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# Hotel Valuation Techniques

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## Hotel Valuation Techniques

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Hotel valuation, like all real estate valuation, must be seen in the context of establishing a point estimate that represents the value of a unique, illiquid asset in an environment with noisyand conflicting information

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## Nội dung Text: Hotel Valuation Techniques

1. Hotel Valuation Techniques By Jan deRoos, Ph.D., and Stephen Rushmore, CHA, MAI Jan deRoos, Ph.D., is the HVS International Professor of Hotel Finance and Real Estate at the Cornell University School of Hotel Administration. On the faculty of the Hotel School since 1988, he has devoted his career to research and teaching in the area of hospitality real estate, with a focus on hotel valuation and investment decision-making. Prior to joining Cornell University, Professor deRoos worked extensively in the hospitality industry. His current research interests concentrate on hotel leases as an alternative to management contracts and the value of goodwill in hotel property. Stephen Rushmore, CHA, MAI, is President and Founder of HVS International, a global hospitality consulting organization with 19 offices worldwide. He directs the global operation of HVS International and is responsible for future office expansion and new product development. HVS International has provided consulting and valuation services for more than 10,000 hotels in all 50 states and more than 60 foreign countries. Mr. Rushmore specializes in complex issues involving hotel feasibility, valuations, and financing. He was one of the creators of the Microtel concept, and has written numerous books and articles on hotel feasibility studies, appraisals, and other aspects of hotel investing.
2. IN THIS CHAPTER, we provide a thorough overview of lodging valuation models. Hotel valuation, like all real estate valuation, must be seen in the context of establishing a point estimate that represents the value of a unique, illiquid asset in an environment with noisy and conflicting information. This gives rise to the use of multiple approaches that must be reconciled. Appraisers are charged with estimating market value1 using the classic troika of the cost approach, the sales comparison approach, and the income approach. Appraisers use "market" indicators of return requirements and other valuation parameters to produce their estimates. Investors, on the other hand, wish to estimate investment value,2 which includes the effects of income taxes, the investor's unique cost of capital, and other investor-specific conditions. Investors typically rely on a modified income approach tailored to their circumstances, augmented with recent transaction information, to estimate value and form their bidding strategy. Three Approaches to Hotel Valuation In valuing hotels, there are three approaches from which to select: the income capitalization, sales comparison, and cost approach. Although all three valuation approaches are generally given consideration, the inherent strengths of each approach and the nature of the hotel in question must be evaluated to determine which approach will provide supportable value estimates. In addition, there is a set of rules of thumb that are used to provide a rough estimate of value. Since hotel investors typically give more weight to it, the income capitalization approach will be emphasized in this chapter. In jurisdictions where ad valorem taxes are based on market value of real estate, hotel owners are concerned with separately estimating the real property component (real estate) and the personal property component (both tangible and intangible personal property). Income Capitalization Approach The income capitalization approach is based on the principle that the value of a property is indicated by its net return, or what is known as the "present worth of future benefits." The future benefits of income-producing properties, such as hotels, are the net income estimated by a forecast of income and expense along with the anticipated proceeds from a future sale. These benefits can be converted into an indication of market value through a capitalization process and discounted cash flow analysis. The forecast of income and expense is expressed in nominal or inflation-adjusted dollars for each of three years. The stabilized year is intended to reflect the anticipated operating results of the property over its remaining economic life, given any or all applicable stages of build-up, plateau, and decline in the life cycle of a hotel. Thus, income and expense estimates from the stabilized year forward exclude from consideration any abnormal relationship between supply and demand, as well as any nonrecurring conditions that may result in unusual revenues or expenses. As stated in the textbook entitled Hotels and Motels: Valuations and Market Studies, "Of the three valuation approaches available to the appraiser, the income capitalization approach generally provides the most persuasive and supportable conclusions when valuing a lodging facility."3 The text goes on to state that using a ten-year forecast and an equity yield rate "most accurately reflects the actions of typical hotel buyers, who purchase properties based on their leveraged discounted cash flow."4 The simpler procedure of using a ten-year forecast and a discount rate (total property yield) is "less reliable because the derivation of the discount rate has little support. Moreover, it is difficult to adjust the discount rate for changes in the cost of capital."5 Because of this difficulty, the procedure is not illustrated in this chapter. A third income valuation technique is the "band of investment using one stabilized year." This technique is appropriate when the local hotel market is not expected to experience any significant changes in supply and demand, so it can be assumed that the subject property's net income has stabilized. Sales Comparison Approach While hotel investors are interested in the information contained in the sales comparison approach, they usually do not employ this approach in reaching their final purchase decisions. Factors such as the lack of recent sales data, the numerous insupportable adjustments that are necessary, and the general inability to determine the true financial terms and human motivations of comparable transactions often make the results of this technique questionable. The sales comparison approach is most useful in providing a range of values indicated by prior sales and in establishing an indicator of pricing momentum; however, reliance on this method beyond the establishment of broad parameters is rarely justified by the quality of the sales data. The market-derived capitalization rates sometimes used by appraisers are susceptible to the same shortcomings inherent in the sales comparison approach. Cost Approach The cost approach may provide a reliable estimate of value in the case of new properties, but as buildings and other improvements grow older and begin to deteriorate, the resultant loss in value becomes increasingly difficult to quantify accurately. Most knowledgeable hotel buyers base their purchase decisions on economic factors such as projected net income and return on investment. Because the cost approach does not reflect these income-related considerations and requires a number of highly subjective depredation estimates, this approach is given minimal weight in the hotel valuation process. However, it is useful in establishing a benchmark for buy versus build decisions and for relative pricing over time. Valuation for Assessment Purposes The question arises of whether to separately estimate a hotel's real property and personal property components in the interest of reducing the tax burden on the property. Such a practice it is hoped would not only reduce property taxes, but take advantage of much shorter depreciation periods for goodwill as opposed to real property. There is no question that some portion of cash flows generated by a hotel must be used to support the unique characteristics of the hotel investment, such as large continuing investment in furniture, fixtures, and equipment (FF&E) and the need to employ specialized management to realize a property's potential. However, because there is a significant financial incentive to attribute a portion of the going-concern value to intangible personal property, valuation of the intangible property component of a hotel is contentious.6 Valuation of the real property and personal property components generally proceeds by establishing the overall net income before any deductions for property taxes, FF&E funding, management fees, and franchise fees. Deductions are made for income attributable to the business or going concern and tangible personal property, leaving what is generally called "net income" attributable to the real estate. This remainder is capitalized at a capitalization rate to establish the value of the real estate component.
3. We focus our discussion in this chapter on three valuation techniques and three income approaches to estimate a hotel's value. Within the income approaches we present two variants of the traditional mortgage-equity model that estimates the market value of individual hotels: (1) an after-tax model that estimates investment value, and (2) an income capitalization technique used to value hotels owned by publicly traded lodging companies. In addition, two alternatives for the sales comparison approach and the cost approach will be considered. Finally, we explore separately the valuing of the real property component of a hotel asset. We conclude with a discussion of all of the techniques. Each method is illustrated by a unified case study that allows for meaningful comparison of the techniques. Case Study Example and Valuation Techniques The Major City Edgemore Hotel is a 250-room upscale property in an urban market catering to the needs of business travelers and moderate-size groups. The property is part of the large national franchise network of Edgemore Hotels. The Edgemore was constructed in 1995 in a growing area of Major City, located in the southern half of the United States. The property has a restaurant and deli with 180 seats, and a club and lobby bar with 90 combined seats. Meeting space totals 15,000 square feet and includes a grand ballroom, two executive boardrooms, several breakout rooms, and a business center. Recreational facilities include an indoor/ outdoor pool with whirlpool and an adjoining fitness center with locker facilities. The property was constructed using superior materials and workmanship and has been maintained in average to above-average condition. The property was recently renovated and shows no signs of distress or deferred maintenance. We assume the current date to be January 1, 2004. The Edgemore has traditionally been an above-average competitor, achieving average daily rates virtually in the middle of its competitive set and above-average occupancy. The property consistently achieves 105 percent RevPAR penetration. The Edgemore achieved solid occupancy and average daily rates during the 1996-2000 period, but suffered after the events of September 11,2001. In addition, one new hotel and a conversion from a mid-price to upscale hotel opened in 2002, increasing the number of rooms in the upscale sub-market by 20 percent. These factors combined to produce a significant drop in market occupancy as the new properties gained their fair share of the upscale market. The occupancy situation is expected to improve rapidly, with no new supply in the pipeline and with demand expected to grow quickly over the next three years. Pro-Forma Financial Projections Exhibit 1 presents a historical statement of the Edgemont Hotel's income and expense for 2003, as well as projections for 2004 through 2008. The current date is assumed to be January 1,2004. The projections account for an increase in the room demand and changes in the relative competitive position of the Edgemore. The operating expenses for the property include all charges normally associated with the operation of the property, including franchise and royalty fees, a management
4. fee, and a capital expenditure (CapEx) reserve. Thus, the net income figure represents the cash available to service debt, provide an equity dividend, and pay income taxes. The projection shows a rapid decline in occupancy as new supply comes into the market, with average daily rate increasing with inflation over the projection period. This combination produces a net income that increases rapidly and then peaks in 2007. The year 2003 obtains a net income of $2.38 million, while the third projection year of 2007 obtains a net income of$4.64 million, an increase of 94.6 percent over the three years. Both projection years are inappropriate for use as an estimate of stabilized net income, with the 2004 figure being too small and the 2007 figure too large. It is thought that long-term stabilized occupancy will average 71 percent. Hence, a stabilized net income of $4.107 million is used.7 Basis of Projections. Room rates are projected to increase by 3 percent for all years. All other revenues and expenses are projected to increase by 3 percent per year. Other assumptions used in the valuation techniques are: Debt Parameters Loan-to-Value Ratio8 60% Amortization 25 years Mortgage Interest 8.75% Yearly Mortgage Constant9 0.098657 Percent of Loan Paid in 10 Years9 17.7403% Equity Parameters Before Tax Equity Dividend Rate 13.0% Before Tax Equity Yield 18.0% After-Tax Equity Yield 14.0% Tax Considerations/or Investment Value Estimates Ordinary Income Tax Rate 35% Capital Gains Income Tax Rate 17.5% Depreciation Parameters (straight line assumed) Building Tax Life 39 years Building Basis 70% of value and 30% of CapEx reserve FF&E Tax Life 7 years FF&E Basis 10% of value and 70% of CapEx reserve Land Basis 20% of value (land is not depreciated) Public Company Information Cost of Debt 8.0% Debt to Total Value Ratio 60% (Giving a 40% Equity to Value Ratio) Public Company Equity Parameters Risk Free Rate 5.0% Equity Market Premium 8.0% C-Corp. Beta 0.80 Public Company Tax Rate 35% Other Valuation Parameters Terminal Capitalization Rate 11.25% Selling Expenses (Broker & Legal) 3.0% of selling price Valuation Technique 1: Band of Investment Using One Stabilized Year Instead of projecting net income over an extended period of time, a single, stabilized estimate of net income can be capitalized at an appropriate rate. The stabilized net income estimate is intended to reflect a representative year for the subject property in terms of occupancy, average rate, and net income. As just mentioned, the stabilized net income for the Edgemore Hotel is estimated to be$4,107,000. The next step in evaluating the Edgemore using the "band of investment using one stabilized year" technique is to develop a rate to capitalize the stabilized net income into an estimate of value. The band of investment, also known as the "weighted average cost of capital," or WACC, is based on the premise that most hotel investors purchase their properties 'using a combination of debt and equity capital. Both of these capital sources are seeking a specific rate of return on their invested capital as well as the return of their invested capital. The appropriate rate for the debt component is called the mortgage constant, which combines the return on capital (interest rate) with the return of capital (sinking fund factor) into a single rate. The proper rate of return for the equity component is the equity dividend rate. The appropriate overall capitalization rate is therefore the weighted average cost of capital from these two sources. The calculations that follow show how the band of investment using one stabilized year technique is used to estimate the value of the Edgemore Hotel: Mortgage Finance Terms: Mortgage Interest Rate: 8.75% Mortgage Amortization: 25 years Mortgage Constant: Loan-to-Value 0.098657 60% Ratio: Equity Dividend Rate (Before Tax) 13% Weighted Average Cost of Capital Calculation:
5. Percent Of Value Rate of Return Weighted Average Mortgage 60% x 0.098657 = 0.059194 Equity 40% x 0.130000 = 0.052000 Overall Capitalization Rate = 0.111194 The stabilized net income is divided by the capitalization rate to calculate the capitalized value: $4,107,000 - 0.111194 =$36,935,333, say $36,935,000 The value can be supported with the following calculations: 60% Mortgage$22,161,000 x 0.098657 = $2,186,000 40% Equity$14,774,000 x 0.130000 = $1,921,000 Total$36,395,000 $4,107,000 These calculations show that the$36,395,000 value can be divided into a mortgage portion of $22,161,000 and an equity portion of$14,774,000. The yearly mortgage payment, consisting of interest and amortization, is calculated by multiplying the original mortgage balance ($22,161,000) by the mortgage constant (0.098657), which results in an annual debt service of$2,186,000. The equity dividend is established by multiplying the equity investment ($14,774,000) by the anticipated equity return (.130), which yields$1,921,000. The annual debt service plus the equity dividend equals the stabilized net income of $4,107,000. Essentially, the band-of-investment technique works backward, using the projected stabilized net income to calculate the value that will meet the demands of both the debt and equity investors. Valuation Technique 2: Room-Rate Multiplier The lodging industry has a well-known rule of thumb known as the average daily rate (ADR) rule, which states that a property is worth 1,000 times its average daily rate on a per-room basis. The rule is essentially a RevPAR multiplier, setting value per room at 3.5 to 4.5 times annual room revenues, depending on occupancy.10 More formally: Value = Average Daily Rate x Number of Rooms x 1,000 One of the questions that immediately arises when implementing the rule is which ADR to use: a "trailing" or historical ADR, ADR in the first projection year, or the stabilized year ADR. Since the rule's origins are clouded in lodging folklore, a generally accepted standard must be used when applying the rule. Extensive research by Corgel and deRoos revealed that practitioners generally use the current year's expected ADR when applying the rule to existing hotels, but apply a stabilized ADR when applying the rule to properties under development." This inconsistency is a source of confusion and inaccuracy. For our purposes, we take the position that the rule should be consistently applied to a stabilized ADR. The stabilized ADR in 2006 (year 3) for the Edgemore Hotel is$182.09, or $171.64 expressed in 2004 dollars.12 Applying the room-rate formula results in the following value:$171.64 x 250 x 1000 = $42,910,000 For the Edgemore, the room-rate multiplier technique produces an estimate of value significantly in excess of all of the other techniques, indicating that this technique is subject to error. Valuation Technique 3: The Coke™-Can Multiplier Another valuation rule-of-thumb used in the lodging industry is that each room of a hotel is worth 100,000 times the price of a Coke™ in the on-floor vending machine or in-room mini-bar. More formally: Value = Coke™ price x Number of Rooms x 100,000 The Edgemore Hotel sells cans of soda for$1.50 in the room mini-bars. Thus, the value of the Edgemore by this "precise" valuation method is: $1.50 x 300 x 100,000 =$37,500,000 We urge market participants to use this technique judiciously, as some properties seriously "misprice" soda in relation to property value. Income Approaches to Value Valuation Technique 4:10-Year Discounted Cash Flow Using Mortgage and Equity Rates of Return Valuation technique 4 is appropriate in dynamic hotel markets where supply and demand is constantly changing and the subject property's occupancy, rate, and net income has not stabilized. The projection of income and expenses reflect changing market conditions and extends over a five- to ten-year time frame. Traditionally, hotel investors use a ten-year projection period. To convert the projected income stream into an estimate of value, the anticipated net income is allocated to the mortgage and equity components based on market rates of return and loan-to-value ratios (similar to the band-of-investment). The total of the mortgage component and the equity component equals the value of the property. The process is described as follows: 1. The terms of typical hotel financing are set forth, including interest rate, amortization term, and loan-to-value ratio. 2. An equity yield rate of return is established. Many hotel buyers base their equity investments on a desired equity yield rate or, equivalently, a desired internal rate of return on invested equity capital. This rate takes into account ownership benefits such as periodic cash-flow distributions, residual sale or refinancing distributions that return any property appreciation and mortgage amortization, income tax benefits, and various non-financial considerations such as status and prestige. 3. The value of the equity component is calculated by first deducting the annual debt service from the projected net income before debt service, leaving the net income to equity for each year. The net income as of the 11th year is capitalized into a reversionary value. After deducting the mortgage balance at the end of the tenth year and the typical brokerage and legal costs, the equity residual is discounted back to the date of value at the equity yield rate. The net income to equity for each of the ten projection years is also discounted to the present value. The sum of these discounted values equates to the value of the equity component. Adding the equity component to the initial mortgage balance yields the overall property value. The mortgage and the debt service amounts are unknown because they depend on the value of the property, which in turn depends on the amounts of the mortgage and debt service. This is the classic simultaneous valuation conundrum. However, since
6. the loan-to-value ratio was determined in Step 1, the preceding calculation can be solved through an iterative process or by use of an algebraic equation that solves for the total property value using a ten-year mortgage and equity technique. This technique was developed by Suzanne R. Mellen, MAI.13 4. The value is proven by allocating the total property value between the mortgage and equity components and verifying that the rates of return set forth in Steps 1 and 2 can be met from the projected net income. This process can be expressed in two algebraic equations that set forth the mathematical relationships between the known and unknown variables using the following symbols: V = Value N1 = Net income available for debt service M = Loan-to-value ratio f = Annual debt service constant n = Number of years in the projection period de = Annual cash available to equity dr = Residual equity value b = Brokerage and legal cost percentage P = Fraction of the mortgage paid off during the projection period fp = Annual debt service constant required to amortize the entire loan during the projection period Rr = Overall terminal capitalization rate that is applied to net income to calculate the total property reversion (sales price at the end of the projection period) 1 /Sn = Present worth of a $1 factor (discount factor) at the equity yield rate (Ye) Using these symbols, the following formulas can be used to express some of the components of this mortgage and equity valuation process. Debt Service. A property's debt service is calculated by first determining the mortgage amount that equals the total value (V) multiplied by the loan-to-value ratio (M). Debt service is derived by multiplying the mortgage amount by the annual debt service constant (f). The following formula represents debt service: f x M x V = Debt Service Net Income to Equity (Equity Dividend). The net income to equity (de) is the property's net income before debt service (N1) less debt service. The following formula represents the net income to equity: N1 - (f x M x V) = de Reversionary Value. The value of the hotel at the end of the tenth year is calculated by dividing the llth-year net income before debt service (NI11) by the terminal capitalization rate (Rr). The following formula represents the property's tenth-year reversionary value: (NP11Rr) = Reversionary Value Brokerage and Legal Costs. When a hotel is sold, certain costs are associated with the transaction. Normally, the broker is paid a commission and the attorney collects legal fees. In the case of hotel transactions, brokerage and legal costs typically range from 1 to 4 percent of the sale price. Because these expenses reduce the proceeds to the seller, they are usually deducted from the reversionary value in the mortgage and equity valuation process. Brokerage and legal costs (b), expressed as a percentage of reversionary value (N^/Rr), are calculated by application of the following formula: b x (N11/Rr) = Brokerage and Legal Costs Ending Mortgage Balance. The mortgage balance at the end of the tenth year must be deducted from the total reversionary value (debt and equity) in order to estimate the equity residual. The formula used to determine the fraction of the loan remaining (expressed as a percentage of the original loan balance) at any point in time (P) takes the annual debt service constant of the loan over the entire amortization period (f) less the mortgage interest rate (i) and divides it by the annual debt service constant required to amortize the entire loan during the ten-year projection period (fp) less the mortgage interest rate. The following formula represents the fraction of the loan paid off (P): (f - i)/(fp - i) = P If the fraction of the loan paid off (expressed as a percentage of the initial loan balance) is P, then the remaining loan percentage is expressed as (1 - P). The ending mortgage balance is the fraction of the remaining loan (1 - P) multiplied by the initial loan amount (M x V). The following formula represents the ending mortgage balance: (1 - P) x M x V = Ending Mortgage Balance Residual Equity Value. The value of the equity upon the sale at the end of the projection period (dp) is the reversionary value less the brokerage and legal costs and the ending mortgage balance. The following formula represents the residual equity value: (NI11/Rr) - (b x (NI11/Rr)) - ((1 - P) x M x V) = dr Annual Cash Flow to Equity. The annual cash flow to equity consists of the equity dividend for each projection year plus the equity residual at the end of the tenth year. The following formula represents the annual cash flow to equity: N11 - (f x M x V) = de1 NP² - (f x M x V) = de2 NI10 - (f x M x V) = de10 Value of the Equity. If the initial mortgage amount is calculated by multiplying the loan-to-value ratio (M) by the property value (V), then the equity value is one minus the loan-to-value ratio multiplied by the property value. The following formula represents the value of the equity: (1 - M) x V 7. Discounting the Cash Flow to Equity to the Present Value. The cash flow to equity in each projection year is discounted to the present value at the equity yield rate (1 / S"). The sum of these cash flows is the value of the equity: (1 - M) x V. The following formula represents the calculation of equity as the sum of the discounted cash flows: (de1 x 1/S1) + (de2 x 1/S²)+ ...+ (de10 x 1/S10 )+(dr x 1/S10) = (1 - M) x V Combining the Equations: Annual Cash Flow to Equity and Discounting the Cash Flow to Equity to the Present Value. The last step is to arrive at one overall equation that shows that the annual cash flow to equity plus the yearly discounting to the present value equals the value of the equity: ((NI1 - (f x M x V)) x 1/S1)+ ((NI2 - (f x M x V)) x 1/S2) +... ((NI10 - (f x M x V)) x 1/S10 )+ {[(NI11/Rr) - (b x (NI11/Rr)) - ((1 - P) x M x V)] x 1/S10}= (1 - M) x V Because the only unknown in this equation is the property's value (V), it can be solved readily. Solving for Value Using the Simultaneous Valuation Formula. In the case of the subject property (the fictional Edgemore Hotel), the following known variables have been determined: Annual Net Income NI For our purposes here, the 2006 NI is considered the stabilized net income. The net incomes for 2007-2013 are assumed to increase at 3.0 percent per year Loan-to-Value Ratio M 60.0% Mortgage Interest Rate i 8.75% Exhibit 2 Present Worth of$1 Factor at the Equity Yield Rate Year Present Worth of $1 Factor at 18.0% 1 0.847458 2 0.718184 3 0.608631 4 0.515789 5 0.437109 6 0.370432 7 0.313925 8 0.266038 9 0.225456 10 0.191064 Debt Service Constant f 0.098657 Equity Yield Ye 18.0% Brokerage and Legal Fees b 3.0% Terminal Capitalization Rate Rr 11.25% Exhibit 2 illustrates the present worth of$1 factor at the 18-percent equity yield rate. Using these known variables, the following intermediary calculations must be made before applying the simultaneous valuation formula. The fraction of the loan paid off during the projection period is:14 P = 0.177403 The annual debt service is calculated as (f x M x V): (f x M x V) = 0.098657 x 0.60 x V = 0.059194V Inserting the known variables into the hotel valuation formula produces the following: (3,007,000 - 0.059194 x V) x 0.847458 + (3,664,000 - 0.059194 x V) x 0.718184 + (4,357,000 - 0.059194 x V) x 0.608631 + (4,488,000 - 0.059194 x V) x 0.515789 + (4,622,000 - 0.059194 x V) x 0.437109 + (4,761,000 - 0.059194 x V) x 0.370432 + (4,904,000 - 0.059194 x V) x 0.313925 +
8. (5,051,000 - 0.059194 X V) X 0.266038 + (5,202,000 - 0.059194 x V) x 0.225456 + (5,359,000 - 0.059194 x V) x 0.191064 + {[(5,519,000/0.1125) - (0.03 x (5,519,000/0.1125)) - ((1 - 0.17740) x 0.60 x V)] x 0.191064} = (1 - 0.60) V Like terms are combined as follows: $28,102,819 - 0.360326V = (1 - 0.60) V$28,102,819 = 0.639674 V V = $28,102,819/0.636974 V =$36,961,542 = say $36,962,000 The value is proven by calculating the yields to the mortgage and equity components during the projection period. If the mortgagee achieves a yield of 8.75 percent and the equity yield is 18 percent, then$36,962,000 is the correct value by the income capitalization approach. Using the assumed financial structure set forth in the previous calculations, the market value can be allocated between the debt and equity as follows: Proof of Value Mortgage Component (60%) $22,177,000 Equity Component (40%) 14,785,000 Total$36,962,000 The annual debt service is calculated by multiplying the mortgage component by the mortgage constant: Mortgage Component $22,177,000 Mortgage Constant 0.098657 Annual Debt Service$2,187,914 The net income (or cash flow) to equity is calculated by deducting the debt service from the projected income before debt service (see Exhibit 3). The equity residual at the end of the tenth year is calculated as follows: Reversionary Value ($5,555,000/0.1125)$49,061,000 Less: Brokerage and Legal Fees (3.0%) 1,472,000 Less: Mortgage Balance 18,243,000 Net Sale Proceeds to Equity $29,346,000 The overall property yield (before debt service), the yield to the lender, and the yield to the equity position have been calculated by computer, with the results shown in Exhibit 4. Exhibit 3 Forecast of Net Income to Equity Net Income Available for Net Income to Year Debt Service Debt Service Equity 2004$3,007,000 - $2,188,000 =$ 819,000 2005 3,664,000 - 2,188,000 = 1,476,000 2006 4,357,000 - 2,188,000 = 2,169,000 2007 4,488,000 - 2,188,000 = 2,300,000 2008 4,622,000 - 2,188,000 = 2,434,000 2009 4,761,000 - 2,188,000 = 2,573,000 2010 4,904,000 - 2,188,000 = 2,716,000 2011 5,051,000 - 2,188,000 = 2,863,000 2012 5,202,000 - 2,188,000 = 3,015,000 2013 5,359,000 - 2,188,000 = 3,171,000
9. Exhibits 5 through 7 demonstrate that the property receives its anticipated yields, proving that the $36,962,000 value is correct based on the assumptions used in this approach. Valuation Technique 4a: 10-Year Discounted Cash Flow Using Mortgage-Equity Model and Debt Coverage Ratio Valuation technique 4a uses a loan-to-value ratio to link the amount of the initial mortgage balance with the property's value. In many instances, the mortgage lender is also interested in the relationship between the hotel's net income and annual debt service (interest plus amortization). The debt coverage ratio (Net income - Annual Debt Service) is used in these cases instead of the loan-to- value ratio. Instead of establishing a maximum loan amount to constrain value, the debt coverage ratio requires that the annual net income "cover" the debt service by a specific amount. For example, a debt coverage ratio of 2.0 means the lender requires$2.00 in net income for each $1 in annual debt service. This establishes a maximum annual debt service, and hence a maximum loan amount, given market interest rates and the amortization period. Many lenders employ both the loan-to-value ratio and debt coverage ratio and will lend funds based on the constraint that results in the smallest loan. In determining the debt coverage ratio, one must decide which net income to use. Since net income typically increases over time, the most conservative lenders base the loan on historical or the smallest of the projected net incomes. Other lenders use a "stabilized" net income, especially when lending to recovering properties or to new properties that take time to reach a stable occupancy rate (typically a stable occupancy rate is reached in the second, third, or fourth year of operation). Using the notation from technique 4, the following formulas express the components of this mortgage and equity valuation process. The only new symbols are "DCR," which stands for "debt coverage ratio," and "NP," which represents the "stabilized" net income used by lenders to size the loan. 10. Debt Service. A property's debt service is calculated by dividing the stabilized net income by the debt coverage ratio: NIs/DCR = Debt Service Mortgage Amount. Without the loan-to-value ratio to size the mortgage, the size of the mortgage is calculated by applying the mortgage constant to the annual debt service. Since the mortgage constant is the annual debt service for each$1 of mortgage, the mortgage can be estimated by dividing the debt service by the mortgage constant: NIs-/(DCR x f) = Mortgage Amount Net Income to Equity (Equity Dividend). The net income to equity (de) is the property's net income before debt service (N1) less debt service. The following formula represents the net income to equity: NI - (NIs-/DCR) = de Reversionary Value. The value of the hotel at the end of the tenth year is calculated by dividing the llth-year net income before debt service (NI11) by the terminal capitalization rate (Rr). The following formula represents the property's tenth-year reversionary value: (NP'/Rr) = Reversionary Value Brokerage and Legal Costs. When a hotel is sold, certain costs are associated with the transaction. Normally, the broker is paid a commission and the attorney collects legal fees. In the case of hotel transactions, brokerage and legal costs typically range from 1 to 4 percent of the sale price. Because these expenses reduce the proceeds to the seller, they are usually deducted from the reversionary value in the mortgage and equity valuation process. Brokerage and legal costs (b), expressed as a percentage of reversionary value (NI11/Rr), are calculated using the following formula: b x (NI11/Rr) = Brokerage and Legal Costs Ending Mortgage Balance. The mortgage balance at the end of the tenth year must be deducted from the total reversionary value (debt and equity) in order to estimate the equity residual. The formula used to determine the fraction of the loan remaining (expressed as a percentage of the original loan balance) at any point in time (P) takes the annual debt service constant of the loan over the entire amortization period (f) less the mortgage interest rate (i), and divides it by the annual constant required to amortize the entire loan during the ten-year projection period (fp) less the mortgage interest rate. The following formula represents the fraction of the loan paid off (P): (f - i)/(fp - i) = P If the fraction of the loan paid off (expressed as a percentage of the initial loan balance) is P, then the remaining loan percentage is expressed as 1 - P. The ending mortgage balance is the fraction of the remaining loan (1 - P) multiplied by the initial loan amount: NIs/(DCR x f). The following formula represents the ending mortgage balance: (1 - P) x (NIs /DCR x f)) = Ending Mortgage Balance Equity Residual Value. The value of the equity upon the sale at the end of the projection period (dp) is the reversionary value less the brokerage and legal costs and the ending mortgage balance. The following formula represents the equity residual value: (NI11/Rr) - (b x (NI11/Rr)) - ((1 - P) x NIs /(DCR x f)) = dr Annual Cash Flow to Equity. The annual cash flow to equity consists of the equity dividend for each projection year plus the equity residual at the end of the tenth year. The following formula represents the annual cash flow to equity: NI1 - (NIs/DCR) = de1 NI 2 - (NIs/DCR) = de2 : : NI10 _ (NIs /DCR) = de10 Value of the Equity. The cash flow to equity in each projection year is discounted to the present value at the equity yield rate (1/Sn). The sum of these cash flows is the value of the equity. The following formula represents the calculation of equity as the sum of the discounted cash flows: (de1 x 1/S1) + (de2 X 1/S2) +...+ (de10 X 1/S10) + (dr X 1/S10) = Value of Equity Combining the Equations: Annual Cash Flow to Equity and Discounting the Cash Flow to Equity to the Present Value. The last step is to arrive at one overall equation that shows that the annual cash flow to equity plus the yearly discounting to the present value equals the value of the equity: ((NI1 - NIs/DCR) x 1/S1) + ((NI2- NIs /DCR) x 1/S2) + ... ((NI10 - NIs/DCR) x 1/S10) + {[(NI11/Rr) - (b x (NI11/Rr)) - ((1 - P) x (NIs/(DCR x f)))] x 1/S10} + NIs /(DCR x f) = V Because the only unknown in this equation is the property's value (V), it can be solved readily. Note that this technique does not have the "simultaneous" problem of valuation technique 4. For the Edgemore Hotel, we estimate market value using a debt coverage ratio of 1.90 on the year 3 stabilized income of
11. $4,357,000 to estimate annual debt service and hence the mortgage amount. The value can be estimated by: (3,007,000 - (4,357,000/1.90) x 0.847458 + (3,664,000 - (4,357,000/1.90) x 0.718184 + (4,357,000 - (4,357,000/1.90) x 0.608631 + (4,488,000 - (4,357,000/1.90) x 0.515789 + (4,622,000 - (4,357,000/1.90) x 0.437109 + (4,761,000 - (4,357,000/1.90) x 0.370432 + (4,904,000 - (4,357,000/1.90) x 0.313925 + (5,051,000 - (4,357,000/1.90) x 0.266038 + (5,051,000 - (4,357,000/1.90) x 0.225456 + (5,359,000 - (4,357,000/1.90) x 0.191064 + {[(5,519,000/0.1125) - (0.03 x (5,519,000/0.1125)) – ((1-0.177403) x (4,357,000/(1.90 x 0.098657)))] x 0.191064} + (4,357,000/(1.90 x 0.098657)) = V For the sake of brevity, the intermediate calculations are omitted. The value of the property is calculated to be$37,387,667, say $37,388,000. Valuation Technique 5: After-Tax Investment Model As demonstrated by deRoos and Rushmore,15 the hotel valuation formula can be extended to incorporate the effects of income taxes on value. Used in this fashion, the model is formally known as an "investment value model" because it reflects the unique characteristics of a particular investor, and it ceases to serve as a market value model because the valuation parameters are no longer derived totally from market information. In addition to the lender criteria, investor criteria, and property information used in technique 4, the after-tax investment technique needs information on depreciation rates, tax rates, and the projected amounts of the annual reserve for replacement. The parameters in the after-tax investment model are: V = Value of the property NIj = Net income available for debt service, indexed by year (i.e., NI1 for year one net income) NIn = Net income used for the reversion calculation, usually specified as the N1 in year n+1 n = Number of years in the projection period Ye = The equity yield, on an after-tax basis M = Loan-to-value ratio Rr = Overall terminal capitalization rate applied to NI11 to calculate the property reversion, on an after-tax basis b = Selling expenses i = Mortgage interest rate m = Mortgage amortization term f = Annual debt service constant tl = Ordinary income- tax rate t2 = Capital gains tax rate P = Fraction of the mortgage paid off during the projection period RFRj = A set of cash flows that will be spent on improving the property over time (i.e., the reserve for replacement, indexed by year) LI = Depreciable life of the building (years) L2 = Depreciable life of the furniture, fixtures, and equipment (FF&E) (years) IB = Proportion of total value attributable to the building for depreciation purposes IBr = Proportion of RFR spent on improvements to the building for depreciation purposes IF = Proportion of total value attributable to FF&E for depreciation purposes IFr = Proportion of RFR spent on replacement of FF&E for depreciation purposes 12. Valuation of Hotel Properties Owned by a Publicly Listed Company Public company techniques focus on whether an acquisition adds value to a publicly traded firm. Recognizing the firm's unique cost of capital, the technique we present is a relatively simple but powerful measure of the impact of a purchase on share value. The model is based on a popular tool from managerial accounting known as "economic value added," or EVA. We apply the EVA methodology to the valuation question faced by a c-corporation. The EVA technique is classified an investment value model because it relies on the unique investment parameters of a given firm, not market derived data for the valuations. Valuation Technique 6: Economic Value Added While EVA is traditionally used to value entire firms, we believe the methodology can be extended to single asset valuation as well. EVA is defined as the excess return on investment available to shareholders after deducting the risk-adjusted cost of capital. More formally: EVA = A-T Earnings - (WACC x Property Investment) "A-T Earnings" are the after-tax accounting definition of earnings generated by the investment. "WACC" is the weighted-average cost of capital, on an after-tax basis. The valuation is performed by realizing that the most the firm would pay for a property is that amount which makes EVA equal to zero. Given this relationship, one can rearrange the terms to arrive at the following equality: Thus, in the end, this technique becomes a specialized form of a capitalized-in-come technique. The numerator (A-T Earnings) is the accounting measure of long-run income, and the denominator (WACC) recognizes the unique capital structure of a given firm, including its cost of debt and beta. If the property can be acquired for less than the value indicated by the EVA technique, it is accretive and adds value to the firm. To implement the model, one must transform the traditional real estate definition of net income into A-T Earnings and determine the weighted-average cost of capital. A-T Earnings are defined as follows, starting from the traditional definition of net income used in a typical lodging pro-forma like Exhibit I:17 Net income Less: Building depreciation Less: Income Taxes A-T Earnings --Income taxes are calculated by deducting depreciation and interest expense from net income to determine taxable income. Taxable income is then multiplied by a tax rate to arrive at the income tax. Since both the interest expense and depreciation charges are a function of property value, this problem is simultaneous and must be solved iteratively to estimate the proper value. The weighted-average cost of capital is derived as follows:18 Debt Component: Debt Rate (%) x (1 - Tax Rate (%)) x Debt to Value Ratio (%) = Debt Component Equity Component: {Risk Free Rate (%) + (Equity Market Premium (%) x Firm Beta)} x Equity to Value Ratio (%) = Equity Component WACC = Debt Component + Equity Component The equity component is a direct application of modem portfolio theory. A firm's equity cost of capital is derived as a premium over the risk-free rate. The premium is a function of the overall market premium for equity investments, times the unique beta of the subject firm. The debt to value and equity to value ratios are determined from the firm's capital structure, as the percentage of total firm value attributable to debt and equity, respectively. To perform the EVA valuation on the Edgemore Hotel, we start with the calculation of A-T earnings, using the assumptions listed previously: Stabilized Net Income 4,107,000 Less: Building Depreciation 662,500 Less: Income Tax (see calc. below) 609,800 A-T Earnings 2,834,700 Income Tax Calculation: Stabilized Net Income 4,107,000 Add: Stabilized CapEx reserve 597,000 Equals: EBITDA 4,614,000 Less: Depreciation 1,189,800 Less: Interest Expense 1,771,700 Equals: Taxable Income 1,742,200 Times: Tax Rate ___35% Income Tax 609,800 13. Depreciation consists of two pieces, building and FF&E. Building depreciation is based on a 39-year life, with 70 percent of total value as the building basis; FF&E depreciation is based on a 7-year life, with 10 percent of total value as the FF&E basis. With value established at$36,911,000, the depreciation calculation is: Building Portion $36,911,000 x 70% ÷ 39 years = 662,500/year FF&E Portion$36,911,000 x 10% ÷ 7 years = 527,300/year Total Depreciation $l,189,800/year Interest expense is calculated as: Interest Expense = Value x Debt Cost (%) x Debt to Value Ratio =$36,911,000 x 8.0% x 60% = $1,771,700 Next, the WACC is calculated: Debt Component = 8.0% x (1 - 35%) x 60% = 3.120% Equity Component = {5.0% + (8.0% x 0.80)} x 40% = 4.560% WACC = 7.680% Value = A-T Earnings ÷ WACC =$2,834,700 ÷ 0.07680 = $36,910,720, say$36,911,000 Valuation Technique 7: Sales Comparison Approach Three property transactions in 2002 and 2003 were found in the HVS International Major Sales Transactions database of hotels that were considered comparable to the Edgemore Hotel. Exhibit 9 summarizes these transactions. The sales comparison approach takes information from these sales and adjusts the data to arrive at an estimate of value for the Edgemore. The most fundamental adjustment is to derive a sale price per room, as shown in Exhibit 9. Other adjustments are made for the condition of the physical plant, relative market strength, brand affiliation, age, below-market financing, and the mix of facilities offered. In the case of the Edgemore Hotel, Sale No. 1 needs to be adjusted upward 3 percent, due to the timing of the sale. Sale No. 2 needs to be adjusted upward 5 percent, due to the superior physical condition of the Edgemore. Sale No. 3 needs to be adjusted downward 5 percent, due to the Edgemore's inferior location. This gives final adjusted sale prices per room, and indicated property values for the Edgemore, as follows: Sale No. Adjusted Sale Indicated Value of Price Per Room The Edgemore 1 $154,500$38,625,000 2 $146,580$36,645,000 3 $146,775$36,693,750 The indicated values provide a range of values that would be expected in a competitive selling environment. In most cases, the adjustments are part of the appraiser's art, because the appraiser must use his or her judgment in their
14. application.19 In the case of the Edgemore Hotel, sale 3 is more representative of market transactions than are sales 1 and 2, which are generally understood to have been "tainted" by the events of September 11, 2001. We will use the range of $36,345,000 to$38,625,000 as a reasonable estimate of the property value via the sales comparison approach. Valuation Technique 8: Market-Derived Capitalization Rate The information in Exhibit 10 can be used to derive market-based capitalization rates. It is important to develop these rates consistently; we recommend using the net income in the 12 months prior to sale as the basis for the calculations. The capitalization rate is simply the trailing 12-month net income divided by the sale price. The table below details the rates for the three comparable sales shown in Exhibit 9: Market Derived Sale No. 1998 Net Income Sale Price Capitalization Rate 1 $3,040,000$48,000,000 6.33% 2 $2,105,000$32,806,000 6.42% 3 $2,927,000$44,805,000 6.53% The market-derived capitalization rates range from a low of 6.33 percent to a high of 6.53 percent, with an average of 6.43 percent. These rates are based on historic net income, which does not mirror the future path of expected earnings, which is expected to rise rapidly. It is therefore important to adjust these rates if it is expected that future conditions will make these rates higher or lower. We therefore will adjust the average market-derived capitalization rate down to 6.4 percent to reflect this future increase in earnings. Dividing the Edgemore's 2003 net income by the 6.4 percent capitalization rate obtains the following estimate of value: $2,383,000 - 0.064 =$37,234,375, say $37,234,000 Note that the market-derived capitalization rates are low by historical standards. The reason is that the income in the 12 months preceding was exceptionally low, significantly impacted by the combined impact of dilution due to new supply and a drop in travel to Major City for a period following September 11,2001. As the market moves to equilibrium, capitalization rates will revert to the 15. historical equilibrium range of 10.0 to 12.0 percent. Valuation Technique 9: Cost Approach—Age-Life Method The cost approach typically involves the use of a cost estimating guide or guides to arrive at a cost to determine the replacement cost of the property. Appraisers often use a unit-in-place method to estimate value of the building and FF&E. Data for many prototype chain properties is generally available; the more unique the property, the more difficult it is to establish the replacement cost. Use of the Marshal and Swift™ estimating guide and consultation with local architects produce data for the Edgemore as follows: Land$7,500,000 (5 acres® 1,500,000/acre) Building & improvements, including soft costs $30,500,000 ($122,000/room) FF&E $3,500,000 ($14,000/room) Preopening & working capital $1,350,000 ($5,400/room) Total replacement cost $42,850,000 ($176,400/room) The total represents the replacement cost, without deductions for depreciation, the physical deterioration, functional obsolescence, and external obsolescence of the property. As shown below we use the age-life method to make an overall depreciation adjustment, reasonable for an eight-year-old property with little external obsolescence. We assume a physical life for the building of 50 years and an average physical life for the FF&E of ten years. We then add back the reserve for replacement invested in the property over the eight-year life of the property. Total replacement cost: $42,850,000 Less: Building depreciation$4,880,000 (8/50 of the replacement cost) FF&E depreciation $2,800,000 (8 /10 of the replacement cost) Add: CapEx reserve$ 3,000,000 (8 years of net new investment) Adjusted total $38,170,000 The use of age-life method is a reasonable approach with a relatively young building and one that is built with conventional means and methods. The principal criticism is that it relies heavily on accurate replacement cost data; there is also the problem of ad hoc adjustments for depreciation. Valuation for Assessment Purposes We submit that the income approach is appropriate for the question of valuation for assessment purposes. Further, adjustments to the net income can isolate the cash flow to the real property component, which can then be capitalized to produce an estimate of the real property distinct from the personal property. In general, the net income of a property must be adjusted for three factors to isolate the cash flow to the real property: 1. The return on and return of funds invested in FF&E 2. Any fees paid to the manager and fees paid for brand affiliation 3. Any net income that might accrue as a result of superior management20 In the case of the Edgemore, there is the question of whether the 105-percent RevPAR penetration of the property is attributable to the real property or to the personal property component. For purposes of clarity, assume that all of the superior performance is due to superior management. To make the adjustment, we take the following steps: • Establish 2006 as the benchmark, stabilized year. The stabilized net income at a 105-percent RevPAR penetration is$4,107,000. • Establish the financial performance of the property with a 100-percent RevPAR penetration, by reducing occupancy. • Make any reductions for the costs of management and brand that could be achieved with the change in RevPAR penetration. The result of these steps produces an adjusted stabilized net income of $3.743 million as opposed to the$4.107 million figure noted. From this figure, we must deduct the annual flows appropriate for items 1 and 2 in the foregoing list. To properly handle the return on and return of investment for the FF&E, we proceed as follows: • Add the CapEx reserve to net income. This is done to avoid a duplicate deduction via the return calculations performed in the second step. • Deduct the appropriate amounts for the returns on and the return o/the FF&E, using the capitalization rate for the property. The appropriate return on the FF&E is simply the capitalization rate times the original investment of $3,500,000 used in the cost approach technique. The appropriate return of the FF&E is a sinking fund factor (SFF) using the capitalization rate and the average physical life of the FF&E. We use the capitalization rate derived in valuation technique 1, or 11.119 percent. Return on FF&E 11.119% Return of FF&E 5.946% (SFF for 11.194%, 10-year average physical life) Total 17.065% Annual return on and return of FF&E =$3,500,000 x 0.17698 = $597,000/year No adjustment is necessary for management and brand fees, as they are deducted from the definition of net income used in Exhibit 1 and from the stabilized income. The summary of adjustments follows: 16. Unadjusted 2004 stabilized net income$ 4,107,000 21 Add: CapEx reserve (stabilized) 615,000 Equals: Unadjusted 2004 stabilized net income before CapEx reserve 4,722,000 Less: Adjustment for superior management __364,000 Adjusted stabilized net income 4,358,000 Less: Return on and return of FF&E $597,000 Less: Management and brand fees ____-0- Adjusted net income Add: attributable to the ______ Equals: real property$ 3,761,000 The adjusted net income attributable to the real property is divided by the capitalization rate to calculate the capitalized value of the real property component. $3,761,000 - 0.111943 =$33,823,663, say $33,824,000 Summary —————————————————————————— The results of the nine approaches to valuing the Edgemore Hotel are summarized below: Rules of Thumb Band of Investment (technique 1)$36,395,000 Room-Rate Muliplier (technique 2) $42,910,000 Coke™-Can Multiplier (technique 3)$37,500,000 Income Approaches to Value Hotel Valuation Formula—LTV version (technique 4) $36,962,000 Hotel Investment Formula—DCR version (technique 4a)$37,388,000 Hotel Valuation Formula—After-Tax (technique 5) $37,017,000 Valuation by a Publicly Listed Company Economic Value-added (technique 6)$36,911,000 Sales Comparison Approaches Sales Comparison Approach (technique 7) $36.345-38.625 million Market-Derived Capitalization Rate (technique 8)$37,234,000 Cost-Approach (technique 9) $38,170,000 With the exception of room-rate multiplier (technique 2), the techniques produce values that are in a very narrow range, from$36,395,000 to $38,625,000. The room-rate multiplier produces a value that is high, because it is based on a single revenue metric and does not account for the operating characteristics of the property. The question, "What is a property worth?" has given rise to an increasingly broad set of methods to estimate value. The rules of thumb give a rough indication of value, but should not be relied on as definitive. They are simple, single-dimension models that do not incorporate the collective actions of market participants. They do, however, give a quick "ballpark" estimate. Appraisers are charged with estimating market value, using the classic troika of the cost approach, the sales comparison approach, and the income approach. The market-derived capitalization rate is a hybrid approach; it uses an income approach to value the property, with the capitalization rate derived from comparable sales. Investors, on the other hand, wish to estimate investment value, which includes the effects of income taxes, the investor's unique cost of capital, and other investor-specific conditions. To estimate value, investors typically rely on a modified income approach tailored to their circumstances. Publicly traded real estate firms have a unique form of investment-value question. Managers in this environment seek to invest when it can be demonstrated that the investment adds to the value of the firm; this is known as a accretive investment. The cost approach provides a physically oriented estimate of value. The difficulty in applying this technique is making the proper adjustments for obsolescence and depreciation. These adjustments require judgment in three areas: the amount of physical, economic, and functional obsolescence. For instance, it is easy to identify that a given location has problems, but it is difficult to quantify the impact of these problems on the property's value. Due to the difficulties in applying these adjustments/this technique has not been included in this chapter. The sales comparison approach, properly applied to homogenous properties -in thickly traded markets such as single-family homes, is a sound tool for gauging value based on actual market transactions. The difficulties in applying the technique to income- property markets include the paucity of sales, obtaining sales that are truly comparable to the subject, and making accurate adjustments. The most effective use of this tool is establishing a reasonable range of value, based on actual sales transactions. The income approaches evolve from two different manners of thinking. The "cap-rate" techniques—band of investment, market- derived capitalization, and EVA—are single-period models that implicitly account for growth in income. The "yield" or "discount- rate" technique—the hotel valuation formula—(both before and after tax) is a multi-period model using explicitly calculated cash flows over a holding period to arrive at value. Each set has its strengths. The cap-rate models are easy to implement and easy to understand, while the yield-based model is not. On the other hand, with high-quality input data, yield-based models produce more accurate valuations than cap-rate models. It is important to support all of the income approaches using the best available data. It is difficult in many cases to determine the returns required by equity participants. Market-value models must be supported by the 17. analyst's reasonable expectations of investor behavior and a thorough understanding of market conditions. Firm- or investor-specific data' is available for the investment value models, and thus the parameters used in these models are easy to support. Market participants should select the most appropriate models for their own use. Use of a variety of methods is encouraged. For instance, a potential seller would not only .wish to know market value, but also buyer-specific valuations, such as the value to a specific public company or to partnerships. In this case/ a classic "three-approaches" appraisal plus the EVA and after-tax SVF are the appropriate models; these produce a most-likely value via appraisal as well as establish estimates of bids by potential buyers. 18. Endnotes 1. Market value is defined as "The most probable price that a property should bring in a competitive and open market under all conditions requisite to a fair sale, the buyer and seller each acting prudently and knowledgeably, and assuming the price is not affected by undue stimulus. From Uniform Standards of Professional Appraisal Practice (Appraisal Foundation, 1997). 2. Investment value is defined as "... the value of an investment to a particular investor based on his or her investment requirements." From The Appraisal of Real Estate, 11th Ed. (The Appraisal Institute, 1996), p. 26. 3. Rushmore, Stephen, and Eric Baum. Hotels and Motels: Valuations and Market Studies (Chicago, ffl.: The Appraisal Institute, 2001), p. 356. 4. Rushmore and Baum, Hotels and Motels. 5. Rushmore and Baum, Hotels and Motels. 6. For further information, see Rubin, Karen, "Hotel and Real Estate Tax Valuation: Current Issues, Real Estate finance, Fall 1998. 7. The 2006 (year 3) net income of$4.357 million, discounted for two years at 3 percent, equals $4,106,890 or$4.107 million, when expressed as 2004 dollars. The year 3 projection, with an occupancy of 71 percent is estimated to represent a stabilized operation. 8. Value based on a debt coverage ratio will be discussed separately. 9. Based on monthly amortization. 10. For an extended discussion, see Corgel, John B., and Jan A. deRoos, "The ADR Rule-of-Thumb as a Predictor of Lodging Property Values," International Journal of Hospitality Management, Vol. 12, No. 4, Winter 1993, pp. 353-365. 11. Corgel and deRoos, "The ADR Rule-of-Thumb." 12. The 2006 ADR of $182.09 discounted for two years at the 3 percent inflation rate equals$171.64 in 2004 dollars. 13. Mellen, Suzanne R., MAI. "Simultaneous Valuation: A New Technique," Appraisal Journal, April, 1983. 14. The calculation is made using a monthly interest rate (8.75% + 12 in this case) and the number of months in the respective terms (300-month amortization term, 120-month projection period). 15. deRoos, J. A., and Stephen Rushmore, "Lodging Property Valuation Models: The Effects of Taxes and Alternative Lender Criteria," The Cornell Hotel and Restaurant Administration Quarterly, December 1995, pp. 62-69. 16. Readers who would like more information about how the after-tax investment model works with a spreadsheet should contact Professor deRoos at: Cornell University, School of Hotel Administration, 435 Statler Hall, Ithaca, NY 14853-6902. 17. Note that this definition differs from a traditional accounting definition of earnings, which would deduct depredation. However, the net income definition used in this chapter has a deduction for the CapEx reserve, which is the true economic measure of the costs necessary to maintain the FF&E, whereas depredation is an approximation, driven by the tax code. Hence, we believe that the CapEx reserve is an improvement over the traditional FF&E depredation deduction. Note that we have deducted building depredation to determine A-T earnings. 18. There are many ways to derive a firm's WACC in the context of the EVA paradigm. Any of these can be substituted for the WACC derivation presented in this chapter. 19. For an expanded discussion, see Fisher, Jeffrey D., and Robert S. Martin, Income Properly Valuation (Chicago, Dl.: Dearborn Financial Publishing, 1994), chapter 6. 20. It is extraordinarily difficult to distinguish between excess cash flows due to superior management and excess cash flows due to superior real estate such as a location advantage or an exceptional facility. This opens a real debate as to whether to attribute the excess cash flow to real property or to intangible personal property. 21. The 2006 (year 3) CapEx reserve of $652,000, discounted for two years at 3 percent equals approximately$615,000.


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# How much is owed?

#### moredard

There is an $85.00 recovery amount... For the month of March,$400.00 was charged at 25% - $85.00 =$15.00... However, rather than doing that; the following was done: $400.00 - 85 =$315.00 x .25 = $78.75 … How much was overpaid or how much is owed? #### Jomo ##### Elite Member Before I help with this problem you first have to right the math correctly. 25% -$85.00 = $15.00 makes no sense at all! You can subtract dollars from a percent.$400.00 - 85 = $315.00 x .25 =$78.75. That first equal sign is not valid, unless $315 =$78.15! Yes, $400-$85 equals $315 BUT you wrote$400.00 - 85 = $315.00 x .25! Note that the left hand side of the equal sign is$315 while the right hand side of the equal sign is $78.75 Please clean this up. And remember that ALL equal signs MUST be valid. #### moredard ##### New member So$85.00 was to be recovered from a past overpayment...

The initial billing was $400.00 and 25% to be paid of that$400.00 minus the $85.00 that should be recovered to give a total of$15.00

$400.00 x .25 =$100.00 - $85.00 =$15.00

However, rather than doing that this was done:

$400.00 - 85 =$315.00 (took the "overpaid amount from the grand total billed initially)

$315.00 x .25 =$78.75

How much was overpaid?

I hope that makes a bit more sense.

