Time Value of Money

1. Time Value of Money Bus 512- Time Value of Money | Dr. Menahem Rosenberg

2. Notation: CF => Cash Flow CF0 => Cash flow now CF1 => Cash flow one period ahead CFt => Cash flow t period ahead PV => Present Value FV => Future Value Bus 512- Time Value of Money | Dr. Menahem Rosenberg

3. Simple Interest • FV = PV + Interest • FV = PV*(1 + i) Bus 512- Time Value of Money | Dr. Menahem Rosenberg

4. Value of Investing $1 • Continuing in this manner you will find that the following amounts will be earned: Bus 512- Time Value of Money | Dr. Menahem Rosenberg

5. Value of $5 Invested • More generally, with an investment of $5 at 10% we obtain Bus 512- Time Value of Money | Dr. Menahem Rosenberg

6. Generalizing the method • Generalizing the method requires some definitions. Let • i be the interest rate • n be the life of the lump sum investment • PV be the present value • FV be the future value Bus 512- Time Value of Money | Dr. Menahem Rosenberg

7. FV with growths from 0% to +6% 3,500 6% 3,000 2,500 Future Value of $1000 4% 2,000 1,500 2% 0% 1,000 0 2 4 6 8 10 12 14 16 18 20 Years Future Value of a Lump Sum Bus 512- Time Value of Money | Dr. Menahem Rosenberg

8. Present Value of a Lump Sum Bus 512- Time Value of Money | Dr. Menahem Rosenberg

9. Lump Sums Formulae • You have solved a present value and a future value of a lump sum. There remains two other variables that may be solved for • interest, i • number of periods, n Bus 512- Time Value of Money | Dr. Menahem Rosenberg

10. Solving Lump Sum Cash Flow for Interest Rate Bus 512- Time Value of Money | Dr. Menahem Rosenberg

11. Solving Lump Sum Cash Flow for Number of Periods Bus 512- Time Value of Money | Dr. Menahem Rosenberg

12. The Frequency of Compounding • Deposit $1,500 in a saving account with 6% annual interest and semi-annual compounding. • What will you have in the account at the end of the year ? Bus 512- Time Value of Money | Dr. Menahem Rosenberg

13. The Frequency of Compounding • Assume m microperiods in a macroperiod and a nominal rate i per macroperiod compounded micro-periodically. That is the effective rate is i/m per microperiod. Bus 512- Time Value of Money | Dr. Menahem Rosenberg

14. The Frequency of Compounding • We can write r as the microperiod rate such that r=i/m and one macro period future value is (1) FV = PV*(1+r)m • Or (2) FV = PV (1+i/m)m Bus 512- Time Value of Money | Dr. Menahem Rosenberg

15. The Frequency of Compounding • When there are n macroperiods (1) FV = PV*(1+r)m*n • Or (2) FV = PV (1+i/m)m*n Bus 512- Time Value of Money | Dr. Menahem Rosenberg

16. The Frequency of Compounding • When we are presented with an APR and m compounding periods. • EAR = (1 + APR/m)m Bus 512- Time Value of Money | Dr. Menahem Rosenberg

17. Effective Annual Rates of an APR of 18% Bus 512- Time Value of Money | Dr. Menahem Rosenberg

18. The Frequency of Compounding • Note that as the frequency of compounding increases, so does the annual effective rate • What occurs as the frequency of compounding rises to infinity? Bus 512- Time Value of Money | Dr. Menahem Rosenberg

19. The Frequency of Compounding • The effective annual rate that’s equivalent to an annual percentage rate of 18% is then e 0.18 - 1 = 19.7217% • While more precision in the daily compounding will produce an EAR = 19.1764% Bus 512- Time Value of Money | Dr. Menahem Rosenberg

20. Multiple Cash Flows • Value a promise for $100 one year from today, and $200 two years from today. Given 10% annual rate. • Time line :CF $0 $100 $200Time 0 1 2 Bus 512- Time Value of Money | Dr. Menahem Rosenberg

21. Multiple Cash Flows • Generalizing the method. Let • i be the interest rate • t time periods counter • T time period of the last cash flow • CFt be cash flow at time t • PV be the present value Bus 512- Time Value of Money | Dr. Menahem Rosenberg

22. Multiple Cash Flows • Present value of multiple cash flows Bus 512- Time Value of Money | Dr. Menahem Rosenberg

23. Net Present Value (NPV) • NPV = - PV(All outflows) + PV(All inflows) • If NPV > 0 (inflows exceed outflows) -- Accept the project • If NPV < (inflows are less than outflows) -- Reject the project Bus 512- Time Value of Money | Dr. Menahem Rosenberg

24. Perpetuity • A stream of cash flows the last forever. • A constant cash flow: Bus 512- Time Value of Money | Dr. Menahem Rosenberg

25. Perpetuity • A g – constant growth cash flow, growth after the first period and g < i: Bus 512- Time Value of Money | Dr. Menahem Rosenberg

26. Annuities • a sequence of equally spaced identical (or constantly growing) cash flows • regular annuity with its first cash flow one period from now • annuity due with its first cash flow today Bus 512- Time Value of Money | Dr. Menahem Rosenberg

27. Annuities Four period annuity replication with two perpetuity. $ $ $ $ $ $ $+0 1 2 3 4 5 6 7 0 0 0 0 $ $ $-0 1 2 3 4 5 6 7 $ $ $ $ 0 0 0 = 0 1 2 3 4 5 6 7 Bus 512- Time Value of Money | Dr. Menahem Rosenberg

28. Annuities • Annuity Formula Notation • PV the present value of the annuity • I interest rate to be earned over the life of the annuity • n the number of payments • pmt the periodic payment Bus 512- Time Value of Money | Dr. Menahem Rosenberg

29. PV Annuity Formula: Payment Bus 512- Time Value of Money | Dr. Menahem Rosenberg

30. Annuity Formula: PV Annuity Due Bus 512- Time Value of Money | Dr. Menahem Rosenberg

31. Growing Annuities • Annuity cash flows that grow at a constant rate (g) after the first cash flow: Bus 512- Time Value of Money | Dr. Menahem Rosenberg

32. PV Annuity Formula: Number of Payments Bus 512- Time Value of Money | Dr. Menahem Rosenberg