Sensitivity of the FMRR technique in a fluctuating market.

Imagine that a survey asked respondents to assume that an investment firm has requested an analysis of the value of a certain property using rates of return. Because of the real estate market's volatility at the time of this survey, an appraiser must take into consideration the fluctuating market when developing assumptions for the valuation techniques to be used. In addition, he or she must decide which method best produces an accurate estimate of the property's value. Before the widespread use of computer spreadsheets and advanced calculators, the model for valuation would have been the direct capitalization approach as opposed to the discounted cash flow (DCF) techniques.

While some authors have maintained that the direct capitalization approach is more reliable than other income methods such as DCF analysis,(1) in a fluctuating market, the possibility of negative income cash flows reduces the value of this technique. This has become extremely important in the 1990s, as declining rental rates in some markets have produced real estate investments with periods of negative cash flows. If a property is expected to have negative cash flows during the initial years, the direct capitalization rate may calculate a negative value for the property. In addition, "capitalizing a single year's income stream into perpetuity does not produce a result that is either rational or logical to expect, unless the income stream is fixed by lease at a flat rate over a long time."(2)

For these reasons, the DCF model and the calculation of internal rate of return (IRR) and financial management rate of return (FMRR) are examined here. Net present value (NPV) will not be analyzed because the focus of this article revolves around rates of returns, not total monetary value added. The sensitivity of the FMRR model in a fluctuating market will be compared with the IRR model.

FMRR

Appraisers have traditionally used the IRR and NPV models. Then in 1975, the FMRR was introduced to overcome some of the significant shortcomings, especially in the IRR.(3) Messner and Findlay(4) wrote about the problems encountered in the IRR and explained how the development of the FMRR attempts to correct these shortcomings:

* Multiple IRRs. The first problem with the IRR directly relates to fluctuating cash flows. Multiple sign changes in the net cash flow stream could lead to the calculation of multiple or zero IRRs. The FMRR model removes this inconsistency by converting any unconventional cash flow streams into a conventional form. This is accomplished by using any prior cash inflows to cover any expected cash outflows.

* Reinvestment Rate. The IRR has an assumption that any size cash flow generated from the project could be invested at the project's reinvestment (IRR) rate. It is very unlikely that the investor would be able to reinvest each year's cash flow in the same project or in differing projects that happen to have the same rate of return. For example, if the IRR of a project is 32% and it produces a cash flow of $10,000 in the third year, this cash flow is expected to be reinvested in a similar investment to yield the same 32% return. This overstatement of the re-investment rate would tend to upwardly bias the IRR.

In the FMRR, funds can be invested or withdrawn, as if from a savings account, at a safe rate ([i.sub.L]). In addition, a base amount is established, and accumulated cash flows that are above this figure can be reinvested in other projects of comparable risk at an after-tax rate ([i.sub.R]) that is above the safe rate ([i.sub.L]). This establishment of separate rates assures the investor that cash flows are being reinvested at a rate that is more realistic with actual investment opportunities.

* Discounting Cash Flows. Within the framework of the IRR method, negative cash flows are discounted at the same internal rate as the positive cash flows. This implies that the investor will be able to borrow at the internal rate during any of the cash flow periods. The following example is similar to the one used by Messner and Findlay.(5) Here an initial investment of $100,000 was made with debt service payments of $200,000 after purchase. When sold at time period 3, the investment yields a net cash flow of $800,000.

Time                Cash Flow

0                   ($100,000)
1                   ($200,000)
2                   ($200,000)
3                    $800,000
IRR = 28.5%

This cash flow scenario develops an IRR of approximately 28.5%. Using the IRR method, it is now assumed that the $200,000 in future cash outlays for time periods 1 and 2 are discounted to a present cost of approximately $276,600 by using 28.5% as a discount rate. Is it a valid assumption that the $276,600 invested at Time 0 would generate the required after-tax yield necessary to cover the outlays of $200,000 during each of the next two time periods (years)? If the reinvestment cannot cover the cash outlays, the calculated IRR will overstate the actual return.

By using a safe rate ([i.sub.L]) and a reinvestment rate ([i.sub.R]), the FMRR model assures that the cash flows are reinvested at a more realistic rate. Assume that the initial cash flow from the previous example is not high enough to be invested at [i.sub.R] (bracket amount set at $500,000 for this example) and is, therefore, invested at Time 0 at [i.sub.L] (7% is used in this example). The new calculation of cash flow will be as follows:

Time                  Adjusted Cash Flow         Adjusted Cash Flow

0                     ($100,000) + ($361,604)       ($461,604)
1                               0                       0
2                               0                       0
3                            $800,000                $800,000
Adjusted IRR = 20%

This adjustment to the IRR is lower because of the more realistic investment rate used in the FMRR model.

FMRR Process

The following summary of the FMRR process was described by Tenzer and Tarantello.(6) First, a simplified cash flow scenario is established:

Time                     Cash Flow

0                        ($25,000)
1                        ($50,000)
2                         $25,000
3                        ($12,500)
4                         $40,000
5                        $100,000

Safe rate ([i.sub.L]), 7%; reinvestment rate ([i.sub.R]), 13%;
bracket amount, $50,000.

1. All future cash flows are removed by using prior cash flows when possible. the negative cash flow at time 3 could be covered by investing cash flows received during time 2 at the safe rate ([i.sub.L] = 7%). The present value of $12,500 in time 3 discounted back to time 2 will be $11,682.

2. All cash flows that remain are discounted to the present by using the safe rate ([i.sub.L]). The same format used in the example above is used for all remaining cash flows. Both of these processes will leave the following:

Time            Cash Flow                Cash Flow

0            ($25,000) + ($46,729) =     ($71,729)
1                  0                        0
2             $25,000 + ($11,682) =       $13,318
3                  0                        0
4                   $40,000               $40,000
5                  $100,000              $100,000

3 Compound the positive cash flows forward at the correct rate. In this example, the $13,318 at time 2 is compounded forward to time 4 using the safe rate because the cash flow is below the bracket amount of $50,000. This compounded figure is now added to the $40,000 at time 4. Because the new figure of $55,247 is above the bracket amount, it will be compounded at the reinvestment rate ([i.sub.R]) of 13%. The new cash flow is as follows:

Time                  Cash Flow         Cash Flow

0                     ($71,729)         ($71,729)
1                        0                  0
2                        0                  0
3                        0                  0
4                        0                  0
5              $100,000 + $62,429        $162,429

4. Finally, the FMRR equation is used to calculate the FMRR.

[Mathematical Expression Omitted]

where,

n = number of time period

[D.sub.n] = adjusted cash flow at time 0

[T.sub.n] = adjusted cash flow at time n

[Mathematical Expression Omitted]

A sensitivity analysis is used to compare the rates of returns found in different types of possible and complex scenarios. This type of analysis cannot cover all possible expected investment opportunities, but an actual cash flow stream from an existing property can be used as a base value. It should be noted, however, that cash flows for years 1 through 5 are actual figures, while years 6 through 10 are estimations based on the current owner's five-year plan. Actual figures were used to acknowledge the cash flow fluctuations that occur in real-life scenarios.

The use of actual cash flows presents the model in a more realistic scenario for comparisons. The numerical values for the base assumptions are provided in Table 1. It should be noted that expenses are held constant throughout the sensitivity analysis. While it is evident that fluctuating income streams caused [TABULAR DATA FOR TABLE 1 OMITTED] by increases and decreases in occupancy will vary expense figures, for the sake of simplicity, they will not be altered in this model.

We will also calculate the purchase date of the property at December 31, 1989, at the actual purchase price of $4,317,000. The sales price will be fixed at the amount calculated by the investment firm that currently owns the property. This price of $6,505,223 was not calculated using a cap rate because of the expectation of increased tenant improvement work in the final years of the analysis. A market survey was performed for the immediate area, and estimations of sales prices were based on these values. It should also be noted that the sales price remains fixed throughout the sensitivity analysis process. While this will tend to bias shorter holding periods upwardly, the cash flow fluctuations in this paper would vary the sales price and increase the difficulty in determining rate changes caused directly from the variables chosen for the sensitivity analysis.

FIXED ASSUMPTIONS FOR EVALUATION

Several variables will impact the overall return rates for the two methods compared in this paper. For calculation of the FMRR, a safe rate ([i.sub.L]) of 5.0% and a reinvestment rate ([i.sub.R]) of 10.0% will be used. Both rates were chosen as conservative returns for a typical investor. In addition, the holding period will remain constant at 10 years unless being used as a variable within the sensitivity analysis.

As mentioned, the actual cash flows from an existing investment property will be used as the basis of the analysis. While holding all expense assumptions constant, the revenue stream will be modified to present these five basic models used throughout the sensitivity analysis:

1. Actual: Actual figures used from existing property

2. Increase: 6% per year revenue increase starting after year 2

3. Decrease: 5% per year revenue decrease starting after year 2

4. Spike: 8% per year revenue increase starting after year 2, followed by a 9% per year revenue decrease from year 7 through year 10

5. Dip: 4% per year revenue decrease starting after year 2, followed by a 13% per year revenue increase from year 7 through year 10

The comparison of IRR and FMRR was completed using three different variables to analyze their effect on the calculated rates of return. Each assumption was processed in a similar fashion by using a software program called "planEASe."(7) In the following section, we will observe the differences in rates between the two processes under similar changes in variables discussed below.

The planEASe program was developed to perform financial analysis and cash flow projections of any income-producing property. Internal rates of return, financial management rates of return, and net present values can be computed before and after taxes. The program was chosen for this analysis because of its ease of use in performing both sensitivity analysis and Monte Carlo risk analysis.

HOLDING PERIOD

In appraisals, a 10-year analysis is used to calculate value. Some appraisal reports have stated that investor surveys consider a 10-year period appropriate. But a 1991 investor survey from Cushman & Wakefield found that typical holding periods for the 61 respondents ranged from 5 years to 30 years.(8) "In the absence of a required divestiture year, the reversionary year should be selected because at that point value is maximized."(9) For this reason, the holding period shall be treated as a variable. With this change, the sensitivity of the change in rates of return for an investment opportunity will be observed.

All five of the possible revenue scenarios have been analyzed by holding all assumptions constant and varying the holding period. Table 2 below compares the IRR and FMRR calculations while the holding period is varied from 4 through 10 years.

As expected, the rates of returns for both IRR and FMRR will decrease as the number of holding years are increased. This is mainly because the sales value of the property remains unchanged throughout the analysis.

In the three scenarios based on increases in cash flow (actual, increase, and spike) the FMRR remains slightly below the IRR throughout all seven holding period changes. This is due to the reinvestment of cash flows in the FMRR at the reinvestment rate ([i.sub.R]) of 10% instead of the calculated IRR. It should be noted that in all three scenarios, the total rate of return remains above 10%.

In the two scenarios based on decreases in cash flow (decrease and dip), the total calculated rate of return falls below the 10% reinvestment level during the change in holding periods. After this point, the FMRR model would be reinvesting cash flow at a higher rate and should, therefore, expect a higher total return. As can be seen in Figure 1, the rate of return drops below 10% at approximately a seven-year holding period. [TABULAR DATA FOR TABLE 2 OMITTED] After this point, the FMRR will be greater than the IRR.

REVERSION VALUE

Appreciation has always been an important variable in the real estate valuation equation. "In the 1980s, many investors looked on appreciation as the principal source of return."(10) In many instances, investors were content with a break-even cash flow if the expected reversion value was large enough to generate their required return. This reliance on the sales proceeds as the major source of return places significant importance on calculating a proper reversion value, especially as the holding period is increased. A method, which produces the lowest variance in rate changes when the reversion values are altered, is used to compare return rates.

Again, all five scenarios are subjected to a change in reversion values holding all other variables constant. The holding period for this analysis remains at 10 years. The reversion values shall be varied from a low of $4 million to a high of $8 million. Sales costs would normally be included in the final calculation of reversion values, but they will be disregarded in this analysis (see Table 3).

With the increase in sales proceeds, this table shows an expected trend of increasing rates in all scenarios. Another interesting trend is observed when graphs are developed for each different comparison. The slope of the line representing the FMRR is flatter than that of the IRR. This implies that each increase or decrease in the reversion value will have less of an effect on the FMRR than it will on the IRR [ILLUSTRATION FOR FIGURE 2 OMITTED].

The more inelastic curve found in the FMRR method is shown to be more stable under adverse assumptions. For example, assume under the "dip" scenario that the investment property is actually sold at a final value of $5.0 million. If the investor had estimated a reversion value at $5.5 million, the difference in estimated-to-actual value would be -11.94% under the IRR method. The same scenario would produce a difference of only -8.33% under the FMRR method.

REINVESTMENT RATE

One of the main differences between the IRk and FMRR is the FMRR's use of an established reinvestment rate. Instead of reinvesting all cash flows at the rate established for the project being analyzed, the FMRR takes into consideration the fact that, in most instances, excess cash flows will not be reinvested into the real estate investment, but placed into alternative investment opportunities. [TABULAR DATA FOR TABLE 3 OMITTED] If the property requires no additional capital, it is illogical to use a model that continues to reinvest even when the opportunity is not viable. The actual scenario for the final sensitivity analysis will be used and the reinvestment rate will be varied from 6% to 15%, with other items constant (see Table 4).

The trend shows exactly what is expected when comparing both IRR and FMRR methods. If the reinvestment rate is less than the IRR, the calculated FMRR will also fall below the IRR. When the reinvestment rate is greater than the IRR, the calculated FMRR will be above the IRR. As the reinvestment rate approaches the calculated IRR, so too does the FMRR as seen in the comparison of the "actual" scenario. As the reinvestment rate reaches 12%, the IRR and FMRR become the same. When each scenario is observed within a graph, it can be seen that the FMRR line crosses the IRR line at the point where the reinvestment rate equals the IRR [ILLUSTRATION FOR FIGURE 3 OMITTED].

If an investor has an accurate measurement of the return available from alternative investment opportunities, the FMRR would better state the real return on the individual investment property. The arrows in Figure 3 depict the difference between methods when a reinvestment rate is specified under FMRR. Thus, if the investor actually knew that he or she could only reinvest at 7%, the IRR would show an overstated return of 11.9% when the actual return would be closer to the FMRR of 10.7% because the IRR is reinvesting at a lower amount. This becomes extremely important for investment opportunities in which high cash flows are projected.

Risk Analysis

The aforementioned scenarios all show how the IRR and FMRR can vary with a single change in one of the variables at a time. Because a comparison of the two methods under more realistic situations is being demonstrated, a Monte Carlo simulation is performed using the actual, dip, and spike scenarios. One thousand trials were used for each simulation to gain a more reliable set of calculations. Using the planEASe program, an average return and a standard deviation for each different scenario were calculated. The standard deviation between the two methods will be used as a measure of risk. As the original assumptions are changed randomly, the method having the greatest change in rate of return (largest standard deviation) will be of greater risk to the investor.

The Monte Carlo simulation for risk analysis was performed for each of the aforementioned scenarios while utilizing the holding period and then the reversion amount as the variable to be randomly generated. Finally, the actual scenario was analyzed while varying the holding period, reversion amount, and reinvestment rate (FMRR only) at the same time. The calculations for each risk analysis are presented below:

[TABULAR DATA FOR TABLE 4 OMITTED]

In tables a and b, the IRR method produces an average rate of return that is higher in the actual and spike scenarios, but a lower rate of return in the dip scenario compared with the FMRR method. On the other hand, the standard deviation for every scenario in all the tests was lower for the FMRR method than that for the IRR method. As a measure of risk, the FMRR method produces a rate of return that is less likely to fluctuate with the changes in each of the different variables that could vary in a real-life investment situation.

In a real-life investment situation, there are many different reasons for variances in the assumptions of both income and expenses. Rental concessions, percentage rent, expense recovery, and vacancy/collection losses can unexpectedly change the expected income stream for an investment opportunity. In the same fashion, insurance costs, capital expenditures, tenant improvement costs, and government regulations could greatly vary the expense figures compared with the initial assumptions.(11) For this reason, numbers were randomly generated for all three of the variables under the parameters of the Monte Carlo simulation in the planEASe program.

(a)

Holding Period                 Actual        Dip         Spike

IRR average return             11.8%         8.1%         10.8%
IRR standard deviation          1.3%         1.2%          1.7%
FMRR average return            11.4%         8.4%         10.7%
FMRR standard deviation         1.1%         0.9%          1.4%

Holding period in years: lowest value (5), most likely value (10),
and highest value (15).

(b)

Sales $ Parameter               Actual        Dip         Spike

IRR average return              11.9%         8.2%         10.6%
IRR standard deviation           0.5%         0.6%          0.5%
FMRR average return             11.4%         8.5%         10.4%
FMRR standard deviation          0.4%         0.5%          0.4%

Sales price parameter (reversion amount): lowest value
($5.5 million), most likely ($6,505,223), and highest value
($7.5 million).

(c)

Multiple Scenario                   IRR         FMRR

Average result                    11.7%         11.5%
Standard deviation                 1.4%          1.2%

Reinvestment rate (FMRR only): lowest value (7%), most likely
(10%), and highest value (15%). Holding period and sales price
parameter (same as above.)

The multiple scenario randomly varies the holding period, sales price parameter, and reinvestment rate (FMRR only). The actual scenario was used so as to produce the most realistic real estate investment situation. It was interesting to note that the FMRR method again produced a standard deviation below that of the IRR method using the same parameters. The FMRR method contained the additional variable of the reinvestment rate but continued to vary less with fluctuations in the assumption values than the IRR method. Using the FMRR method, the rate produced is less likely to fluctuate than the IRR and should, therefore, be the less risky method.

CONCLUSION

In evaluating real estate investment opportunities, an appraiser should use the method of calculating a return that produces the most realistic result while taking into consideration all possible changes from the original assumption. In this analysis, it was shown that the FMRR method produces results that fluctuate less with changes in the holding period and selling price parameter compared with the IRR method. The FMRR method also gives a more realistic return when the reinvestment rate is known by an investor because the excess cash generated from the prospective purchase is reinvested at this calculated reinvestment rate rather than at an artificial rate generated by the real estate investment itself.

In the analysis that concentrated on a single variable, the FMRR varied less than the IRR under changing holding periods. Also, the FMRR slope is flatter than the IRR slope when reversion values are varied, showing that the FMRR will vary less when there are differences between actual and estimated sales prices.

While the differences between the average returns and standard deviations were not extremely large, in the analysis of an investment opportunity, a change of 0.5% could mean the difference between acceptance or rejection. Although the FMRR method contains more complicated calculations than the IRR method, the prudent investor is not excused from using FMRR in an investment analysis. With the use of new computer programs, calculating FMRR is no more difficult than calculating IRR. The main drawback is that the FMRR process is not as accepted throughout the investment community as the IRR method. The excuses of complicated calculations or acceptability should not discourage an investor from using an investment tool like the FMRR method. Packaged software programs enable investors and appraisers to perform complex calculations with ease and speed.

With real estate cash flows fluctuating as much as they are in current times, it would benefit any investor to use a method of rate calculation that best reflects the actual return to be received during the life of the investment. This should be a rate that also varies the least as the original variables are changed under actual investment conditions. Between the two rates of return, the FMRR method is more accurate and reliable than the IRR method in fluctuating scenarios such as those used in this article and should, therefore, not be overlooked as a useful tool for real estate investment.

1. Robert Plattner, "Income Capitalization Problems," The Appraisal Journal (October 1992): 549-555.

2. Lloyd D. Hanford, Jr., "Discounting the Cash Flow Models," The Appraisal Journal (July 1991): 314.

3. Gary M. Tenzer and R. Tarantello, "FMRR: A Programmable Calculator Implementation," The Real Estate Appraiser and Analyst (November-December 1979): 11-16.

4. Stephen D. Messner, and M. Chapman Findlay, III, "Real Estate Investment Analysis: IRR Versus FMRR," The Real Estate Appraiser (July-August 1975): 5-20.

5. Ibid., 13-17.

6. Tenzer and Tarantello, 13-14.

7. Analytic Associates; 4817 Browndeer Lane; Rolling Hills Estates, California 90275. (800) 959-EASe.

8. Cushman & Wakefield, Inc., Real Estate Outlook (June 1991).

9. Lloyd D. Hanford, Jr., "Discounting the Cash Flow Models," The Appraisal Journal (July 1991): 316.

10. Charles E. Edwards and James P. Gaines, "Reversion Value Estimates and Real Estate Investment Returns," The Real Estate Appraiser and Analyst (Fall 1982):40

11. Vernon Martin, III, "Reviewing Discounted Cash Flow Analyses," The Appraisal Journal (January 1990): 83-87.

Lawrence F. Sherman, PhD, is an associate professor of finance and real estate at California State University, Long Beach, and president of L. F. Sherman & Company, Inc., Laguna Hills, California. He received his PhD at the University of Illinois, Champaign-Urbana, in finance and economics.

Keith D. Walters is a senior property manager at Charles Dunn Real Estate Services, Inc. He specializes in the operation and management of high-rise office and industrial properties. He received his MBA from California State University, Long Beach, with an emphasis in finance.