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The math underlying discounting and the calculation of present value is the exact flip side to compounding and future value. Discounting is the process of going from an expected future value to a present value. Discounting also means to multiply by a number less than 1.0, over a number of time periods. Present value is what an investment is worth today, if an expected future value is discounted at a certain rate and for a period of time. Let s look at an example to see how discounting and present value works. Today s your birthday, and Uncle Larry promises you a present $100 to be given to you at some time in the next five years. What is the present value of Uncle Larry s $100 promise This is a tough one. Uncle Larry is usually reliable, but he s a tad overweight and is not in great health. There s a slight chance that he may die before he pays off, so risk is involved. You re also not sure exactly when that $100 is going to flow to you. With the uncertainty of both risk and timing, this sounds like a problem for a present value table. In Table 2-6 we lay out the possibilities, with the number of years, n, ranging from 0 to 5, and the discounting rates ranging from 6 percent to 8 percent to 10 percent. The numbers listed in Table 2-6 are known as discount factors. These factors give the present value of a dollar that you expect to receive at some time, n, in the future, discounted at a rate, r. As you can see in Table 2-6, as n increases and the payment is expected further into the future, the present value factor decreases meaning that the

write_clock: high NJU_IO ' Set for write. high NJU_CE ' Select the chip. for digit = 0 to 10 ' Write 11 digits. shiftout DATA_,CLK,lsbfirst,[DTG(digit)\4] ' Shift out a digit. ' next ' Next digit. low NJU_CE ' Deselect the chip. return ' Return to program. show_date: serout tpin,bdmd,[CR,HEX DTG(Mo10s), HEX DTG(Mo1s),"/",HEX DTG(D10s), HEX DTG(D1s),"/",HEX DTG(Y10s), HEX DTG(Y1s)] return show_time: serout tpin,bdmd,[CR,HEX DTG(H10s), HEX DTG(H1s),":",HEX DTG(M10s), HEX DTG(M1s),":",HEX DTG(S10s), HEX DTG(S1s)] return gosub_chan_1: high M0 low M1 pause 3000 gosub read_clock gosub show_date serout tpin,bdmd,[10,13] gosub show_time serout tpin,bdmd,[10,13] serout tpin,bdmd,10, [" ALERT CHANNEL 1 NOW ACTIVATED ",10,13] serout tpin,bdmd,10, [" BURGLAR ALARM ACTIVATION ",10,13] serout tpin,bdmd,10, [" TAKE ACTION IMMEDIATELY ",10,13] goto fin_1 gosub_chan_2: high M0 low M1 pause 3000 gosub read_clock gosub show_date serout tpin,bdmd,[10,13] gosub show_time serout tpin,bdmd,[10,13] serout tpin,bdmd,10, ["ALERT CHANNEL 2 NOW ACTIVATED",10,13] serout tpin,bdmd,10, [" TEMPERATURE ALARM ",10,13] serout tpin,bdmd,10, [" TAKE ACTION IMMEDIATELY ",10,13] goto fin_2 gosub_chan_3: high M0 low M1 pause 3000 gosub read_clock gosub show_date serout tpin,bdmd,[10,13] gosub show_time serout tpin,bdmd,[10,13] serout tpin,bdmd,10, ["ALERT CHANNEL 3 NOW ACTIVATED",10,13] serout tpin,bdmd,10, [" MOTION ALARM ",10,13] serout tpin,bdmd,10, [" TAKE ACTION IMMEDIATELY ",10,13]

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Table 30-1.

Binary number value 1 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000 100000000000 1000000000000 10000000000000 100000000000000 1000000000000000

Listing 21-2 Data-Alert Program 2 (Continued).

TABLE 2-6

Logic signals 557 In general, the nth place from the right has the decimal value 2n The total decimal value, for a given binary number, is the sum of the decimal values of each of the places Converting decimal numbers to binary form can be done using Table 30-1 Suppose, for example, that you want to convert 35912 to binary notation First, find the largest decimal number on the table that is no greater than the decimal number you wish to convert In this case, it is 32768 215 From this, you know that there will be 16 digits in the binary representation of this number, one for each place 20 through 215 Mark off 16 slots or spaces on a sheet of paper (quadrille graph paper is perfect for this), and place a digit 1 in the left-most space, representing 215.

goto fin_3 gosub_chan_4: high M0 low M1 pause 3000 gosub read_clock gosub show_date serout tpin,bdmd,[10,13] gosub show_time serout tpin,bdmd,[10,13] serout tpin,bdmd,10, ["ALERT CHANNEL 4 NOW ACTIVATED",10,13] serout tpin,bdmd,10, [" FIRE ALARM MESSAGE ",10,13] serout tpin,bdmd,10, [" TAKE ACTION IMMEDIATELY ",10,13] goto fin_4 fin_1:fin_2:fin_3:fin_4: pause 2000 serout tpin,bdmd,10,["+++"] pause 2000 serout tpin,bdmd,10,["ATH",10,13] pause 100 high M0 high M1 pause 10000 low Tele low LEDPin low Siren low Lamp low M0 pause 100 return end

Now, use Table 30-1 to determine which number can be added to 32768 to get the largest decimal number that doesn t exceed 35912 It happens to be 2048 211 Place digits 0 in the slots for 214, 213, and 212 Then place 1 in the space for 211 If you continue this process, you ll ultimately get the binary number 1000110001001000 This 16-digit binary number is equivalent to the decimal 35912 8 64 1024 2048 32768 23 26 210 + 211 215 The slots for exponents 3, 6, 10, 11, and 15 each are filled with binary digit 1; the others are filled with binary digit 0 It is possible to have fractional values in binary notation, just as it is in decimal notation.

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