Business schools teach students that risk is the possibility that an investment s

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1 features Property Risk Assessment: A Simulation Approach by Barrett A. Slade, PhD, MAI abstract Simulation analysis leads to greater insights into the risk characteristics of investment properties by explicitly recognizing that many input Business schools teach students that risk is the possibility that an investment s actual return may be different than is expected, including the possibility that some or all of the original investment may be lost. Therefore, prudent investment analysis requires that this uncertainty be examined so that appropriate steps may be taken to mitigate the potential loss. Risk assessment in real estate investment requires both qualitative as well as quantitative analysis, which is often referred to as the art and science of real estate investment analysis. Although the art or human judgment aspect of investment analysis is an important part of the investment decision, the objective of this article is to illustrate the advantages of simulation analysis so that the science of risk assessment can be improved. Historically this topic has received little attention in the appraisal literature; however, more recent articles by Weaver and Michelson; 1 Kummerow; 2 Kelliher and Mahoney; 3 and Li 4 suggest that understanding and analyzing dimensions of risk in real property investment and valuation are becoming increasingly important. With continued advances in software design and increased familiarity with spreadsheet modeling, market participants are requesting and expecting market analysts and appraisers to assist them in risk assessment. In their 2003 article, Weaver and Michelson recognize that simulation analysis is an advanced risk assessment tool. However, they elect to present simple two- and three-dimension spreadsheets for examining risk, rather than simulation analysis, because they suggest that appraisers do not have the statistical skills necessary to properly use the simulation software. Although this concern may have been valid assumptions are uncertain. It allows the analyst to realistically model this uncertainty and estimate the likelihood of an investment outcome. Compared to traditional risk assessment techniques, simulation analysis allows the investment decision maker to be much more informed. By applying modern simulation analysis to the investment analysis of an apartment property, this article illustrates the comparative advantages of modern risk assessment techniques over traditional methods. 1. William Weaver and Stuart Michelson, A Practical Tool to Assist in Analyzing Risk Associated with Income Capi talization Approach Valuation or Investment Analysis. The Appraisal Journal (October 2003): Max Kummerow, A Statistical Definition of Value, The Appraisal Journal (October 2002): Charles F. Kelliher and Lois S. Mahoney, Using Monte Carlo Simulation to Improve Long-Term Investment Decisions, The Appraisal Journal (January 2000): Ling Hin Li, Simple Computer Applications Improve the Versatility of Discounted Cash Flow Analysis, The Appraisal Journal (January 2000): property risk assessment: a simulation approach 347

