that s/he expects to receive dividends
  and hopefully benefit from a price increase, as well.
 In other words, P0 = PV(D1) + PV(P1)
 However what determines P1?
 Again, using the previous logic, we must say that
  it’s the expectation of a dividend in period 2 and
  hopefully a further price rise. Continuing, in this
  vein, we see that the stock price must be the sum of
  the present values of all future dividends.

                      P.V. Viswanath                  11
                  Dividend Mechanics

 Declaration date: The board of directors declares a payment
  Record date: The declared dividends are distributable to
  shareholders of record on this date.
  Payment date: The dividend checks are mailed to
  shareholders of record.
 Ex-dividend date: A share of stock becomes ex-dividend on
  the date the seller is entitled to keep the dividend. At this
  point, the stock is said to be trading ex-dividend.




                          P.V. Viswanath                      12
            Dividend Discount Model

 What is the price of a stock on its ex-dividend date?
 Using the previous logic, we see that it’s simply
          D1       D2                Dn
   P0                    ...              ...
         1  k (1  k ) 2
                                  (1  k ) n


 where k is the appropriate discount rate to discount
  the dividends consistent with their riskiness.
 We assume that the one-period ahead discount rate
  is the same for all periods. That is, we use the same
  rate to discount D1 to time 0, as we use to discount
  D2 to time 1.
                      P.V. Viswanath                  13
              Gordon Growth Model

 If we assume that the dividend is growing at a rate
  of g% per annum forever, this formula simplifies
  to:
                          D1
                    P0 
                         kg
 We see that the price of a stock is higher, the higher
  the level of dividends, the higher the growth rate of
  dividends and the lower the required rate of return
  or the discount rate, k.


                      P.V. Viswanath                    14
   Earnings and Investment Opportunities

 Dividends = Earnings – Net New Investment
 Hence a firm’s stock price cannot be the present
  value of discounted earnings!
      Unless the firm needs no new investment to maintain its
       earnings.
 What determines the price of the stock of a
  company that reinvests part of its earnings?




                          P.V. Viswanath                     15
         Reinvestment and Stock Price

 Suppose a firm starts out with a certain stock of
  investment capital at the beginning of period 1 (end
  of period 0).
 Assume that it earns a return, ROE, on this capital,
  so as to assure it of earnings of $E1 each period
  forever.
 Assume, furthermore, that this firm pays out all of
  these earnings as dividends, each period.
 Then, its stock price today, P0 will be equal to E1/k.

                       P.V. Viswanath                 16
           Reinvestment and Stock Price

 Now assume, in addition to its existing investments, that the
  firm expects to have at t=1, an investment opportunity with
  a t=1 value of $M1 (that is, this is the present value at t=1 of
  all the future cashflows that will be generated by this
  investment opportunity.
 Implementation of this idea requires additional capital of
  DI1, which the firm raises from the marketplace.
 If the capital market is efficient, the firm will have to pay for
  this additional capital with promises of future cashflows
  with a present value equal to the amount of additional
  capital raised.

                           P.V. Viswanath                        17
           Reinvestment and Stock Price

 Hence the t=1 value that will accrue to the firm’s
  shareholders is only M1- DI1.
 Denote by NPV1, the t=0 value of M1- DI1. That is, NPV1=
  (M1- DI1)/(1+k).
 Taking this additional investment opportunity into account,
  the firm’s stock price will not just be E1/k, but E1/k + NPV1.
 Similarly, let NPVi represent the t=0 value of investment
  opportunities that the firm expects to have a time t=i, for
  each future time period.
 Proceeding thus, we see that P0 = E1/k + NPVGO, where
  NPVGO = S NPVi for all i = 2,…


                          P.V. Viswanath                       18
           Reinvestment and Stock Price
 Upto this point, we have assumed that the firm has raised
  this additional capital from other investors in the market
  place.
 Suppose, however, that the firm raises the additional capital
  in period 1 from its own shareholders, by reducing the
  amount of dividends that it pays. That is, D1 = E1 – DI1.
 This reduction in dividends will cause the stock price to
  drop by an amount equal to the present value of DI1.
  However, the firm will no longer have to pay the outside
  investors future compensation for the contribution of the
  additional capital, DI1.
 These two quantities will cancel each other out. We, see,
  therefore, that P0 = E1/k + NPVGO.


