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Statistical & Risk Rated Income Approach Method

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Austin

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On another thread I described my method of integrating the income approach and stistical methods and I was asked by one person to e-mail them a sample of what I am doing. I couldn’t e-mail it for technical reasons so here is an example you can work on your own computer.
The significant point of this method is that in times such as these with interest and equity rates making huge shifts over short time periods, it appeared to me that using the conventional income approach format was misleading because of the significance of the rate structure and the elasticity of rates versus NOI. In summary: Rates are so much more elastic than NOI changes that in this environment point estimates of value results are questionable. A better method is to show a most probable price range under a wide range of reasonable scenarios. What this indicates to me is that the elasticity of rates is so significant and expenses so unstable and relatively less elastic that it is a waste of time and misleading to fine tune operating expenses and give point estimates of value.
To begin, take any pro forma operating income statement or use this example. Say we are appraising an apartment complex of five units. We estimate the NOI at $10,000 and the operating expense ratio of 25%.
At t his point on your spreadsheet make a column labeled operating expense ratios. Label the adjoining column to the right NOI’s. Bracket the extracted operating expense ratio with a ratio of 20% and 30%, so now you have a column of expense ratios of 20%, 25%, and 30%. Next in the adjoining column put the corresponding NOI’s for that expense ratio by multiplying 1-Expense ratio times the EGI.
Now, start a new box in another location on the spreadsheet. The first column put the historic cap rate range for that type property. For example, I have done a lot of apartment complexes over the years and the cap rate has never been below 8% or above 11%, so anything outside that range is possible but not probable. Make a column of rates at 8% thru 11%.
Now label the adjoining three columns value @ OER 20%, value @ OER 25%, value @ OER 30%. Then under each column put a capitalized price for each cap rate. For example, under value @ OER 20%, cap the NOI at OER 20% for each cap rate 8-11%. Then do the same for each column for each expense ratio.
At this point you have a left hand column of cap rates from 8-11%, and the adjoining three columns show a range of indicated prices at each rate.
Now graph the results using cap rates on the X-axis and prices Y-axis all on the same graph. If you want to, highlight one set of data points and calculate the trend line and formula but that is not necessary.
Use your best estimate of most probable cap rate range giving consideration to the past and future possible rates as reflected by interest and equity rates. In this example, say 8.5 to 10.5% cape rate. Now using your drawing or line art function, draw a box that covers all possible prices encompassed inside the trend lines between 8.5% and 10.5%. With interest and equity rates at their present low level, the indicted point estimate of price is in the upper left corner of the box, but under any foreseeable and reasonably expected rate changes, the price will most likely not be outside the box. This box represents the most probable price range under any reasonable and foreseeable situation. Also, note that a 1% change in the cap rate will make a big difference in price but a 5% change in the expense ratio makes a much less difference. Point being: Under these conditions why spend hours estimating expense items that are as unstable as a feather in the wind.
In my view it would be misleading to report a point estimate of price in the upper right hand corner of the box because of the artificially lot interest and equity rates at present. If you used the band of investment method to support a cap rate I don’t even have a guess what the equity cap or yield rate would be, but they would be on the low end resulting in an artificially low cap rate and resulting in an artificially inflated price estimate.
If interest and equity rates and expense items were fairly stable, this would not be necessary, but in this type environment given the sensitivity of cap rates and unstable expense items, like for instance casualty insurance and utilities, in my view this is the best an appraiser can report. In a risky environment you have to use methods that account for the risk. If I were a client and saw this box it would give me a feel of the gravity and dynamics of the market forces. For example, if six months from now interest rates go up 1.5% and the stock market returns to something reasonable, I know my property is not likely still worth what it was six months ago and I have an idea what is worth in today’s market.
 

Terrel L. Shields

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. A better method is to show a most probable price range under a wide range of reasonable scenarios.

Seems if you had enough data, you could arrive at a point value by using scenario analysis, a.k.a. Monte Carlo Simulation...which does dozens of iterations for each variable and ending up with a curve (which should for real estate data be a relatively uniform Bell Curve of values)

You still could report it as a range, but I might would rather say XX dollars with a margin of error of X %.

I say this because for estate purposes, the CPA is going to need a point value in calculating tax burdens. A range invites the IRS to take up the high value as fair market value (the highest probable price) as opposed to market value (the most probable price)
 

Austin

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Terrell: If you did the graph as I described and used middle of the range income and expense items in the operating income statement, then the middle trend line or the 25% ratio price trend line is the center of gravity and the point or most probable estimate of price is the center of that line at the center of the box based on historical market supported cap rates. It is sort of like a three-dimensional bell curve that occurs over time. Time is the key element because the lack of rate and economic price stability over time is what makes an unqualified point estimate of value potentially misleading. That is the point estimate I report but I point out that as of date of appraisal due to existing rate levels the existing rate numbers indicate a higher price but the historical market rates over time reflect a lower price. The market as of appraisal date has not caught up to reflect the new number situation. When we see market evidence of a shift in the range of cap rates we can move the numbers up, but it is extrapolating to do so without market support.
The thing I like about this system is that it gives you such a feel of the gravity of the problem. Did you notice on the graph that it takes better than a 9% movement in the operating expense ratio to equal a 1% movement in the cap rate? If interest rates are political numbers subject to the range of movements we are experiencing, it seems to me that the scope of the appraisal changes and methods have to reflect to scope. We now have a lag period between old and new rate numbers to deal with.
 

Fred

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Austin,
1. It is very hard to follow your instructions. Post the spreadsheet. Use Code
2. Elasticity has a definition- for example, on page 313 of Spencers, Contemporary Economics. It is percentage-change over percentage-change. Going from 8 to 9 is not as you say a percentage change of 1. As a percentage change this is 1/8 or 12.5%. Go to your 12C, second row of keys down, fourth key from the left. The percentage change key has triangle and a percent sign. Hit 8, then ENTER, then 9, the PERCENTAGE CHANGE. Your display should read 12.5, which means 12.5%. The formula for percentage change is (A-B)/A, not A-B as you are saying. You are equating disproportional changes. That is why their effect on results is so disproportional.

3. Below is the spreadsheet I made. The first three columns show the effects of changing OARs. The next three columns show the effects of changing NOIs. The baseline is in the row in the middle of the table containing the "correct" numbers - $10 NOI, 10% OAR and $100 Value., Each successive row above that increases (error) the OAR by 5% in Column 2 and decreases (error) the NOI by 5% in column 5. The bottom half of the spread sheet contines the same pattern. The top line of the spreadsheet shows the effects of 8 successive 5% errors to OAR and NOI, measuring elasticity (as defined in Economics). Please note in the top row that column 3 shows the value changed from the correct $100 to $67.68 by OAR errors. Column 6 shows that the value changed from the correct $100 to $66.34. So, yes, the results changed more with OAR, but not by very much more. Not that inch versus mile stuff you were coming up with.

Code:
NOI	OAR-Err	V Chg	NOI-ERR	OAR	V Chg

$10	14.77%	$67.68	$6.63	10.00%	$66.34

$10	14.07%	$71.07	$6.98	10.00%	$69.83

$10	13.40%	$74.62	$7.35	10.00%	$73.51

$10	12.76%	$78.35	$7.74	10.00%	$77.38

$10	12.16%	$82.27	$8.15	10.00%	$81.45

$10	11.58%	$86.38	$8.57	10.00%	$85.74

$10	11.03%	$90.70	$9.03	10.00%	$90.25

$10	10.50%	$95.24	$9.50	10.00%	$95.00

$10	10.00%	$100.00	$10.00	10.00%	$100.00

$10	9.50%	$105.26	$10.50	10.00%	$105.00

$10	9.03%	$110.80	$11.03	10.00%	$110.25

$10	8.57%	$116.64	$11.58	10.00%	$115.76

$10	8.15%	$122.77	$12.16	10.00%	$121.55

$10	7.74%	$129.24	$12.76	10.00%	$127.63

$10	7.35%	$136.04	$13.40	10.00%	$134.01

$10	6.98%	$143.20	$14.07	10.00%	$140.71

$10	6.63%	$150.73	$14.77	10.00%	$147.75
 

Austin

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Certified General Appraiser
State
Virginia
Steven: I am not sure if you are talking about the subject of this post or what we were talking about the other day with your example. My definition of elasticity is the math relationship between independent and dependent variable like on a graph. If a change of 1 unit in X ='s a 1 unit change in Y, then that is unitary elasticity. As the slope of the trend line changes elasticity takes place when a 1 unit change in X becomes more or less than 1 unit change in Y. If the slope of the trend line is really sloping the elasticity becomes large so that a 1 unit change in X can equal a 5 unit change in Y.

You asked me to post my spread sheet in code but I assume you mean the one I described in my opening post on this thread. I don't know how it will come out but it is below posted in code. The first box is the OER rates versus corresponding NOI's. The last box is the cap rates vs indicated prices at each OER rate. Now just go into Excell, highlight the cap rate range, then the next three columns and do an X-Y graph. All three trend lines should show up. Note under value 20% column that a 1% change in the cap rate from 8 to 9% ='s a price change of about $15,000. Then notice across the top row of price indications at 8% that a 10% (from 20% to 30%)change in the OER ='s a price change of about $16,000. This is what I mean by elasticity between the independent variable, cap rate, and the dependent variable price. A 1% difference in cap rate wipes out or makes insignificant a relatively large change in the OER. The point of all of this that when interest rates, the key component of the cap rate, are moving in a range of up to 4% and equity rates are at rock bottom but also showing a wide range, cap rates based on these numbers really muddies up the water. To cope with the situation we need a system to put price estimates in a perspective that reflects this sitution.

