Review of NPV AND DISCOUNTING Goal valuing costs vs benefits A decision will increase the market value of the firm when the value of the benefits exceeds the value of the costs But how to value compare costs and benefits that occur at different times that are uncertain or have different risks Main tool NPV Net Present Value compares the present value of cash inflows benefits to the present value of cash outflows costs NPV PV Benefits PV Costs The NPV Decision Rule The NPV rule implies that we should Accept all projects with NPV 0 Reject all projects with NPV 0 accepting them would reduce the value of the firm whereas rejecting them has no cost NPV 0 Allows us to evaluate e g investment decisions Example NPV Would you be willing to pay 5 000 for the following stream of cash flows if the discount rate is 7 The present value of the benefits is 3000 1 05 2000 1 05 2 1000 1 05 3 5 366 91 The present value of the cost is 5 000 because it occurs now The NPV PV benefits PV cost 5 366 91 5 000 366 91 TIME VALUE OF MONEY The Time Value of Money In general a dollar today is worth more than a dollar in one year and a dollar tomorrow is worth less than a dollar today EXAMPLE You are considering investing in a savings bond that will pay 20 000 in twenty years If the interest rate is 5 per year what is the bond worth today PV 20 000 1 05 20 7 537 79 today Valuing a Stream of Cash Flows If there are many cash flows in the future we can discount each one at the appropriate interest rate to get the present value of the sum Example Valuing A Stream of Cash Flows for Alex Rodriguez Alex Rodriguez 275 million contract with the Yankees Year 2008 Salary 27 2009 Signing bonus With no discounting Present value at 10 2 29 0 32 1 30 0 2010 32 1 27 3 2011 31 1 24 0 2012 29 1 20 5 2013 28 1 18 0 2014 25 3 15 8 2015 21 10 8 2016 20 9 3 2017 20 8 5 Total 265 10 275 193 2 The PV of payments to Rodriguez the calculation is made in 2008 275 264 254 245 236 228 220 213 206 199 193 187 182 0 1 2 3 4 5 6 7 8 9 10 11 12 Perpetuities A perpetuity is a stream of equal cash flows that occur at regular intervals and last forever Note The first cash flow does not occur immediately but at the end of the first period Present Value of a Perpetuity PV perpetuity Annuities An annuity is a stream of N equal cash flows paid at regular intervals Difference between an annuity and a perpetuity is that an annuity ends after N payments Present Value of an Annuity PV annuity 1 Growing Perpetuities A growing perpetuity is a stream of cash flows that occur at regular intervals and grow at a constant rate forever For example a growing perpetuity with a first payment of 100 that grows at a rate of 3 has the following timeline C PV growing perpetuity r g Growing Annuities A growing annuity is a stream of N growing cash flows paid at regular intervals N 1 1 g PV C 1 r g 1 r Example Endowing a professor s chair You want to endow a chair for a finance professor at Illinois You d like to attract a prestigious faculty member so you d like the endowment to add 100 000 per year to the faculty member s resources salary conference travel databases etc You expect to earn a rate of return of 4 annually on the endowment How much will you need to donate to fund the chair Solution The timeline of the cash flows looks like this This is a perpetuity of 100 000 per year The funding you would need to give is the present value of that perpetuity From the formula PV C 100 000 2 500 000 r 04 You would need to donate 2 5 million to endow the chair Example An endowed chair with inflation The University has asked you to increase the donation to account for the effect of inflation which is expected to be 2 per year How much will you need to donate to satisfy that request Solution The timeline of the cash flows is now The cost of the endowment will start at 100 000 and increase by 2 each year This is a growing perpetuity PV C 100 000 5 000 000 r 04 02 You would need to donate 5 0 million to endow the chair Example Present Value of Annuity You are the lucky winner of the 30 million state lottery You can take your prize money either as a 30 payments of 1 million per year starting today or b 15 million paid today If the interest rate is 8 which option should you take Solution We need to value the 30 year payments From the formula for an annuity 1 1 1 PV 29 year annuity of 1million 1 million 0 08 1 0829 1 million 11 16 11 16 million today Thus the total present value of the cash flows is 1 million 11 16 million 12 16 million So you should take 15 million now instead THE PRICE OF RISK P 77 81 IN THE TEXTBOOK Risky Cash Flows Assume there is an equal probability of either a weak economy or strong economy next year The one year risk free interest rate is 4 The following are the Cash Flows in of 1 A Risk Free Security 2 A Risky Security Cash flow in one year Weak economy Strong economy Risk free security 1100 1100 Risky security 800 1400 Risky Cash Flows cont d The Risk free Security always pays 1100 We can value it by discounting using the one year risk free interest rate Price risk free security 1100 1 04 1056 For the Risky Security we can calculate the expected cash flow Expected cash flow risky security 800 1400 1100 To get the current value of this security we need to discount the expected cash flow But what is the appropriate discount rate Risk Aversion and the Risk Premium Risk Aversion Investors prefer to have a safe income rather than a risky one of the same average amount Risk Premium The additional return that investors expect to earn to compensate them for a security s risk When a cash flow is risky to compute its present value we must discount the cash flow we expect on average at a rate that equals the risk free interest rate plus an appropriate risk premium Risky Cash Flows cont d The appropriate discount rate is determined by the risk free rate PLUS a risk premium The risk premium depends on how the cash flows of the risky security correlate with other risks that investors are concerned about e g how the cash flows …