Discounted cash flow valuation Chapter 4 Discounted cash flow valuation

4.1 Valuation: the one-period case 10,000 (return of principle) + (0.12 x 10,000) (interest) = 11,200 Future value / compound value Present value PV = C1 / (1+r)

NPV = - cost + PV Is the present value of future cash flows minus the present value of the cost of the investment The selection of the discounted rate (example 4.2)

4.2 The multi-period case 1 x (1+r) Compounding (the process of leaving the money in the financial market and lending it for another year) 1 x (1+r) x (1+r) = 1 + 2r (simple interest) + r2 (interest on interest)

What is the effect of compounding? 1 FV = C0 x (1 + r)t (example 4.3-4.5) Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? FV = 10(1.055)200 = 447,189.84 What is the effect of compounding? Simple interest = 10 + 200(10)(.055) = 120.00 Compounding added $447,069.84 to the value of the investment. 6

FUTURE VALUE AS A GENERAL GROWTH FORMULA 1 FUTURE VALUE AS A GENERAL GROWTH FORMULA Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in year 5? FV = 3,000,000(1.15)5 = 6,034,072 7

Present value & discounting PV x (1.09)2 = 1 Discounting - the process of calculating the present value of a future cash flow

1 Present Values How much do I have to invest today to have some amount in the future? FV = PV(1 + r)t Rearrange to solve for PV = FV / (1 + r)t When we talk about discounting, we mean finding the present value of some future amount. When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value. 9

Present Value – One Period Example 1 Present Value – One Period Example Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today? PV = 10,000 / (1.07)1 = 9,345.79 Calculator 1 N 7 I/Y 10,000 FV CPT PV = -9,345.79 11

Present Values – Example 2 1 Present Values – Example 2 You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? PV = 150,000 / (1.08)17 = 40,540.34 12

Present Values – Example 3 1 Present Values – Example 3 Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? PV = 19,671.51 / (1.07)10 = 10,000 13

1 Discount Rate Often we will want to know what the implied interest rate is in an investment Rearrange the basic PV equation and solve for r FV = PV(1 + r)t r = (FV / PV)1/t – 1 17

Discount Rate – Example 1 You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest? r = (1,200 / 1,000)1/5 – 1 = .03714 = 3.714% Calculator – the sign convention matters!!! N = 5 PV = -1,000 (you pay 1,000 today) FV = 1,200 (you receive 1,200 in 5 years) CPT I/Y = 3.714% 18

Discount Rate – Example 2 1 Discount Rate – Example 2 Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25% 19

Discount Rate – Example 3 1 Discount Rate – Example 3 Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it? r = (75,000 / 5,000)1/17 – 1 = .172688 = 17.27% 20

Finding the Number of Periods 1 Finding the Number of Periods Start with basic equation and solve for t (remember your logs) FV = PV(1 + r)t t = ln(FV / PV) / ln(1 + r) You can use the financial keys on the calculator as well; just remember the sign convention. 21

Number of Periods – Example 1 You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years I/Y = 10; PV = -15,000; FV = 20,000 CPT N = 3.02 years 22

Number of Periods – Example 2 1 Number of Periods – Example 2 Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs? 23

Number of Periods – Example 2 Continued 1 Number of Periods – Example 2 Continued How much do you need to have in the future? Down payment = .1(150,000) = 15,000 Closing costs = .05(150,000 – 15,000) = 6,750 Total needed = 15,000 + 6,750 = 21,750 Compute the number of periods Using the formula t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years Per a financial calculator: PV = -15,000, FV = 21,750, I/Y = 7.5, CPT N = 5.14 years 24

Spreadsheet Example Use the following formulas for calculations 1 Spreadsheet Example Use the following formulas for calculations = FV(rate,nper,pmt,pv) = PV(rate,nper,pmt,fv) = RATE(nper,pmt,pv,fv) = NPER(rate,pmt,pv,fv) The formula icon is very useful when you can’t remember the exact formula 25

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FUTURE VALUE OF MULTIPLE CASH FLOWS 1 FUTURE VALUE OF MULTIPLE CASH FLOWS The time line: Calculating the future value: 1 2 Time (years) $100 CF 1 2 Time (years) $100 CF Future values +108 x1.08 $208 $224.64 27