Code:
OER	NOI 

20% $10,666.67 	

25% $10,000.00 	

30% $9,333.33 	



CAP RATE	VALUE 20% VAL 25%    VAL 30%

8% $133,333.33  $125,000.00  $116,666.67 

9% $118,518.52  $111,111.11  $103,703.70 

10% $106,666.67  $100,000.00  $93,333.33 

11% $96,969.70  $90,909.09  $84,848.48
 

Steve Owen

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Missouri
I'm not sure I see what is new and different here, other than reporting a range rather than a specific point of value. Based on the last graph, it looks like at various different OER's with various different CAP rates, all of the values within the graph are possible (probable). I always knew that, and didn't have to graph it out (other than in my mind) to know it. If you expand the graph to it's most distant possibilities, then you could say "based on my analysis, the market value of the subject property could be any possible amount." My clients would never stand for that, even with the range of values restricted to the probable rather than expanded to the possible. (I am well aware that USPAP makes reporting a range of values allowable.) Without exception, my clients want me to report an opinion of value that is a single number. So, the question is, are you using your trend line to come to a single point of value? If so, how is that reflective of the market? Other than taking the starting point NOI and OER from the actual property, and using a range of possible CAP rates from previous appraisals, I don't see where this analysis is truely reflective of the market.

All of that being said, this analysis appears to be quite logical. However, it seems to me to be a bit misleading. Unless you are using very specific language in your report that specifies this analysis is for a range of probable values under a set of probable interest and expense assumptions, rather than for a specific value under current interest and expense conditions, I don't think the typical reader would understand it.

Why not reflect a more specific reality instead of spelling out all of the probable (possible) values. Take the most recent sales of similar income properties and extract a CAP rate based on NOI/Price, weight those extracted ratios based on which properties are most like the subject and the most recent sales to get a specific CAP rate; then use actual expense experience, modified by changing conditions (such as increased insurance costs), if necessary. If you get your CAP rate from the market and your OER from experience you will actually be able to report a specific opinion of value, which is what most players in the market want an appraiser to do. I don't buy the logic that this is difficult to do in changing times - if it was easy everyone would be an appraiser. The appraisal work I do is for a specific opinion of value as of a specific date, not for a range of possible values based on possible future conditions; that's what the client wants.
 

Austin

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Virginia
Steve:
I think you missed the thrust of the point. My graph does all of the things you just said an appraisal should do and then some. The cap rates are not something I pulled out of the air, they are market derived, as are the expense ratios. The point is, if everything was static none of this would be necessary but the key ingredients of cap rates are anything but static. Nobody would expect cap rates to be the same when interest rates are 9% and equity cap rates 15% versus when interest rates are 5% and equity cap rates 5%. My graph shows a very definite point estimate of value as well as the risk. The point estimate of value is the center of the 25% OER trend line in the dead center of the box. I guess you have to have some experience with probability density functions to get the point. Let me tell you what I see on this graph. The most probably price estimate from the graph of the above data is centered at $105,000. The range is about $125,000 to $85,000. To be at the extremes of the range would really take some unusual circumstances, which suggest a strong center of gravity at the most probable price point, meaning it ain’t likely to get that high or low. The center of gravity is at $105,000 and it is a strong pull. Remember the definition of market value explicitly asks for “the most probable price.” Most probably is a statistical concept that can only mean the price estimate with the highest probability of being achieved. You stated the price could be any value inside that range and it certainly could be but the point is that it is not probable. From a statistical point of view, there is only one most probable price indication and it is $105,000. If you put anything else you have it wrong.
Not only do I know the most probable price estimate, I have a good feeling for the market forces affecting price. On the vertical axis I can gauge the influence of changes in expenses and gauge the gravity of that force. On the horizontal axis I can gauge the influence of cap rate changes, and on the slope of the trend line curves I can gauge the combined influence of the combined forces, expenses and cap rates. This graph gives me a three dimensional picture of the economic model.
I bet over the last five years I have been told a thousand times: “Yea, but can the clients understand it?” My reply is: “Do the clients understand that using existing appraisal methods you are reporting a most probable price estimate by using a procedure that makes it mathematically impossible to answer that question? Do your clients understand that the sequence of adjustments comes from the market but the science of appraisal has no procedure to extract the adjustments from the market in the proper sequence so your sales approach result can’t possibly be correct? Do your clients understand that one of the most significant factors affecting vale estimating is normal random variance, and existing appraisal methods don’t even recognize its existence much less deal with it while at the same time the definition of market value is predicated on normal random variance? Does your client understand that covariance of value factors is a very significant force in the cost and sales comparison approach and that when using existing methods you are completely ignoring its existence so there is no way your outcomes could be mathematically correct? Do your clients understand that the definition of market value has no meaning outside of a statistical definition when not one appraiser in 1,000 understands the concept much less the question, so how can the clients understand the result of the appraisal method while at the same time not knowing exactly what the answer means……”
 

Steve Owen

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The point is, if everything was static none of this would be necessary but the key ingredients of cap rates are anything but static.

