Chapter 4 Discounted Cash Flow Valuation � You want to retire at the age of 60 and you estimate that you will need $100,000 a year for the rest of your life – HOW MUCH DO YOU NEED TO SAVE PER MONTH STARTING TODAY? � If I can afford to pay $1000 a month, WHAT PRICE OF A HOUSE SHOULD I CONSIDER? � When buying a new car, which offer is better – the $3000 rebate or the 1.9% interest rate over 60 months? � How can I value a business/firm? What about corporate or government bonds? 1
} Describe and compute the future value and/or present value of a single cash flow or series of cash flows } Define and calculate the return on an investment } Recognize and compute the impact of compounding periods on the true return of stated interest rates } Use a financial calculator and spreadsheets to solve time value problems } Comprehend and calculate time value metrics for perpetuities and annuities } Familiarization with loan types and amortization To Use DCF We Need To Know Three Things: 1. The Amount Of The Projected Cash Flows 2. The Timing Of The Cash Flows 3. The Proper Discount (Interest) Rate, r (r should reflect current capital market conditions, and generally can include a premium for risk) 2
} Individuals and institutions have different income streams and different intertemporal consumption preferences. } An individual can alter his consumption across time periods through borrowing and lending. ◦ The job of balancing the supply of and demand for loanable funds is taken by the money market. ◦ When the quantity supplied equals the quantity demanded, the market is in equilibrium at the equilibrium price. } Because of this, a market has arisen for money. The price of money is the interest rate. 3
} A dollar today is more valuable than a dollar to be received in the future } Why? ◦ A dollar today is more valuable because: It can be invested to make more dollars It can be immediately consumed There is no doubt about its receipt } If you know your required rate of return and the length of time before cash is harvested, you can calculate some critical metrics: ◦ The value today of a payment to be received in the future This measure is called a “Present Value” ◦ The value in the future of a sum invested today This measure is called a “Future Value” ◦ Present and Future Values can be calculated over single and multiple periods 4-7 4
} If you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500. $500 would be interest ($10,000 × .05) $10,000 is the principal repayment ($10,000 × 1) $10,500 is the total due. It can be calculated as: $10,500 = $10,000 × (1.05) q The total amount due at the end of the investment is call the Future Value ( FV ). } If you were to be promised $10,000 due in one year when interest rates are 5-percent, your investment would be worth $9,523.81 in today’s dollars. $ 10 , 000 $ 9 , 523 . 81 = 1 . 05 The amount that a borrower would need to set aside today to be able to meet the promised payment of $10,000 in one year is called the Present Value ( PV ). Note that $10,000 = $9,523.81 × (1.05). 5
} In the one-period case, the formula for FV can be written as: FV = C 0 × (1 + r ) Where C 0 is cash flow today (time zero), and r is the appropriate interest rate } In the one-period case, the formula for PV can be written as: C 1 PV = 1 r + Where C 1 is the cash flow at date one, and r is the appropriate interest rate } The Net Present Value ( NPV ) of an investment is the present value of the expected cash flows, less the cost of the investment. } Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy? 6
