Chapter 9 Time Value of Money Future and Present Values Loan Amortization, Annuities Financial Calculator All Rights Reserved Chapter 9

INTRODUCTION What is something worth? Two Important Questions In economics and finance, the expectation is that the price we pay ought to be related to the value we receive. We seek to relate time and value Two Important Questions What will a quantity of money invested today be worth tomorrow? What will a quantity of money to be received tomorrow worth today? All Rights Reserved Chapter 9

INTRODUCTION The time value of money (TVM) forms the basis for analysis of value or worth. How saving and/or investing are reltd to wealth. A dollar invested today will earn interest (or dividends) and be worth more tomorrow. All Rights Reserved Chapter 9

Time Value of Money Four Critical Formulas Future Value: value tomorrow of $1 invested today. Present Value: value today of $1 to be received “tomorrow”. Future Value of an Annuity: value several periods from now of a stream of $1 investments. Present Value of an Annuity: value today of a stream of $1 payments to be received for a set number of future periods. All Rights Reserved Chapter 9

Important TVM Concepts A. Future Value 1. What $1 invested today should grow to over time at an interest rate i. 2. FV = future value, P = principal, i = int. rate. a. I = interest (dollar amount), I = P  i 3. Single interest: FV = P + I = P + P(i) = P(1+i) 4. Multiple Interest Periods: FVi,n = P (1+i)n b. (1+i)n = Future Value Interest Factor c. FVi,n = P  FVIFi,n All Rights Reserved Chapter 9

Important TVM Concepts B. Present Value; 1. The value today of $1 to be received tomorrow. 2. Solving the Future Value Equation for PV; a. PV = FV  (1+i) single period discounting. b. PV = FV  (1+i)n multi-period discounting. c. PV = FV  (1+i)-n common form. d. (1+i)-n = Present Value Interest Factor. e. PVIF = 1 / FVIF (and vice-versa for same i, n) All Rights Reserved Chapter 9

Important TVM Concepts Future Value of an Annuity (FVA) e.g. Retirement Funds: IRA, 401(k), Keough 1. A series of equal deposits (contributions) over some length of time. 2. Contributions are invested in financial securities; stocks, bonds, or mutual funds. 3. The future value of accumulation is a function of the number and magnitude of contributions, reinvested interest, dividends, and undistributed capital gains. FVA = PMT * FVIFA All Rights Reserved Chapter 9

Important TVM Concepts Present Value of an Annuity (PVA) Insurance Annuities a. Provide recipient with a regular income (PMT) for a set period of time. The present value (PV) of the payments to be received is the price of the insurance annuity. PVA = PMT * PVIFA 2. Types of Annuities: a. Ordinary Annuity: payments received at end-of-period. b. Annuity Due: payments received at beginning-of- period All Rights Reserved Chapter 9

Important TVM Concepts Annuitize Investment Accumulations We have accumulated a sum of money and now desire to begin a series of [N] regular payouts: e.g. monthly checks We assume accumulated funds will continue to earn some rate of return (I/YR) The accumulation is treated as the present value (PV). How much income (PMT) will a certain accumulated amount produce? All Rights Reserved Chapter 9

Computing FVA FVA formula: 1. FVA = P  ([(1+i)n - 1]  i) = P  FVIFA [(1+i)n - 1]  i = future value interest factor for an annuity or FVIFAi,n. 1. Assumption; steady return rate over time and equal dollar amount contributions. All Rights Reserved Chapter 9

Computing PVA PVA formula: 1. PVA = P  ([1 - (1+i)-n]  i) = P  PVIFA [1 - (1+i)-n ]  i = present value interest factor for an annuity or PVIFAi,n. 1. Assumption; constant return rate over time and equal dollar amount distributions. All Rights Reserved Chapter 9

TVM Problems Question #1: How much will $1,000 grow to if left on deposit for 10 years in a savings account that pays 5% per annum compounded monthly? Question #2: How much is $10,000 to be received 5 years from now worth today if we assume a discount rate of 9% per annum compounded quarterly? Question #3: How many months will it take to double our money if we assume 6% per annum, compounded monthly? All Rights Reserved Chapter 9

TVM Problems How much must a person save each month in order to accumulate the $ 250,000 in 15 years if they can invest at 12 % per annum, compounded monthly (P/Y, C/Y = 12) All Rights Reserved Chapter 9

TVM Problems Each month Fred will invest $200 in stocks recommended by his stockbroker and will hold them in a self-directed IRA plan. After doing some research on the stock market you find out that the stock market has returned an average of 15% per annum for the last 20 years. If Fred earns 15% per annum on his stock investments (compounded monthly), how much should he have in his portfolio at the end of 30 years? All Rights Reserved Chapter 9

TVM Problems Ms. Jonas has $750,000 in a mutual fund IRA, is 60 years old and wants to retire. His idea is to purchase an insurance annuity that will provide him with a steady, guaranteed income should he desire to retire early. You consult an actuarial table and estimate that a person retiring at age 60 can expect to live another 25 years. The insurance annuity plan will make monthly payments and will guarantee 5.25% per annum, compounded monthly. How much will those monthly payments? All Rights Reserved Chapter 9

TVM Problems You are interviewing for a job as a bank financial analyst and the interviewer wants to test your ability to analyze a mortgage problem. She gives you the following information. The principal amount of the mortgage is $ 160,000 and will be amortized monthly over a 30-year period. The interest rate is 6.75 percent per annum. How much is the monthly payment? Prepare an Amortization table for the first three payments All Rights Reserved Chapter 9

TVM Problems How do you compute Annual Percentage Rate (APR)? Enter Nominal rate (annual rate) Up arrow Set value of C/Y Compute EFF = All Rights Reserved Chapter 9

Interest Rates Cost of Credit I-rate is the cost of borrowing 4 Factors influence rates Investment opportunities (macro environment) Time preferences for consumption (today vs. tomorrow) Riskiness of investment choices Inflation All Rights Reserved Chapter 9

Interest Rates Term Structure of Interest Rates Other Factors that influence rates Federal Reserve Policy Foreign Interest rates (demand for money flows) Business decisions >>> Capital Investment Term Structure of Interest Rates Normal: upward sloping to right The greater the risk, the greater the expected return Inverted: when short-term risk or inflation is greater than long-term All Rights Reserved Chapter 9

Interest Rates Composition of Interest rates Nominal Rate = r* + IP + DP + LP Real Rate (r*) Inflation Premium (IP) Default risk premium (DP) Liquidity Premium (LP) All Rights Reserved Chapter 9

Homework Questions: How does the rate of interest affect the future value on the money left on deposit in the bank? How does the rate of interest affect the present value of money to be received at some point in the future? Describe briefly the purpose of interest factors. All Rights Reserved Chapter 9

Homework Problems: If you invest $10,000 today in a certificate of deposit paying 5.3% per annum compounded quarterly, how much will your investment grow to in five years? How much interest income will you earn? How much must you save each month in order to accumulate $10,000 in 4 years? Your investment plan earns 4.5 percent per annum compounded monthly. All Rights Reserved Chapter 9

Homework 3. How long will it take for $ 2,000 to quadruple (grow to $8,000) if you invest it at 15 percent per annum compounded weekly?   4. How much must a person save each month in order to accumulate the $ 250,000 in 15 years if they can invest at 12 % per annum, compounded monthly (P/Y, C/Y = 12) All Rights Reserved Chapter 9

Homework 5. Mr. Jonas has $750,000 in a mutual fund IRA, is 60 years old and wants to retire. His idea is to purchase an insurance annuity that will provide him with a steady, guaranteed income should he desire to retire early. You consult an actuarial table and estimate that a person retiring at age 60 can expect to live another 25 years. The insurance annuity plan will make monthly payments and will guarantee 5.25% per annum, compounded monthly. How much will those monthly payments? All Rights Reserved Chapter 9