Lecture 3 Managerial Finance FINA 6335 Ronald F. Singer

2-2 In Lecture 1 we concluded that the object of a manager is to attempt to make an individual's " wealth" as great as possible. Wealth

2-3 Consumption next year After Investment Notice: The investment increases wealth by $140. Wealth

2-4 If Benefits > Costs P.V. Future > Current Reduction Income In income P.V. Future > Investment Income P.V. Future - Investment > 0 Income NPV > 0 Net Present Value

2-5 Net Present Value = Present Value - Required Investment = Additional x Discount - Required Future Amount Factor Investment Example: Suppose there is a project requiring a $500 initial investment returning $800 in one year's time. At a discount rate of 25%, what is the Net Present Value of that investment? = 800 x = 800 x [0.80] = = 140 The individual's wealth increases by the net present value of investment Alternatively, Net Present Value shows the change in investor’s wealth Net Present Value

2-6 To maximize Stockholders' Wealth, take all Projects that increase Stockholders' Wealth, and reject all Projects that decrease Stockholders' wealth. Positive Net Present Value: Projects increase Stockholders' Wealth Negative Net Present Value projects Projects decrease Stockholders' Wealth Take all positive Net Present Value projects Reject all positive Net Present Value This is called the Net Present Value Rule Net Present Value Rule

2-7 Separation Principle Now consider the Separation Principle Regardless of their individual preferences, All stockholders will agree on the Best Investment Decisions The Best Investment Decision is one that maximizes the stockholders' wealth. Given his/her maximum wealth each stockholder can borrow and lend in the capital markets to arrive at the most preferred consumption pattern The Investment Decision is separate from the consumption decision - Thus the Separation Principle Why do we care about the Separation Principle?

2-8 Basic Concepts of Time Value of Money What is the time value of money? If I offered you either $6,000 or $6,500 which one would you choose? If I offered you $6,000 today or $6,500 in two years, which one would you choose? The first problem is easy: It involves two different amounts received at the same time. The second problem is more difficult, as it involves different amounts at different periods of time. The interesting part of finance is that it involves cash flows that are received at different points in time. We must devise a way of "comparing" these two different amounts to be able to make a choice between them, (or to add them up).

The Timeline A timeline is a linear representation of the timing of potential cash flows. Drawing a timeline of the cash flows will help you visualize the financial problem.

The Timeline (cont’d) Assume that you loan $10,000 to a friend. You will be repaid in two payments, one at the end of each year over the next two years.

The Timeline (cont’d) Differentiate between two types of cash flows –Inflows are positive cash flows and up arrows. –Outflows are negative cash flows, which are indicated with a – (minus) sign and down arrows.

The Timeline (cont’d) Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments. The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow. Timelines can represent cash flows that take place at the end of any time period.

Three Rules of Time Travel Financial decisions often require combining cash flows or comparing values. Three rules govern these processes.

2-14 Present Value of a Lump Sum and the Discounting Process There are three ways of computing the Present Value: 1. Use the Formula: Where R is the discount rate T is the number of periods to wait for the Cash Flow 2. Use a spreadsheet 3. Use a Financial Calculator

2-15 Future Value of a Lump Sum and the Compounding Process There are three ways of computing the Future Value: 1. Use the Formula: Where: R is the discount rate T is the number of periods to wait for the Cash Flow 2. Use a Spreadsheet 3. Use a Financial Calculator

2-16 The Relationship Between Present and Future Value PV = FV T ( 1 ) = FV T x (1/(1+R) T ) ( 1 + R) T FV =PV(1+R) T =

2-17 Financial calculators recognize the formulas and relationship above so that they calculate present and future values by balancing the above equations. Typical layout: N I/YR PV PMT FV Now the idea here is that given N (number of periods) and I/YR the interest rate per period, then the equation PV = FV N * (1/(1+I/YR) T must hold. Example Using Calculator

2-18 In the above example perform the following operations: Enter Press calculator shows 10 N I/YR FV PV Similarly if you want the future value of 1000 after 5 years at 8% 5 N PV FV Financial Calculator

2-19 Examples What is the Present Value of $1 received five years from today if the interest rate is 12%? Using the formula: Using the Spreadsheet =PV(0.12, 5, 0,1) Using the Calculator: 5 N 12 I/Y  FV PV

2-20 Examples What is the Future Value of $1 in five years if the interest rate is 12% ? Using the formula: Using the spreadsheet formula FV(RATE, NPER, PMT,PV) = FV(.12,5,0,1) Using the calculator: 1 PV FV

2-21 Future Value of a Lump Sum and the Compounding Process What is the future value of $100 in 3 years if the interest rate is 12% ? Approach 1: Keep track of dollar amount being compounded: Period 1 $ $100.00(0.12) OR $100(1.12) = $ $ $112.00(0.12) OR $100(1.12) 2 = $ $ $125.44(0.12) OR $100(1.12) 3 = $ Approach 2: Keep track of number of times interest is earned: Period 1 $100(1.12) $100(1.12) = $ $100(1.12)(1.12) $100(1.12) 2 = $ $100(1.12)(1.12)(1.12) $100(1.12) 3 = $ Notice that the process earns interest on interest. This is called compounding. The further out in the future you go the more important is the effect of compounding

2-22 Simple Vs. Compound Interest Simple Interest: Is the amount earned on the original principal Compound Interest: Is the amount earned as interest on interest earned. Note: Future Value(R%,2) = Present Value (1+R) 2 = Present Value (1 + 2R + R 2 ) Original PrincipalSimple Interest Compound Interest Future Value(R%,3) = Present Value (1 + 3R + 3R 2 + R 3 )

2-23 Valuing Any Financial Security First: What is a "financial security?“ A Financial Security is a promise by the issuer (usually a firm or government agency, but could be an individual) to make some payments, to the holder of the security, under certain conditions, over some specific period of time in the future.

2-24 Example Suppose there is a financial security promising to make specific payments over this coming year. At the "appropriate" interest rate of 10%, you determine that the Present Value of these payments is $110. Suppose that you can purchase this security at the current price of 100. Is this a "good" buy? Do you need some additional information? Does it depend on the individual's feelings and desires? (i.e. utility function) Efficient Market The Purchase = -100 Receive = +110 NPV = 10 c1 c0

2-25 Present Value of an Annuity

2-26 Example 4.7 (cont'd)

2-27 Example 4.7 (cont'd) Future Value of an Annuity

2-28 Example 4.7 Financial Calculator Solution Since the payments begin today, this is an Annuity Due. –First: –Then: $15 million > $12.16 million, so take the lump sum.

2-29 Finding the Present Value of an Uneven Cash Flow Stream Typically, a security will have an uneven cash flow stream over time, and the problem is to determine the present value of that cash flow stream. Suppose we have the following cash flow stream, and that the "interest rate" is 10%: Time Line: There are several ways of finding this present value:

2-30 Method 1: The Sum of the Present Values of each payment: └─────┴─────┴─────┴─────┴────┘ $800 X (.909) = $727.20──┘ $300 X (.826) = ────────┘ $200 X (.751) = ──────────────┘ $200 X (.683) = ────────────────────┘ $200 X (.621) = ─────────────────────────┘ Present Value $1,386.00

2-31 Method 2: Recognize that this is a combination of two lump sums and an annuity that begins two periods in the future and lasts for three periods. That is: └─────┴─────┴─────┴─────┴────┘ is equivalent to Plus └─ ─┴──┴──┴───┴───┘ └─ ─┴──┴──┴───┴───┘ Which in turn is equivalent to:

2-32 Present Values: └─────┴─────┴─────┴─────┴────┘ PV(.10, 1, 0, 800) $ PLUS PV(.10, 2, 0, 300) PLUS PLUS └─────┴─────┴─────┴─────┴────┘ PV(.10, 5, 200 ) MINUS PV(.10,2,200) └─────┴─────┴─────┴─────┴────┘ TOTAL $1, Using Financial Calculator: $1, (Hewlett Packard)

2-33 Method 3: Treat this as two lump sum payments plus an annuity that begins in period └─────┴─────┴─────┴─────┴────┘ Is equivalent to └─────┴─────┴─────┴─────┴────┘ PV(10%,3,200) Total $1,386.28

2-34 Perpetuities Some Securities last "forever," and generate the equivalent of a perpetual cash flow. Clearly, we cannot evaluate these perpetual cash flows in the conventional manner. We do however, have formulas which allows us to evaluate these cash flows. A Perpetuity is a series of equal payments that continues forever └────┴────┴────┴────┴────┴── ──┴────┴── ──┘ The Present Value of a Perpetuity is: How much would you pay for this bond?