2 in the past, recent advances in software design and documentation as well as increased course offerings in statistics and spreadsheet modeling suggest that many analysts and appraisers have, or are obtaining, the skills necessary for proper implementation of simulation analysis. Therefore, an important aspect of this article is to provide the interested reader with an introduction into the characteristics and benefits of simulation analysis for assessing risk in real property investments. This article is organized in five parts. First, an Excel spreadsheet is introduced, depicting the expected cash flows from a prospective investment property. Second, a traditional scenario analysis is performed by analyzing the best-case, most-likely-case, and the worst-case outcomes. Third, a discussion regarding the limitations of what-if analysis is presented. Fourth, a simulation analysis is introduced, and fifth, a conclusion is provided that summarizes the main objectives and findings of this article. Spreadsheet Construction Conventional risk assessment techniques generally begin with the construction of a spreadsheet that presents the expected future cash flows from an investment property, including rental revenue, vacancy loss, operating expenses, debt service, capital expenses, and reversionary cash flows. The spreadsheet in Table 1 illustrates the expected cash flows from an apartment complex that will be held by an investment management firm for a tax-exempt client, i.e., a pension fund. In Table 1, the figures shown for Free & Clear Cash Flows After Fees assume no debt financing, while the figures for Leveraged Equity Cash Flows After Fees assume debt financing. The total cash flows with and without debt include the initial outlay as well as the reversion, providing the basis for comprehensive investment analysis. Table 2 presents the reversion calculation. Table 3 shows important spreadsheet assumptions that are applied in Tables 1 and 2, including the annual growth rate assumptions; acquisition and reversion assumptions; and mortgage loan assumptions. These assumptions result from market research conducted by the analyst. Table 4 includes three important return measures critical to the overall acquisition decision, including the going-in cap rate; free and clear internal rate of return (IRR) after fees; and the leveraged equity IRR after fees. Traditional Scenario Analysis Generally, the first step in risk analysis is to perform traditional scenario analysis that includes examining the best-case, most-likely-case, and worst-case scenarios of the property as forecast by the analyst. Although the single-point estimates shown in Tables 1 and 2 reflect what the analyst generally considers the most likely case, familiarity with market-driven forecasts suggests that each of the input variables is uncertain and a range of values is possible for each. This knowledge of the relevant range of each input variable allows the analyst to examine the possible extremes of investment outcome, specifically, the best-case and worst-case scenarios. Table 5 reflects the relevant range of values for the input variables, and Table 6 displays the first year net operating income (NOI) and the results for the return measures from each scenario. As observed in Table 6, the worst-case and best-case scenarios identify the relevant extremes, highlight potential concerns, and signal possible opportunities. However, the extremes reveal only a range of outcomes, not likelihoods which naturally would be much more insightful for the decision maker. For instance, in the scenario analysis illustrated in Table 6, Leveraged Equity IRR After Fees ranges from 2.22% to 30.44%. In this traditional analysis there is no insight into the likelihood or probability of actually obtaining these respective outcomes, which is suboptimal for the decision maker. What-If Analysis Limitations In an attempt to overcome the limitations observed in scenario analysis, the analyst frequently begins performing what-if analysis, the process of examining changes in output values resulting from changes in one or more input values. The general approach is to hold all input variables constant except one. The resulting marginal change in output values provides the analyst with information on the impact or sensitivity of the output values resulting from a marginal change in an input value. Although this approach may provide the decision maker with some insight into the sensitivity of the output values to an input value, the cost of such an exercise may be extreme. One obvious shortfall with this approach is that the process is frequently random in nature, with the analyst spending considerable time generating and examining each output change based on a marginal change of an input variable. The volume 348 The Appraisal Journal, Fall 2006

3 Table 1 Apartment Complex Property Cash Flow Projections 10-Year Analysis Year Revenue Gross Market Rent 5,882,357 6,058,828 6,240,593 6,427,810 6,620,645 6,819,264 7,023,842 7,234,557 7,451,594 7,675,142 7,905,396 Vacancy/Concessions/Bad Debt -294, , , , , , , , , , ,270 Other Income 649, , , , , , , , , , ,333 Gross Effective Revenue 6,238,081 6,425,224 6,617,980 6,816,520 7,021,015 7,231,646 7,448,595 7,672,053 7,902,215 8,139,281 8,383,459 Expenses Payroll 445, , , , , , , , , , ,365 Utilities 308, , , , , , , , , , ,996 Administrative 54,320 55,950 57,628 59,357 61,138 62,972 64,861 66,807 68,811 70,875 73,002 Advertising & Marketing 104, , , , , , , , , , ,626 Repairs & Maintenance 46,764 48,167 49,612 51,100 52,633 54,212 55,839 57,514 59,239 61,016 62,847 Redecorating Costs 67,292 69,311 71,390 73,532 75,738 78,010 80,350 82,761 85,243 87,801 90,435 Contract Services 196, , , , , , , , , , ,843 Management Fee 167, , , , , , , , , , ,083 Insurance 115, , , , , , , , , , ,458 Real Estate Taxes 755, , , , , , , , , ,483 1,015,048 Total Operating Expenses 2,262,568 2,330,445 2,400,358 2,472,369 2,546,540 2,622,936 2,701,625 2,782,673 2,866,153 2,952,138 3,040,702 Net Operating Income 3,975,513 4,094,779 4,217,622 4,344,151 4,474,475 4,608,709 4,746,971 4,889,380 5,036,061 5,187,143 5,342,757 Mortgage Interest Expense 2,041,250 2,041,250 2,041,250 2,041,250 2,041,250 2,041,250 2,041,250 2,041,250 2,041,250 2,041,250 2,041,250 Net Property Income 1,934,263 2,053,529 2,176,372 2,302,901 2,433,225 2,567,459 2,705,721 2,848,130 2,994,811 3,145,893 3,301,507 Capital Expenditures Specified Capital Improvements 218, , , , , , , , , , ,241 Replacements 182, , , , , , , , , , ,945 Total Capital Expenditures 400, , , , , , , , , , ,186 Cash Flow After Capital Expenditures 1,533,802 1,641,054 1,751,523 1,865,306 1,982,503 2,103,215 2,227,549 2,355,613 2,487,519 2,623,382 2,763,321 Asset Management Fees (5% of NOI) 198, , , , , , , , , , ,138 Free & Clear Cash Flow After Fees 3,376,276 3,477,565 3,581,892 3,689,348 3,800,029 3,914,030 4,031,451 4,152,394 4,276,966 4,405,275 4,537,433 Leveraged Equity Cash Flows After Fees 1,335,026 1,436,315 1,540,642 1,648,098 1,758,779 1,872,780 1,990,201 2,111,144 2,235,716 2,364,025 2,496,183 Total Free & Clear Cash Flows (54,700,000) 3,376,276 3,477,565 3,581,892 3,689,348 3,800,029 3,914,030 4,031,451 4,152,394 4,276,966 76,624,614 Total Leveraged Equity Cash Flows (19,200,000) 1,335,026 1,436,315 1,540,642 1,648,098 1,758,779 1,872,780 1,990,201 2,111,144 2,235,716 39,083,364 Numbers rounded to the nearest dollar. See Table 2 for reversion calculations for the most likely case. property risk assessment: a simulation approach 349

4 Table 2 Reversion Calculation Analysis without Debt Equity Contribution $54,700,000 Gross Sale Price 73,693,203 Less Selling Costs (1,473,864) Less Mortgage Balance 0 Net Sales Proceeds 72,219,339 Analysis with Debt Equity Contribution $19,200,000 Reversion Gross Sale Price 73,693,203 Less Selling Costs (1,473,864) Less Mortgage Balance (35,500,000) Net Sales Proceeds $36,719,339 Table 3 Important Spreadsheet Assumptions Annual Compound Growth Rate Assumptions Market Rent Growth Rate 3.00% Other Income Growth Rate 3.00% Operating Expense Growth Rate 3.00% Vacancy/Concessions/Bad Debt 5.00% Acquisition and Reversion Assumptions Purchase Price $54,700,000 Reversionary Cap Rate (Applied to 11th-Yr. NOI) 7.50% Selling Expenses 2.00% Mortgage Loan Assumptions Mortgage Loan Amount $35,500,000 Mortgage Interest 5.75% Table 4 Important Return Measures Going-in Cap Rate 7.27% Free & Clear Internal Rate of Return After Fees 9.03% Leveraged Equity Internal Rate of Return After Fees 13.70% Table 5 Relevant Range of Input Assumptions Worst-Case Inputs Most-Likely-Case Inputs Best-Case Inputs Market Rent Growth Rate 1.00% 3.00% 6.00% Vacancy Rate 7.00% 5.00% 0.00% Other Income Growth Rate 1.00% 3.00% 6.00% Operating Expense Growth Rate 4.00% 3.00% 1.00% Purchase Price $55,500,000 $54,700,000 $53,500,000 Reversion Cap Rate 8.00% 7.25% 6.50% Selling Expenses 2.00% 2.00% 2.00% Mortgage Amount $32,500,000 $35,500,000 $38,500,000 Mortgage Interest Rate 6.25% 5.75% 5.25% 350 The Appraisal Journal, Fall 2006