#### Jomo

##### Elite Member
The initial billing was $400.00 and 25% to be paid of that$400.00 minus the $85.00 that should be recovered to give a total of$15.00

$400.00 x .25 =$100.00 - $85.00 =$15.00
It is true that $400.00 x .25 =$100.00, BUT you did not write that. You wrote $400.00 x .25 =$100.00 - $85.00. That equation is NOT correct, do you see why? The lhs equals 100, while the rhs equals 15! #### moredard ##### New member It is true that$400.00 x .25 = $100.00, BUT you did not write that. You wrote$400.00 x .25 = $100.00 -$85.00. That equation is NOT correct, do you see why? The lhs equals 100, while the rhs equals 15!
Right;

$400.00 x .25 =$100.00

$100.00 -$85.00 = $15.00$15.00 is what should have been paid correctly.

BUT instead the payment was a such:

$400.00 - 85 =$315.00 (took the "overpaid amount from the grand total billed initially)

$315.00 x .25 =$78.75

How much was overpaid?

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# Math Help - related rates of change

1. ## related rates of change

2. What are the formulas for the surface area A and volume V of a sphere with radius r?

You are given that dA/dt = 8. You are asked to find dV/dt when A = 60.

Differentiate the area formula, plug in the known value, and solve for dr/dt.

Differentiate the volume formula, plug in the known values, and solve for dV/dt.


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# Design the diode circuit

1. Feb 7, 2015

### Zondrina

1. The problem statement, all variables and given/known data

Design the circuit in the following figure so that $V_o = 3V$ when $I_L = 0$ and $V_o$ changes by $20mV$ per $1mA$ of load current.

(a) Use the small-signal model of the diode to find the value of $R$.
(b) Specify the value of $I_S$ (saturation current) of each of the diodes.

2. Relevant equations

3. The attempt at a solution

I don't understand how part (a) is to be accomplished. Assuming a diode current $I_D$ flows from top to bottom, the current can be found from:

$$I_D = \frac{15V - 3V}{R}$$

The small signal resistance is given by:

$$r_d = \frac{V_T}{I_D}$$

The answer is listed as $2.4k$, but I am unsure how to proceed.

Last edited: Feb 7, 2015
2. Feb 7, 2015

### Staff: Mentor

Okay, so you have an expression relating the operating point current $I_D$ and the resistance R. You should also be able to determine what the diode voltage $V_D$ has to be in order for the output to be at 3 V under no load. (It looks pretty high for a silicon diode that must be passing a fairly modest current, so we can anticipate that the diode saturation current is going to be pretty darn small).

Next I think you'll want to turn to the small signal model. Maybe write a nodal equation at the Vout node. The voltage there will be the Δv that results from a load current Δi. See if you can't arrange the equation to equate Δv/Δi to something involving the unknown resistances $r_d$ and R since you're given some information about what you want Δv/Δi to be. Your expression for $r_d$ in your post will let you bridge these results back to the large signal stuff by getting rid of $r_d$ using $V_T/I_D$.

See where that takes you.

3. Feb 7, 2015

### Zondrina

If I wanted $V_o = 3V$, would I not say $V_D = 0.75V$ for each diode in series?

The small signal model consists of a source $v_s$ in series with the resistor $R$ and the small signal resistance $r_d$ (the series combination of the diodes). I don't think I can write a node equation for this series circuit, did you mean a voltage divider?

4. Feb 7, 2015

### Staff: Mentor

Yup.
The only source in the small signal model will be the load current. The 15 V DC source is suppressed, and it never had an AC signal component.

5. Feb 8, 2015

### Zondrina

Ah, short circuit the voltage and open circuit the current, sorry it's pretty early. So the DC analysis has already been done.

The small signal model has a resistor $R$ in series with $r_d$ and a common ground.

Since $I_L = 0$, the voltage at the node above $r_d$ is $V_o = 3V$. Does that mean the current across $r_d$ is $i_d = \frac{V_o}{r_d}$?

I'm still a bit lost, i'll try after getting some sleep.

6. Feb 8, 2015

### Staff: Mentor

In the small signal model the Vo = 3 V disappears. That's a DC operating point condition and doesn't enter into the small signal model. The load current is of interest in the small signal model though, it is the "signal" in the small signal model. So what you're left with is the resistance R, the diode small signal resistances rd, and a load current as the signal. I know it seems strange to have the load as a signal, but it's what is varying according to the problem. Here's what we're looking at:

7. Feb 8, 2015

### Zondrina

I thought about replacing the diodes with $4 r_d$ in the model before I slept, but I didn't know if it had any merit. So it turns out the load current IS the signal in the small signal model.

So writing a nodal equation at the only node in the circuit:

$$I_R - I_{r_d} - \Delta i = 0 \Rightarrow \frac{0 - \Delta v}{R} - \frac{\Delta v - 0}{4 r_d} = \Delta i \Rightarrow - \frac{\Delta v}{R} - \frac{\Delta v}{4 r_d} = \Delta i$$

So we can conlcude:

$$\Delta v \left[- \frac{1}{R} - \frac{1}{4 r_d} \right] = \Delta i \Rightarrow \frac{\Delta v}{\Delta i} = - \frac{1}{\left[ \frac{1}{R} + \frac{1}{4 r_d} \right]} \Rightarrow \frac{\Delta v}{\Delta i} = - \frac{4 R r_d}{R + 4r_d}$$

Substituting in for $r_d = \frac{V_T}{I_D} = \frac{V_T}{\frac{12 V}{R}} = \frac{(25 \times 10^{-3} V)R}{12 V} = 0.002083 R$:

$$\frac{\Delta v}{\Delta i} = - \frac{4 R (0.002083 R)}{R + 4(0.002083 R)} = - 0.00826315 R$$

So we know $\frac{\Delta v}{\Delta i} = \frac{20 mV}{1 mA} = 20 \Omega$, hence:

$$R = \frac{20 \Omega}{0.00826315} = 2420.4 \Omega = 2.42 k \Omega$$

The only thing bothering me at this point is why there's a negative sign, which I had to ignore (since $|R| \geq 0$). Is this relevant?

Sleeeep <3

EDIT: I forgot to mention the saturation current:

$$I_S = \frac{I_D}{e^{\frac{V_D}{V_T}}} = \frac{\frac{12 V}{2.42 \times 10^3 \Omega}}{e^{\frac{0.75 V}{25 \times 10^{-3} V}}} = 4.64 \times 10^{-16} A$$

Last edited: Feb 8, 2015
8. Feb 8, 2015

### Staff: Mentor

Well done! Just a little niggle:

Do a little thinking about Δv/Δvi. The problem statement says that Vout changes by 20 mV per 1 mA of load current. It doesn't say it which direction it changes. Reflect on what should happen to Vout as the load draws more current.

9. Feb 8, 2015

### Zondrina

Ah I see, so as the load current $I_L$ increases by $1mA$ at a time, the output voltage $V_o$ is going to decrease. Hence why the negative sign was there.

10. Feb 8, 2015

Yes indeed!


text:
## Forum archive 2000-2006

### Thomas Hagedorn - Problem Library, redux

by Arnold Pizer -
Number of replies: 0
 Problem Library, redux topic started 7/21/2004; 5:52:29 PMlast post 7/23/2004; 3:32:51 PM
 Thomas Hagedorn - Problem Library, redux  7/21/2004; 5:52:29 PM (reads: 1114, responses: 4) Hi, We're *still* having difficulty getting any problems to show up in our Webwork2 system. We're using the version of Webwork2 from 7/13/04, so almost the last version before the release last week. We are still using gdbm and not mySQL. We have global.conf version 1.91 and we have changed the command $coursefiles{problibs}=""; to$coursefiles{problibs}= {rochester => "Rochester"} in global.conf and added it to course.conf for the course test_course. We have put a link in /local/webwork2/courses/test_course/templates named rochester which points to the problems which are stored in the directory /local/webwork2/templates using the command ln -s /path/to/templates rochester (ln -s /path/to/local/webwork2/templates rochester doesn't help either) Finally, all the problems (sorted into directories) are in the directory /local/webwork2/templates It looks like we have all our ducks in a row, but still we don't see any *mathematical* problems when using the "Library Browser" command. Any ideas on what we should do next? Thanks, Tom <| Post or View Comments |>

 Michael Gage - Re: Problem Library, redux  7/21/2004; 9:17:38 PM (reads: 1326, responses: 0) Let's check the link first. In the directory /local/webwork2/courses/test_course/templates use this command ln -s /local/webwork2/templates rochester To check this do cd rochester you should arrive in the directory for the rochester problems. In the Library Browser, a button labeled "Rochester" should appear on the second line. The first line has the buttons Problem Library, "local Problems" and "From this course". Click on "select a problem collection" and you should see many entries of the form "rochester/setAlgebra.... etc. If this is not working, reset the variable in global.conf \$coursefiles{problibs}= true; Now those entries "rochester/setAlgebra..." should appear under the "local Problems" (press "select a problem collection" button to see them) The Problem Library button won't work until you get mysql installed. The other thing to check is to make sure that the rochester problem library directory can be read by the webserver. Hope this helps. Let us know how it goes. -- Mike <| Post or View Comments |>

 Thomas Hagedorn - Re: Problem Library, redux  7/22/2004; 1:07:12 AM (reads: 1298, responses: 0) Thanks Mike, I can now see the directory of problems. It was in fact the link that wasn't working. I'm pretty mystified as I thought I had used your ln -s /local/webwork2/templates rochester command earlier today and it too didn't work. But it did the trick and now the "Rochester" button shows up when you use the "Library Browser" command. I can see the individual problem sets, but alas, not the problems as it looks like we have a permission problem with webwork not being able to write to /local/webwork2/logs /local/webwork2/tmp I'll have our root user change those permissions tomorrow and hopefully, we'll be able to see the problems. Thanks again, -Tom <| Post or View Comments |>

 Michael Gage - Re: Problem Library, redux  7/22/2004; 8:07:23 AM (reads: 1306, responses: 0) It's possible that you may need to create those directories as well. If so, let us know on the list so we can add that instruction to the setup. (We could also make it automatic, but we are trying to be a bit conservative about the disk access available to the server.) The server needs read and write privileges inside those directories. --Mike <| Post or View Comments |>

 Thomas Hagedorn - Re: Problem Library, redux  7/23/2004; 3:32:51 PM (reads: 1314, responses: 0) Changing the permissions solved the problem. The problem was that the system administrator set up some directories as "root" and webwork couldn't write to them. But now it's working and we can see problems. Thanks, Tom <| Post or View Comments |>


text:
# zbMATH — the first resource for mathematics

## Dambrine, Michel

Compute Distance To:
 Author ID: dambrine.michel Published as: Dambrine, M.; Dambrine, Michel External Links: ORCID
 Documents Indexed: 38 Publications since 1993
all top 5

