## Insight Into The Analysis

It is time to engage in some decomposition, looking behind the equations and the spreadsheet icons into the inner workings of the process. Previously, we referred to the variables as "deterministic" because we can determine the outcome—the net present value—by choosing values for particular variables.1 The outcome changes every time the values of the variables change. With any change in the nominal value of a variable, we explicitly cause a change in return, as measured by the net present value. But there is usually also a corresponding implicit change in the risk. Understanding the inner workings of the variables provides a more explicit view of risk and an insight into the bargaining process. Seeing dependencies at the general level allows us to ask ''if-then'' type questions about the entire process, not just about a single acquisition.

To illustrate the concept of dependency in a very simple case, we begin by looking at the deterministic inputs that affect the gross rent multiplier. We know this equation as value .

gross scheduled income

Hence, it would seem that grm is simply dependent upon two variables, the value and the gross income. Because value is defined in our example as a combination of two other deterministic variables, down payment and initial loan, the expression ''grm'' actually depends on variables which are the antecedent primitives that make up value.

g gsi

1Many of the relationships described in this section are dependent on the way our sample project is described. The most general approach would be independent of the construction of any particular example. Our purpose here is to strike a balance between theory and practice by using a stylized example and highlighting aspects of the process to illuminate its general meaning.

So we see that, given how we have defined the variables, three things determine the grm, not the two we originally thought.

Perhaps the decomposition of grm is too obvious. One can easily see what determines grm. More difficult and complex examples exist at the other extreme. When we look at what affects after-tax cash flow, cf0, we find a really ugly equation that incorporates all of the inputs leading to this output.

12 i (1 + if initln ( , 1 cfo = gsi--:--7--txrt I initln H-----r

(initln ((1 + i)12 -(1 + i)'(1 + 12i) + dprt(-1 + (1 + if )(-1 + land)) + dpdprt (-1 + (1 + i)')(-1 + land))+(-1 + exprt) gsi (-1 + vacrt) + exprt gsi (-1 + vacrt) - gsi vacrt

Ugly as Equation (4-3) may seem, it is really nothing more than a fairly long algebraic equation. One could, with some difficulty, construct such an equation from the formulae underlying the cells of a spreadsheet program.

Sometimes we can gain more useful insight by giving fixed, numeric values to some of the variables. This has the beneficial effect of eliminating some of the variables as symbols in favor of constants. One approach is to substitute real numbers for those variables out of the owner's control. For instance, income tax rates, depreciation rates, and land assessments are handed down by government. Taking the relevant data from Table 4-1, in Equation (4-4) we reproduce Equation (4-3), providing fixed values for tax rates and land assessments, thereby reducing the number of symbolic variables to cap rate, loan amount, interest rate, expense and vacancy rates, and the gross scheduled income.2 Do these affect cash flow? They certainly do, and the owner has some influence on them.

Suppose we have already decided to purchase the property or we already own it. Under those conditions we may know the income, loan details, and expense and vacancy factors. Inserting these values as numbers, Equation (4-5) shows us that our cash flow is related to some constants and the interest rate. This permits us to consider explicitly the risk of variable interest rate loans. We also get a feel for the meaning of what is sometimes referred to as ''positive leverage.'' Using capitalization rate > loan constant as the

2Note that some of the constants combine into other numbers not shown in Table 4-1 because Equation(4-4) has been simplified.

definition of positive cfo gsi----—t—h exprtgsi (—1 + vacrt) — gsi vacrt

— 0.35 (—0.0225806 dp + gsi + 0.977419 initln---———

(1 + i)l(1 — (1 + i)12—t) initln --^ y--^ exprtgsi (—1 + vacrt) — gsi vacrt)

leverage, we know that if leverage is positive then cash flow must be positive. (If you don't know that then you have just discovered an important reason to use symbolic analysis.) As the first constant term in Equation (4-5) is the net operating income, the aggregate of everything after that term must be smaller than that number for cash flow to be positive. This is, of course, critically dependent on the interest rate.3

10500000i

0.35

By varying the loan interest to a rate above and below the going-in capitalization rate, cr4, Table 4-3 shows first positive leverage then negative leverage, this time using capitalization rate > interest rate as our definition. Note the difference in cash flow.

