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## Discounted Cash Flow Valuation

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**Discounted Cash Flow Valuation**Chapter 4**Time Value of Money**News Reports: Teixeira, Yankees finalize $180M deal The Associated Press released some of the details of the contract that Mark Teixeira signed: December 2008: $5 million signing bonus 2009 - $20 million 2010 - $20 million 2011 - $22.5 million 2012 - $22.5 million 2013 - $22.5 million 2014 - $22.5 million 2015 - $22.5 million 2016 - $22.5 million Is this really worth $180m?**Future Value**• General formula: FV = C0×(1 + r)T Where C0 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.**Future Value**• Suppose that you invest $500 in a savings account the earns 3%. • What will you account be worth in five years? FV = C0×(1 + r)T**Compounding**• 500*1.03 = 515 • 515*1.03 = 530.45 = 500*(1.03)2 • 530.45*1.03 = 546.36 = 500*(1.03)3 • 546.36*1.03 = 562.75 = 500*(1.03)4 • 562.75*1.03 = 579.64 = 500*(1.03)5**Present Value**• General formula: Where PV is at t=0 CT is cash flow at date T, r is the appropriate interest rate, and T is the number of periods over which the cash is invested. {Note: PV/FV are same formula!}**Present Value**• You need to have $40,000 in 6 years to use as a down payment on a house. The money will earn about 8%. How much do you need to invest today?**How Long is the Wait?**If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?**What Rate Is Enough?**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?**Multiple Cash Flows**• 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 Multiple Cash Flows Present Value < Cost → Do Not Purchase! NPV is negative!**Compounding Periods**Compounding an investment m times a year for T years provides for future value of wealth: If you invest $50 for 3 years at 12% compounded semi-annually, what will your investment grow to?**Effective Annual Rate (aka APY)**A reasonable question to ask in the above example is what is the effective annual rate of interest on that investment? The Effective Annual Interest Rate (EAR) is the annual rate that would give us the same end-of-investment wealth after 3 years:**Effective Annual Rate**Investing at 12.36% compounded annually is the same as investing at 12% compounded semiannually**Effective Annual Rate**• General formulaWherer is the quoted or stated rate (APR)**Effective Annual Rate**• Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly.**Continuous Compounding**• General formula FV = C0×erT Where C0 is cash flow at date 0 r is the stated annual interest rate T is the number of periods over which the cash is invested e is a transcendental number approximately equal to 2.718. {ex is a key on your calculator} Example: You invest $1,000 at a continuously compounded rate of 10% for 2 years. How much will your investment be worth?**Perpetuity**A constant stream of cash flows that lasts forever. The formula for the present value of a perpetuity is:**Perpetuity: Example**What is the value of a British consol that promises to pay £15 each year, every year until the sun turns into a red giant and burns the planet to a crisp? The interest rate is 10%.**Growing Perpetuity**A growing stream of cash flows that lasts forever. The formula for the present value of a growing perpetuity is:**Growing Perpetuity: Example**The expected dividend next year is $1.30 and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream?**Annuity**A constant stream of cash flows with a fixed maturity. The formula for the present value of an annuity is:**Annuity Intuition**An annuity is valued as the difference between two perpetuities: one perpetuity that starts at time 1 less a perpetuity that starts at time T + 1**Annuity: Example**If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?**Annuity: Delayed Annuity**• What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?**Annuity: Annuity Due**• You just won the lottery! You will get paid $50,000 a year with the first payment being made immediately. You will receive 20 payments total. Assume a discount rate of 8% What is the real value to you?**Growing Annuity**A growing stream of cash flows with a fixed maturity. The formula for the present value of a growing annuity:**Growing Annuity**• You are evaluating an income property that is providing increasing rents. Net rent is received at the end of each year. The first year's rent is expected to be $8,500 and rent is expected to increase 7% each year. Each payment occur at the end of the year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%?**Growing Annuity**• On Calc • CF0=0, CF1=8500, CF2=9095, CF3=9731.65, CF4=10412.87, CF5=11141.77, I=12, NPV=?**Growing Annuity**A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?**What is a Firm Worth?**• Economics • Price of any asset is equal to PV of expected future cash flows • The tricky part is determining the size, timing and risk of those cash flows • Example • A firm is expected to generate net CFs of $5,000 in Yr1, $2,000 for each of the next 5 years. The firm can be sold for $10,000 in Yr 7. The owners expect a 10% return. What is the value of the firm? • Second, suppose you could buy the firm today for $12,000. Should you acquire it?**Net Present Value**• The formula for the net present value of an investment that pays $C for N periods is: