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**Chapter 4: Discounted cash flow valuation**

Corporate Finance Ross, Westerfield, and Jaffe

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**Outline 4.1 Future value 4.2 Present value 4.3 Other parameters**

4.4 Multiple cash flows 4.5 Comparing rates 4.6 Loan types

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**Definitions Present value (PV): earlier money on a time line.**

Future value (FV): later money on a time line. Interest rate (i), e.g., discount rate, required rate, cost of capital: exchange rate between earlier money and later money. The number of time periods on a time line (N). PV FV: “time value of money” via the exchange rate, i.e., interest rate, i.

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**End-of-period cash flows**

By default, in this class cash flows occur at the end of each period. If cash flows occur at the beginning of each period, it will be explicitly specified.

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**One equation; one solution**

In general, we have one equation: 0 = f (PV, FV, i, N). Since we have only one equation, we can only allow for one unknown parameter (variable). That is, if we’d like to calculate the value of a parameter, say FV, the values of the remaining parameters, i.e., PV, i, and N, need to be known.

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FV example I Suppose that we buy a 12-month CD at 12% annual interest rate for $10,000. FV = PV (1 + i)N = $10,000 (1 + 12%)1 = $11,200.

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**Do not compare apples with oranges**

Why N = 1 while the CD matures in 12 months? The key is that: The time frequency of i and N must be the same. If we use annual interest rate, then we need to measure the investment period using the unit of year. In this case, 12 months equal a year; so N = 1. What is the value of N if the example provided us monthly interest rate, say 0.96% per month? Any volunteer?

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Compounding Of course, the previous formula, FV = PV (1 + i)N, is based on the notion of compounding. Compounding: the process of accumulating interest on an investment over time to earn more interest. Earn interest on interest. Reinvest the interest. A popular method.

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FV example II Deposit $50,000 in a bank account paying 5%. How much will you have in 6 years? Formula: FV = PV (1 + i)N = $50,000 (1 + 5%)6 = $67,000. Financial table (Table A.3): FV = $50,000 = $67,000. Financial calculator: 6 N; 5 I/Y; PV; CPT FV. The answer is FV = -67, Ignore the negative sign.

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**Texas Instruments BAII Plus (keys)**

FV: future value. PV: present value. I/Y: period interest rate. Interest is entered as a percent. N = number of time periods. Clear the registers (CLR TVM, i.e., 2nd FV) after each calculation; otherwise, your next calculation may come up with a wrong answer.

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FV example, III Jacob invested $1,000 in the stock of IBM. IBM pays a current dividend of $2 per share, which is expected to grow by 20% per year for the next 2 years. What will the dividend of IBM be after 2 years? Formula: FV = PV (1 + i)N = $2 (1 + 20%)2 = $2.88. Table A.3: FV = $2 = $2.88. Calculator: 2 PV; 20 I/Y; 2 N; CPT FV. The answer is

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Discounting Discounting: the process of calculating the present value of future cash flows. We call i the discount rate when we try to solve for present value. Depending on the question, this rate can be interest rate, cost of capital, or opportunity cost.

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PV example, I Suppose that you need $4,000 to pay your tuition. 1-year CD interest rate is 7%. How much do you need to put up today? Formula: PV = FV / (1 + i)N = $4,000 / (1 + 7%)1 = $3,738.3. Table A.1: PV = $4,000 = $3,738.4. Calculator: 4000 FV; 7 I/Y; 1 N; CPT PV. The answer is -3,

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PV example, II Suppose that you are 21 years old. Your annual discount (return) rate is 10%. How much do you need to invest today in order to reach $1 million by the time you reach 65? Formula: PV = FV / (1 + i)N = $1,000,000 / (1 + 10%)44 = $15,091. Table A.1 does not have the present value factor for N = 44. This is the limitation of using a financial table. Thus, we will focus on the other 2 methods in the following discussions. Calculator: FV; 10 I/Y; 44 N; CPT PV. The answer is -15,

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PV relationship, I Holding interest rate constant – the longer the time period, the lower the PV. What is the present value of $5,000 to be received in 5 years? 10 years? The discount rate is 8% 5 years: 5 N; 8 I/Y; 5000 FV; CPT PV. The answer is PV = -3, 10 years: 10 N; 8 I/Y; 5000 FV; CPT PV. The answer is PV = -2,

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PV relationship, II Holding time period constant – the higher the interest rate, the smaller the PV. What is the present value of $5,000 received in 5 years if the interest rate is 10%? 15%? 10%: 10 I/Y; 5 N; 5000 FV; CPT PV. The answer is PV = -3, 15%: 15 I/Y; 5 N; 5000 FV; CPT PV. The answer is PV = -2,

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**The other parameters Recall that 0 = f (PV, FV, i, N).**

We can find the value of i or N as long as we know about the values of the other parameters. The easiest way is to use a financial calculator. They are formulas, i.e., analytical solutions, for i and N as well. But these are not the focus of the course.

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Interest rate example Suppose that you deposit $5,000 today in a bank account paying interest rate i per year. If you reach $10,000 in 10 years, what rate of return are you being offered? Calculator: 5000 PV; FV; 10 N; CPT I/Y. The answer is I/Y = Note that for entering FV, this is the sequence: /– FV.

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Time period example Suppose that you have $10,000 today. You want to retire as a millionaire. The annual rate of return that you can earn on the market is 10%. In how many years can you retire? Calculator: PV; FV; 10 I/Y; CPT N. The answer is: N =

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Multiple cash flows When there are multiple cash flows need to be discounted or compounded, the PV or FV of multiple cash flows are simply the sum of individual PV’s or FV’s, respectively.

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**Multiple cash flow example**

Dennis has won the Kentucky State Lottery and will receive $2,000 (cash flow 1)in a year and $5,000 (cash flow 2) in 2 years. Dennis can earn 6% in his money market account, so the appropriate discount rate is 6%. PV = PV1 + PV2 = $2,000 / (1 + 6%)1 + $5,000 / (1 + 6%)2 = $6,337. That is, Dennis is equally inclined toward receiving $6,337 today and receiving $2,000 and $5,000 over the next 2 years.

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**Multiple cash flow example, Excel**

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Annuity (Ordinary) Annuity: a level of stream of cash flows for a fixed period of time (multiple, equal cash flows). Same dollar amount per period, making calculation much easier. FV = C { [ (1 + i)N – 1] / i }. PV = C { [ 1 – 1 / (1 + i)N ] / i }. C is the fixed periodical payment.

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Annuity PV example Suppose that you want to buy a car. You can afford to pay $632 per month for the next 48 months. You borrow at 1% per month for 48 months. How much can you borrow? Formula: PV = C { [ 1 – 1 / (1 + i)N ] / i } = $632 { [ 1 – 1 / (1 + 1%)48 ] / 1% } = $24,000. Calculator: 632 PMT; 1 I/Y; 48 N; CPT PV. The answer is: PV = -23, In the solution manual (textbook), PVIFA (PVIA) stands for the PV of an annuity. PVIFA(i,N) = [ 1 – 1 / (1 + i)N ] / i .

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Let’s work on this Questions and Problems #28

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Annuity FV example Suppose that you put $3,000 per year into a Roth IRA. The account pays 6% per year. How much will you have when you retire in 30 years? Formula: FV = C { [ (1 + i)N – 1] / i } = $3,000 { [ (1 + 6%)30 – 1] / 6% } = $237, Calculator: 3000 PMT; 6 I/Y; 30 N; CPT FV. The answer is: FV = -237,

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**Other parameters for annuity**

An insurance company offers to pay you $10,000 per year for 10 years if you will pay $67,100 up front. What is the rate of return? Calculator: PV; PMT; 10 N; CPT I/Y. The answer is: I/Y =

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Annuity due Annuity due: an annuity for which the cash flows occur at the beginning of the period. For calculating PV and FV of an annuity due, we can use the following formula: Annuity due value = ordinary annuity value (1 + i).

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Annuity due example You are going to rent an apartment for a year. You have 2 choices: (1) pay the monthly rent, $500, at the beginning of the month, or (2) pay the entire year’s rent, $5,000, today. Suppose that you can earn 1% every month. Which is the better choice? Ordinary PV: 500 PMT; 1 I/Y; 12 N; CPT PV. The answer is: PV = -5, Annuity due PV = ordinary PV (1 + i) = $5, 1.01 = $5, You would want to pay $5,000 today if you can.

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Growing annuity Growing annuity: a finite number of growing cash flows, where the constant growth rate is g. PV = C { [ 1 – ((1 + g) / (1 + i))N ] / (i – g) }.

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**Growing annuity example**

Emily has just been offered a job at $80,000 a year. She anticipates her salary increasing by 9% a year until her retirement in 40 years. Given an interest rate of 20%, what is the present value of her lifetime salary? PV = C { [ 1 – ((1 + g) / (1 + i))N ] / (i – g) } = $80,000 { [ 1 – ((1 + 9%) / (1 + 20%))40 ] / (20% – 9%) } = $711,

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**Perpetuity Perpetuity: a constant stream of cash flows without end.**

PV = C / i.

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Perpetuity example Preferred stock promises the buyer a fixed cash dividend every period (usually every quarter) forever. Suppose that VTinsurance Inc. wants to sell preferred stock. The quarterly dividend is $1 per share. The required rate of return for this issue is 2.5% per quarter. What is the fair value of this issue? PV = C / i = $1 / 2.5% = $40 (per share).