$ 10 , 000 NPV $ 9 , 500 = − + 1 . 05 NPV $ 9 , 500 $ 9 , 523 . 81 = − + NPV $ 23 . 81 = The present value of the cash inflow is greater than the cost. In other words, the Net Present Value is positive, so the investment should be purchased. In the one-period case, the formula for NPV can be written as: NPV = – Cost + PV If we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5 percent, our FV would be less than the $10,000 the investment promised, and we would be worse off in FV terms : $9,500 × (1.05) = $9,975 < $10,000 7
} The general formula for the future value of an investment over many periods can be written as: FV = C 0 × (1 + r ) T Where C 0 is cash flow at date 0, r is the appropriate interest rate, and T is the number of periods over which the cash is invested. } Suppose a stock currently pays a dividend of $1.10, which is expected to grow at 40% per year for the next five years. } What will the dividend be in five years? FV = C 0 × (1 + r ) T $5.92 = $1.10 × (1.40) 5 8
} Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40- percent on the original $1.10 dividend: $5.92 > $1.10 + 5 × [$1.10 × .40] = $3.30 This is due to compounding . 5 $ 1 . 10 ( 1 . 40 ) × 4 $ 1 . 10 ( 1 . 40 ) × 3 $ 1 . 10 ( 1 . 40 ) × 2 $ 1 . 10 ( 1 . 40 ) × $ 1 . 10 ( 1 . 40 ) × $ 1 . 10 $ 1 . 54 $ 2 . 16 $ 3 . 02 $ 4 . 23 $ 5 . 92 0 1 2 3 4 5 9
} How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%? PV $20,000 0 1 2 3 4 5 $ 20 , 000 $ 9 , 943 . 53 = 5 ( 1 . 15 ) } Examples thus far have offered the time and interest rate and solved for PV or FV } Keep in mind that there are four variables: ◦ PV ◦ FV ◦ T ◦ R } If you have any three you can solve for the fourth } The math can become cumbersome ◦ Financial Calculators and Spreadsheets are very helpful 10
If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000? T T FV C ( 1 r ) $ 10 , 000 $ 5 , 000 ( 1 . 10 ) = × + = × 0 $ 10 , 000 T ( 1 . 10 ) 2 = = $ 5 , 000 T ln( 1 . 10 ) ln( 2 ) = ln( 2 ) 0 . 6931 T 7 . 27 years = = = ln( 1 . 10 ) 0 . 0953 Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child’s education? About 21.15%. T FV C ( 1 r ) 12 $ 50 , 000 $ 5 , 000 ( 1 r ) = × + = × + 0 $ 50 , 000 12 ( 1 + r ) 10 1 12 ( 1 + r ) 10 = = = $ 5 , 000 10 12 1 r 1 1 . 2115 1 . 2115 = − = − = 11
} Consider an investment that pays $200 one year from now, with cash flows increasing by $200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows? } If the issuer offers this investment for $1,500, should you purchase it? 0 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93 Present Value < Cost → Do Not Purchase 12
First, set your calculator to 1 payment per year. Then, use the cash flow menu: 12 CF0 0 CF3 600 I 1,432.93 CF1 200 F3 1 NPV 800 F1 1 CF4 400 CF2 F4 1 F2 1 } All examples thus far have assumed annual compounding } Instances of other compounding schedules abound: ◦ Banks compound interest quarterly, monthly or daily ◦ Mortgage companies compound interest monthly } Yet, almost all interest rates are expressed annually } If a rate is expressed annually, but compounded more frequently, then the effective rate is higher than the stated rate } This concept is called the Effective Annual Rate or EAR 13
Compounding an investment m times a year for T years provides for the future value of wealth: mT r ⎛ + ⎞ FV C 1 = × ⎜ ⎟ 0 m ⎝ ⎠ For example, if you invest $50 for 3 years q at 12% compounded semi-annually, your investment will grow to 2 3 × . 12 ⎛ + ⎞ 6 FV $ 50 1 $ 50 ( 1 . 06 ) $ 70 . 93 = × = × = ⎜ ⎟ 2 ⎝ ⎠ 14
A reasonable question to ask in the above example is “what is the effective annual rate of interest on that investment?” . 12 2 3 6 FV $ 50 ( 1 ) × $ 50 ( 1 . 06 ) $ 70 . 93 = × + = × = 2 The Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-of-investment wealth after 3 years: 3 = $ 50 ( 1 EAR ) $ 70 . 93 × + 3 = FV $ 50 ( 1 EAR ) $ 70 . 93 = × + $ 70 . 93 3 = ( 1 + EAR ) $ 50 1 3 $ 70 . 93 ⎛ ⎞ EAR 1 . 1236 = − = ⎜ ⎟ $ 50 ⎝ ⎠ So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually. 15
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