                          P.V. Viswanath                      19
  Fundamental Determinant of Growth Rate
 What are the determinants of growth in a firm’s earnings?
 Earnings in any period depends on the investment base, as
  well as the rate of return that the firm earns on that
  investment base:
 Et+1 = (It)ROE
      = (It-1 + DIt)(ROE), where DIt is the increment in
  investment in period t over and above that in period t-1.
      = (It-1)ROE + (DIt)(ROE)
      = Et + (DIt)(ROE);
 Hence Et+1 - Et = (DIt)(ROE)
 Dividing both sides by Et , we get gt = (Retention
  Ratio)(ROE), assuming that the additional investment is
  made possible by retaining part of the firm’s earnings.

                         P.V. Viswanath                       20
           Reinvestment and Stock Price

 We see from the previous demonstration that retention of
  earnings by a firm for reinvestment will not increase in a
  higher stock price if that additional investment has a zero
  NPV
 That is, if it earns a return no greater than the rate of return
  required by the market on financial investments of similar
  risk, already available to investors in the marketplace.
 We see, furthermore, that it is not the firm’s dividend policy
  that causes the firm’s stock price to be higher, but rather the
  availability of positive NPV investment opportunities.
 This can be seen clearly in the following example.


                           P.V. Viswanath                        21
       Example of Dividend Irrelevance

 Stellar, Inc. has decided to invest $10 m. in a new
  project with a NPV of $20 m., but it has not made
  an announcement.
 The company has $10 m. in cash to finance the
  new project.
 Stellar has 10 m. shares of stock outstanding,
  selling for $24 each, and no debt.
 Hence, its aggregate value is $240 m. prior to the
  announcement ($24 per share).


                      P.V. Viswanath                    22
       Example of Dividend Irrelevance

Two alternatives:

One, pay no dividend and finance the project with
   cash.
   The value of each share rises to $26 following the
   announcement. Each shareholder can sell 0.0385
   (= 1/26) shares to obtain a $1 dividend, leaving
   him with .9615 shares value at $25 (26 x 0.9615).
   Hence the shareholder has one share worth $26, or
   one share worth $25 plus $1 in cash.

                      P.V. Viswanath                23
           Example of Dividend Irrelevance

Two, pay a dividend of $1 per share, sell $10m. worth of new
   shares to finance the project.
 After the company announces the new project and pays the $1
   dividend, each share will be worth $25.
 To raise the $10 m. needed for the project, the company must
   sell 400,000 (=10,000,000/25) shares. Immediately following
   the share issue, Stellar will have 10,400,000 shares trading for
   $25 each, giving the company an aggregate value of 25 x
   10,400,000 = $260m.
 If a shareholder does not want the $1 dividend, he can buy
   0.04 shares (1/25).
 Hence, the shareholder has one share worth $25 and $1 in
   dividends, or 1.04 shares worth $26 in total.
                            P.V. Viswanath                     24
    Assumptions for Dividend Irrelevance

1. The issue of new stock (to replace excess dividends) is
   costless and can, therefore, cover the shortfall caused by
   paying excess dividends.
2. Firms that face a cash shortfall do not respond by cutting
   back on projects and thereby affect future operating cash
   flows.
3. Stockholders are indifferent between receiving dividends
   and price appreciation.
4. Any cash remaining in the firm is invested in projects
   that have zero net present value. (such as financial
   investments) rather than used to take on poor projects.

                        P.V. Viswanath                    25
     Implications of Dividend Irrelevance

 A firm cannot resurrect its image with stockholders
  by offering higher dividends when its true prospects
  are bad.
 The price of a company's stock will not be affected
  by its dividend policy, all other things being the
  same. (Of course, the price will fall on the ex-
  dividend date.)




                      P.V. Viswanath                26
            Dividends in the Real World

 In practice, dividends are taxed higher than capital gains.
  Hence investors may prefer that the firm retain funds for
  new investment rather than raise it from the financial
  marketplace.
 On the other hand, managers have to justify the need for
  additional funds in order to get them from investors. This
  ensures a greater check on managers. Hence, the
  marketplace might prefer that managers raise funds
  externally.
 In practice, firms have to take both factors into account and
  craft the best dividend policy.



                          P.V. Viswanath                      27

				
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