Another awful looking equation is what goes into the witches brew we call the equity reversion, shown in Equation (4-6). Note that since the loan is assumed to be paid off at the time of sale, the equation contains a constant, the final loan balance. This would certainly be a constant when the loan has a fixed interest rate. If the loan carried a variable rate of interest, an equation

3Further analysis, left to the reader as an exercise, will disclose under what conditions our definition of positive leverage is a stronger or weaker constraint than the alternate definition for positive leverage, capitalization rate > interest rate.

TABLE 4-3 Initial Cash Flow with Loan Interest above and below the Capitalization Rate cri = .0936

would replace the constant. 1

(cro (-843381 + cgrt (dp + initln)(1 + dprt k (-1 + land)) — ppmt) + (1 + g)kgsi (1 — scrt — vacrt + scrt vacrt + exprt (—1 + scrt + vacrt — scrt vacrt) + cgrt (—1 + scrt + vacrt — scrt vacrt + exprt (1 — scrt — vacrt + scrt vacrt))))

The capital gain in Equation (4-7) is a little more accessible. Note that it is, not surprisingly, quite dependent on the going-out capitalization rate.

+ (— 1 + exprt)(1 + g)kgsi( 1 + scrt)(— 1 + vacrt) , N

If we are interested in what drives before-tax cash flow, Equation (4-8) shows that it is, of course, heavily dependent on the loan terms and net operating

, . 12 i (1 + iYinitln , btcf =---1 +(/+ i)t +(-1 + exprt) gsi (-1 + vacrt) (4-8)

A look at the variables that influence the tax consequence is the result of subtracting the symbolic expression for before-tax cash flow (btcf) from the symbolic expression for after-tax cash flow (cf0 in the initial year). Note the recognizable components in Equation (4-9). The large term inside the parentheses multiplied by the tax rate is the taxable income from operating the property. Inside the parenthesis we see the components of real estate taxable income. If you stare at it long enough, you will see the components of the net operating income, the interest deduction, and the depreciation deduction.

f , f , , 12 iinitln (1 -(1 + i)12-t) initln cf0 - htcf = - txrt (gsi + initln - 1 -(1 + .)-t--1 -(1 + i)-t-

— dprt (dp + initln - (dp + initln) land) - gsi vacrt

Returning to an exceedingly simple term, as we learned in Chapter 3 the net operating income (or debt free before-tax annual cash flow) in Equation (4-10) is really only a function of the gross income and two rates, vacancy and expenses.

Of course the debt service, ds (the annualized monthly loan payment), is a function of the interest rate, the term, and the amount borrowed. Note in Equation (4-11) the constant 12 multiplies out the monthly factor. This is necessary when the input data provide the interest rate and amortization period in monthly form.

Some readers will recall the Ellwood tables. The equations underlying these are easily provided. Equation (4-12) is the factor from Ellwood Table #6—the payment necessary to amortize a dollar. To produce this we divide out the 12 in Equation (4-11) and make initln equal to 1.

For museum curators and those who still own Ellwood tables, inserting numeric values for i and t produce one of the numbers found in the tables. This same number is more usually found with a hand calculator with financial function keys. Using i = 0.10/12 as the interest rate and t = 360, Equation (4-12) returns a monthly payment of 0.00877572 for a loan of \$1.

In considering a variable interest rate loan, it can be useful to ask what happens to cash flow if interest rates rise. In Equation (4-13), note the second term, the fraction with the i variables in it. Of course, this term is monthly debt service (all the other variables sum to noi). Remembering what a negative exponent in the denominator means, we observe this function rising with interest rates. The entire term is negative, so as it gets bigger, btcf grows smaller.

, r 12 i initln btcf = gsi - -—^ + t - gsi vacrt - exprt (gsi - gsi vacrt) (4-13)

Some equation decomposition is unproductive. For instances, suppose the vacancy increases. What does this do to after-tax cash flow? Notice in Equation (4-14) that it affects only the last term in the equation for first year cash flow. This is not too helpful as that last term also has the tax rate in it, something that has nothing to do with vacancy.

(initln ((1 + i)12 -(1 + i)'(1 + 12i) + dprt(-1 + (1 + i)')(-1 + land))

+ (-1 + exprt) gsi (-1 + vacrt)) + exprt gsi (-1 + vacrt) - gsi vacrt

We have covered just a few examples showing how insight into the process can be gained by dissecting the equations in component parts and looking at dependencies. Symbolic analysis is rather sterile and too abstract for some. Let's combine this approach with the sample project and see how it may be applied in practice.

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