(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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While equity growth rates, \ capitalization rates, interest rates and changes in operating inputs (rents, \ expenses, vacancy, etc.) are routinely varied, holding period is often \ assumed to be fixed. Modeling an acquisition based on a 5 or 10 year holding \ period is common. It would seem that the analysis would be improved by \ \"tuning\" holding period to market cycles. The technique used here will \ consider the direction of capitalization rate movement (up or down) during \ holding period as the index of the market cycle.\n\nVariable definitions are:\ \n\ner\t=\tequity reversion (full value as we assume no debt) calculated as \ "], StyleBox[Cell[BoxData[ \(TraditionalForm\`cf\_\(k + 1\)\/cr\_o\)]]], StyleBox[ "\nnpv \t=\tnet present value\npd\t=\tinitial investment, made at t=0. We \ initially assume that property is debt free\n"], StyleBox[Cell[BoxData[ \(TraditionalForm\`cf\_0\)]]], StyleBox["\t=\tinitial cash flow to be received at the end of year 1\n"], StyleBox[Cell[BoxData[ \(TraditionalForm\`cf\_n\)]]], StyleBox[ "\t=\tsubsequent year cf, compounded annually based on growth rate, g\ncr"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\_i\)]]], StyleBox["\t=\t\"going in\" capitalization rate\n"], StyleBox[Cell[BoxData[ \(TraditionalForm\`cr\_o\)]]], StyleBox[ "\t=\t\"going out\" capitalization rate\nr\t=\tdiscount rate (when pv = 0 \ this is the internal rate of return)\ng\t=\tgrowth rate for cash flow, as \ described above\nk\t=\tlength holding period (year of sale, presumed to be at \ the end of the year)\n\nCompounding of cash flows is offset one year. The \ idea is that an investor on January 1 makes a decision to purchase based on \ his expectation of the income to be received at the end of the year. Thus, "], StyleBox[Cell[BoxData[ \(TraditionalForm\`cf\_\(0\ \)\)]]], StyleBox[ " is the initial cash flow to be received at the end of year one. The first \ year of compounding is 12 months later at the end of year two, hence only n-1 \ years of "], StyleBox["compounding", FontSlant->"Italic"], StyleBox[ " cash flows are in a holding period. This matter has repercussions when \ considering the equity reversion at the time of sale. Value still compounds \ for the full n periods. The rationale is that at the time of sale, the next \ investor anticipates the end-of-year income (to be received 12 months after \ his acquistion) in making his acquisition decision. So the first investor \ collects the first cash flow without any compounding and n-1 years of \ compounding cash flows but prices the property on the basis of n compounded \ cash flows.\n\nThe examples below assume a going-in cap rate of 8%, a ten \ year holding period, collection of a first year "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\((cf\_0)\)\)]]], StyleBox[ " cash flow and 9 years of cash flows compounding at the rate of 4%, \ followed by a sale based on the next year's (yet-to-be-collected) cash flow \ at a going-out cap rate of 10%. The discount rate is 10%. The examples are \ then modified to show what happens when the direction of cash flows reverses, \ when growth rates are cyclical and when holding period is extended.\n\nFor \ each illustration below, several outputs are provided: The equation for \ present value is illustrated without specifying values for r or k, then \ inserting a value for r, then inserting a value for both r and k. For \ illustration in two dimensions, all but two must be fixed. For illustration \ in three dimensions, all but three must be fixed. For simplicity initial cash \ flows are all shown at $500"] }], "Text", PageWidth->PaperWidth, CellDingbat->None], Cell[CellGroupData[{ Cell[BoxData[{ \(\n\(cfn = cf0\ \((1 + g)\)\^\(n - 1\);\)\), "\n", \(\(er = cf0\ \((1 + g)\)\^k/cro;\)\), "\n", \(\(pd = cf0/cri;\)\), "\n", \(npv := er\/\((1 + r)\)\^k + \((\[Sum]\+\(n = 1\)\%k Evaluate[ cfn\/\((1 + r)\)\^n])\) - pd\), "\n", \(npv1 = npv /. {g \[Rule] .04, cf0 \[Rule] 500, cri \[Rule] .08, cro \[Rule] .1}\), "\n", \(npv1a = npv /. {g \[Rule] .04, cf0 \[Rule] 500, cri \[Rule] .08, cro \[Rule] .1, r -> .1}\), "\n", \(npv1b = npv /. {g \[Rule] .04, cf0 \[Rule] 500, cri \[Rule] .08, cro \[Rule] .1, r -> .1, k -> 10}\), "\n", \(npv1c = npv /. {g \[Rule] .04, cf0 \[Rule] 500, cri \[Rule] .08, cro \[Rule] .1, r -> .1, k \[Rule] 10, n \[Rule] 1}\)}], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(-6250.`\) + 5000.`\ 1.04`\^k\ \((1 + r)\)\^\(-k\) + \(500\ \((\(-1\) + 1.04`\^k\ \ \((1\/\(1 + r\))\)\^k)\)\)\/\(\(\(0.04`\)\(\[InvisibleSpace]\)\) - r\)\)], \ "Output", PageWidth->PaperWidth], Cell[BoxData[ \(\(-6250.`\) + 5000.`\ 0.9454545454545454`\^k - 8333.333333333332`\ \((\(-1\) + 0.9454545454545454`\^k)\)\)], "Output", PageWidth->PaperWidth], Cell[BoxData[ \(181.00583077772217`\)], "Output", PageWidth->PaperWidth], Cell[BoxData[ \(181.00583077772217`\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "Below, the limit of the sum of discounted cash flows is merely the sum of \ the uncompounded cash flows when the growth in income is equal to the \ discount rate because the factor ", Cell[BoxData[ \(TraditionalForm\`\((1 + r)\)\^\(n - 1\)\)]], " appears in both the numerator and the denominator, thus canceling and \ leaving only ", Cell[BoxData[ \(TraditionalForm\`\[Sum]cf\_0\)]], " * k\n\nThe message here, not surprisingly, is that r and g must be \ different to make this process meaningful. Of course, they always are because \ they are quite different concepts. \"g\" is the result of local \ macroeconomic factors, the microeconomy of the neighborhood and [central to \ our story] the entrepreneurial effort of the owner to deal with site specific \ risk. \"r\" is the cost of capital influenced by many national factors such \ as expected inflation, interest rates, the return on other investments such \ as financial assets, etc." }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(\(?Limit\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \("Limit[expr, x->x0] finds the limiting value of expr when x approaches \ x0."\)], "Print", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\[Sum]\+\(n = 1\)\%k Evaluate[cfn/\((1 + r)\)\^\(n - 1\)], r -> g]\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(cf0\ k\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell["\<\ Checking the cash flows individually, here is the 9th cash flow in our \ example. For Excel users it may be comforting to set this example up in \ columns and rows, use Excel functions and arrive at the same answers as the \ next two below:\ \>", "Text", PageWidth->PaperWidth, FontSize->13], Cell[CellGroupData[{ Cell[BoxData[ \(cfn /. {cf0 -> 500, g -> .04, n -> 9}\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(684.284525202637`\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell["\<\ Checking the example's reversion and its discounted present value:\ \>", "Text", PageWidth->PaperWidth, FontSize->13], Cell[CellGroupData[{ Cell[BoxData[{ \(cf0\ \((1 + g)\)\^k/cro /. {cro -> .1, cf0 -> 500, g -> .04, n -> 10, k -> 10}\), "\n", \(%/\((1 + r)\)\^k /. {r -> .1, k -> 10}\)}], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(7401.221424591723`\)], "Output", PageWidth->PaperWidth], Cell[BoxData[ \(2853.491253833416`\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell["\<\ Returning to the general, we plot present values against various levels of r \ and k:\ \>", 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.25996 .71624 .27391 Metetra .698 .654 .802 r .73154 .10688 .67363 .0991 .66636 .18043 .72381 .19119 Metetra .454 .446 .76 r .58969 1.02959 .5382 .97174 .53306 1.02842 .58408 1.0911 Metetra .475 .465 .767 r .59541 .9668 .54344 .91365 .5382 .97174 .58969 1.02959 Metetra .496 .484 .773 r .60124 .90269 .54878 .85412 .54344 .91365 .59541 .9668 Metetra .517 .502 .779 r .60719 .83723 .55424 .79311 .54878 .85412 .60124 .90269 Metetra .538 .52 .784 r .61325 .77038 .55982 .73057 .55424 .79311 .60719 .83723 Metetra .558 .538 .789 r .61943 .70209 .56551 .66646 .55982 .73057 .61325 .77038 Metetra .577 .556 .794 r .62573 .63234 .57132 .60074 .56551 .66646 .61943 .70209 Metetra .596 .572 .797 r .63216 .56107 .57725 .53335 .57132 .60074 .62573 .63234 Metetra .614 .589 .801 r .63872 .48825 .58331 .46426 .57725 .53335 .63216 .56107 Metetra .632 .604 .804 r .64542 .41382 .5895 .39341 .58331 .46426 .63872 .48825 Metetra .649 .619 .806 r .65225 .33774 .59582 .32075 .5895 .39341 .64542 .41382 Metetra .665 .633 .808 r .65923 .25996 .60228 .24622 .59582 .32075 .65225 .33774 Metetra .681 .647 .81 r .66636 .18043 .60888 .16978 .60228 .24622 .65923 .25996 Metetra .695 .66 .811 r .67363 .0991 .61563 .09136 .60888 .16978 .66636 .18043 Metetra .444 .451 .772 r .5382 .97174 .48752 .91753 .48293 .96995 .53306 1.02842 Metetra .466 .47 .779 r .54344 .91365 .49222 .86363 .48752 .91753 .5382 .97174 Metetra .488 .49 .785 r .54878 .85412 .49702 .80821 .49222 .86363 .54344 .91365 Metetra .509 .508 .791 r .55424 .79311 .50192 .75121 .49702 .80821 .54878 .85412 Metetra .53 .527 .796 r .55982 .73057 .50694 .6926 .50192 .75121 .55424 .79311 Metetra .551 .545 .801 r .56551 .66646 .51207 .63232 .50694 .6926 .55982 .73057 Metetra .571 .562 .805 r .57132 .60074 .51731 .57032 .51207 .63232 .56551 .66646 Metetra .59 .579 .808 r .57725 .53335 .52267 .50655 .51731 .57032 .57132 .60074 Metetra .609 .595 .811 r .58331 .46426 .52816 .44096 .52267 .50655 .57725 .53335 Metetra .627 .611 .814 r .5895 .39341 .53377 .37349 .52816 .44096 .58331 .46426 Metetra .645 .625 .816 r .59582 .32075 .5395 .30408 .53377 .37349 .5895 .39341 Metetra .661 .64 .818 r .60228 .24622 .54537 .23268 .5395 .30408 .59582 .32075 Metetra .677 .653 .819 r .60888 .16978 .55138 .15922 .54537 .23268 .60228 .24622 Metetra .692 .666 .819 r .61563 .09136 .55753 .08365 .55138 .15922 .60888 .16978 Metetra .434 .457 .784 r .48752 .91753 .43758 .86668 .43359 .91532 .48293 .96995 Metetra .456 .477 .791 r .49222 .86363 .44168 .81651 .43758 .86668 .48752 .91753 Metetra .479 .496 .797 r .49702 .80821 .44587 .76476 .44168 .81651 .49222 .86363 Metetra .501 .515 .803 r .50192 .75121 .45016 .7114 .44587 .76476 .49702 .80821 Metetra .523 .533 .808 r .50694 .6926 .45456 .65635 .45016 .7114 .50192 .75121 Metetra .544 .551 .812 r .51207 .63232 .45907 .59958 .45456 .65635 .50694 .6926 Metetra .564 .569 .816 r .51731 .57032 .46368 .54102 .45907 .59958 .51207 .63232 Metetra .584 .585 .82 r .52267 .50655 .46841 .48061 .46368 .54102 .51731 .57032 Metetra .604 .601 .822 r .52816 .44096 .47325 .41831 .46841 .48061 .52267 .50655 Metetra .622 .617 .824 r .53377 .37349 .47821 .35403 .47325 .41831 .52816 .44096 Metetra .64 .632 .826 r .5395 .30408 .4833 .28773 .47821 .35403 .53377 .37349 Metetra .657 .646 .827 r .54537 .23268 .48851 .21933 .4833 .28773 .5395 .30408 Metetra .674 .659 .828 r .55138 .15922 .49385 .14877 .48851 .21933 .54537 .23268 Metetra .689 .672 .828 r .55753 .08365 .49933 .07598 .49385 .14877 .55138 .15922 Metetra .423 .463 .797 r .43758 .86668 .3883 .81889 .38495 .8642 .43359 .91532 Metetra .447 .483 .804 r .44168 .81651 .39175 .77204 .3883 .81889 .43758 .86668 Metetra .47 .502 .81 r .44587 .76476 .39528 .7236 .39175 .77204 .44168 .81651 Metetra .492 .521 .815 r .45016 .7114 .39891 .67351 .39528 .7236 .44587 .76476 Metetra .514 .54 .82 r .45456 .65635 .40264 .62172 .39891 .67351 .45016 .7114 Metetra .536 .558 .824 r .45907 .59958 .40647 .56816 .40264 .62172 .45456 .65635 Metetra .557 .575 .828 r .46368 .54102 .4104 .51278 .40647 .56816 .45907 .59958 Metetra .578 .592 .831 r .46841 .48061 .41443 .45551 .4104 .51278 .46368 .54102 Metetra .597 .608 .833 r .47325 .41831 .41857 .39628 .41443 .45551 .46841 .48061 Metetra .617 .624 .835 r .47821 .35403 .42283 .33503 .41857 .39628 .47325 .41831 Metetra .635 .638 .836 r .4833 .28773 .4272 .27168 .42283 .33503 .47821 .35403 Metetra .653 .652 .837 r .48851 .21933 .43168 .20617 .4272 .27168 .4833 .28773 Metetra .669 .666 .838 r .49385 .14877 .43629 .13841 .43168 .20617 .48851 .21933 Metetra .686 .678 .837 r .49933 .07598 .44103 .06834 .43629 .13841 .49385 .14877 Metetra .413 .47 .81 r .3883 .81889 .33961 .77392 .33693 .81627 .38495 .8642 Metetra .437 .49 .816 r .39175 .77204 .34237 .73003 .33961 .77392 .3883 .81889 Metetra .46 .509 .822 r .39528 .7236 .34521 .68455 .34237 .73003 .39175 .77204 Metetra .483 .528 .827 r .39891 .67351 .34813 .63743 .34521 .68455 .39528 .7236 Metetra .506 .547 .832 r .40264 .62172 .35114 .5886 .34813 .63743 .39891 .67351 Metetra .528 .565 .836 r .40647 .56816 .35424 .53799 .35114 .5886 .40264 .62172 Metetra .55 .582 .839 r .4104 .51278 .35743 .48555 .35424 .53799 .40647 .56816 Metetra .57 .599 .842 r .41443 .45551 .36072 .43119 .35743 .48555 .4104 .51278 Metetra .591 .615 .844 r .41857 .39628 .3641 .37485 .36072 .43119 .41443 .45551 Metetra .61 .63 .846 r .42283 .33503 .36759 .31645 .3641 .37485 .41857 .39628 Metetra .629 .645 .847 r .4272 .27168 .37118 .25592 .36759 .31645 .42283 .33503 Metetra .648 .659 .847 r .43168 .20617 .37488 .19318 .37118 .25592 .4272 .27168 Metetra .665 .672 .847 r .43629 .13841 .3787 .12815 .37488 .19318 .43168 .20617 Metetra .682 .685 .847 r .44103 .06834 .38263 .06073 .3787 .12815 .43629 .13841 Metetra .403 .477 .823 r .33961 .77392 .29143 .73154 .28946 .77128 .33693 .81627 Metetra .427 .497 .829 r .34237 .73003 .29347 .69029 .29143 .73154 .33961 .77392 Metetra .45 .516 .835 r .34521 .68455 .29558 .64748 .29347 .69029 .34237 .73003 Metetra .474 .535 .84 r .34813 .63743 .29776 .60303 .29558 .64748 .34521 .68455 Metetra .497 .553 .844 r .35114 .5886 .30002 .5569 .29776 .60303 .34813 .63743 Metetra .519 .571 .848 r .35424 .53799 .30235 .509 .30002 .5569 .35114 .5886 Metetra .541 .589 .851 r .35743 .48555 .30476 .45926 .30235 .509 .35424 .53799 Metetra .563 .605 .854 r .36072 .43119 .30725 .40762 .30476 .45926 .35743 .48555 Metetra .584 .621 .855 r .3641 .37485 .30983 .35399 .30725 .40762 .36072 .43119 Metetra .604 .637 .857 r .36759 .31645 .3125 .2983 .30983 .35399 .3641 .37485 Metetra .623 .652 .857 r .37118 .25592 .31526 .24045 .3125 .2983 .36759 .31645 Metetra .642 .666 .858 r .37488 .19318 .31811 .18037 .31526 .24045 .37118 .25592 Metetra .66 .679 .857 r .3787 .12815 .32106 .11797 .31811 .18037 .37488 .19318 Metetra .677 .692 .856 r .38263 .06073 .32412 .05315 .32106 .11797 .3787 .12815 Metetra .392 .485 .835 r .29143 .73154 .24373 .69155 .24248 .72899 .28946 .77128 Metetra .416 .504 .841 r .29347 .69029 .24502 .65266 .24373 .69155 .29143 .73154 Metetra .441 .523 .847 r .29558 .64748 .24637 .61223 .24502 .65266 .29347 .69029 Metetra .464 .542 .852 r .29776 .60303 .24778 .57021 .24637 .61223 .29558 .64748 Metetra .488 .56 .856 r .30002 .5569 .24924 .52653 .24778 .57021 .29776 .60303 Metetra .51 .578 .86 r .30235 .509 .25077 .48111 .24924 .52653 .30002 .5569 Metetra .533 .595 .862 r .30476 .45926 .25235 .43388 .25077 .48111 .30235 .509 Metetra .555 .612 .865 r .30725 .40762 .25401 .38476 .25235 .43388 .30476 .45926 Metetra .576 .628 .867 r .30983 .35399 .25573 .33368 .25401 .38476 .30725 .40762 Metetra .597 .644 .868 r .3125 .2983 .25753 .28054 .25573 .33368 .30983 .35399 Metetra .617 .658 .868 r .31526 .24045 .2594 .22526 .25753 .28054 .3125 .2983 Metetra .636 .672 .868 r .31811 .18037 .26135 .16774 .2594 .22526 .31526 .24045 Metetra .654 .686 .867 r .32106 .11797 .26339 .10789 .26135 .16774 .31811 .18037 Metetra .672 .699 .866 r .32412 .05315 .26551 .04561 .26339 .10789 .32106 .11797 Metetra .363 .509 .87 r .14951 .61805 .10292 .58421 .10397 .6162 .14978 .65163 Metetra .386 .527 .876 r .14926 .58311 .10187 .55093 .10292 .58421 .14951 .61805 Metetra .41 .546 .881 r .14902 .54674 .10081 .51628 .10187 .55093 .14926 .58311 Metetra .434 .563 .885 r .1488 .50889 .09975 .48021 .10081 .51628 .14902 .54674 Metetra .458 .581 .889 r .1486 .46947 .09868 .44264 .09975 .48021 .1488 .50889 Metetra .481 .598 .893 r .14842 .42842 .0976 .4035 .09868 .44264 .1486 .46947 Metetra .504 .615 .896 r .14826 .38564 .09653 .3627 .0976 .4035 .14842 .42842 Metetra .527 .632 .898 r .14812 .34107 .09545 .32016 .09653 .3627 .14826 .38564 Metetra .549 .648 .899 r .14802 .29461 .09437 .2758 .09545 .32016 .14812 .34107 Metetra .571 .663 .9 r .14794 .24616 .09329 .22952 .09437 .2758 .14802 .29461 Metetra .593 .678 .9 r .14789 .19565 .09221 .18122 .09329 .22952 .14794 .24616 Metetra .614 .692 .899 r .14787 .14295 .09114 .1308 .09221 .18122 .14789 .19565 Metetra .634 .706 .898 r .1479 .08798 .09008 .07815 .09114 .1308 .14787 .14295 Metetra .654 .719 .897 r .14796 .03061 .08902 .02316 .09008 .07815 .1479 .08798 Metetra .372 .501 .859 r .19644 .65378 .14951 .61805 .14978 .65163 .19594 .68917 Metetra .396 .52 .865 r .19697 .61697 .14926 .58311 .14951 .61805 .19644 .65378 Metetra .42 .538 .87 r .19753 .57869 .14902 .54674 .14926 .58311 .19697 .61697 Metetra .444 .556 .875 r .19813 .53886 .1488 .50889 .14902 .54674 .19753 .57869 Metetra .468 .574 .879 r .19878 .49741 .1486 .46947 .1488 .50889 .19813 .53886 Metetra .491 .592 .882 r .19946 .45427 .14842 .42842 .1486 .46947 .19878 .49741 Metetra .514 .609 .885 r .20019 .40936 .14826 .38564 .14842 .42842 .19946 .45427 Metetra .537 .625 .887 r .20097 .36259 .14812 .34107 .14826 .38564 .20019 .40936 Metetra .559 .641 .888 r .2018 .31389 .14802 .29461 .14812 .34107 .20097 .36259 Metetra .58 .657 .889 r .20268 .26317 .14794 .24616 .14802 .29461 .2018 .31389 Metetra .601 .672 .889 r .20362 .21032 .14789 .19565 .14794 .24616 .20268 .26317 Metetra .622 .686 .889 r .20461 .15527 .14787 .14295 .14789 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