Panos Parpas

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Stocks and Their Valuation

381 Computational Finance

• Panos Parpas

Imperial College London
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Topics Covered

• Stocks and their valuation
• zero dividend growth
• constant dividend growth
• non-constant dividend growth
• earning and sales based valuation methods

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Stocks

• Shares, securities or equities an ownership in
  part of a company
• You are entitled to a portion of companys
  profits and any voting rights attached to stock
• A company
• issues the stocks shares the profits with stock
  holders
• sells some ownership in company (in form of
  stocks) to raise money (equity financing) to use
  for upgrading equipment, marketing, expansion
• Profits are paid out in dividends the more
  shares you own, the larger profits you own
• Common Stock
• yields higher return than other forms of
  investment securities
• ownership in a company a portion of profits
  (variable dividends)
• voting rights
• in case of bankrupt, shareholders receive no
  money until rest of stockholders is paid
• Preferred Stock
• some degree of ownership in a company
• no voting rights
• guaranteed a fixed dividend forever
• may be callable company has an option to
  purchase shares from shareholders at anytime for
  a premium
• paid off before common shareholders

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Stock Prices

• The price of a stock shows what investors feel
  the company is worth.
• Stock prices change everyday by the market
  risky many factors
• Buyers and sellers cause prices to change
  because of supply and demand
• If more people want to
• buy a stock than sell it, then price moves up
• sell a stock, there would be more supply than
  demand, price would start to fall
• Many factors drive stock prices earnings
  (profit) of a company
• For securities issued by the company, there is
  uncertainty and also
• no maturity date,
• no dividend rate,
• need to estimate expected selling price to
  calculate current price
• Discounted Dividend Model
• Stock price is PV (future cash flows to be
  received by investor)

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Valuation of Stocks

• If investor buys a stock, he is entitled to
  receive all future dividends and can sell the
  stock in the future
• The cash flow of common stock consists of
  dividends plus a future sale price
• The stock discount rate re is the rate of
  return that investors expect to earn on
  securities with similar risk
• Time 0 1 2
  T
• Cash-Flow D1 D2 DT
• They buy stock for P0 and sell for P1 after one
  year and receive dividend D1,
• The next buyer also sells after one year
• At time period T,

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Valuation of Stocks

• Using recursive substitution, the current price
  of the stock is
• Expression for expected price can be neglected
  for a large time horizon
• The current value of the stock is the present
  value of all future cash flows dividends and
  expected selling price
• the value of T is determined by the investor
• dividends are uncertain
• need to estimate the expected selling price

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Required Returns

• The return on equity is sum of dividend yield
  and expected capital gain
• D1 is not known since it is an expected value
  about future dividend
• dividend yield percentage return for the
  dividend annual dividend per share is divided by
  price per share
• historic or trailing dividend yield
• prospective dividend yield

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Stocks Value Estimation

• The Zero Dividend Growth Model
• If the dividend is expected to stay constant over
  time,
• shares are valued like perpetual bonds
• expected return on equity is equal to dividend
  yield.
• Do not reflect reality because of constancy of
  dividends

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Constant Dividend Growth Model

• an amount grows at a constant rate forever is
  called a growing perpetuity
• stock with a constant dividend growth is a
  growing perpetuity
• let g be a constant dividend growth rate
• current stock price

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Constant Dividend Growth Model

• If , then
• The growth rate is
• If the company in steady state where dividends
  are expected to grow at a constant rate g, the
  stock price grows at the same constant rate.
• Example Next year dividends per share for
  Company X is expected to be 0.95. The dividends
  are expected to grow at 14 per year in the
  future. What should be the current price if the
  required rate of return is 16 per year?

growing perpetuity
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Non-constant Dividend Growth Model

• If the company is not a steady state, not
  possible to use the previous model
• Define sub periods with different growth rates
• Estimate the value of stock by considering each
  sub period with their discounting
• PV of non-constant growth dividends at each
  period
• PV of constant growth dividends PV(Pt)
• Value of the stock is

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Example
The next three years dividends for Company Y are
expected to be 0.50, 1.00, 1.50. Then the
dividends are expected to grow at a constant 5
forever. If the required return is 10, then what
is the value of the stock?

• Constant dividend growth based on 3rd dividend

• Non-constant dividend growth

• The current price of the stock

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Example

• Consider a company pays a dividend of 0.75 per
  share. Demand for this companys product is
  growing at 2 per year and inflation averages
  2.5 per year. The company expects its profits
  and dividends to grow at about 4.55 per year
  (1.02X1.025 1.0455). Stockholders require a 10
  rate of return. What is the market price of this
  companys stock?
• The dividend next period is

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Price-Earnings (P/E) Ratio

• used to price equities a fair value of stock
  can be determined with the P/E multiple
• the earning yield E1 / P0 where E1 is the
  earnings per share.
• dividends and earnings are related via the
  companys pay out policy
• explained by the pay out ratio p
• required return on equity is related to earnings
  yield
• If companies have the same pay out ratio,
  discount rate, growth rate, they have the same
  P/E ratio

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Price-Earnings (P/E) Ratio

• Analysts often report historical price earning
  ratio P0 /E0
• When dividends and earnings grow at the same
  constant rate g from now on,
• The required return and P / E ratio are