# Bridging the gap between discount rate theory and investor surveys.

Abstract

This article is intended to assist appraisers in reconciling the differences between the theoretical relationship of discount rates and overall capitalization rates and the published market information in investor surveys. The study explains several disparities in the methodology used to estimate terminal value and define income stream that account for much of the gap between theory and secondary market data in the form of investor surveys.

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Investor surveys reflect a differential between discount rates and overall capitalization rates that is substantially smaller than basic capitalization theory would indicate. Much of this difference between theory and the survey data can be understood by examining the estimates of terminal value and how the income stream is capitalized.

Theoretical Relationship between Discount Rates and Cap Rates

In appraisal textbooks, the relationship between discount rates and overall capitalization rates, or cap rates, is a simple one: Discount rate = cap rate + income growth rate or

[Y.sub.o] =[R.sub.o] + CR

where:

[Y.sub.o] = overall yield rate, or discount rate

[R.sub.o] = overall capitalization rate, or cap rate

CR = rate of change for income

Discount Rate

A discount rate, also known as an overall yield rate, is an interest rate or yield rate used to convert anticipated future payments or receipts into present value. The resulting present value represents the amount of capital to be invested so that the investor's expected yield equals the specified discount rate. The overall yield rate ([Y.sub.o]) is the rate of return on the total capital invested. It considers all changes in income over the investment projection period as well as the reversion at the end of the projection period. It does not, however, consider the effect of debt financing. Rather, it is calculated as if the property were purchased with no debt capital. It is calculated in the same way the internal rate of return is calculated. (1)

Overall Capitalization Rate

An overall capitalization rate ([R.sub.o]), also known as an overall rate, a cap rate, or a going-in cap rate, is an income rate for a total real property interest that reflects the relationship between a single year's net operating income expectancy and the total property price or value. It is used to convert net operating income into an indication of overall property value. (2) The equation used to reflect the overall capitalization rate relationship to income and value is [R.sub.o] = [I.sub.o]/[V.sub.o]. It is considered the combination of a return of investment and a return on investment.

Income Growth Rate

The income growth rate (CR) is the constant rate of growth of net operating income on an annual basis. This requires consideration of both revenue and expense growth rates over the holding period. However, the assumptions necessary to support this relationship are often not well understood.

The necessary assumptions built into the theory of "discount rate equals overall capitalization rate plus income growth" are as follows:

* The income growth rate must be constant throughout the holding period.

* The income stream is net operating income only, i.e., without other cash flows typically reflected in discounted cash flow analysis, such as vacancy/lease turnover costs, leasing commissions, tenant improvement allowances, and replacement reserves.

* Terminal value is estimated using a terminal cap rate equal to the overall capitalization rate. This terminal value assumption applies only to the simplified model discussed earlier. In the real-world marketplace (as reflected in published investor surveys), a terminal cap rate is equal to the overall capitalization rate plus a risk increment for the greater risk of estimating cash flows several years into the future.

* Terminal value is estimated without deduction of sale commissions or closing costs.

Each of these assumptions is built into the analysis presented later in Table 1.

Questions Raised by Investor Surveys

A review of investor surveys frequently shows that most investors forecast future revenue and expense growth for many properties that would support a 3% annual income growth rate. With that type of forecast, it would be reasonable for investor surveys to reflect a roughly 3% differential between the discount rate and the going-in cap rate for these same property categories. However the real differential between discount rate and overall capitalization rate in investor surveys seems to be in the 1% to 2% range rather than the approximately 3% range that theory would indicate. The question is, why the difference? This study provides some significant insight into that question, but cannot resolve every difference precisely. There will always be some unexplained differences between appraisal theory and the real mechanics of the marketplace. Nevertheless, unlike previous works cited, this article presents tools to quantify the major differences between theory and investor surveys.

Comparing Cash Flow Differences

The theoretical discount rate just discussed assumes that the cash flow is net operating income only. In contrast, the discount rate considered in market surveys for office, retail, industrial, and apartment properties typically assumes a significantly different income stream. The discount rate reflected in published investor surveys, such as the Korpacz Real Estate Investor Survey and the Real Estate Research Corporation (RERC) Real Estate Report, is applied to net operating income minus the following cash flow items: (1) periodic costs that occur when leases are renewed or new leases are executed, (2) leasing commissions, (3) tenant improvement allowances, and (4) replacement reserves. To account for these additional cash flows, the income stream being discounted is reduced. To achieve the same value for the same property, a reduced discount rate must be applied to the lower income stream from cash flow rather than net operating income. Because investor surveys are primarily intended for sophisticated investors, the additional cash flow items mentioned here are not explicitly spelled out as being part of the income stream discounted for office and retail properties. However, most of these items are regularly addressed as estimates within such surveys as the Korpacz Real Estate Investor Survey. The analytical problem is made more complex because the definition of net operating income is not universal among investors surveyed. In the table "Income Capitalized in Direct Capitalization" that regularly is presented within the Korpacz Real Estate Investor Survey, approximately 70% or more of investors use a definition exclusive of leasing commissions, tenant improvement allowances, replacement reserves, and lease turnover vacancy.

This type of logic applies to retail, office, industrial, and apartment properties. For skilled-nursing, assisted-living, and hotel properties, these cash flow elements do not cause a difference. Leasing commissions and tenant improvement allowances are not applicable to these property types. Replacement reserves are already fully accounted for in the income streams used for both the overall capitalization rate and the discount rate.

Comparing Terminal Value Estimate Differences

In the theoretical model for the discount rate (overall capitalization rate plus income growth rate), the terminal value estimate is highly simplified. The terminal value is simply estimated by calculating terminal year sale price, which is done by dividing the terminal year net operating income by the terminal cap rate. The terminal cap rate is equal to the overall capitalization rate. As discussed earlier, having the terminal cap rate equal to the overall capitalization rate is not consistent with how the marketplace typically views the relative risk of current cash flow versus future cash flow. Greater risk is generally associated with estimating cash flows several years into the future. The impact of that difference should be considered and, in fact, is considered in investor surveys.

In the methodology reflected in investor surveys, two changes are made that better represent what is done in the marketplace: (1) the sale proceeds are reduced by closing costs (commission and escrow costs) often in the 5% to 7% range, and (2) the terminal cap rate is adjusted upward for the additional risk of estimating cash flows eleven years into the future.

Quantifying the Differences between Theory and Market Practices

Common Parameters

For the analyses presented in Tables 1-3, the assumed subject property is an office building with a market value of \$10,000,000. In all three tables, the net operating income is the same: \$700,000 or 7.0% of market value, where the market overall capitalization rate (going-in cap rate) is 7.0%. Each analysis assumes stabilized occupancy. The holding period is ten years, which is consistent with what current investor surveys indicate about the typical holding for most property types. The differential between the estimated discount rate and the overall capitalization rate is determined by simply subtracting the overall capitalization rate from the estimated discount rate.

The tables presented in this analysis are for illustrative purposes only. The variables should be changed to reflect actual market conditions and property type.

Simplified Framework

Table 1 shows the theoretical model for determining the discount rate, using a simplified method of estimating terminal value.

Discount Rate Change

As changes are made to the simplified method of estimating terminal value in Table 1, the discount rate must change to generate the same market value. This is because the subject property, the market overall capitalization rate, and the Year I net operating income that it generates are unchanged. The changes are illustrated in Table 2.

Table 3 builds on Table 2. In Table 2, cash flow rather than net operating income is capitalized. As changes are also made to the income stream to be capitalized (brought to a present value), the discount rate must change to generate the same market value. Again, this is because the subject property, the market overall capitalization rate, and the Year 1 net operating income that it generates are unchanged. This is illustrated in Table 3.

In Tables 2 and 3, the estimated discount rate is determined by calculating the internal rate of return (IRR) of the cash flows, with a market value of \$10,000,000 as the Year Zero negative cash flow.

Results

Table 1 shows the discount rate derivation from the theoretical model of the discounting process. The result is a 7.0% overall capitalization rate, an estimated 10.0% discount rate, and the expected differential of 5.0% (the annual income growth rate) between the discount rate and the overall capitalization rate. The primary assumptions for the analysis shown in Table 1 include (1) a market value of \$10,000,000; (2) a market overall capitalization rate of 7.0%; (5) an annual income growth rate of 3.0%; and (4) a terminal cap rate of 7.0%, the same as the overall capitalization rate or going-in cap rate.

In Table 2, the impact of differences in estimating terminal value is shown. The analysis supports a 2.04% differential between the estimated 9.04% discount rate and the 7.0% going-in cap rate rather than the 5.0% theoretical difference. The primary assumptions for the analysis shown in Table 2 are the same as those for Table 1 with the following exceptions:

* The terminal cap rate is 7.5%, 0.5% higher than the overall capitalization rate, to account for the higher risk of forecasting terminal cash flow eleven years into the future. In contrast, a going-in cap rate reflects forecasting in the near term.

* Commission and closing costs are deducted from the terminal year sale price in order to reflect a more realistic picture of sale proceeds typically used by market participants.

The impact of these changes to terminal value estimate methodology is far less significant if the appraiser is working on a long-term analysis because the present value factor at the end of a twenty-, thirty-, or forty-year holding period is minimal. Such long holding periods can be necessary to analyze leased fee interests on long-term leases. Ground leases would be one such example.

The reverse is also true. At shorter holding periods, i.e., five to seven years, the impact of these changes to the terminal value estimate methodology is far more significant. The present value factor at the end of a five- or seven-year holding period is much larger than the present value factor at the end of the assumed ten-year holding period. A change to a seven-year analysis with the same assumptions would reduce the estimated difference between discount rate and overall capitalization rate by approximately an additional 0.5%.

The 0.5% risk increment for the terminal cap rate is somewhat typical of what is seen in investor surveys. However these can vary greatly depending on how investors see the risk of future cash flows several years into the future. During weak economic cycles, the distant future looks more positive, and the risk increment for the terminal cap rate is reduced, bringing it closer to zero. During strong economic cycles, the long-term future is less certain, and the risk increment for the terminal cap rate is increased. The risk increment also increases with the length of the holding period.

Table 3 shows the impact of the differences in the terminal value estimate and the inclusion of leasing commissions, tenant improvement allowances, and replacement reserves in determining cash flow. The analysis supports a 1.31% differential between the estimated 8.31% discount rate and the 7.0% going-in cap rate rather than the 3.0% theoretical difference. The primary assumptions for the analysis shown in Table 3 include the first three assumptions for Tables 1 and 2, namely (1) a market value of \$10,000,000; (2) a market overall capitalization rate of 7.0%; and (3) an annual income growth rate of 3.0%. The assumptions specific to Table 2 also apply to Table 3, namely (4) the terminal cap rate is 0.5% higher than the overall capitalization rate, and (5) commission and closing costs are deducted from the terminal year sale price.

In addition, Table 3 has its own set of assumptions that differ from those in Table 2. There are now cash flow line items included in the income stream (cash flow) to be capitalized:

* Leasing commissions are at 5% of effective gross income. Net operating income is assumed to be 35% of effective gross income. The turnover of leases is estimated to be 15% annually, assuming that an average lease ranges from six to eight years.

* A tenant improvement allowance is set at \$7.00 per square foot, assuming the building is stabilized and the improvements are second-generation tenant improvements. Again, the turnover of leases is estimated to be 15% annually, assuming an average lease range of six to eight years.

* Replacement reserves are set at \$0.20 per square foot per year.

* Expenses increase at 3.0% per year, which applies to both the tenant improvement allowance and the replacement reserves.

* The subject property is a 40,O00-square-foot office building.

Limitations of the Analysis

This analysis is primarily intended to improve appraisers' understanding of the conceptual basis of major published investor surveys, such as the Korpacz Real Estate Investor Survey and the RERC Real Estate Report. However, it does not attempt to address other differences that may be built into the methodology of deriving discounted cash flow reflected in investor surveys, specifically turnover vacancy, the impact of forecasting rent spikes, and revenue and expense growth rates that are substantially different.

Although not a focus of this article, it is appropriate to mention the impact of differing revenue and expense growth rates. In general, when expected revenues grow faster than expected expenses, both the overall capitalization rate and the discount rate tend to be lower. The reverse is true when expected expenses grow faster than expected revenues-both the overall capitalization rate and the discount rate are increased.

In practice, the cap rate movement is more complex and does not consistently conform to theory and modeling. As reported in the article "The Impact of Market Variation on Cap Rates," (3) both investors and appraisers are not consistently looking forward when estimating cap rates and, in fact, may be locked more into historical conditions than forward expectations. For that reason, cap rates do not fully adjust to forward income change expectations. In other words, cap rates do not behave as expected based on income growth expectations.

Other factors not addressed within this analysis that may significantly influence discount rates can include such things as tenant quality, quality and condition of the property, occupancy rate, relationship between contract and market rent, typical holding period, improving or declining markets, remaining term of existing leases, and geographical location.

Conclusion

There is a sound basis for investor surveys to reflect a differential between discount rates and overall capitalization rates that is substantially smaller than basic capitalization theory would indicate. In the marketplace, a discount rate is not simply the overall capitalization rate plus the income growth rate. The analysis presented indicates that much of the difference between theory and investor survey data can be explained by differences in estimating terminal value and defining the income stream capitalized. Perusal of this analysis should give greater confidence and understanding to the appraiser reviewing and considering investor surveys.

Sevelka, Tony. "Where the Overall Cap Rate Meets the Discount Rate." The Appraisal Journal (Spring 2004): 155-146.

Vincott, D. Richard, Kevin A. Hoover, and Terry V. Grissom. "Capitalization Rates, Discount Rates and Reasonableness" Real Estate Issues (August 1996): 11-15.

(1.) Appraisal Institute, The Appraisal of Real Estate, 13th ed. (Chicago: Appraisal Institute, 2008), 463.

(2.) Ibid., 462.

(3.) Petros Sivitanides et al., "The Impact of Market Variations on Cap Rates: Understanding the Idiosyncrasies of Appraisal-Based Pricing," Property (Winter 2002), http://www.property-mag.com/property/Winter02/ investment_fundamentals.html.

Donald Sonneman, ASA, is the president of AblePlus Valuations in Santee, California. He has an appraisal practice that includes commercial real estate appraisal, partial interest valuation, and business valuation. His real estate appraisal work has included virtually all commercial property types, including nursing homes and assisted-living facilities. Sonneman received a bachelor's degree in mechanical engineering from the Illinois Institute of Technology and a CFP (certified financial planner) degree from the Denver College for Financial Planning. He has previously authored four articles for The Appraisal Journal and contributed to the recent Appraisal Institute book Appraising industrial Properties. Contact: valexcel@lnetworld.nnt
```Table 1 Theoretical Model of Discounted Cash Flow:
Discount Rate = Overall Rate + Income Growth Rate

Net Present
Operating Value Present
Year Income Factor Value

1 \$700,000 0.9091 \$636,370
2 \$721,000 0.8264 \$595,834
3 \$742,630 0.7513 \$557,938
4 \$764,909 0.6830 \$522,433
5 \$787,856 0.6209 \$489,180
6 \$811,492 0.5645 \$458,087
7 \$835,837 0.5132 \$428,952
8 \$860,912 0.4665 \$401,615
9 \$886,739 0.4241 \$376,066
10 \$913,341 0.3855 \$352,093
Terminal year \$940,741
Present value of cash flows \$4,818,568
Terminal net operating income \$940,741
Terminal cap rate 7.0%
Future value sale proceeds \$13,439,157
Present value factor for
proceeds 0.3855
Terminal value \$0,000,005
Market value \$9,999,363
Rounded \$10,000,000
Assumptions:
Overall rate
(also known as going-in
cap rate) 7.0%
Annual income growth 3.0%
Discount rate 10.0%
Terminal cap rate
(Going-in cap rate) 7.0%
Market value \$10,000,000
Market cap rate 7.0%
Net operating income \$700,000

Table 2 Discounted Cash Flow with Changes to Terminal Value Estimate

Net Present
Operating Value Present
Year Income Factor Value

1 \$700,000 0.9091 \$636,370
2 \$721,000 0.8264 \$595,834
3 \$742,630 0.7513 \$557,938
4 \$764,909 0.6830 \$522,433
5 \$787,856 0.6209 \$489,180
6 \$811,492 0.5645 \$458,087
7 \$835,837 0.5132 \$428,952
8 \$860,912 0.4665 \$401,615
9 \$886,739 0.4241 \$376,066
10 \$913,341 0.3855 \$352,093
Terminal year \$940,741

Present value of cash flows \$4,818,568
Terminal net operating income \$940,741
Terminal cap rate 7.5%
Future sale price \$12,543,213
Less commission & closing @ 6% \$752,593
Future value sale proceeds \$11,790,620
Present value factor for
sale proceeds 0.3855
Terminal value \$4,545,284
Market value at theoretical
discount rate \$9,363,852
Rounded \$9,360,000

Theoretical discount rate 10.00%
Required discount rate 9.04%
(to generate market value
from direct capitalization equal to \$10,000,000)

Discount Cap
Rate Rate Difference

Theoretical rates 10.0% 7.0% 3.0%
Required rates 9.0% 7.0% 2.0%

Assumptions:
Going-in cap rate 7.0%
Annual income growth
(CPI equivalent) 3.0%
Theoretical discount rate 10.0%
Terminal cap rate
(Going-in cap rate plus 0.5%) 7.5%
Market value \$10,000,000
Market cap rate 7.0%
Net operating income \$700,000

Cash
Year Flows
Year 0 (\$10,000,000) Market value

1 1 \$700,000
2 2 \$721,000
3 3 \$742,630
4 4 \$764,909
5 5 \$787,856
6 6 \$811,492
7 7 \$835,837
8 8 \$860,912
9 9 \$886,739
10 10 \$12,703,961 Yr. 10 cash
Terminal year flow plus ter-
