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# Modeling

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```									   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.
 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
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%
 The final concept incorporated in CAPM is the
   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
 If the asset being valued is riskier than the
average asset, an investor should require a
 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.
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?

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