Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg 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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg Simple Interest FV = PV + Interest FV = PV*(1 + i) 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg Value of Investing $1 Continuing in this manner you will find that the following amounts will be earned (10% interest rate): 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg Value of $5 Invested More generally, with an investment of $5 at 10% we obtain 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Future Value of a Lump Sum FV with growths from 0% to +6% 1,000 1,500 2,000 2,500 3,000 3,500 2 4 6 8 10 12 14 16 18 20 Years Future Value of $1000 6% 4% 2% 0% 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Present Value of a Lump Sum 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg 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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Solving Lump Sum Cash Flow for Interest Rate 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Solving Lump Sum Cash Flow for Number of Periods 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

The Frequency of Compounding When we are presented with an APR and m compounding periods. EAR = (1 + APR/m)m 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Effective Annual Rates of an APR of 18% 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

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? 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg 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 $200 Time 0 1 2 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg 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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg Multiple Cash Flows Present value of multiple cash flows 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg Perpetuity A stream of cash flows the last forever. A constant cash flow: 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg Perpetuity A g – constant growth cash flow, growth after the first period and g < i: 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg 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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg 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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg 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 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

PV Annuity Formula: Payment 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Annuity Formula: PV Annuity Due 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg Growing Annuities Annuity cash flows that grow at a constant rate (g) after the first cash flow: 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

PV Annuity Formula: Number of Payments 10/28/201910/28/2019 Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg