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Chapter 4: Discounted cash flow valuation

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Chapter 4: Discounted cash flow valuation

Corporate Finance

Ross, Westerfield, and Jaffe

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

- 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.

- 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.

- 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.

- 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.

- 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?

- 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.

- 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 1.3401 = $67,000.
- Financial calculator: 6 N; 5 I/Y; 50000 PV; CPT FV. The answer is FV = -67,004.7820. Ignore the negative sign.

- 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.

- 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 1.4400 = $2.88.
- Calculator: 2 PV; 20 I/Y; 2 N; CPT FV. The answer is -2.8800.

- 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.

- 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 0.9346 = $3,738.4.
- Calculator: 4000 FV; 7 I/Y; 1 N; CPT PV. The answer is -3,738.3178.

- 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: 1000000 FV; 10 I/Y; 44 N; CPT PV. The answer is -15,091.1332.

- 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 = -3,402.9160.
- 10 years: 10 N; 8 I/Y; 5000 FV; CPT PV. The answer is PV = -2,315.9674.

- 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 = -3,104.6066.
- 15%: 15 I/Y; 5 N; 5000 FV; CPT PV. The answer is PV = -2,485.8837.

- 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.

- 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; -10000 FV; 10 N; CPT I/Y. The answer is I/Y = 7.1773.
- Note that for entering -10000 FV, this is the sequence: 10000 +/– FV.

- 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: 10000 PV; -1000000 FV; 10 I/Y; CPT N. The answer is: N = 48.3177.

- 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.

- 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.

- (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.

- 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 = -23,999.5424.
- In the solution manual (textbook), PVIFA(PVIA) stands for the PV of an annuity.
- PVIFA(i,N) = [ 1 – 1 / (1 + i)N ] / i .

- Questions and Problems #28

- 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%} = $237,174.56.
- Calculator: 3000 PMT; 6 I/Y; 30 N; CPT FV. The answer is: FV = -237,174.5586.

- 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: -67100 PV; 10000 PMT; 10 N; CPT I/Y. The answer is: I/Y = 8.0003.

- 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).

- 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 = -5,627.5387.
- Annuity due PV = ordinary PV (1 + i) = $5,627.5387 1.01 = $5,683.8141.
- You would want to pay $5,000 today if you can.

- 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) }.

- 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%) } = $711,730.71.

- Perpetuity: a constant stream of cash flows without end.
- PV = C / i.

- 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).

- 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.

- 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.

- 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.

- 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.

- 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%.

- What is the APR if the monthly rate is 1%?
- APR = 1% 12 = 12%.
- What is the monthly (period) rate if the APR is 6% with monthly compounding?
- Period (monthly) rate = 6% / 12 = 0.5%.

- FV = PV × eAPR×the number of years , where e has the value of 2.718.
- 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 = $1,105.20. So, EAR = 10.52%.

- 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.

- 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)

- 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 = $9,345.79.

- 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.

- 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?

- Down payment = $20,000 – $5,500 = $14,500.
- Loan = $154,500 – $14,500 = $140,000.
- Calculator: 140000 PV; 0.5 I/Y; 360 N; CPT PMT. The answer is: PMT = -839.3707.
- PMT/income = $839.3707 / ($50,000 / 12) = 20.14%.
- Banks usually do not want to see this ratio to be higher than 25%.

- 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.

- 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.

- Concept #3, p. 123. 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?

- 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.

- Concept questions: 1-5.
- Questions and problems: 1-28, 30-43 and 45-50 (also excluding those questions with variable/differential discount rates).