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## Time Value of Money

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**Time Value of Money**Lecture No.3**Chapter 2Time Value of Money**• Interest: The Cost of Money • Economic Equivalence • Interest Formulas – Single Cash Flows • Equal-Payment Series • Dealing with Gradient Series • Composite Cash Flows. Power-Ball Lottery**What Do We Need to Know?**• To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different point in time. • To do this, we need to develop a method for reducing a sequence of benefits and costs to a single point in time. Then, we will make our comparisons on that basis.**Time Value of Money**• Money has a time value because it can earn more money over time (earning power). • Money has a time value because its purchasing power changes over time (inflation). • Time value of money is measured in terms of interest rate. • Interest is the cost of money—a cost to the borrower and an earning to the lender**End-of-Period Convention**Interest Period 0 1 End of interest period Beginning of Interest period 0 1**Methods of Calculating Interest**• Simple interest: the practice of charging an interest rate only to an initial sum (principal amount). • Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.**Simple Interest**• P = Principal amount • i= Interest rate • N = Number of interest periods • Example: • P = $1,000 • i = 8% • N = 3 years**Simple Interest Formula**F = P+(iP)N where P: principal amount i: simple interest rate N: number of interest periods F: total amount accumulated at the end of period N**Compound Interest**• Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.**Compound Interest**• P = Principal amount • i= Interest rate • N = Number of interest periods • Example: • P = $1,000 • i = 8% • N = 3 years**Compounding Process**$1,080 $1,166.40 0 $1,259.71 1 $1,000 2 3 $1,080 $1,166.40**$1,259.71**2 1 0 3 $1,000**Some Fundamental Laws**The Fundamental Law of Engineering Economy**Practice Problem: Warren Buffett’s Berkshire Hathaway**• Went public in 1965: $18 per share • Worth today (August 22, 2003): $76,200 • Annual compound growth: 24.58% • Current market value: $100.36 Billion • If he lives till 100 (current age: 73 years as of 2003), his company’s total market value will be ?**Market Value**• Assume that the company’s stock will continue to appreciate at an annual rate of 24.58% for the next 27 years.**Example**• In 1626 the Indians sold Manhattan Island to Peter Minuit • of the Dutch West Company for $24. • If they saved just $1 from the proceeds in a bank account • that paid 8% interest, how much would their descendents • have now? • As of Year 2003, the total US population would be close to • 275 millions. If the total sum would be distributed equally • among the population, how much would each person receive?**Excel Solution**=FV(8%,377,0,1) = $3,988,006,142,690**Practice Problem**• Problem Statement If you deposit $100 now (n = 0) and $200 two years from now (n = 2) in a savings account that pays 10% interest, how much would you have at the end of year 10?**Solution**F 0 1 2 3 4 5 6 7 8 9 10 $100 $200**Practice problem**• Problem Statement Consider the following sequence of deposits and withdrawals over a period of 4 years. If you earn 10% interest, what would be the balance at the end of 4 years? ? $1,210 1 4 0 2 3 $1,500 $1,000 $1,000**?**$1,210 0 1 3 2 4 $1,000 $1,000 $1,500 $1,100 $1,000 $2,981 $1,210 $2,100 $2,310 + $1,500 -$1,210 $1,100 $2,710