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Growing perpetuity Growing perpetuity: an infinite cash flow stream that grows at a constant rate, g. PV = C1 / (i – g), C1 is the cash flow at time 1.

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**Growing perpetuity example**

Toyota is expected to pay a dividend (annual dividend) of $3 per share in a year. Investors also anticipate that the annual dividend will rise by 6% per year forever. The applicable discount rate is 11%. What is the present value of future dividends? PV = C1 / (i – g) = $3 / (11% – 6%) = $60 per share.

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**Comparing rates, I Rates are quoted in many different ways.**

Tradition. Legislation. Effective annual rate (EAR): the actual rate paid (or received) after accounting for compounding that occurs during the year. When comparing two alternative investments with different compounding frequencies, one needs to compute the EARs and use them for reaching a decision.

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Comparing rates, II Annual percentage rate (APR) or stated annual interest rate (SAIR): the annual rate without consideration of compounding. APR = period rate the number of periods per year, m. EAR = [1 + (APR / m)]m – 1.

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Rate example, I You went to a bank to borrow $10,000. You were told that the rate is quoted as “8% compounded semiannually.” What is the amount of debt after a year? FV = PV (1 + i)N = $10,000 (1 + 4%)2 = $10,816. EAR = [1 + (APR / m)]m – 1= [1 + (8% / 2)]2 – 1 = 8.16%.

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**Rate example, II What is the APR if the monthly rate is 1%?**

What is the monthly (period) rate if the APR is 6% with monthly compounding? Period (monthly) rate = 6% / 12 = 0.5%.

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**Continuously compounding**

FV = PV × eAPR×the number of years , where e has the value of Suppose that you invest $1,000 at a continuously compounded rate of 10% for a year. FV = PV × eAPR×the number of years = $1,000 × e10%×1 = $1, So, EAR = 10.52%.

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APR vs. EAR in real life By Trust-in-saving law, banks need to disclose EAR ( or called annual percentage yield (APY), or effective annual yield (EAY)). So you get the correct rate when you save. By Trust-in-lending law, banks need to disclose APR, the stated (quoted) rate. So you get a seemingly low rate when you borrow.

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Extra In residential mortgage markets, “APR” is the cost of credit that includes the quoted rate and transactions costs. This APR is higher than the quoted rate. If it is 0.75%-1% higher than the quoted rate, the financial charges and fees are most likely too high. Example: a quote from quickloans.com: rate 3.625% (3.8% APR)

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Pure discount loans Pure discount loans: the borrower receives money today and repays a single lump sum at some time in the future. Treasury bills: U.S. government borrows money and promises to repay a fixed amount at some time less than one year. Suppose that the maturity is 12 months. The face value is $10,000. The market discount rate is 7%. How much do you need to pay for the T-bill? PV = FV / (1 + i)N = $10,000 / (1 + 7%)1 = $9,

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Amortized loans Amortized loans: the loans that are paid off by making regular principal reductions. Payment per period = interest + a portion of principal. The most common type of amortized loans require borrowers make a single, fixed payment every period, i.e., annuity.

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Buying a house, I You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be $5,500. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. You are able to buy the house at $154,500. What is the monthly payment? Suppose that you have an annual salary of $50,000. What is the ratio of the mortgage payment to your monthly income?

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**Buying a house, II Down payment = $20,000 – $5,500 = $14,500.**

Loan = $154,500 – $14,500 = $140,000. Calculator: PV; 0.5 I/Y; 360 N; CPT PMT. The answer is: PMT = PMT/income = $ / ($50,000 / 12) = 20.14%. Banks usually do not want to see this ratio to be higher than 25%.

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Interest-only loans Interest-only loans: borrower pays interest each period and repay the entire original principal at some time in the future. Example: bonds. This serves as a launch point for next topic: Chapters 8 & 9: How to value bonds and stocks.

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**Review: let us work on this one**

Q11, P. 125: Conoly Co. Has identified an investment project with the following cash flows. If the discount rate is 10%, what is the PV? Year 1: $960. Year 2: $840. Year 3: $935. Year 4: $1,350. Answer: $3,

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**Review: let us work on this one**

Concept #3, p Suppose that two athletes sign 10-year contracts for $80 million. In one case, we are told that the $80 million will be paid in 10 equal installments. In the other case, we are told that the $80 million will be paid in 10 installments, but the installments will increase by 5% per year. Who got the better deal? Why?

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Assignment Please analyze the following statement: “on average, pursuing a 4-year college degree is a value-added proposition.” Must utilize TVM concept. Understanding Questions and Problems #28 (p.127) will be beneficial. A typed report; due in a week. Individual assignment.

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**End-of-chapter Concept questions: 1-5.**

Questions and problems: 1-28, and (also excluding those questions with variable/differential discount rates).

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