The point is, my clients want an opinion of value that is a specific number as of a specific date. At that point in time all of these figures are rather static. Reporting a set of ranges based on possible changes in CAP rates or OER would be misleading. I don't see anything wrong with your statistical method, I just don't see anything new or different here and I don't see any point in confusing clients with possibilities when what they want is an opinion of value. I have used similar methods in the past in conjunction with standard appraisal techniques to come to an opinion of value.

As for your last paragraph and all your denigrating remarks about appraisal methods. Well, most of that is pure hog wash. The math still cannot "see" that one apartment complex has a higher degree of risk (for whatever reason) and therefore deserves a slightly different CAP rate than the next. A certain amount of subjective jugement will always be called for in this profession.
 

Fred

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Austin,
I meant to write in my last post that you seem to have your own definition of elasticity and this "unit elasticity" thing is indeed your own creation. I was operating under the classic definition of elasticity, which is why I could not figure how you got to your conclusions. As I have been say, our disagreement was in part semantics. Also, you changed the basis of comparison a bit from the original discussion, which was which effects Vo more - change in OAR versus change in NOI. Now you are measuring change in OAR versus change in OER and that changes the way the numbers look in unit elasticity.

In the classic use of the term elasticity, the analyst uses one of two formats:
1. two rates of change figured the exact same way (delta% divided by delta%)
2. two rates of change put over a base of 1 to create elasticity ratios (or coefficients).
In this format, the analysis results in two quantities born of the same proportionality or two quantities with a common denominator.

You "unit" elasticity statistically speaking is comparing apples and oranges. The classic axiom that I have heard to express your idea is “a 1% change in cap rate results in about a 10% change in value.’ This is relevant to your cap rate range of 8-11. The problem with that axiom is that just because you can stick a percent sign after two numbers does not mean that are related in the way that is critical to any particular analysis. In this axiom, the 1% and 10% are not percentages of the same common denominator and as rates of change they are not derived the same way – so in no way whatsoever do the denote the 10-to-1 proportion (or elasticity) that they seem to imply.

You must control the experiment. On my spread sheet each successive degree of error to NOI and OAR is equally proportional. Only when you vary the two independent variables in the same proportion can you measure their relative effect on the dependent independent variable Vo. If not, it is apples and oranges, because your "unit" is actually two separate units.

I would like to say, I agree with Austin about 90% of the time. The other 10% of the time is when he is wrong. :lol:
 

Austin

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Steven:
I would like to say, I agree with Austin about 90% of the time. The other 10% of the time is when he is wrong.

I will take 90% any time. The other 10% is actually semantics.
My definition of elasticity is not my own creation. I was taught that definition in a micro economics theory class taught by a very prominent Harvard educated professor. Unitary elasticity means no elasticity; I just used that phrase to illustrate the difference. I agree that when I compare OAR and OER rates is comparing apples and oranges, but there is a reason to compare apples and oranges. I am not looking at the graph the same way you apparently are. I am looking at it as a measure of the gravity of these two value factors. It tells me how critical operating expenses are in influencing price and it tells me how critical the cap rate is, which is important in an unstable environment. If interest and equity cap rates are making quantum leaps in the short run, then an appraiser cannot ignore the consequences. The consequences can only logically be a widening range of cap rates for a specific type of property over the short run. This is happening as we speak. Look at the published data on yield and cap rates over the last few years and they all show a range and an average. If interest and equity rates were static then the differences would be attributable to different levels of risk in the properties or just general random variance. But, in the environment we are in now, the range is being driven by unstable interest and cap rates, which adds to existing risk. If you asked ten appraisers to review a data set and extract a cap rate and OER's, you would come up with ten different numbers. I think you said that you can log into the AI’s site, so if you can, go over their and find the latest edition of “The Appraisal Journal” October 2002, turn to page 407 and read “A Statistical Definition of Value" by Max Kummerow.” The reason we are having semantic problems is that we have a completely different underlying concept of the definition of market value and the general statistical nature of appraising. I can say the same about Steve Owens' comments. It is like to computer operating systems trying to communicate. Same words and symbols mean different things because the underlying concepts are different. That atricle will show you my mind set because it reflects what I have believed for many years.
PS: Don't forget that the reason interest and equity rates are low is because of a depressed economy. This can only add to the risk factor thus further increasing the cap rate range over the short run.
 
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