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Time Value Notes

Time Value Notes - TIME VALUE OF MONEY Fire 314 Fall 2009...

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TIME VALUE OF MONEY Fire 314 – Fall 2009 David E. Upton, Ph.D., CFA ® , AIFA ® The concept of the time value is basic to the area of finance. Time value is the relationship between time, rate of return, and dollar amounts. Given the rate of return, time value allows the translation of dollar amounts to different times; given dollar amounts at different times it provides a measure of return. Time value is presented here as a series of questions in order to develop an intuitive understanding of the time value concept. Although we will “do the math,” that is merely a means to another end – understanding the underlying relationships TIME VALUE OF MONEY Each time value concept can be thought of as answering a question. These questions will be presented in terms of $1 amounts, if the amount is $X, simply multiply the $1 factor by X. If you get confused as to what you are doing, go back to these questions! FUTURE VALUE: If I put away $1 today at rate I, how much will I have at the end of N periods? Time Line: 0 1 2 3 N-1 N -$1 $0 $0 $0 $0 $0 +? We will follow the convention that money being paid in / invested / going away from you will have a negative sign, money being removed / received / coming toward you will have a positive sign. At the end of the first period (N=1), you will have your original dollar plus the interest on that dollar for one period at rate, or $(1+I). At the end of the second period, you have the (1+I) that you had at the end of the first period plus the interest on the (1+I) over the second period at rate I, or a $ amount of (1+I) + I(1+I). Note that this is (1+I)(1+I) (1+I) 2 . At the end of the third period, you have the (1+I) 2 that you had at the end of the second period plus the interest on the (1+I) 2 for the third period at rate I, or a $ amount of (1+I) 2 + I (1+I) 2 . Note that this is equal to (1+I)(1+I) 2 = (1+I) 3 . Thus, at the end of each period, the amount increases by a factor (1+I). The extension to N periods is intuitively obvious. N $ Amount Amount if I = 6% 1 FV 1 = 1 + 1(I) = (1+I) 1(1.06) = 1.06 2 FV 2 = (1+I) + I(1+I) = (1+I)(1+I) = (1+I) 2 (1.06) 2 = 1.1236 3 FV 3 = (1+I) 2 + I(1+I) 2 = (1+I)(1+I) 2 = (1+I) 3 (1.06) 3 = 1.191016 . . Time Value - 1
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. . N FV N = (1+I) N If N = 40, FV N = (1.06) 40 = 10.285718 NB : The subscript on FV is usually omitted, I inserted it here simply to show that the FV applied to different periods. The subscript will (usually) not appear from here on. Almost all finance texts have tables giving future values. These tables are awkward, slow to use, and only give values for integer years and interest rates. If you have a problem like 10 years 8 months at 11.57%, you would have to use interpolation. Interpolation is easier than it sounds, but it is even easier to use a financial calculator. Examples: On a Texas Instruments BA II Plus calculator, you can find the $1 FV factor by entering (for 40 periods at 6%): 40 N; 6 %I; 1 +/- PV; 0 PMT; CPT FV => 10.29.
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