5 Table 6 Results from Scenario Analysis Worst-Case Inputs Most-Likely-Case Inputs Best-Case Inputs First Year NOI $3,857,866 $3,975,513 $4,269,631 Going-in Cap Rate 6.95% 7.27% 7.98% Free & Clear IRR After Fees 3.39% 9.03% 16.53% Leveraged Equity IRR After Fees -2.22% 13.70% 30.44% of data and output can frequently overwhelm and confuse the analyst, resulting in no greater insight into the likelihood of the outcomes. In an effort to be sensitive to the reader, a what-if analysis has not been performed for this article. Simulation Analysis Scenario and what-if analyses both may shed some light on the potential risk of the investment, but each lacks the capacity to estimate the likelihood or probability of a specific outcome. This limitation is overcome by Monte Carlo simulation analysis. In this setting, the analyst can simultaneously examine thousands of alternative scenarios and subsequently estimate the probability of an outcome. Simulation analysis recognizes that each input variable has a relevant range and a probability distribution, which of course, more closely reflects reality. In short, the analyst uses probabilistic input assumptions to generate forecasts or outcomes that can be analyzed statistically. This leads to greater insight into investment risk, which in turn leads to better property acquisition decisions. The recent advancements in simulation software, both in technology and economy, have made simulation analysis much more practical for real estate investment risk analysis. Although there are a number of vendors that provide reliable simulation software, the author has found Crystal Ball software easy to learn and implement, reliable, and economical. 5 Crystal Ball is a Microsoft Excel add-in; therefore, prior to implementation, a basic spreadsheet such as found in Tables 1 and 2 must be constructed in Microsoft Excel. Once Crystal Ball has been started within Microsoft Excel, taking the following four basic steps will complete the simulation analysis. Steps to Complete Simulation Analysis For a simulation analysis, the analyst first must determine which input values are uncertain and then define the distributional assumptions associated with these values. On a practical basis, Crystal Ball allows the analyst to examine a gallery of distributions and select the one that is most applicable for the particular input variable being analyzed. This software also allows the analyst to examine the distributional characteristics of empirical data so that much of the guesswork is eliminated. The analyst s knowledge of the relevant range or market attributes for the input variable is also required. For example, assume that market rents are forecast to grow from 1% to 6% annually and that the most likely growth rate is 3%. Assume that the distributional characteristics of rent growth rates are unknown, so a triangular distribution is used. A triangular distribution allows for identification of the minimum, maximum, and most likely values. Therefore, in this case, the minimum input value is set at 1%, the maximum input value is set at 6%, and the most likely input value is set at 3%, as depicted in Figure 1. As Figure 1 shows, the area of the triangle is much larger around the most likely value of 3% compared with the minimum and maximum values of 1% and 6% respectively. In other words, the values near 3% have a greater probability of occurrence; therefore, when the simulation begins to run, values near 3% will be selected more frequently than the values near the minimum and maximum of 1% and 6%, respectively. To illustrate another input value with an alternative distribution, assume that the mortgage interest rate is expected to follow a normal distribution with a mean of 5.75% and a standard deviation of 0.50%. 5. Crystal Ball is a product of Decisioneering, Inc. Single-user site licenses are available; for more information on this software see property risk assessment: a simulation approach 351

