Finance, Financial Markets, and NPV

1. Finance, Financial Markets, and NPV First Principles

2. Finance • Most business decisions can be looked at as a choice between money now versus money later. • Finance is all about how special markets, the financial markets, help people make themselves better off by moving money across time. • As simple as this sounds, the related concepts seem complex and the markets appear complicated enough to require some introduction. • We will develop the important concept that helps us “keep score” in an honest way while we think about moving money across time.

3. Example • Suppose right now I have $100 but I am not planning to use it until dinner tomorrow. • You on the other hand have the good fortune of having a dinner date tonight but the misfortune of not getting paid until tomorrow. Oh the humiliation. • You offer a solution to the terrible problem. • You suggest I give you the $100 today and tomorrow night you give me $100 back. • This makes you better off by avoiding the humiliation and allowing you to engage in a desired activity, but what about me?

4. Example cont… • One thought is of course that idiot professors don’t really matter in the face of your humiliation. But let’s put that aside for now. • How can we make it so we are both better off and what do we call such an arrangement? • What if you can’t find me or someone as kind and generous? • How can we make the process easier? • Can we keep lots of people from wasting lots of time looking for partners? • Can we balance those who want to borrow and those who want to lend?

5. Example Concluded • Simple as it was our example enabled us to introduce the following fundamental ideas. • Financial market. • Time value of money. • Interest rate (the price in this market). • Financial Intermediaries. • Market clearing. • Equilibrium interest rates.

6. Maintained Assumptions • For the moment, in order to simplify the analysis, we will assume: • Perfect certainty • Perfect capital markets • Information freely available to all participants. • Equal access. • All participants are price takers. • No transactions costs or taxes. • Investors are rational. • We are in a one-period world. • The last assumption will be dropped quickly with the first to follow soon.

7. Money & Time • An important message of the example is that money must be thought of as having two “units.” • Currency ($, £, ¥) is of course the commonly identified unit but time (date received) must also be established before we can determine value. • Example: Your employer offers you a bonus for excellent performance. You may choose between $10,000 today or $12,500 in one year (after the firm does its IPO and has more liquidity). • Compare future values.

8. Present Value • Compare $100 today versus $107 in one year if you can earn 6% interest. • Compare them today instead of in one year. • Rearrange this to find:

9. Present Value Examples • You just won the new Colorado lottery scratch game. The lottery office offers you $50,000 today or $55,000 if you wait a year. The current interest rate is 7%, what do you do? How much money (present value) will a poor choice cost you? • Your rather odd uncle Ralph has set up a trust in your name that will pay you $1,300,000 in one year. How much can you borrow against this trust if the current interest rate is 9%?

10. The First Principle • The financial markets provide information that enables us to evaluate choices (investment opportunities) both as individuals and as corporations. • An investment opportunity is a way that individuals or firms adjust their consumption (spending) across time. • Since financial markets also represent a way to accomplish this important task we know that: • An investment project can be worth undertaking only if it represents a better option than is available in the financial markets. • Close substitutes provide the basis for comparison. • Lottery example and opportunity costs.

11. Net Present Value • In order to determine whether you are better off making an investment or not we can use the idea of discounting future cash flows and comparing the present value of the future cash in-flows to the current cost. • This is net present value. It is a powerful decision making tool. If NPV is positive what does that tell us? If it is negative? • The interest rate that sets the NPV equal to zero is called the internal rate of return or the yield of the investment. (More on this later.)

12. Net Present Value Example • Do you take a riskless investment that requires $217 to undertake and will payout $230 in one year if the bank is offering you a 5% CD? • NPV: -$217 + $230/1.05 = $2.05 ($, time 0) > 0. • What if you put the $217 in the CD: $217(1.05) = $227.85 so the comparable alternative has a lower future payout. • Comparing directly: $230 - $227.85 = $2.15 ($, time 1). • Note: $2.05(1.05) = $2.15, i.e., the approaches are making exactly the same comparison, NPV does it at time zero. Comparing future values just compares value at time one.

13. The Two-Period Case • One payment two years from now: • We talked about getting cash next year, what if it doesn’t come till two years from now? • One illustration: if is the time 0 value of a cash flow at time 1, is the time 1 value of a time 2 cash payment. We already know how to change a time 1 value to a time 0 value:

14. A Two-period Example • A second view: • If you have $100 cash today, a bank will give you 7% interest per year, and you leave the money in the bank for two years, how much will you have? • Answer: $100(1.07)(1.07) = $114.49. So $114.49 is the future (2 year) value of $100 of current cash (at 7%). Algebra tells us that the present value of the future $114.49 must be $100. Calculate this as $114.49/(1.07)2 = $100. • Notation: PV(C2) = C2/(1+r)2 • Generally: PV(Ct) = Ct/(1+r)t and FVt(C0) = C0(1+r)t

15. Multi-period Examples • If you invest $15 for 20 years at 9% with no withdrawals what will be the final balance (future value)? $15(1.09)20 = $84.07 • If you will receive $25,000 in 6 years and the relevant interest rate is 11%, what is the present value of this future payment?

