Private Real Estate Investment: Data Analysis and Decision Making_5 potx

THE NET PRESENT VALUEIn order to determine net present value, we need a function, Equation 4-1, investor’s required rate of return, r, sums these present values adds the total to Figure

THE NET PRESENT VALUE

In order to determine net present value, we need a function, Equation (4-1),

investor’s required rate of return, r, sums these present values adds the total to

Figure 4-8 displays the results of Equation (4-1) for the sample project

We certainly went to a lot of trouble only to learn that the property has anegative net present value For our sample project this means its return doesnot support its cost of capital Note that all ‘‘Yields’’ in Figure 4-8 are less thanthe investor’s required rate of return, r Therefore, the potential buyer/investor will reject the project Simply stated, a negative net present valuemeans this project is a bad deal from the buyer’s perspective

VALUE 1250000 1287500 1326125 1365909 1406886 1449093 1492565 1537342 LOANS 875000 871061 866667 861764 856294 850191 843381 835784 EQUITY 375000 416439 459458 504145 550592 598902 649184 701559 ACCRUED DEPRECIATION 31818 63636 95455 127273 159091 190909 222727 SALE COST PERCENT:

ACB AT SALE 1314744 1285823 1256989 1228244 1199591 1171033 1142573 CAPITAL GAIN − 27244 40302 108920 178642 249502 321532 394769 REAL GAIN − 59063 − 23334 13466 51370 90411 130623 172042 TAX RATE 15.00% 15.00% 15.00% 15.00% 15.00% 15.00% 15.00% 15.00% RECOVERY RATE 25.00% 25.00% 25.00% 25.00% 25.00% 25.00% 25.00% 25.00%

REVERSION CALCULATION:

B-TAX SALES PROCEEDS 319876 359999 401701 445076 490220 537242 586258 TAX 905 − 12409 − 25883 − 39524 − 53334 − 67321 − 81488 AFTER TAX EQ REVERSION 320781 347590 375818 405552 436886 469921 504770

FIGURE 4-7 Sale computations for sample project.

INSIGHT INTO THE ANALYSIS

It is time to engage in some decomposition, looking behind the equations andthe spreadsheet icons into the inner workings of the process Previously, wereferred to the variables as ‘‘deterministic’’ because we can determine the

The outcome changes every time the values of the variables change With anychange in the nominal value of a variable, we explicitly cause a change inreturn, as measured by the net present value But there is usually also acorresponding implicit change in the risk Understanding the inner workings

of the variables provides a more explicit view of risk and an insight into thebargaining process Seeing dependencies at the general level allows us to ask

‘‘if–then’’ type questions about the entire process, not just about a singleacquisition

To illustrate the concept of dependency in a very simple case, we begin bylooking at the deterministic inputs that affect the gross rent multiplier Weknow this equation as

Hence, it would seem that grm is simply dependent upon two variables, thevalue and the gross income Because value is defined in our example as acombination of two other deterministic variables, down payment and initialloan, the expression ‘‘grm’’ actually depends on variables which are theantecedent primitives that make up value

FIGURE 4-8 Net present value and IRR computations for sample project.

1 Many 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 thingsdetermine the grm, not the two we originally thought.

Perhaps the decomposition of grm is too obvious One can easily seewhat determines grm More difficult and complex examples exist at the

we find a really ugly equation that incorporates all of the inputs leading tothis output

tinitln

to some of the variables This has the beneficial effect of eliminating some ofthe variables as symbols in favor of constants One approach is to substitutereal 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) wereproduce Equation (4-3), providing fixed values for tax rates and landassessments, thereby reducing the number of symbolic variables to cap rate,loan amount, interest rate, expense and vacancy rates, and the gross

owner has some influence on them

Suppose we have already decided to purchase the property or we alreadyown it Under those conditions we may know the income, loan details, andexpense 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 interestrate 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

2 Note 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

cf0gsi12 i ð1 þ iÞ

tinitln

tinitln

ð1 þ iÞtð1  ð1 þ iÞ12tÞinitln

ð4-4Þ

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 touse symbolic analysis.) As the first constant term in Equation (4-5) is the netoperating income, the aggregate of everything after that term must be smallerthan that number for cash flow to be positive This is, of course, critically

@

1CAð4-5Þ

By varying the loan interest to a rate above and below the going-in

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 callthe equity reversion, shown in Equation (4-6) Note that since the loan isassumed 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 afixed interest rate If the loan carried a variable rate of interest, an equation

3 Further 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.

would replace the constant.

croðcro ð843381 þ cgrt ðdp þ initlnÞ ð1 þ dprt k ð1 þ landÞÞ  ppmtÞ

ð1 þ scrt þ vacrt  scrt vacrtÞ þ cgrt ð1 þ scrt þ vacrt

ð4-6ÞThe 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

cg ¼ 

cro ðdp þ initlnÞ ð1 þ dprt k ð1 þ landÞÞ

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

tinitln

A look at the variables that influence the tax consequence is the result ofsubtracting the symbolic expression for before-tax cash flow (btcf ) from

Note the recognizable components in Equation (4-9) The large term insidethe parentheses multiplied by the tax rate is the taxable income fromoperating the property Inside the parenthesis we see the components of realestate taxable income If you stare at it long enough, you will see the

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

cr i ¼ 0936

i ¼ 09 cf 0 ¼ 28,921

i ¼ 095 cf 0 ¼ 26,652

components of the net operating income, the interest deduction, and thedepreciation deduction.

ð1  ð1 þ iÞ12tÞinitln

Returning to an exceedingly simple term, as we learned in Chapter 3 thenet 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 andexpenses

Of course the debt service, ds (the annualized monthly loan payment), is afunction of the interest rate, the term, and the amount borrowed Note inEquation (4-11) the constant 12 multiplies out the monthly factor This isnecessary when the input data provide the interest rate and amortizationperiod in monthly form

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

For museum curators and those who still own Ellwood tables, insertingnumeric 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 finan-cial function keys Using i ¼ 0.10/12 as the interest rate and t ¼ 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 whathappens to cash flow if interest rates rise In Equation (4-13), note the secondterm, 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 anegative exponent in the denominator means, we observe this function risingwith interest rates The entire term is negative, so as it gets bigger, btcf growssmaller.

Some equation decomposition is unproductive For instances, suppose thevacancy increases What does this do to after-tax cash flow? Notice inEquation (4-14) that it affects only the last term in the equation for first yearcash 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

tinitln

AN ILLUSTRATION OF BARGAINING

Most of the foregoing examples all have to do with isolating one deterministicvariable Does the change in one variable affect another? What about interestrates and capitalization rates or vacancy and expenses? Are these related? Yes,they are How about gross income and vacancy? What happens when two ofthese change? Let’s take a simple example When rents increase vacancyshould also increase Below, we see they both affect net operating income Thekey question is: How much of the increase in vacancy will neutralize theincrease in income? This is the sort of thing that sensitivity testing does Weare interested in knowing how sensitive tenants are to rent increases Will a

4 Economists call this price elasticity.

buyer for the property in our example, and using the required rent raise (rrr)idea introduced in Chapter 3, we will look at how this process enters into thenegotiations with the seller.

We will assume our building is in a market where the equilibrium grm is 6

We know from Table 4-2 that the grm for our building is 6.25 We see that thebuilding is, not surprisingly, offered for sale above the equilibrium GRM Onecannot blame the seller for trying Recalling that Equation (3-1) fromChapter 3 tells us what rent raise is necessary to bring the acquisition toequilibrium, inserting the grm for our project and our market-based rule forequilibrium grm, we find that our required rent raise is 4.1667% After acareful rent survey in the neighborhood, we conclude that the tenants will paythe new rent without excessive turnover or increased vacancy

We now modify our data to consider the higher rents to see what happens

to npv We modify the input data, increasing gsi from the original data by therrr, given the equilibrium grm of 6 This means the buyer will have to institute

an immediate rent raise upon the transfer of title to him

The npv given this new information is 10,353 Even with thisimprovement we still do not have a positive after-tax net present value.Something else has to change We believe we have extracted the most out ofthe tenants in the form of increased current rent, so our second change willaffect future rent We assume a higher annual growth rate, 3.25% rather than3% in Table 4-1, on rents The two changes together produce a positive npv of

$84.85, essentially zero

With a barely positive net present value we know that the project has anIRR just above the 13% hurdle rate But these modifications to thedeterministic variables have the buyer taking all the risk Why? It is thebuyer who must raise current rents It is the buyer who must depend for hisrequired return on a higher future rate of growth in rent The assumption of ahigher growth in rent means the buyer is required to raise future rents faster.How might we transfer some of the risk to the seller? The simple answer is

to offer a lower price A buyer refusing to pay a certain price is simultaneouslyrefusing to take a certain level of risk for the reward offered Our lastmodification restores the old 3% growth rate for rent but reduces the down

project in the positive npv range without having to make the assumption of3.25% future rent increases

Note how this change improves first year performance as measured by therules of thumb in Table 4-4 With the loan amount constant, the ltv is higher,

an indication of increased risk, but at the same time the dcr has increased, anindication of reduced risk One wonders if these perfectly offset How we

5 In practice, it may be that price reduction is shared between loan amount and down payment.

reconcile them to determine if, on balance, the overall risk is more or lessthan before will be left for Chapter 5.

Note in Table 4-5 that the npv is positive But for this to be true, thecapitalization rate must decline over the holding period This is another riskfactor that we will address later

The payoff for undertaking symbolic analysis begins to take shape Thepositive npv outcome for the last set of inputs produces the same approxi-mately 13% IRR as the earlier set of inputs (where npv was approximately

$85) But is the second 13% IRR the same 13%? By now we should recognizethat the two IRRs, though nominally the same, are, in fact, not equivalent Itshould be evident that the risk of the project must be different under the lastset of conditions than under the prior set, even though npv is approximatelyzero in both cases and the IRR is essentially the same

The internal rate of return is the number that solves Equation (4-1) for

r when npv is set to zero Mathematicians consider this a problem offinding the ‘‘root’’ of the equation, an IRR of 0.132001 when npv is zero usingdata that produced the npv of $3,306.97 when investor required rate ofreturn was 13%

The focus of this discussion as regards npv and IRR has been from thestandpoint of the negotiation between two parties over a specific property,what might be termed a ‘‘micro’’ approach There is a larger, ‘‘macro’’ view thatasks the broader underlying question: Where do discount rates come from?Entire books are written in response to this question, and it seems an injustice

to summarize them in a few phrases, but here is a way of thinking about them

TABLE 4-4 Rules of Thumb for Sample Project with Revised Down Payment Cap rate 0.0987

that fits in our context In general, discount rates are the aggregate of all thenegotiations that take place every day with all the buyers and sellers in amarket They encapsulate the combined expectations of a large number ofpeople who compete with one another to acquire business opportunities thathave uncertain outcomes During times of positive business conditionscharacterized by solid growth, low inflation, high employment, and lowinterest rates, discount rates will be lower than they are during the oppositetimes of negative macroeconomic news when buyers demand morecompensation in the form of higher discount rates for undertaking riskwhen the horizon is comparatively dark.

ANOTHER GROWTH FUNCTION

The above, quite standard discounted cash flow (DCF) analysis implies a fixedholding period terminating in a taxable sale The model also depends on theunrealistic assumption that the change in income and value over the holdingperiod is constant and positive Not only is this unlikely because of variableeconomic conditions, due to the owner’s active management, the propertycould undergo a dramatic transformation in the early years, resulting in arapid change in value in those years, after which slower, ‘‘normal,’’appreciation takes place To represent this we choose a modified logisticgrowth function, val(n) in Equation (4-15), which exhibits two phases ofvalue change, an early entrepreneurship phase with high appreciation,followed by a stabilized normal appreciation phase The dependent variable,

n, means that value is dependent on time But the specific functional form ofval(n) is chosen such that the change in early years is different from changesoccurring in later years

Figure 4-9 illustrates how val(n) changes over ten years

It is helpful to examine this function a little closer Let us focus on the firstterm on the rightside of equation (4-15) Note that as n grows larger, thesecond term in the denominator approaches zero, making the entiredenominator approach unity; hence the entire term approaches the numerator

as a limit (n!1) Thus, the value selected for the numerator, which we namethe logistic constant (‘‘lc’’), is the answer to the question ‘‘how high is up’’ inthe near term It is this number that represents the upper limit of valueimprovement over the short run due to entrepreneurial effort in the early

years of the holding period In the field this is sometimes known as the

‘‘upside’’ or ‘‘value added’’ potential Figure 4-10 displays alternatives usingdifferent values for lc and keeping the denominator the same

We now focus on the second constant in the first term in Equation (4-15)

It appears in the denominator as the exponent of e, operating on n We willcall it the acceleration factor (‘‘af ’’) This answers the question ‘‘how fast’’ as itdetermines how quickly the limit is reached It may be viewed as the efficiency

of the entrepreneurial effort Thus, the larger this constant, the more rapidlythe limit is reached Compare the value of the x-axis at the sharp bend foreach of the four alternative plots in Figure 4-11

The last term in the function involves what might be considered normalgrowth (‘‘g’’), stabilized after the early year ‘‘turnaround period.’’

In figure 4-12 we compare two entrepreneurs, both in possession ofproperties with the same upside potential One is more efficient, having an

0.8

1.21.41.61.8Value

FIGURE 4-9 Modified logistic growth function.

0.751.251.51.752Value

acceleration factor of 4 contrasted to af ¼ 1 for the less efficient owner Thefilled area in Figure 4-12 represents the additional growth reaped in the earlyyears for the more efficient owner The two converge after about six years Butone might surmise that the more efficient party would not hold the originalproperty for the full six years, choosing instead to repeat the process once ortwice in six years.

Suppose the acceleration rate is influenced by institutional factorsdiscussed in Chapter 2 We now take one investor as he considers twoprojects, one with an upside lc of 1.2 in a community that imposesburdensome regulation constraining his entrepreneurial ability to an artificial

af ¼ 2 A second property has greater upside of lc ¼ 1.5, but is located in acommunity that allows him to fully exercise his entrepreneurial skills,represented by af ¼ 4, relatively unfettered by regulatory interference InFigure 4-13 we see that the two growth rates do not converge in 20 years

0.8

1.21.41.61.8Value

FIGURE 4-12 Difference in gain for owners with different efficiency.

This has implications for communities interested in attracting the real estateequivalent of incubator companies, developers who specialize in urbanrenewal and infill projects in older neighborhoods that benefit the community

by raising the tax base

One can smooth out an irregular growth rate to create an average over theholding period Returning to the original growth function, we encountered inEquation (4-15) with fixed values for lc ¼ 1.5 and af ¼ 2, value increasesabout 60% in the first three years, achieving a value 1.616 times the original.After ten years, val[10] ¼ 1.9, not quite double the original, representing aflattening of the curve in the last seven years For the sake of comparison, wecan look at what sort of continuous return would be necessary to producethe same outcome if a constant rate were earned over the same ten years.This involves solving for r in Equation (4-16), producing a continuouscompounding return of 0.0642 over ten years

Figure 4-14 displays a three-dimensional plot over the range of lc and afvalues suggested in all the examples above that shows all the outcomes overall the possible combinations in those ranges

FIGURE 4-13 Property in different jurisdictions, with one constraining the owner’s activities.