Chapter 8 Time Value of Money Part I: Future and Present Value of Lump Sums
Learning Objectives • Explain the relationship between the time value of money and inflation. • Distinguish between effective rate and stated rate. • Calculate the future value lump sum and present value lump sum factors that are used to solve time value of money problems.
Learning Objectives (continued) • Compare bank discount and simple interest. • Calculate the internal rate of return with respect to the present value of a lump sum and future value of a lump sum. • Integrate the present value of a lump sum and the future value of a lump sum to solve real-life financial problems.
Learning Objectives (continued) • Use financial tables to solve time value of money problems. • Use financial calculators to solve time value of money problems.
Simple Interest • Simple interest is the amount of interest earned on the principal amount stated. • Principal amount stated is the base amount that we borrow or save.
Simple Interest (Examples) • Interest on $1,000 borrowed for one year at 8%: • Interest on $1,000 borrowed for six months at 8%:
Total Due on Simple Interest Loans • The total amount due (maturity amount) is equal to principal plus interest: Where
Manipulating Simple Interest • If we know any three of the four variables: • Solving for principal • Solving for time
Bank Discount • The bank discount is an amount of interest that is deducted from the amount you wish to borrow: Where
Bank Discount (continued) • Proceeds are the amount the bank actually provides to the borrower after deducting the discount from the amount intended to be borrowed.
Federal Treasury Bills • There are situations in which the entrepreneur can actually perform the function of a bank. • What better source of investing than to lend the government of the United States money for a short period of time? • The government issues discounted treasury bills in denominations of $10,000 for three months, six months, and one year.
Compound Interest • Compound interest is earned or charged on both the principal amount and on the accrued interest that has been previously earned or charged.
Compound Interest (continued) • We can bypass the multiple individual steps in computing compound interest by using the following compound interest formula to determine future value: where
Effective Rate • The stated or quoted rate is the rate of interest that is listed, normally on an annual basis, and it disregards compounding. • The effective annual rate is the actual rate that is paid by the borrower or earned by the investor after compounding is taken into consideration.
Effective Rate (continued) • Example: A bank quotes 8 percent annual rate. The bank wants monthly payments, so it compounds monthly.
Future Value of a Lump Sum • What is the future value of a lump sum amount for n periods and at i rate of return? Where
Future Value of a Lump Sum (Examples) • You save $10,000 at 5 percent interest for 10 years compounded annually. What is the future value of this investment after 10 years?
Future Value of a Lump Sum (Examples) • If a wedding costs $20,000 today, how much will the wedding cost 10 years from now if inflation averages 4% a year? • What is the future value of $100,000 if money is compounded monthly at 6% for 18 years? Note: The answer below was obtained by using a calculator. If you use tables, the answer is $293,680.
Present Value of a Future Lump Sum • What is the present value of a future lump sum amount for n periods at an i rate of return? Where
Present Value of a Lump Sum (Examples) • How much do you have to deposit in an account today that will have a value of $10,000,000 in 7 years if annual interest is 6% compounded annually? Note: If tables are used rather than a calculator, the answer will be $6,651,000.
Present Value of a Stream of Unequal Payments • An athlete is offered a $20 million contract over 5 years with a $4 million signing bonus. The contract consists of $2 million for year 1, $3 million for year 2, $3 million for year 3, $3 million for year 4, and $5 million for year 5. What is the present value of the $20 million contract if money can earn 5 percent annual interest?
Present Value of a Stream of Unequal Payments (continued) • This requires us to build a table which is illustrated below:
Internal Rate of Return • Internal Rate of Return (IRR) is the actual rate of return that equates a dollar invested now with a dollar received in the future.
Internal Rate of Return (continued) • The IRR is found by using a calculator and the following formula:
IRR Problem • In January 2002, you bought 10,000 shares of a stock at $2 per share. In January 2006, you sold the 10,000 shares at $3 a share. What is the internal rate of return?
Rule of 72 • We can also find an approximation of the amount of time that it takes a present sum of money to double by dividing the number 72 by the interest rate earned on an investment. This procedure is known as the rule of 72. Example: How long will it take $1,000 to double if it can be invested at 12%?
Rule of 72 (continued) • We can also find the interest required if we know how long it takes an investment to double. • Example: We want $1,000 to double in eight years. What interest to we have to earn on our investment?