Modeling

							   Modeling

FIN415 Week 4 Slides
A Painting is a Model
Artists use Color to Create a Model
We use Mathematical Entities
           Movement through Time

 Our most basic mathematical entity.
 (1+r)t
 Future Value = Amount * (1+r)t
 Present Value = Amount / (1+r)t
             FV Example

 Current Amount = $1000
 Interest Rate = 8%
 Time = 10 years
 FV = $1000 (1+0.08)10
 FV = $1000 (1.08)10
 FV = $1000 (2.15892)
 FV = $2,158.92
            PV Example
 Future Amount = $1000
 Interest Rate = 8%
 Time = 10 years
 PV = $1000 / (1+0.08)10
 PV = $1000 / (1.08)10
 PV = $1000 / (2.15892)
 PV = $463.19
                 Payment

 Principle Amount * Rate = Payment
      Example of Payment

 Principle Amount: $1000
 Rate: 8%
 Payment = $1000 * 0.08 = $80
                  Perpetuity
 A Perpetuity is a financial instrument which pays
  a set payment every period forever.
 A Payment equals Principle * Rate (PMT=PV * r)
 Therefore PV = PMT/r
 The present value of a Perpetuity is the payment
  divided by the rate.
       Example of a Perpetuity

 A Perpetuity which pays $80 per year forever if
 the prevailing interest rate is 8% is as follows:

 PV = $80/0.08 = $1000
                    Annuity
 An Annuity is a financial instrument which pay a
  specific payment for a specific period.
 Annuity = PV of a Perpetuity – FV of a Perpetuity
  at the end of the Annuity period.
 PV Perpetuity – FV Perpetuity = Annuity
           Example of Annuity
 Calculate PV of an Annuity which pays $80 per
  year for 10 years.
 PV of Perpetuity which pays $80 forever, if the
  interest rate is 8% = $1000.
 FV of the Perpetuity at any point in time equals
  $1000, because from any point in time it will pay
  $80 per year forever.
 Thus, the PV of a Perpetuity which begins in 10
  years and pays $80 forever after that point is the
  PV of $1000.
  Example of Annuity Continued
 Annuity = (PMT/r) – ((PMT/r)/(1+r)t)
 ANN = ($80/0.08) – (($80/0.08)/(1+0.08)10)
 ANN = ($1000) – (($1000)/(1.08)10)
 ANN = ($1000) – (($1000)/(2.15892))
 ANN = ($1000) – ($ 463.19)
 ANN = $536.81
                      Bond
 A bond is a right to a stream of payment plus a
  lump sum payment (the principle) at the end.
 In other words a bond is an annuity plus the PV of
  the final payment.
                Bond Equation
 Annuity Equation + PV of Principle Payment
 BOND = (PMT/r) – ((PMT/r)/(1+r)t) +
    ((PRIN)/(1+r)t)
   BOND = ($80/0.08) – (($80/0.08)/(1+0.08)10) +
    ($1000 / (1+0.08)10)
   BOND = ($1000) – (($1000)/(1.08)10) +
    ($1000/(1.08)10)
   BOND = ($1000) – ($1000)/(2.15892) +
    ($1000/2.15892)
   BOND = ($1000) – ($ 463.19) + ($ 463.19)
   BOND = $1000
        Net Present Value (NPV)
 We can simplify everything down to (1+r)t
 We can use Excel to value each cash flow
    automatically.
   A Perpetuity (PMT/r) is the limit as the number of
    payments goes to infinity.
   We can approximate this with NPV in Excel by
    selecting a large number of payment (which
    become smaller as they go into the future).
   NPV 100 payments of $80 at 8% =$999.55
   Our estimate is only $0.45 off after 100 payments.
            Annuity with NPV

 Our Annuity equals 10 annual cash flows of $80.
 Using NPV Excel provides and answer of
 $536.81 (Identical to our original calculation)
              Bond with NPV

 Our Bond is simply nine cash flows of $80 plus a
  final cash flow of $1080.
 Again Excel provides the identical answer using
  NPV or $1000.
        Gordon Growth Model

