2. Introduction To Valuation: The Time Value Of Money Chapter 4

3. 2 Topics 1. How To Determine The Future Value Of An Investment 2. How To Determine The Present Value Of Cash To Be Received At A Future Date 3. How To Find The Return On An Investment

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5. 4 Fundamental Financial Concept A dollar received today is worth more than a dollar received later This is because of interest  This is because of the discount rate 

6. 5 Definitions: Simple interest Interest earned only on the original principal amount invested Compound interest Interest earned on both the initial principal and the interest reinvested from prior periods Interest on interest Interest earned on the reinvestment of pervious interest payments Compounding The process of accumulating interest in an investment over time to earn more interest

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8. 7 Future value: The amount an investment is worth after one or more periods

9. 8 Future Values: (Textbook formula) Textbook FV = PV(1 + r) t FV = future value PV = present value r = period interest rate, expressed as a decimal T = number of periods Future value interest factor = (1 + r) t

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13. 12 How To Determine The Future Value Of An Investment Suppose you invest $1000 for one year at 5% per year, compounded yearly. What is the future value in one year? Interest = 1000(.05) = 50 Value in one year = principal + interest = 1000 + 50 = 1050 Future Value (FV) = 1000(1 +.05) = 1050 Suppose you leave the money in for another year. How much will you have two years from now? FV = 1000(1.05)(1.05) = 1000(1.05) 2 = 1102.50

14. 13 Effects of Compounding Simple interest Compound interest Consider the previous example FV with simple interest = 1000 + 50 + 50 = 1100 FV with compound interest = 1102.50 The extra 2.50 comes from the interest of.05(50) = 2.50 earned on the first interest payment

15. 14 Future Values – Example 2 Suppose you invest the $1000 from the previous example for 5 years, compounded yearly. How much would you have? FV = 1000(1.05) 5 = 1276.28 The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1250, for a difference of $26.28.)

16. 15 Future Values – Example 3 Suppose you had a relative deposit $10 at 5.5% interest 200 years ago, compounded yearly. 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) = 210.55 Compounding added $446,979.29 to the value of the investment

17. 16 Present Value How much should you put in the bank today in order to receive a future value amount after one or more periods The current value of future cash flows discounted at the appropriate rate If you know the future amount you would like, assume an interest rate, and take all the interest that you will need to earn out of the future value amount

18. 17 Present Values (Textbook formula) How much do I have to invest today to have some amount in the future? Textbook 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.

19. 18 Definitions: Discount Rate The rate used to calculate the present value of future cash flows Discount Calculate the present value of some future amounts Discounted Cash Flow (DCF) valuation Calculating the present value of future cash flows to determine its value today

20. 19 How To Determine The Present Value Of Cash To Be Received At A Future Date You want to begin saving for you 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, compounded yearly, how much do you need to invest today? PV = 150,000 / (1.08) 17 = 40,540.34

21. 20 Present Values – Example 2 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, compounded yearly, how much did your parents invest? PV = 19,671.51 / (1.07) 10 = 10,000

22. 21 PV – Important Relationship I For a given interest rate – the longer the time period, the lower the present value What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%, compounded yearly. 5 years: PV = 500 / (1.1) 5 = 310.46 10 years: PV = 500 / (1.1) 10 = 192.77

23. 22 PV – Important Relationship II For a given time period – the higher the interest rate, the smaller the present value What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? (both compounded yearly). Rate = 10%: PV = 500 / (1.1) 5 = 310.46 Rate = 15%; PV = 500 / (1.15) 5 = 248.58

24. 23 Figure 4.3

25. 24 How To Find The Return On An Investment If you invest $100 today in an account that compounds interest yearly and in 8 years you have $200, what is the interest rate? Rule of 72 = A reasonable estimate for the required rate to have an investment double = 72/(i/n) = Number of periods

26. 25 How To Find The Number Of Periods Required For An Investment If you want to buy an asset that cost $100,000 and you have $50,000 to invest now, at a rate of 12%, compounded annually, how many years must you wait?

27. 26 Example: Spreadsheet Strategies Use the following formulas for TVM calculations FV(rate,nper,pmt,pv) PV(rate,nper,pmt,fv) RATE(nper,pmt,pv,fv) NPER(rate,pmt,pv,fv)

28. 27 Real assets/ Financial assets Present value and future value are fundamental to finance Most instruments: Real assets Buildings, trucks Financial assets Debt, Equity, Preferred stock, derivatives Can be analyzed using DCF valuation techniques