NPV AND DISCOUNTINGReview ofGoal: 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 ev aluate e.g. investment decisionsExample: 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 = 366.91TIME VALUE OF MONEYThe Time Value of MoneyIn 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?2020,000$7,537.79 today1.05PV Valuing a Stream of Cash FlowsIf 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 sumExample: Valuing A Stream of Cash Flows (for Alex Rodriguez)Alex Rodriguez “$275 million” contract with the YankeesYear Salary"Signing bonus"With no discountingPresent value (at 10%)2008 272 29.02009 321 30.02010 321 27.32011 311 24.02012 291 20.52013 281 18.02014 253 15.82015 2110.82016 209.32017 208.5Total 26510 275 193.2The PV of payments to Rodriguez(the calculation is made in 2008) $275$264$254$245$236$228$220$213$206$199$193$187$1820% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%Perpetuities• A perpetuity is a stream of equal cash flow s 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: (growing perpetuity) CPVrgGrowing Annuities• A growing annuity is a stream of N growing cash flows, paid at regular intervalsN1111gPV= Cr - g rExample: Endowing a professor’s chair• You want to endow a chair for a finance professor at Illinois• You’d like to attr act 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:• You would need to donate $2.5 million to endow the chair.C $100,000PV $2,500,000r.04 Example: An endowed chair with inflation• The University has asked you to increase the donation to account for the eff ect 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.C $100,000PV $5,000,000r.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 tak e 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 tak e?Solution PV( 29-year annuity of $1million)  $1 million 10.081 11.0829 $1 million  11.16 $11.16 million todayFrom the formula for an annuity,Thus, the total present value of the cash flows is $1 million + $11.16 million = $12.16 million. So you should tak e $15 million now instead. We need to value the 30‐year payments: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•