Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- Time Value of Money | Dr. Menahem Rosenberg Simple Interest FV = PV + Interest FV = PV*(1 + i) 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- 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: 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- Time Value of Money | Dr. Menahem Rosenberg Value of $5 Invested More generally, with an investment of $5 at 10% we obtain 8/6/20198/6/2019 Fin 301- 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 8/6/20198/6/2019 Fin 301- 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% 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Present Value of a Lump Sum 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Solving Lump Sum Cash Flow for Interest Rate 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

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 ? 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

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. 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

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 8/6/20198/6/2019 Fin 301- 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 8/6/20198/6/2019 Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Effective Annual Rates of an APR of 18% 8/6/20198/6/2019 Fin 301- 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? 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

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% 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- Time Value of Money | Dr. Menahem Rosenberg Multiple Cash Flows Present value of multiple cash flows 8/6/20198/6/2019 Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- Time Value of Money | Dr. Menahem Rosenberg Perpetuity A stream of cash flows the last forever. A constant cash flow: 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- Time Value of Money | Dr. Menahem Rosenberg Perpetuity A g – constant growth cash flow, growth after the first period and g < i: 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- 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 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

PV Annuity Formula: Payment 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Annuity Formula: PV Annuity Due 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

Fin 301- Time Value of Money | Dr. Menahem Rosenberg Growing Annuities Annuity cash flows that grow at a constant rate (g) after the first cash flow: 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg

PV Annuity Formula: Number of Payments 8/6/20198/6/2019 Fin 301- Time Value of Money | Dr. Menahem Rosenberg