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" # $ % & n = FV n *(1 + r ) ' n 8 PV t = 0 = FV n (1 + r ) n = FV n *(1 + r ) ! n
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Money Can Travel Through Time! H The Three Rules of Time Travel: 1. You can move the value of a cash flow forward in time by compounding it. 2. You can move the value of a cash flow backward in time by discounting it. 3. To combine or compare values of cash flows that occur at different points in time, you have to bring them to the same point in time by compounding or discounting them. Where: PV 0 = present value (sometimes denoted by P for “Principal”) FV n = future value n periods from now n = the number of time periods over which interest accrues r = the effective periodic interest rate I = total amount of interest 9 FV n = PV 0 *(1 + r ) n PV 0 = FV n (1 + r ) n = FV n *(1 + r ) ! n
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Example: Discounting H You think you will need a new computer in three years. H You expect the computer to cost $1,000 in three years. H If the annual interest rate you earn on your savings is 4%, how much money do you have to set aside today to be able to buy the computer in three years? H Timeline: 0 1 2 3 Period |-------|-------|-------|-------|---- ? $1,000 Cash Flows 10 PV 0 = 1,000 (1 + 0.04) 3 = $1,000*(1.04) ! 3 = $889.00
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iClicker Example: Discounting H You have $980 in the bank right now. You want to have $1000 three years from now to buy a new computer. How much money can you spend today if the annual interest rate is 7%? a) $816.30 b) $163.70 c) $180.03 d) $106.56 e) None of the above Timeline: 0 1 2 3 Periods ---|---------------|---------------|---------------|----------- PV 0 = ? $1,000 $816 $873 $935 $1,000 CFs correct answer is b) First: How much do you have to invest today to have $1,000 in 3 years? PV of $1000 in 3 years is $1000*(1.07) -3 = $816.30 You can spend $980 – $816.30 = $163.70 today 11
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Example: The Third Rule of Time Travel (1) H Suppose we plan to save $1,000 today, and $1,000 at the end of each of the next two years. If we can earn a fixed 10% interest rate on our savings, how much will we have three years from today? 12
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Example: The Third Rule of Time Travel (2) H The time line would look like this: 13
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Example: The Third Rule of Time Travel (3) 14
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Example: The Third Rule of Time Travel (4) 15
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Multiple Cash Flows H Present values are additive (and addictive). H To calculate the present value of a sum of future payments, we just add together the individual present values H Example The present value of a payment of $1,000 one year from now and another $1,000 two years from now if the annual interest rate is 10% is: You could also calculate FV 3 and then discount it back to today.
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