2-35 Perpetuities Example: A British Government Bond pays 100,000 pounds a year forever (Consul). The market rate of interest is 8%. How much would you pay for this bond? PV = Cash Flow = 100,000 = 1,250,000 of perpetuity r 0.08 How much is the bond worth if the first coupon is payable immediately? PV of Bond = PV Immediate Payment Plus Value of Perpetuity = 1,250, ,000 = 1,350,000

2-36 Growing Perpetuity If the cash flow grows at a constant rate, then the perpetuity is called a growing perpetuity where CF 1 = Cash flow next year r = Market rate interest g = Constant Growth rate How much would you pay for the previous bond if the cash flows grow at 5% starting at 105,000 next year PV of growing = 105,000 = 3,500,000 perpetuity = 105,000 = 3,500,

2-37 Annuity Recall: An annuity provides equal cash flows for a fixed number of periods C 1 C 2 C C N │__________│___________│________│_________│ 0 Notice that the first payment starts next period. The value of an annuity is the difference in the value of two perpetuities, one that starts now and one that starts N-periods from now. Present Value Present Value Present Value of N-period = of Perpetuity - of Perpetuity Annuity That starts now that starts N- Periods from now What is the Equation for an Annuity?

2-38 A Clarification on Different Compounding Periods We have assumed that we are dealing with compounding only once a year. But what happens when the compounding is done more than annually? Given the periodic interest rate, you can use the tables to find the present value of a single payment, the present value of a periodic annuity, as well as the future values. Example: Suppose you will receive $1,000 per month for 12 months. at an annual (simple) interest rate of 18%, compounded monthly, what is the present value of this cash flow?

2-39 Definition of Rates Periodic Interest Rate: the interest earned inside the compounding period. Example: 18% compounded monthly has a periodic rate of 1.5% Nominal (Simple) Interest Rate: interest is not compounded. the amount you would earn, annually, if the interest were withdrawn as soon as it is received. (This is the APR (Annual Percentage Rate) you find on credit card and bank statements) Example: Invest $1,000 today at 18%, APR paid monthly. you would have $1,180 at the end of one year.

2-40 Definition of Rates Effective Interest Rate: the annual amount you would have if the interest is allowed to compound. (This is the actual interest earn over the year allowing for compounding) Example: invest $1,000 today at 18%, compounded monthly. Then the periodic interest rate is 1.5% per month. The nominal rate is 18%. Then, allowing for compounding, the effective rate is: ┴─── 1 ┴─── 2 ┴─── 3 ┴─── 4 ┴─────────────┴───┘ (1+.015) = 19.56% thus if you invested $1,000 at 18% compounded monthly you would have $1, To convert from simple rates to effective rates, use the formula: effective rate = (1 + r/m) m – 1r is the simple rate m is the number of compounding periods per year.

2-41 Now the present value of monthly cash flow over one year is calculated as the present value of an annuity, received for 12 periods at a periodic rate of 1.5%. Thus you want to use 12 as N and 1.5% as I%YR in the calculator. (Alternatively, if your calculator has an option to set the payments per year you could set it to 12 but this is not recommended. There is a tendency to forget to reset it to annual payments for the next problem, and what to use for N gets confusing)

2-42 What if, at the same compounding interval, you received only 2 cash flows of 6,000 each in month 6 and month 12? │ │ │ │ 0───1───2───3───4───5────6───────-- 11────12 Finally, what if the compounding of 18% occurs only twice per year? effective rate present value

2-43 To pay an Annual Interest Rate r, compounded m times during the year means pay r/m for m-times in a year Example, to pay 10% compounded quarterly means 2.5% is paid 4 times a year The Effective Interest Rate is = ( ) 4 – 1 = or 10.38% The Effective Interest Rate exceeds 10% since interest is paid on interest. When the compounding interval approaches zero, we have continuous compounding (1 + r/m) m - 1 = e r - 1 = (2.7183) r - 1.

2-44 If 1 dollar is continuously compounded at rate r, at the end of the year 1 dollar will grow to e r where e = (e is the base of the natural log) after n years, 1 dollar will grow to = e nr Example: ABC bank offers 10.2% compounded quarterly. XYZ bank offers 10.1% interest continuously compounded. which is better for you? in ABC bank 1 dollar deposit grows to, after 1 year, = ( /4) 4 = in XYZ bank 1 dollar deposit after 1 year grows to = e.101 = therefore, even though XYZ only pays 10.1%, continuous compounding makes XYZ interest a better deal. notice that both of these offers are better than 10.5% simple