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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. 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?

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posted: | 6/28/2012 |

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