 Growing Perpetuity
 You receive regular payments, but they are
  increasing at a specified rate.
 PMT/(r-g)
 Original payment amount divided by discount rate
  minus the growth rate.
           Growing Business
 Gordon Growth Model might apply to cash flows
  from a growing business producing regular cash
  flows.
 Assume that the current discount rate is 8%, but
  you expect the $80 per year cash flows from the
  business will grow 3% per year.
 PV of Business = $80/(0.08-0.03) or
  $80/0.05=$1600
              Terminal Value
 Future cash flows become harder to estimate the
  farther they are from the present.
 If you feel you have good estimates for the next 3
  years, you can apply NPV to the 3 years of cash
  flows and then add the PV of a growing perpetuity
  for all cash flows after the first three years.
 Terminal Value =((Estimate Annual Cash Flow for
  Year 4)/(r-g))/(1+r)4
        Estimating Cash Flows
 First determine what you are really trying to value.
 If you want to value an asset (say a factory) you
  need to determine the actual cash flows
  generated by the asset.
 You must subtract out non-cash expenses like
  depreciation (because no cash was actually paid)
 You must subtract expenses such as debt
  payments which are independent of the asset
  itself. In other words the value of the asset is
  independent of the debt associated with the
  asset.
               Valuing Equity
 If you are valuing an equity interest in a business,
  then you must look at the amount of cash which
  will actually flow to equity.
 Again, you must eliminate non-cash expenses
  like depreciation.
 However, since the entity owes the debt, in this
  case you are looking at the free cash flows after
  debt payments are made.
               Discount Rate

 The discount rate must reflect the movement
  through time. For example, inflation and
  opportunity cost make a dollar in the future less
  valuable than a dollar today.
 The discount rate must reflect the Risk inherent in
  the project being valued.
    Capital Asset Pricing Model
 CAPM was developed by William Sharpe in 1964.
 He received the Nobel Prize in 1990.
CAPM
               Risk Free Rate
 Rf is a proxy from the drift as value travels
  through time.
 The most common figure used is the 10 year
  Treasury Bond Rate.
 Should try to match up time durations. If a short
  duration 90 day Treasury Bill might be better.
 Represents inflation and the opportunity cost of
  money related to a risk free investment.
        Average Market Return
 Rm is the average market return.
 This is impossible to actually calculate.
 It would require knowledge of the return on all
  risky assets in the economy.
 Proxy must be used as a “sample” of market
  returns.
 S&P 500 commonly used as the proxy.
 Consider whether the S&P 500 is a good proxy.
                      Beta β

 β is the risk factor in the equation.
 β is thus a very important concept for this class.
 β is actually a measure of the risk of the asset we
  wish to value compared to the average risk in the
  economy as a whole.
              Risk = Volatility
 For purposes of CAPM, Risk equals volatility.
 Volatility is thought to be the Standard Deviation
  of an assets returns from the mean.
 The greater the swing in return, the more risky an
  assets is thought to be.
β
                   Example of β
 Assume that the volatility associated with the asset
    you are trying to value is 14% (we will use annual
    volatility).
   Assume that the average volatility of the S&P 500,
    which you are using as your market proxy is 9%
   β = (0.14*0.09)/(0.09)2=1.55
   In other words the asset you are valuing is riskier than
    the average risky asset in the market. If the average
    asset were to increase in value by 10%, you would
    expect that your asset would increase by 15.5%.
   However, if the average asset fell 10% in value, you
    would expect this asset to decline by 15.5%
                 Risk Premium
 The final concept incorporated in CAPM is the
    concept of Risk Premium.
   The Risk Premium is (rm – rf)
   This is the average market rate of return minus
    the risk free rate of return.
   In other words, on average this is the additional
    returns that investors in the economy are
    demanding in order to invest in risky assets.
   If the average market return is 9% and the risk
    free rate is 4%, then investors are demanding a
    risk premium of 5%.
  Risk Premium on Valued Asset
 If the asset being valued is riskier than the
  average asset, an investor should require a
  higher risk premium.
 If the asset being valued is less risky, then an
  investor should require a lower risk premium.
 By multiplying the market risk premium by β we
  get the risk premium associated with the asset.
 Add that risk premium to the risk free rate to
  obtain the discount rate for the asset.
             Example of CAPM

 Rf = 4%
 Rm = 9%
 β = 1.55
 RA=0.04 + 1.55(0.09 – 0.04) = 0.04 + 1.55(0.05) =
  0.04 + 0.075 = 0.115 or 11.5% discount rate
 This is the rate you would use in your NPV
  calculation.
         Option Pricing Models

 Binomial Model
 Black-Scholes Option Pricing Model
 Monte Carlo Simulations
  Binomial Option Pricing Model
 John C. Cox
 Stephen A. Ross
 Mark Rubenstein
 Option Pricing: A Simplified Approach (1979)
Binomial Option Pricing Model
              Black-Scholes
 Fischer Black
 Myron Scholes
 Robert C. Merton
 Paper published in 1973
 Scholes and Merton Awarded Nobel Prize in
 1997.
Black-Scholes
Monte Carlo Simulations
       Monte Carlo Simulations

 Rather than finding an equation that will provide
  an answer, we can use the brute force or
  computers to simulate the world.
 Typically, we run the simulation 100,000 times to
  determine the probabilities of certain outcomes.
 You can run your own simulations at:
  http://www.myonlineforecast.com/
   How Does Your Mind Model the
             World?

 What do you see in your mind when you try to
  remember a number?
 What do you see in your mind when thinking of
  the passage of time?
 What do you see in your mind when you think of
  cash flows?