16. Simple vs. Compound Interest • Suppose that I have had some finance training and I know better than to stuff my $100,000 under my mattress. Instead I put it in the bank for 12 years at an 8% interest rate. Not having stayed till the end of the course, however, at the end of each year I withdraw the interest I earn and stuff it under my mattress. How much will I have at the end of the 12 years? • I’ll still have my $100,000 of principal and at the end of each of the 12 years I will have put $100,000(.08) = $8,000 under the mattress, leaving $100,000 + 12*$8,000 = $196,000. • If I made no withdrawals during the 12 years I’d have $100,000(1.08)12 = $251,817.01 • What drives the $55,817 difference?

17. The Present Value of a Series of Future Cash Flows • What happens if we have an investment that provides cash flows at many future dates? • Its very easy, discount each of the future cash flows to the present, then just add them up. • We can (and should) do this because once we have discounted them, their present values all represent cash values today. Since all the values are as of the same date they can be directly compared (added). • In other words, proper discounting restates the future cash flows as their equivalent amounts at a common point in time. They are (only) then directly comparable.

18. Present Value of a Series of Future Cash Flows • Those are the words, here are the symbols: • For NPV the adjustment is obvious:

19. Multi-Period NPV Example • Ralph, your brother-in-law, has offered you an investment opportunity. For an investment of $117,000 you will own half of a ferret ranch located outside of Flagstaff AZ. • You are convinced that the ranch will generate enough income to payout a total of $80,000 in one year, $95,000 in two years and $150,000 in three years. • The current rate of interest is 5%. • What is the wise investment decision?

20. Alternate Compounding Periods • Interest is sometimes “compounded” over periods other than a year. In terms of “bank account” examples, this simply means that interest is credited to the account more frequently than once a year. • Caveat: All of the time value of money formulas we will see use the implicit assumption that the compounding interval is the same as the payment interval. e.g.: • Mortgage loans call for monthly payments. • Bonds make coupon payments semiannually. • If this is not true you must make adjustments.

21. Alternate Compounding Periods (Cont.) • Let m denote the number of compounding intervals per year, n the number of years, and r be the stated annual rate of interest. • The relation between present and future values is given as: • FVn = PV(1 + r/m)n×m • E.g., if PV = 1000, r = .12 and m = 1 then FV2 is: • FV2 = 1000(1 + .12)2×1 = $1254.40, • while if m = 4 (quarterly compounding), then • FV2 = 1000(1 + .12/4)2×4 • FV2 = 1000(1 + .03)8 = $1266.77

22. Example • Find the PV of $500 to be received in 5 years, with: • 12% stated annual rate, annual compounding,. • 12% stated annual rate, semiannual compounding, • 12% stated annual rate, quarterly compounding,

23. Stated And Effective Annual Rates • Notice that the use of more frequent compounding acts as if to (or effectively) increase(s) the interest rate. • The Effective Annual Rate (EAR) is the annual interest rate that would produce the same answer, with annual compounding, as is obtained with more frequent compounding. It can be obtained by: • EAR = (1 + r/m)m - 1 • so if r = .12 and m = 4, then EAR = (1.03)4 - 1 = .1255. • The effective annual rate is the rate that if you earned it for a year with annual compounding, you end up with the same money as you would under frequent compounding. • Note the appropriate discount rate in any application is always an effective rate. At times this may also be a stated rate.

24. Example • A bank quotes a mortgage rate of 8% (the stated annual rate), but will compute monthly loan payments using standard time value formulas. This implies monthly compounding. What is the effective annual interest rate on the loan? So the loan effectively costs you 8.30% per year for every dollar you borrow for a year.

25. Valuing Streams of Structured Future Cash Flows • Now we are going to discuss the valuation of certain highly structured cash flow streams. • The resulting valuation formulas are useful for simplifying the analysis of certain situations. • Pay attention to the exact timing of the cash flows, the formulas don’t work unless you get this right. • Drawing diagrams of the cash flows can be useful.  • These formulas can make life easier and so are worth understanding.

26. C C C … 0 1 2 3 Perpetuity • A stream of equal payments, starting in one period, and made each period, forever. Forever?? • Remember, this gives the value of this stream of cash flows as of time 0, one period before the first payment arrives.

27. C1 C2 = C1(1+g) C3 = C1(1+g)2 … 0 1 2 3 Growing Perpetuity • A growing perpetuity is a stream of periodic payments that grow at a constant rate and continue forever. • The present value of a perpetuity that pays the amount C1 in one period, grows at the rate g indefinitely, when the discount rate is r, is:

28. … … Examples Perpetuity: $100 per period forever discounted at 10% per period $100$100$100 0 1 2 3 PV = C/r = $100/0.10 = $1,000 Growing perpetuity: $100 received at time t = 1, growing at 2% per period forever and discounted at 10% per period $100 $102 $104.04 0 1 2 3 PV = C1/(r –g ) = $100/(0.10 – 0.02) = $1,250

29. Verification of the Perpetuity Example Answers Place the present value in a bank account, and recreate the payments. Let’s stop at 4 years since “forever” would take a while. Note that the account balance is growing. At what rate? Why must this happen?

30. 2 2 3 3 1 1 0 0 C C C C C Annuities • An annuity is a series of equal payments, starting next period, and made each period for a specified number (3) of periods. • If payments occur at the end of each period (the first is one period from now) it is an ordinary annuity or an annuity in arrears. • If the payments occur at the beginning of each period (the first occurs now) it is an annuity in advance or an annuity due. C

31. Valuing Annuities • We can do a lot of grunt work or we can notice that a T period annuity is just the difference between a standard perpetuity and one whose first payment comes at date T+1. • The present value of a T period annuity paying a periodic cash flow of C, when the discount rate is r, is: • If we have an annuity due instead, the net effect is that every payment occurs one period sooner, so the value of each payment (and the sum) is higher by a factor of (1+r). • Or we can add C to the value of a T-1 period annuity.

32. Annuity Example • Compute the present value of a 3 year ordinary annuity with payments of $100 at r = 10%. or,

33. Annuity Due Example • What if the last example had the payments at the beginning of each period not the end? • Or, • Or,

34. Example: A five year annuity paying $2000 per year, with r = 5%. • Valuing the payments individually we get: • Using the annuity formula we get:

35. Alternatively, suppose you were given $8,658.95 today instead of the annuity • Notice that you can exactly replicate the annuity cash flows by investing the present value to earn 5%. • This again demonstrates that present value calculations provide a literal equality, in that the future cash flows can be converted into the present value and vice versa, if (and only if) the selected discount rate is representative of actual capital market conditions.

36. C1 C1(1+g) C1(1+g)T-2 C1(1+g)T-1 … 0 1 2 T-1 T Growing Annuities • A stream of payments each period for a fixed number of periods where the payment grows each year at a constant rate.

37. 500 500(1.02) 500(1.02)18 500(1.02)19 … 0 1 2 19 T=20 Example • What is the present value of a 20 year annuity with the first payment equal to $500, where the payments grow by 2% each year, when the interest rate is 10%?

38. A Valuation Problem What is the value of a 10-year annuity that pays $300 a year at the end of each year, if the first payment is deferred until 6 years from now, and if the discount rate is 10%? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 300 300 300 • • • • • • 300 The value of the annuity payments as of five years from now is: Now discount this equivalent payment back 5 years to time zero:

39. Application: Retirement Planning • You have determined that you will require $2.5 million when you retire 25 years from now. Assuming an interest rate of r = 7%, how much should you set aside each year from now till retirement? • Step 1: Determine the present equivalent of the targeted $2.5 million. PV = $2,500,000/(1.07)25 PV = $2,500,000/5.42743 = $460,623 • Step 2: Determine the annuity that has an equivalent present value:

40. Retirement Planning cont… • Now suppose that you expect your income to grow at 4% and you want to let your retirement contributions grow with your earnings. How large will the first contribution be? How about the last?

41. A College Planning Example – Outside Class • You have determined that you will need $60,000 per year for four years to send your daughter to college. The first of the four payments will be made 18 years from now and the last will be made 21 years from now. You wish to fund this obligation by making equal annual deposits at the end of each of the next 21 years. You expect to earn 8% per year on the deposits. • Step 1: Determine the t = 17 value of the obligation. • Step 2: Determine the equivalent t = 0 amount.

42. College Planning cont… • Step 3: Determine the 21-year annuity that is equivalent to the stipulated present value.

43. Present Value Homework Problem • Your child will enter college 5 years from now. Tuition is expected to be $15,000 per year for (hopefully) 4 years (t=5,6,7,8). • You plan to make equal yearly deposits into an account at the end of each of the next 4 years (t=1,2,3,4) to fund tuition. The interest rate is 7%. • How much must you deposit each year? • What if tuition were growing over the 4 years? • Think about: • How to decide whether/when to refinance your house?

44. Leasing vs. Buying a Car – Outside Class • Saab 9-3 five-door/five-speed, CD, Air, Prestige • Lease Terms (Source WSJ 8/6/98) • Up front fees $1,748 including down payment (due at t=0). • Refundable security deposit of $300 • 38 monthly payments of $299 (t=1, 2, …, 38) • Residual value of $16,454 • Annual interest rate: 8%

45. T=1 T=2 T=35 T=38 T=0 ... -1,748 -300 -2,048 -299 -299 -299 -299 -299 -299 -16,454 300 -16,154 Payments are an annuity: Present value of residual value and security deposit:

46. Lease vs. Buy cont… • The present value of the lease payments is: -$2,048 + (-$10,001) + (-$12,534) = -$24,583 • What does this number mean? • If we could purchase the car for less than $24,583 we are better off buying. • When considering the alternative of purchasing the car, does whether we pay cash or borrow to make the purchase affect the lease/buy decision?