Multiple Cash Flows –Future Value Example 1 Multiple Cash Flows –Future Value Example Suppose you deposit $4000 at end of next 3 years in a bank account paying 8% interest. You currently have $7000 in the account. How much you will have in year 3, in year 4? 1 Time (years) $4000 $7000 CF 2 3 4 28

Multiple Cash Flows –Future Value Example 1 Multiple Cash Flows –Future Value Example Find the value at year 3 of each cash flow and add them together. Today (year 0): FV = 7000(1.08)3 = 8,817.98 Year 1: FV = 4,000(1.08)2 = 4,665.60 Year 2: FV = 4,000(1.08) = 4,320 Year 3: value = 4,000 Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 + 4,000 = 21,803.58 Value at year 4 = 21,803.58(1.08) = 23,547.87 29

Multiple Cash Flows – FV Example 1 Multiple Cash Flows – FV Example Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in year 2? FV = 500(1.09)2 + 600(1.09) = 1,248.05 1 2 Time (years) $600 $500 30

Multiple Cash Flows – Example Continued 1 Multiple Cash Flows – Example Continued How much will you have in year 5 if you make no further deposits? First way: FV = 500(1.09)5 + 600(1.09)4 = 1,616.26 Second way – use value at year 2: FV = 1,248.05(1.09)3 = 1,616.26 1 2 Time (years) $600 $500 3 4 5 31

Multiple Cash Flows – FV Example 1 Multiple Cash Flows – FV Example Suppose you plan to deposit $100 into an account in year 1 and $300 into the account in year 3. How much will be in the account in year 5 if the interest rate is 8%? FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97 1 2 3 4 5 Time (years) $100 $300 32

4.3 Compounding periods Stated annual interest rate/annual percentage rate By definition APR = period rate times the number of periods per year Consequently, to get the period rate we rearrange the APR equation: Period rate = APR / number of periods per year Effective annual rate (EAR): (1+r/m)m - 1

Computing APRs What is the APR if the monthly rate is .5%? 1 Computing APRs What is the APR if the monthly rate is .5%? .5(12) = 6% What is the APR if the semiannual rate is .5%? .5(2) = 1% What is the monthly rate if the APR is 12% with monthly compounding? 12 / 12 = 1% 35

1 Things to Remember You ALWAYS need to make sure that the interest rate and the time period match. If you are looking at annual periods, you need an annual rate. If you are looking at monthly periods, you need a monthly rate. If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly 37

Computing EARs - Example 1 Computing EARs - Example Suppose you can earn 1% per month on $1 invested today. What is the APR? 1(12) = 12% How much are you effectively earning? FV = 1(1.01)12 = 1.1268 Rate = (1.1268 – 1) / 1 = .1268 = 12.68% Suppose you put it in another account, you earn 3% per quarter. What is the APR? 3(4) = 12% FV = 1(1.03)4 = 1.1255 Rate = (1.1255 – 1) / 1 = .1255 = 12.55% 38

1 EAR - Formula 39

1 Decisions, Decisions You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? First account: EAR = (1 + .0525/365)365 – 1 = 5.39% Second account: EAR = (1 + .053/2)2 – 1 = 5.37% Which account should you choose and why? 40

Decisions, Decisions Continued 1 Decisions, Decisions Continued Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year? First Account: Daily rate = .0525 / 365 = .00014383562 FV = 100(1.00014383562)365 = 105.39 Second Account: Semiannual rate = .053 / 2 = .0265 FV = 100(1.0265)2 = 105.37 You have more money in the first account. 41

Computing APRs from EARs 1 Computing APRs from EARs If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: 42

1 APR - Example Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? 43

4.4 Simplifications Perpetuity Growing perpetuity Annuity Growing annuity

Annuities and Perpetuities Defined 1 Annuities and Perpetuities Defined Annuity – finite series of equal payments that occur at regular intervals If the first payment occurs at the end of the period, it is called an ordinary annuity If the first payment occurs at the beginning of the period, it is called an annuity due Perpetuity – infinite series of equal payments 45

Annuities and Perpetuities – Basic Formulas 1 Annuities and Perpetuities – Basic Formulas Perpetuity: PV = C / r Annuities: 46