#### Co-Authors

 0 single-authored 15 Richard, Jean-Pierre 5 Dequidt, Antoine 4 Dang, Quoc Viet 4 Lauber, Jimmy 4 Vermeiren, Laurent 3 Belkoura, Lotfi 3 Fridman, Emilia 3 Gouaisbaut, Frédéric 3 Goubet-Bartholoméüs, A. 3 Yeganefar, Nima 2 Blažič, Sašo 2 Borne, Pierre 2 Guechi, El-Hadi 2 Klančar, Gregor 2 Mondié, Sabine 2 Nguyen, Anh-Tu 2 Orlov, Yury V. 2 Perruquetti, Wilfrid 2 Roose, Dirk 2 Tchangani, Ayeley Philippe 1 Engelborghs, Koen 1 Guerra, Thierry Marie 1 Kolmanovskii, Vladimir Borisovich 1 Michiels, Wim 1 Moulay, Emmanuel 1 Orlov, Yuri V. 1 Pepe, Pierdomenico 1 Santos, Omar 1 Tanaka, Kazuo 1 Tchangani, P. A.
all top 5

#### Serials

 3 IEEE Transactions on Automatic Control 3 Mathematics and Computers in Simulation 3 Systems & Control Letters 2 International Journal of Systems Science 2 Automatica 2 IMA Journal of Mathematical Control and Information 2 Dynamic Systems and Applications 1 Journal of the Franklin Institute 1 Fuzzy Sets and Systems 1 Kybernetika 1 Journal of Intelligent & Robotic Systems 1 International Journal of Robust and Nonlinear Control 1 Mathematical Problems in Engineering 1 Asian Journal of Control 1 Nonlinear Analysis. Theory, Methods & Applications
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#### Fields

 26 Systems theory; control (93-XX) 12 Ordinary differential equations (34-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Difference and functional equations (39-XX) 1 Computer science (68-XX) 1 Operations research, mathematical programming (90-XX)

#### Citations contained in zbMATH

32 Publications have been cited 431 times in 375 Documents Cited by Year
Finite-time stability and stabilization of time-delay systems. Zbl 1140.93447
Moulay, Emmanuel; Dambrine, Michel; Yeganefar, Nima; Perruquetti, Wilfrid
2008
Control under quantization, saturation and delay: an LMI approach. Zbl 1179.93089
Fridman, Emilia; Dambrine, Michel
2009
Robust control of delay systems: a sliding mode control design via LMI. Zbl 0994.93004
Gouaisbaut, F.; Dambrine, M.; Richard, J. P.
2002
Limitations of a class of stabilization methods for delay systems. Zbl 1056.93607
Engelborghs, K.; Dambrine, M.; Roose, D.
2001
Stability of perturbed systems with time-varying delays. Zbl 0901.93047
Goubet-Bartholoméüs, A.; Dambrine, M.; Richard, J. P.
1997
On input-to-state stability of systems with time-delay: a matrix inequalities approach. Zbl 1153.93502
Fridman, Emilia; Dambrine, Michel; Yeganefar, Nima
2008
Input-to-state stability of time-delay systems: a link with exponential stability. Zbl 1367.93284
Yeganefar, Nima; Pepe, Pierdomenico; Dambrine, Michel
2008
On second order shape optimization methods for electrical impedance tomography. Zbl 1161.49038
Afraites, L.; Dambrine, M.; Kateb, D.
2008
On identifiability of linear time-delay systems. Zbl 1364.93167
Orlov, Y.; Belkoura, L.; Richard, J. P.; Dambrine, M.
2002
Adaptive identification of linear time-delay systems. Zbl 1039.93013
Orlov, Y.; Belkoura, L.; Richard, J. P.; Dambrine, M.
2003
An extremal eigenvalue problem for the Wentzell-Laplace operator. Zbl 1347.35186
Dambrine, M.; Kateb, D.; Lamboley, J.
2016
Sliding mode control of systems with time-varying delays via descriptor approach. Zbl 1052.93011
Fridman, E.; Gouaisbaut, F.; Dambrine, M.; Richard, J.-P.
2003
Approximation of control laws with distributed delays: a necessary condition for stability. Zbl 1265.93148
Mondié, Sabine; Dambrine, Michel; Santos, Omar
2002
Vector Lyapunov functions: Nonlinear, time-varying, ordinary and functional differential equations. Zbl 1039.34066
Borne, P.; Dambrine, M.; Perruquetti, W.; Richard, J. P.
2003
On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems. Zbl 1207.49055
Dambrine, M.; Kateb, D.
2011
The effect of approximating distributed delay control laws on stability. Zbl 1134.93392
Michiels, Wim; Mondié, Sabine; Roose, Dirk; Dambrine, Michel
2004
Feedback control of time-delay systems with bounded control and state. Zbl 0918.93040
Dambrine, M.; Richard, J. P.; Borne, P.
1995
Stability and stability domains analysis for nonlinear differential- difference equations. Zbl 0807.34089
Dambrine, M.; Richard, J. P.
1994
Stability analysis of time-delay systems. Zbl 0797.34076
Dambrine, M.; Richard, J. P.
1993
Stability, attraction domains, and ultimate boundedness for nonlinear neutral systems. Zbl 1017.34503
Tchangani, A. P.; Dambrine, M.; Richard, J. P.
1998
Stability of nonlinear differential equations with distributed delay. Zbl 0947.34065
Tchangani, A. P.; Dambrine, M.; Richard, J. P.; Kolmanovskii, V. B.
1998
Stability in shape optimization with second variation. Zbl 1416.49046
Dambrine, M.; Lamboley, J.
2019
Nonlinear delay systems: Tools for a quantitative approach to stabilization. Zbl 0918.93041
Richard, J. P.; Goubet-Bartholomeüs, A.; Tchangani, P. A.; Dambrine, M.
1998
Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems. Zbl 1364.93422
Nguyen, Anh-Tu; Tanaka, Kazuo; Dequidt, Antoine; Dambrine, Michel
2017
Robust stabilizing controller design for Takagi-Sugeno fuzzy descriptor systems under state constraints and actuator saturation. Zbl 1381.93060
Dang, Quoc Viet; Vermeiren, Laurent; Dequidt, Antoine; Dambrine, Michel
2017
Simultaneous design of parallel distributed output feedback and anti-windup compensators for constrained Takagi-Sugeno fuzzy systems. Zbl 1347.93166
Nguyen, Anh-Tu; Dambrine, Michel; Lauber, Jimmy
2016
Air-fuel ratio control in a gasoline engine. Zbl 1207.37053
Lauber, J.; Guerra, T. M.; Dambrine, M.
2011
Identifiability and identification of linear systems with delays. Zbl 1063.34074
Belkoura, Lotfi; Dambrine, Michel; Orlov, Yuri; Richard, Jean-Pierre
2004
Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations. Zbl 1252.35136
Dambrine, M.; Kateb, D.
2012
Design and control of force feedback haptic systems with time delay. Zbl 1275.93051
Dang, Quoc Viet; Dequidt, Antoine; Vermeiren, Laurent; Dambrine, Michel
2014
A remark on precomposition on $$\mathrm H^{1/2}(S^1)$$ and $$\varepsilon$$-identifiability of disks in tomography. Zbl 1293.35363
Dambrine, M.; Kateb, D.
2008
On stability analysis in shape optimization: critical shapes for Neumann problem. Zbl 1126.49034
Dambrine, M.; Sokołowski, J.; Żochowski, A.
2003
Stability in shape optimization with second variation. Zbl 1416.49046
Dambrine, M.; Lamboley, J.
2019
Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems. Zbl 1364.93422
Nguyen, Anh-Tu; Tanaka, Kazuo; Dequidt, Antoine; Dambrine, Michel
2017
Robust stabilizing controller design for Takagi-Sugeno fuzzy descriptor systems under state constraints and actuator saturation. Zbl 1381.93060
Dang, Quoc Viet; Vermeiren, Laurent; Dequidt, Antoine; Dambrine, Michel
2017
An extremal eigenvalue problem for the Wentzell-Laplace operator. Zbl 1347.35186
Dambrine, M.; Kateb, D.; Lamboley, J.
2016
Simultaneous design of parallel distributed output feedback and anti-windup compensators for constrained Takagi-Sugeno fuzzy systems. Zbl 1347.93166
Nguyen, Anh-Tu; Dambrine, Michel; Lauber, Jimmy
2016
Design and control of force feedback haptic systems with time delay. Zbl 1275.93051
Dang, Quoc Viet; Dequidt, Antoine; Vermeiren, Laurent; Dambrine, Michel
2014
Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations. Zbl 1252.35136
Dambrine, M.; Kateb, D.
2012
On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems. Zbl 1207.49055
Dambrine, M.; Kateb, D.
2011
Air-fuel ratio control in a gasoline engine. Zbl 1207.37053
Lauber, J.; Guerra, T. M.; Dambrine, M.
2011
Control under quantization, saturation and delay: an LMI approach. Zbl 1179.93089
Fridman, Emilia; Dambrine, Michel
2009
Finite-time stability and stabilization of time-delay systems. Zbl 1140.93447
Moulay, Emmanuel; Dambrine, Michel; Yeganefar, Nima; Perruquetti, Wilfrid
2008
On input-to-state stability of systems with time-delay: a matrix inequalities approach. Zbl 1153.93502
Fridman, Emilia; Dambrine, Michel; Yeganefar, Nima
2008
Input-to-state stability of time-delay systems: a link with exponential stability. Zbl 1367.93284
Yeganefar, Nima; Pepe, Pierdomenico; Dambrine, Michel
2008
On second order shape optimization methods for electrical impedance tomography. Zbl 1161.49038
Afraites, L.; Dambrine, M.; Kateb, D.
2008
A remark on precomposition on $$\mathrm H^{1/2}(S^1)$$ and $$\varepsilon$$-identifiability of disks in tomography. Zbl 1293.35363
Dambrine, M.; Kateb, D.
2008
The effect of approximating distributed delay control laws on stability. Zbl 1134.93392
Michiels, Wim; Mondié, Sabine; Roose, Dirk; Dambrine, Michel
2004
Identifiability and identification of linear systems with delays. Zbl 1063.34074
Belkoura, Lotfi; Dambrine, Michel; Orlov, Yuri; Richard, Jean-Pierre
2004
Adaptive identification of linear time-delay systems. Zbl 1039.93013
Orlov, Y.; Belkoura, L.; Richard, J. P.; Dambrine, M.
2003
Sliding mode control of systems with time-varying delays via descriptor approach. Zbl 1052.93011
Fridman, E.; Gouaisbaut, F.; Dambrine, M.; Richard, J.-P.
2003
Vector Lyapunov functions: Nonlinear, time-varying, ordinary and functional differential equations. Zbl 1039.34066
Borne, P.; Dambrine, M.; Perruquetti, W.; Richard, J. P.
2003
On stability analysis in shape optimization: critical shapes for Neumann problem. Zbl 1126.49034
Dambrine, M.; Sokołowski, J.; Żochowski, A.
2003
Robust control of delay systems: a sliding mode control design via LMI. Zbl 0994.93004
Gouaisbaut, F.; Dambrine, M.; Richard, J. P.
2002
On identifiability of linear time-delay systems. Zbl 1364.93167
Orlov, Y.; Belkoura, L.; Richard, J. P.; Dambrine, M.
2002
Approximation of control laws with distributed delays: a necessary condition for stability. Zbl 1265.93148
Mondié, Sabine; Dambrine, Michel; Santos, Omar
2002
Limitations of a class of stabilization methods for delay systems. Zbl 1056.93607
Engelborghs, K.; Dambrine, M.; Roose, D.
2001
Stability, attraction domains, and ultimate boundedness for nonlinear neutral systems. Zbl 1017.34503
Tchangani, A. P.; Dambrine, M.; Richard, J. P.
1998
Stability of nonlinear differential equations with distributed delay. Zbl 0947.34065
Tchangani, A. P.; Dambrine, M.; Richard, J. P.; Kolmanovskii, V. B.
1998
Nonlinear delay systems: Tools for a quantitative approach to stabilization. Zbl 0918.93041
Richard, J. P.; Goubet-Bartholomeüs, A.; Tchangani, P. A.; Dambrine, M.
1998
Stability of perturbed systems with time-varying delays. Zbl 0901.93047
Goubet-Bartholoméüs, A.; Dambrine, M.; Richard, J. P.
1997
Feedback control of time-delay systems with bounded control and state. Zbl 0918.93040
Dambrine, M.; Richard, J. P.; Borne, P.
1995
Stability and stability domains analysis for nonlinear differential- difference equations. Zbl 0807.34089
Dambrine, M.; Richard, J. P.
1994
Stability analysis of time-delay systems. Zbl 0797.34076
Dambrine, M.; Richard, J. P.
1993
all top 5