minal value
Present value of cash flows IRR 9.04%
Terminal net operating income
Terminal cap rate
Future sale price
Less commission & closing @ 6%
Future value sale proceeds
Present value factor for
sale proceeds
Terminal value
Market value at theoretical
discount rate
Rounded

Theoretical discount rate
Required discount rate
(to generate market value
from direct capitalization equal to \$10,000,000)

Theoretical rates
Required rates

Assumptions:
Going-in cap rate
Annual income growth
(CPI equivalent)
Theoretical discount rate
Terminal cap rate
(Going-in cap rate plus 0.5%)
Market value
Market cap rate
Net operating income

Table 3 Discounted Cash Flow with (1) Changes to Terminal Value
Estimate and (2) Cash Flow Items Not Included in Net Operating
Income

Not Included in Direct Cap NOI

Net Leasing Tenant Replace.
Operating Commission Improvements Reserves
Year Income 5.0% \$7.00/S.F \$0.20/S.F

1 \$700,000 \$15,450 \$43,260 \$08,240
2 \$721,000 \$16,391 \$44,558 \$08,487
3 \$742,630 \$17,389 \$45,895 \$08,742
4 \$764,909 \$18,448 \$47,271 \$09,004
5 \$787,856 \$19,572 \$48,690 \$09,274
6 \$811,492 \$20,764 \$50,150 \$09,552
7 \$835,837 \$22,028 \$51,655 \$09,839
8 \$860,912 \$23,370 \$53,204 \$010,134
9 \$886,739 \$24,793 \$54,800 \$010,438
10 \$913,341 \$26,303 \$56,444 \$010,751
Terminal year \$940,741

Present value of cash flows
Terminal net operating income
Terminal cap rate
Future sale price
Less commission & closing @ 6%
Future value sale proceeds
Present value factor for sale proceeds
Terminal value
Market value at theoretical discount rate
Rounded

Theoretical discount rate 10.00%
Required discount rate 8.31%
(to generate market value
from direct cap equal to \$10,000,000)

Discount Cap
Rate Rate Difference

Theoretical rates 10.00% 7.0% 3.00%
Required rates 8.31% 7.0% 1.31%

Assumptions:
Going-in cap rate 7.0%
Annual income growth 3.0%
Theoretical discount rate 10.0%
Terminal cap rate
(Going-in cap rate
plus 0.5%) 7.5%
Market value \$10,000,000
Market cap rate 7.0%
Net operating income \$700,000

Net operating income as
percentage of Year 1
effective gross income 35.0%
Annual expense increase 3.0%
of leases rolling over/yr. 15.0%
Tenant improvements, \$/s.f. \$7.00
Replacement reserves, \$/s.f. \$0.20
Sq. ft. of building area 40,000

Present
Cash Value Present
Year Flows Factor Value

1 \$633,050 0.9091 \$575,506
2 \$651,564 0.8264 \$538,452
3 \$670,604 0.7513 \$503,825
4 \$690,186 0.6830 \$471,397
5 \$710,320 0.6209 \$441,038
6 \$731,026 0.5645 \$412,664
7 \$752,315 0.5132 \$386,088
8 \$774,204 0.4665 \$361,166
9 \$796,708 0.4241 \$337,884
10 \$819,843 0.3855 \$316,049
Terminal year

Present value of cash flows \$4,344,069
Terminal net operating income \$940,741
Terminal cap rate 7.5%
Future sale price \$12,543,213
Less commission & closing @ 6% \$752,593
Future value sale proceeds \$11,790,620
Present value factor for sale proceeds 0.3855
Terminal value \$4,545,284
Market value at theoretical discount rate \$8,889,353
Rounded \$8,890,000

Theoretical discount rate
Required discount rate
(to generate market value
from direct cap equal to \$10,000,000)

Theoretical rates Cash
Required rates Year Flows

Assumptions: Market value 0 (\$10,000,000.00)
Going-in cap rate 1 \$633,050
Annual income growth 2 \$651,564
Theoretical discount rate 3 \$670,604
Terminal cap rate 4 \$690,186
(Going-in cap rate
plus 0.5%) 5 \$710,320
Market value \$10,000,000 6 \$731,026
Market cap rate 7 \$752,315
Net operating income 8 \$774,204
9 \$796,708
Net operating income as Yr. 10 cash 10 \$12,610,463
percentage of Year 1 flow plus
terminal value
effective gross income IRR 8.31%
Annual expense increase
of leases rolling over/yr.
Tenant improvements, \$/s.f.
Replacement reserves, \$/s.f.
Sq. ft. of building area
```