6 Figure 1 Triangular Distribution 1% 3% 6% In other words, the mortgage interest rate is expected to range between 5.25% and 6.25% about 68% of the time. Thus, when the simulation runs, it will select mortgage interest rates that follow a normal distribution with a mean of 5.75% and a standard deviation of 0.50%. For illustrative purposes, only the triangular and normal distributions are used in this article; however, the Crystal Ball software has numerous continuous and discrete distributions available for analysis. 6 Figure 2 provides the assumptions and relevant ranges for each uncertain input variable. After defining the distributional assumptions for the input variables, the second step requires identification of specific output variables (referred to as forecasts in Crystal Ball). In this context, three output variables are considered: the going-in cap rate; the free and clear IRR after fees; and the leveraged equity IRR after fees. After the output variables are identified, the third step is starting the simulation after specifying the number of trials. Using Monte Carlo simulation, Crystal Ball software will randomly select input values from the relevant ranges and distributions previously specified, then calculate the output or forecast values, and store the results. After completing this process many times, in this case 1000 times the simulation ends. The simulation results are displayed in Figure 3. The fourth and final step is to interpret the statistical results from the simulation analysis. The forecast charts illustrate the likelihood or probability of a particular outcome. For the sake of brevity, only the results from the leveraged equity IRR after fees are discussed here. The forecast chart represents a histogram or frequency chart of the output values from the simulation. The data shows that the minimum and maximum leveraged equity IRR after fees is 4.66% and 24.68% respectively. The mean and median are 15.52% and 15.54% respectively. The proximity of the median to the mean suggests that the distribution is highly symmetrical, making statistical inference more reliable. The percentiles show that there is an 80% probability of achieving a leveraged equity IRR after fees of at least 12.55%. In addition to providing the percentiles, Crystal Ball also can be used to calculate the likelihood or probability of obtaining a specific IRR. As shown in Figure 4, there is a 66.97% probability of obtaining a leveraged equity IRR after fees that is greater than 14%. A sensitivity chart can also be developed, as shown in Figure 5, which identifies the correlation of each input variable with a specific output or forecast variable. The sensitivity chart orders the variables by the level of correlation and indicates whether the correlation is positive or negative. For instance, growth rate of market rents is found to be highly and positively 6. Note that Crystal Ball and most simulation software also allows for direct examination and modeling of correlations between input variables. This feature, although not examined in this introductory article, can enhance the analysis and should be investigated and considered by the serious model builder. Go to to find a list of reference books that provide assistance for advanced model building and simulation analysis. 352 The Appraisal Journal, Fall 2006

7 Figure 2 Distributional Assumptions for Each Input Variable Assumption: Market Rents Growth Rate Minimum 1.00% Likeliest 3.00% Maximum 6.00% Assumption: Mortgage Amount Minimum $32,500,000 Likeliest $35,500,000 Maximum $38,500,000 1% 2% 3% 4% 5% 6% Market Rent Growth Rate Assumption: Mortgage Interest Rates Mean 5.75% Std. Dev. 0.50% $32.5 $33.8 $35.1 $36.4 $37.7 Mortgage Amount (Millions) Assumption: Operating Expense Growth Rate Minimum 1.00% Likeliest 3.00% Maximum 4.00% 5% 6% 7% Mortgage Interest Rate 1% 2% 3% 4% Operating Expense Growth Rate property risk assessment: a simulation approach 353

8 Figure 2 Distributional Assumptions for Each Input Value (Continued) Assumption: Other Income Growth Rate Minimum 1.00% Likeliest 3.00% Maximum 6.00% Assumption: Purchase Price Minimum $53,500,000 Likeliest $54,700,000 Maximum $55,500,000 1% 2% 3% 4% 5% 6% Other Income Growth Rate Assumption: Reversion Cap Rates Minimum 6.50% Likeliest 7.25% Maximum 8.00% $53.5 $54.0 $54.5 $55.0 $55.5 Purchase Price (Millions) Assumption: Vacancy Rate Minimum 0.00% Likeliest 5.00% Maximum 7.00% 7% Reversion Cap Rate 8% 0% 1% 2% 3% 4% 5% 6% 7% Vacancy Rate 354 The Appraisal Journal, Fall 2006

9 Figure 3 Simulation Results Going-in Cap Rate Frequency Forecast Forecast Statistics Values Percentiles Values Trials 1,000 0% 7.05% Mean 7.38% 10% 7.17% Median 7.37% 20% 7.23% Mode 30% 7.28% Standard Deviation 0.17% 40% 7.33% Variance 0.00% 50% 7.37% Skewness* % 7.43% Kutosis % 7.48% Coefficient of Viability % 7.54% Minimum 7.05% 90% 7.62% Maximum 7.92% 100% 7.92% Range Width 0.88% Mean Standard Error 0.01% % 7.40% 7.60% 7.80% Free & Clear IRR After Fees Frequency Forecast Forecast Statistics Values Percentiles Values% Trials 1,000 0% 5.36% Mean 9.99% 10% 7.87% Median 9.97% 20% 8.48% Mode 30% 9.00% Standard Deviation 1.63% 40% 9.49% Variance 0.03% 50% 9.97% Skewness* % 10.41% Kutosis % 10.89% Coefficient of Viability % 11.46% Minimum 5.36% 90% 12.19% Maximum 14.40% 100% 14.40% Range Width 9.04% Mean Standard Error 0.05% % 8.00% 10.00% 12.00% 14.00% Leveraged Equity IRR After Fees Frequency Forecast Forecast Statistics Values Percentiles Values Trials 1,000 0% 4.66% Mean 15.52% 10% 10.97% Median 15.54% 20% 12.55% Mode 30% 13.77% Standard Deviation 3.37% 40% 14.71% Variance 0.11% 50% 15.54% Skewness* % 16.60% Kutosis % 17.46% Coefficient of Viability % 18.55% Minimum 4.66% 90% 19.88% Maximum 24.68% 100% 24.68% Range Width 20.02% Mean Standard Error 0.11% % 12.00% 16.00% 20.00% 24.00% * Skewness is the measure of the degree of deviation of a curve from the norm. The greater the degree of skewness, the more points of the curve lie to either side of the peak of the curve: more to the right for positive skew and more to the left for negative skew. A normal distribution curve, having no skewness, is symmetrical in shape. Kurtosis is the measure of the degree of peakedness of a curve. The higher the kurtosis, the closer the points of the curve lie to the mode of the curve. A normal distribution curve has a kurtosis of 3. Coefficient of variability is a measure of relative variation that relates the standard deviation to the mean. Results can be represented in percentages for comparison purposes. property risk assessment: a simulation approach 355

10 Figure 4 Frequency Chart Leveraged Equity IRR After Fees Frequency % 12.00% 16.00% 20.00% 24.00% of IRR greater than 14%: 66.97% Figure 5 Sensitivity Chart Leveraged Equity IRR After Fees 0.0% 30.0% 60.0% Market Rents Growth Rate 79.5% Mortgage Interest Rate -5.1% Vacancy Rate -4.7% Reversion Cap Rate -3.7% Operating Expense Growth Rate -3.7% Mortgage Amount 2.3% Other Income Growth Rate 0.7% Purchase Price -0.3% 356 The Appraisal Journal, Fall 2006

11 correlated with the leveraged equity IRR, suggesting that as the growth rate of market rents increases, so does the leveraged equity IRR after fees. Compared with the other input variables, the influence of the market rents growth rate is much higher than the other input variables. The mortgage interest rate is second in significance and is negatively correlated with the leveraged equity IRR, suggesting that as the mortgage interest rate increases, the leveraged equity IRR decreases. These insights, provided by simulation analysis, increase the analyst s understanding of the investment property risks with respect to the input assumptions, thereby allowing the decision maker to be much more informed. 7 Conclusion Simulation analysis leads to greater insights into the risk characteristics of investment properties by explicitly recognizing that many input assumptions are uncertain. This allows the analyst to realistically model this uncertainty and estimate the probability or likelihood of an investment outcome. Armed with this information, the decision maker is much more informed, compared with the limited insight provided by traditional risk-assessment techniques. 8 Barrett A. Slade, PhD, MAI, has been employed as a chief appraiser with a major financial institution; has worked for a large appraisal firm; and has owned and managed his own commercial appraisal and consulting business. He received his PhD in real estate from the University of Georgia and is currently an associate professor of finance in the Business Management Department at the Marriott School, Brigham Young University, in Provo, Utah. His work has been published in numerous real estate finance and economics journals, including The Appraisal Journal, Real Estate Economics, Journal of Real Estate Finance and Economics, and the Journal of Real Estate Research. Contact: bslade@byu.edu 7. In addition to the insights illustrated in the actual case study, simulation analysis may be used to aid the user in optimizing loan-to-value ratios; loan rates and terms; and timing of capital improvements. 8. As pointed out by one reviewer of this article, there remain many unknowns that may influence the factors considered, such as the rate of development of competing properties, demographic changes due to large employers shutting down or opening up, a potential flight of capital into securities markets, natural disasters, or unanticipated changes in labor and materials costs. Therefore, care and disclaimers should be included when presenting simulation analysis to a client. property risk assessment: a simulation approach 357

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