#### Cited by 685 Authors

 19 Richard, Jean-Pierre 17 Fridman, Emilia 13 Efimov, Denis V. 11 Dambrine, Marc 11 Dambrine, Michel 10 Pepe, Pierdomenico 10 Perruquetti, Wilfrid 8 Pham Huu Anh Ngoc 7 Polyakov, Andrey 7 Spurgeon, Sarah K. 6 Cao, Jinde 6 Mazenc, Frédéric 6 Niculescu, Silviu-Iulian 6 Trinh, Hieu Minh 6 Yang, Renming 5 Gouaisbaut, Frédéric 5 Han, Qinglong 5 Harbrecht, Helmut 5 Karafyllis, Iasson 5 Kharitonov, Vladimir Leonidovich 5 Li, Xiaodi 5 Mondié, Sabine 5 Nam, Phan Thanh 5 Pathirana, Pubudu Nishantha 5 Vũ Ngoc Phát 4 Han, Xiaoran 4 Ho, Daniel W. C. 4 Kao, Chiu-Yen 4 Nguyen, Anh-Tu 4 Yan, Xinggang 4 Yang, Xinsong 4 Zhang, Qingling 3 Bogoşel, Beniamin 3 Caubet, Fabien 3 Cavallina, Lorenzo 3 Choi, Ho-Lim 3 Cui, Baotong 3 Garcia, Pedro 3 Lam, James 3 Melchor-Aguilar, Daniel 3 Michiels, Wim 3 Mobayen, Saleh 3 Niamsup, Piyapong 3 Niu, Yugang 3 Su, Hongye 3 Sun, Ximing 3 Verriest, Erik I. 3 Wang, Yuzhen 3 Wang, Zidong 3 Xia, Changyu 3 Xu, Jian 3 Yang, Ying 3 Zheng, Gang 2 Albertos, Pedro 2 Alhejaili, Weaam 2 Antic, Dragan S. 2 Chen, Guopei 2 Chu, Jian 2 Debeljkovic, Dragutin Lj. 2 Deng, Zhenghong 2 Dequidt, Antoine 2 Ding, Zhengtao 2 Dion, Jean-Michel 2 Du, Feng 2 Dugard, Luc 2 Eppler, Karsten 2 Fiagbedzi, Yawvi A. 2 Fridman, Ella 2 Fridman, Leonid M. 2 Furtat, Igor B. 2 Gao, Huijun 2 Gomez, Oscar 2 Goubet-Bartholoméüs, A. 2 Gu, Keqin 2 Guo, Rongwei 2 Gutman, Per-Olof 2 He, Shuping 2 He, Yong 2 Hill, David John 2 Huang, Lirong 2 Ibeas, Asier 2 Insperger, Tamás 2 Ivănescu, Dan 2 Ji, Yan 2 Jiang, Zhong-Ping 2 Kao, Yonggui 2 Karimi, Hamid Reza 2 Kateb, Dalila 2 Kateb, Djalil M. 2 Kolmanovsky, Ilya V. 2 Koo, Min-Sung 2 Krstić, Miroslav 2 Lamboley, Jimmy 2 Laurain, Antoine 2 Li, Jianning 2 Liang, Jinling 2 Lim, Jong-Tae 2 Liu, Bin 2 Liu, Fei 2 Liu, Guoping ...and 585 more Authors
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#### Cited in 97 Serials

 69 Automatica 24 Journal of the Franklin Institute 23 Systems & Control Letters 15 Mathematical Problems in Engineering 14 International Journal of Control 11 International Journal of Systems Science 10 Applied Mathematics and Computation 9 International Journal of Robust and Nonlinear Control 9 European Journal of Control 8 Asian Journal of Control 7 Kybernetika 7 Automation and Remote Control 6 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 6 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 6 International Journal of Systems Science. Principles and Applications of Systems and Integration 5 Mathematics and Computers in Simulation 5 SIAM Journal on Control and Optimization 5 Nonlinear Dynamics 5 Abstract and Applied Analysis 4 Journal of Computational and Applied Mathematics 4 Complexity 3 Mathematical Methods in the Applied Sciences 3 Chaos, Solitons and Fractals 3 Circuits, Systems, and Signal Processing 3 Neural Networks 3 International Journal of Adaptive Control and Signal Processing 3 Journal de Mathématiques Pures et Appliquées. Neuvième Série 3 SIAM Journal on Mathematical Analysis 3 Discrete Dynamics in Nature and Society 3 Communications in Nonlinear Science and Numerical Simulation 3 Nonlinear Analysis. Real World Applications 3 Journal of Systems Science and Complexity 3 Inverse Problems in Science and Engineering 3 Nonlinear Analysis. Hybrid Systems 2 Journal of Mathematical Analysis and Applications 2 Fuzzy Sets and Systems 2 Information Sciences 2 Journal of Differential Equations 2 Applied Mathematics Letters 2 Journal of Scientific Computing 2 SIAM Journal on Applied Mathematics 2 Journal of Mathematical Sciences (New York) 2 Journal of Vibration and Control 2 Chaos 2 Journal of Control Theory and Applications 2 Journal of Control Science and Engineering 1 Applicable Analysis 1 Biological Cybernetics 1 Computers & Mathematics with Applications 1 Inverse Problems 1 Journal of Computational Physics 1 Journal of Geometry and Physics 1 Applied Mathematics and Optimization 1 Duke Mathematical Journal 1 Journal of Optimization Theory and Applications 1 Kybernetes 1 Proceedings of the American Mathematical Society 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Optimal Control Applications & Methods 1 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 Applied Numerical Mathematics 1 MCSS. Mathematics of Control, Signals, and Systems 1 Differential Geometry and its Applications 1 The Journal of Geometric Analysis 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 Applied Mathematical Modelling 1 Linear Algebra and its Applications 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Potential Analysis 1 Computational Optimization and Applications 1 International Applied Mechanics 1 Journal of Inverse and Ill-Posed Problems 1 Vietnam Journal of Mathematics 1 Soft Computing 1 Journal of Inequalities and Applications 1 Revista Matemática Complutense 1 International Journal of Applied Mathematics and Computer Science 1 Differential Equations 1 Differentsial’nye Uravneniya i Protsessy Upravleniya 1 Discrete and Continuous Dynamical Systems. Series B 1 Journal of Applied Mathematics 1 Multiscale Modeling & Simulation 1 Fuzzy Optimization and Decision Making 1 Mediterranean Journal of Mathematics 1 Advances in Difference Equations 1 Journal of Industrial and Management Optimization 1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 1 Acta Mechanica Sinica 1 Mathematical Geosciences 1 Science China. Information Sciences 1 Arabian Journal for Science and Engineering 1 Numerical Algebra, Control and Optimization 1 ISRN Applied Mathematics 1 Evolution Equations and Control Theory 1 Journal of Computational Dynamics 1 Mathematics 1 Nonlinear Analysis. Theory, Methods & Applications
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#### Cited in 29 Fields

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# Physical meaning of orbital polarization?

1. Oct 21, 2014

### phys_student1

In this paper, the Orbital Polarization (OP) is defined as:

$$OP=\frac{n_{x^2-y^2}-n_{z^2}}{n_{x^2-y^2}+n_{z^2}}$$

where $$n_i$$ is the occupancy of that given orbital. I would like to understand the physical meaning of this. Also, is there a difference between OP and Orbital Hybridization?

2. Oct 23, 2014

### Staff: Mentor

I am not an expert in the field, so do not take what I write here with any authority. It is simply my interpretation.

From what I get from the article, I take it that under normal conditions, the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals would be equally populated by electrons. The presence of strain will lift that degeneracy, and one orbital will end up with a higher electronic occupancy (probability). This is what orbital polarization "measures."

It does not seem to be related to hybridization, in that it does not consider that the orbitals themselves are modified, only their occupancy. Also, hybrid orbitals are usually constructed from equal proportions of the base orbitals.

I think you should keep in mind that orbitals are basically single electron solutions, and do not take into account the presence of other electrons in the atom, or chemical bonding. It is expected that the actual electronic wave function will be only approximated by considering that electrons occupy disctinct orbitals. Orbital polarization appears as one way to take this into account.

3. Oct 23, 2014

### phys_student1

"The presence of strain will lift that degeneracy, and one orbital will end up with a higher electronic occupancy (probability)"

Does this mean the two orbitals will have different energy? This is what first come to my mind when I see "degeneracy lifting".

4. Oct 23, 2014

### Staff: Mentor

I was paraphrasing the article:
I interpreted that as meaning that in the absence of strain, the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals would have the same energy, and thus be equally populated.

5. Oct 23, 2014

### phys_student1

There must be something changed that causes them to have different occupancy. This thing should be the energy of each orbital. Do you agree with this interpretation?

6. Oct 24, 2014

### Staff: Mentor

Sounds reasonable.

7. Oct 25, 2014

Thanks.


text:
## Counting scattering poles.(English)Zbl 0823.35139

Ikawa, Mitsuru (ed.), Spectral and scattering theory. Proceedings of the Taniguchi international workshop, held at Sanda, Hyogo, Japan. Basel: Marcel Dekker. Lect. Notes Pure Appl. Math. 161, 301-331 (1994).
This survey describes the recent results on estimating the number of scattering poles, obtained by the author and J. Sjöstrand in joint works. For compactly supported perturbations of the Laplacian in $$\mathbb{R}^ n$$, various upper bounds are obtained for the number of scattering poles inside conic-type regions of the complex plane, asymptotically as the diameter of these regions tends to infinity. Many other questions are discussed, such as the distribution of the poles near the real axis, and the obtention of lower bounds on their number. Also, several examples are discussed showing the accuracy (and sometimes the optimality) of the general results.
For the entire collection see [Zbl 0798.00016].

### MSC:

 35P25 Scattering theory for PDEs 81U20 $$S$$-matrix theory, etc. in quantum theory 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations