August 2015

Optimal Deal Flow Management for Direct Real Estate Investments Northfield Information Services Research

Emilian Belev, CFA Richard B. Gold

Introduction

This paper proposes an innovative practical solution to a unique problem confronting direct real estate investments. The problem is rooted in a question as old as investing itself: what to buy and when to sell in order to maximize profit while minimizing risk. The solution to this problem for illiquid investments, such as private equity real estate, however, cannot be the same as publicly traded investments since illiquid assets do not operate under the same conditions as public assets. The proposed solution is a cohesive amalgamation of the benefits of quantitative and fundamental analysis that is both theoretically rigorous and immediately practical. It also assists investors in making incremental investment decisions which are in turn transparent and intuitive to all relevant stakeholders.

Furthermore, the marriage of real estate and publicly traded assets under the parameters of Modern Portfolio Theory has historically been more “War of the Roses” than “When Harry Met Sally”. This has presented a major challenge to those wishing to include real estate on an equal footing with other asset classes in their portfolio analytics. The approach described in this paper is designed in such a way to rectify this conflict.

Background

Directly-owned commercial real estate is an essential part of the typical institution portfolio. It comprises a significant percentage of the portfolios of some of the world’s largest investment institutions such as pension funds, sovereign wealth funds, insurance companies, and family offices. It is not unusual for illiquid assets (real estate, infrastructure, and private equity) to be upwards of 30-35 percent of an institution’s holdings, with real estate typically one third to a half of an investor’s illiquid holdings. The allocation to direct real estate varies from5 to 40 per cent of an investor’s portfolio asset value, with the highest proportion of cases in the 10-20 proportion range.1

1See: Nesbitt, S.L.,”Trends in State Pension Asset Allocation and Performance”: Cliffwater LLC, 2012.

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Optimal Deal Flow Management for Direct Real Estate Investments

There have been a number of reasons that made real estate attractive. Historically real estate was added to the typical investment portfolio because it was thought to be a diversifier.2 Its perceived low correlation with other asset classes and stable yields made it lucrative given the diversification algebra of Modern Portfolio Theory, despite the fact that appraisal-bias leaves this conclusion in doubt.3 The period following the Great Recession entrusted real estate with an additional role in the enterprise portfolios of long-term investors. That is the role of a yield provider, in an environment where other asset classes’ low yield and returns were predominant. Finally, one of the most desirable features of a typical real estate investment has been its clear, understandable, and consistent business model, which is unlike the businesses that underlie most other equity investments, and derivative financial claims in such businesses, like fixed income investments, or outright financial derivatives.

Besides sheer attractiveness, real estate has warranted serious attention by analysts for other reasons. By virtue of large size and indivisible nature compared to other asset classes each incremental deal introduces a high level of idiosyncratic, or non-diversifiable, risk to the portfolio. Due to its “lumpiness”, real estate deals are also highly illiquid which requires that the investor assumes idiosyncratic risks for much longer periods compared to other asset classes. Hence the nature of this “investment marriage” requires a high level of scrutiny of the impact of each investment deal that is about to enter the portfolio.

In contrast, stock portfolios exhibit a level of divisibility, liquidity, and turnover that is incomparably higher than almost any real estate deal (with the exception of REITs). Stocks are bought and sold in multiple numbers and types, every day, hour, minute, second, or even fraction of a second, freeing the investor almost entirely from the burden of monitoring idiosyncratic risks which are easily diversifiable as long as a large basket of stocks is held.

Another distinctive feature of real estate investing is its toolset. The majority of the financial theories and applied research have been entirely related to issues of the public stock markets, with only a tiny fraction of work being dedicated to directly owned real estate, despite its major role and standing in institutional portfolios. Portfolio management theories like CAPM and APT do not explicitly describe how real estate should be treated as an investible asset class because of the heterogeneous nature of the underlying investment deals, and the non- arbitrage free nature of real estate values as embedded appraisal-based valuations.4 Even when a major theory refers to “complete” asset markets, the available estimation methodologies, like regression/time-series econometric analysis, render their use with real estate impractical due to the lack of observable pricing history, as one of the corollaries of being an illiquid asset class.

For all of the above reasons, unlike equity analysis where the quantitative aspect of investment analysis is just as dominant as fundamental analysis, real estate investment deal making has evolved to be almost exclusively based on fundamental investment studies and techniques. The elaborate review of coverage, leverage, and profitability

2 See: Hoesli, M., Lekander, and J, Witkiewicz, W, “International Evidence on Real Estate as a Portfolio Diversifier”: Journal of Real Estate Research, 2004, 26: 2, 161 – 206. 3See Northfield Information Services’ March 2014 Newsletter article (http://northinfo.com/Documents/589.pdf) entitled, “The Pitfalls of an Index-Based Approach to Managing Real Estate Investment Risk” by Emilian Belev, CFA and Richard B. Gold for a more complete discussion of these issues and the implications for investors. 4Appraisal smoothing has long been recognized as an issue in private equity real estate. See: Geltner, D.M., “Smoothing in Appraisal-Based Returns”: Journal of Real Estate Finance and Economics, 1991, 4, 327 – 345.

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Optimal Deal Flow Management for Direct Real Estate Investments

ratios, tenant turnover, vacancy, and others variables can occasionally be supplemented by econometric studies of inputs that determine demand and supply, but essentially the estimation procedures result in the figures that rest on first moment (central tendency) in a distributional sense. Where such work stretches to estimation of the uncertainty around the central tendency estimate (e.g. standard deviation of projected return), eventually it cannot connect to the impact on the overall portfolio, or even to other such estimates within the real estate portfolio but from different geographic or property type markets, due to differences in the employed risk factor set.5

Notwithstanding all these seemingly practical contradictions, there are reasons for optimism. First, as already mentioned, portfolio theory, abstract as it might be, does refer to “complete” markets, encompassing all asset classes. Second, risk models do exist, that span real estate markets while simultaneously measure real estate risk with the same yard stick as other asset classes. Finally, the high quality of the in-depth fundamental analysis by real estate practitioners is an excellent source of information with respect to the asset class’ “profitability”, which in turn plays a central role even in the most involved of “quant” methodologies

The Real Estate Investment Process

Every asset owner has a particular deal allocation process that is guided by some general fixed investment principles but takes a distinct practical form. For example, smaller asset owners, with only one fund or portfolio, may be highly selective and have specific filters by land use, deal size, location, quality, expected return, and leverage to quickly determine their level of interest in a particular deal package. Larger firms, with multiple funds and separate accounts, may have a formal allocation process by which they allocate deals. If there are several bidders for a particular property, a second screening process is likely to determine who will be the “winner”. Often it is as simple as which fund is the next in line but often it may involve some heuristic “goodness of fit” metric.

The number of deal packages any investor receives in any given time period is a function of a number of factors, some of which are not under the investor’s control. The factors include, but are not limited to, the size of the firm, its reputation, and how active it has recently been in the market. Even more important in determining deal flow are both available space and capital market conditions. There are times when even the most active and largest investors may see few, if any deals, go over the transom. For example, during the “Great Recession” transactions levels were constrained because there were few sellers willing to take the necessary valuation haircut to bring a property to market. Even when markets are their most fluid, unlike equity markets, not every building is for sale, and even fewer properties that meet a specific asset owner’s investment criteria are for sale.

Finally, as a byproduct of the heterogeneity of deals, the bidding against a single property suggests that bidders face an additional risk. That is, with multiple candidates, a bidding war is likely to break out and in the absence of a rigorous and clear price limit, a buyer may end up paying more than they initially expected. For unsuccessful bidders

5 Real estate indices currently available to investors are not “actual” in the sense that they are appraisal-based and do not use transactions of the specific asset. Even transaction-based indices are not truly real time in the sense that they require hedonic estimates to fill in gaps, have small sample sizes, and have other shortcomings. For a discussion on using index based data to determine correlation with other assets see: Pedersen, Niels, Sebastien Page, CFA, and Fei He, CFA, Asset Allocation: “Risk Model for Alternative Investments”: Financial Analyst Journal, Vol. 70, No. 3 (May/June 2014), p. 34-45.

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forced to move on to properties further down their acquisition list, not only are they choosing properties with secondary and tertiary investment characteristics, but adding insult to injury, they are also likely paying higher than desired prices on those deals as well, as part of the bidding process, and therefore taking on higher than expected risk for lower return.6

Therefore, it is given these practical considerations that we have approached the issue of the optimal deal flow process for real estate (ODFRE).

Key Conceptual Tools

As with any methodology aimed at ranking investments, here the estimation of profitability plays a central role. The profitability analysis inevitably employs the traditional approach used by real estate investors when estimating a building’s expected cash flow that involves factors such as: current and expected rent levels, lease renewal probability, expected occupancy levels, tenant credit quality, expected capital expenditures, upkeep costs, etc. Even if these are figures predominantly derived from fundamental analysis, the estimated level of profitability can be augmented by forecasting quantitative techniques, and/or expert judgment. A statistical combination of various estimates has the potential to produce a better overall result. Within the framework of the Bayes-Stein’s estimation methodology a quantitative estimate shrunk towards an independent number – e.g. an expert or fundamentally derived subjective prior - will produce a modified estimate that is superior in terms of confidence around the estimate.7

Another cornerstone of our approach is no other than a basic tenet of Modern Portfolio Theory. For multi-class investors, it is paramount that the performance of an investment should be considered in relation to the performance of other assets. For example if an investor has an expectation that a certain asset will produce a return two times that of the existing assets in the portfolio, MPT dictates that the investor should also consider the correlation of this asset with all other assets in the portfolio. If, in fact, adding the asset increases the uncertainty of return due to high volatility and correlation with the rest of the assets to unacceptable levels for the investor, then the asset should not be considered. As a basic and revealing example, a single asset portfolio might seem to benefit of adding a 50/50 mix of the same asset and leverage, as it will have twice the expected return, but it will also have twice the expected standard deviation, and will have perfect correlation with the existing starting “portfolio”, and therefore should be considered with caution. As a consequence, any inferences we can draw about the new investment performance, especially regarding the uncertainty around the mean profitability estimate, should be made in light of the impact on total portfolio performance, rather than the standalone distributional characteristics of the asset’s total return.

6 See: Akin, S. N., Lambson, V. E., McQueen, G. R., Platt, B., Slade, B.A., Wood, J., Rushing to Overpay: The REIT Premium Revisited (http://home.business.utah.edu/finmh/McQueen%27s%20REIT%20Paper%20for%20U%20of%20U.pdf), March 22, 2011. 7 See: Jorion, Philippe, “Bayes-Stein Estimation for Portfolio Analysis”, Journal of Financial and Quantitative Analysis, Vol. 21, No. 3 (Sep., 1986), 279-292

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Another distinctive feature of our approach is the chosen metric of risk. Instead of looking simply at standard deviation, recognizing that the initial purchase price of an asset will have an explicit impact on performance in dollar terms, we view loss performance as a contingency claim (the contingency is investment payoff in present value terms that is lower than the purchase price) the value of which is our chosen measure of risk. This application of the contingency-claim analysis (CCA) is broadly related to the methodology developed by Merton8 for the analysis of credit risk. In Merton’s application ,the downside performance of a debt investment– in particular the event of default – can be viewed as a real put option in the hands of the borrower where the assets owned by the borrower and serve as collateral to the debt are the assets underlying the put option. Likewise, the underperformance of a real estate investment under the purchase price can be viewed as a put option at the hands of the real estate investor, where the strike price is the purchase price and the underlying asset is the value of cash inflows produced by the real estate investment.

Figure 1 Expected loss in the context of contingency claim analysis

Expected Value

Expected Loss Purchase Price adjusted for TVM

The parallel with the Merton model is for illustration purposes only. This model does not refer directly to debt instruments, nor is it directly connected to a real estate property in the role of collateral against any borrowings. Instead, we utilize CCA for two specific reasons – to capture the expectation of downside performance of the investment being analyzed, inasmuch as it places a dollar value on its potential downside loss in present value terms, and it allows investors to identify those underperformance outcomes in absolute terms relative to the expected mean outcome. The latter feature is one of differentiators of this approach with mean-variance optimization (MVO) which does not make a distinction between over- or under-performance in reference to a mean in terms of determining the stochastic dominance of portfolios (even though, as will later be shown there is some degree of similarity between this approach and traditional MVO).

8 See: Merton, R.C., “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates”: Journal of Finance, 29 (1974), pp. 449-470

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Optimal Deal Flow Management for Direct Real Estate Investments

Another critical aspect of this application of CCA in our approach, in light of our previously stated adherence to the tenets of MPT, is that when taking asset volatility as an input in option pricing, the volatility that is considered is not the standalone volatility of the asset itself but the incremental portfolio-level volatility. That is, we are concerned with the incremental volatility which the asset introduces to the total portfolio and, by extension, the proportion of otherwise idle cash that is equal to the purchase price of the new investment.

Implementation

Based on the toolset that has been outlined, practical implementation aligns with intuition. The first step is to determine the expected revenue value of the new deal assuming no potential loss due to outcomes of projected performance below the purchase price. In a later section of the paper, we offer our argumentation as to why this is a proper starting point and how it corresponds with the fundamental investor utility. The simplest measure that reflects performance of an investment and does concern the potential of an unexpected investment loss is the present value (PV) of the cash inflows using a base discount rate – the risk free rate.9

At this point, we need only accept that the next steps properly incorporate the true risk embedded in the investment and act in lieu of a risk-adjusted discount rate. For now we are concerned only with the expectation of only positive net future cash flows which can be rationalized as follows. These expected future cash flows are in essence a “free lunch” with no downside risk. Therefore, considering higher moments, apart from an aggregate measure of the amount received (similar to expected value), is an exercise of re-measuring gain in the light of random assumptions of a particular desired distribution by the recipient, which is not standard practice by investors. By principle of parsimony, in the absence of superior knowledge of the performance of one hypothesis over another we should employ the one offering the simplest and most intuitive explanation. In this case, in the absence of any upfront cost for the uncertain asset inflows (a cost which would introduce a potential loss, hence aversion), the expected value of those cash flows is the simplest aggregate measure of preference.

It is also worth noting why we did not pursue using a risk-adjusted discount rate and rather chose an explicit risk adjustment methodology. Simply stated, the calculation of an economically justified discount rate to derive the present value of an investment is predicated on one of the following conditions:

1. The calculation of expected loss as an explicit measure of penalty 2. The calculation of another criteria of penalty, most often associated with a factor risk premium and exposure to factors to those risk premiums a. This approach requires an assumption of what the risk factor driving the premium, and the relationship between the factor and asset (typically linear) b. If linear, for the linearity in the beta - premium interaction to hold one will need to make some assumptions about the underlying distribution of the factor and the asset

9 The more realistic situation where a term structure of risk free rates is used, is equally well admissible.

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c. Imposes practical challenges in multi-period setting (APT, CAPM, etc. are one period models, that do not work over compounded periods, unless estimated from log-returns in which case returns will not be additive across assets) 3. Availability of discount rates implied by the market.

As we will demonstrate, the first approach is the most clear-cut, least assumption-laden method (as opposed to 2). Furthermore, it does not restrict the results of the model to the alleged correctness or availability of implied metrics (as opposed to 3).

The rationale of the approach can be exemplified as follows. Consider a project that will pay out either $0 or $1000, compared to a project that will pay out either $499 or $501. There is same expected value of strictly positive cash flows, but are obviously much different projects. One can question the wisdom of using the risk free rate in the calculation of the expected present given such a difference. After all, once could object, there might be higher moments at work with the two projects payoff that we do not capture by simply using the risk-free rate.

Let us consider, however, what happens when we introduce a purchase price of e.g. $500 (the midpoint of either case). The $0/$1000 investment would produce an expected loss of 0.5*$500 = $250, and the $499/$501 investment is going to produce an expected loss of 0.5 * $1 = $0.5, which will most definitely capture the extreme difference of the dispersion of outcomes. This is no accident. By construction, the introduction of a cost, which introduces potential downside aversion, will introduce and reflect not only second, but higher moments that accrue to larger expected losses.

From a distributional moments perspective, the higher the purchase price the more and lower order of the central moments will matter for the expected loss and the investor. This is because the probability mass both close below and far below the mean will enter in the expectation calculation. The lower the purchase price the higher order central moments will predominantly matter as only probability mass far below the mean will enter in the expectation calculation. This reasons also well with intuition – if there is a potential only for one but very negative event, a low price might not be enough to entice us into a deal.

Next, we estimate the impact of the new investment on portfolio volatility. This impact can be considered either in regards to the portfolio of this specific investment mandate, the aggregate illiquid asset portfolio level, or the total enterprise portfolio inclusive of illiquid and liquid assets, depending on the investment policy guidance and the benchmark of portfolio manager’s performance. A risk (volatility) model capable of handling this estimation is required for this task. One such model, utilized by a number of major global investment institutions, is described below.10 This is the only model familiar to the authors that has the required breadth by asset class and geographic coverage, but we certainly do not assume that it is unique, and only refer to it to establish the feasibility of the overall process described herein. That is, the methodology we propose applies equally well to any risk model that meets the requirements of global and multi-asset class coverage.

10 See: Belev, Emilian, Dan DiBartolomeo, and Richard B. Gold, “Integrating Physical Real Estate and Infrastructure Assets In Enterprise Risk Management”, Enterprise Risk Management Symposium Monograph, http://www.ermsymposium.org/2014/pdf/erm-2014-paper-belev.pdf, Sep. 29 –Oct 1, 2014

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Besides spanning the divide between real estate risk assessment methods and those used in securities markets, the risk model has a number of other important features. First, it eliminates the need for appraisals and repeat sales methodologies for the purpose of estimating volatility. Second, the model is economical in its factor structure. Third, the model traces the relationship between individual properties and other asset classes such that the impact of risk factors is simultaneously measured across all asset classes and, therefore, the model can also measure the true incremental impact of buying or selling a property on a portfolio.

The liquid instrument coverage of the model spans approximately 4 million publically traded securities globally and an extendable list of over-the-counter derivatives. Even if the overall factor set is larger, each security is exposed to 18 risk drivers that are chosen based on the pertinent security characteristics: regional factor, industry factor, interest rates, oil prices, currency, value/growth, market development, company size, 5 statistical factors, currency of denomination, and 3 yield curve factors. Notably, sector and region factors dominate performance in the equity universe11. Some of the other factors are intended to measure investor confidence and sentiment (e.g. value/growth, market development, and size), as well as macro-economic conditions (interest rates, energy costs, exchange rates). Statistical factors have been added to capture transient market effects. For fixed income securities, the model breaks down yield curve impact into different types of movements of the curve – parallel (“shift”), slope change (“twist”), and curvature change (“butterfly”). With regards to the real estate coverage space, the model decomposes each property’s cash flow into three components that broadly relate to the basic marketable financial claims – stocks and bonds:

I. Steady-state Lease Fixed Income Security: A deterministic cash flow module for existing and expected leases over a building’s expected useful life. This module is based on the properties’ physical characteristics (size and number of leasable units), the quality of its tenants (credit risk) and other lease characteristics (downtime between leases, lease renewal probability, etc.). Rents in the steady-state model are assumed to grow at the rate of inflation. The model employs a Merton approach to calculate the probability of default for a property’s tenants, and resulting cash flow loss, over the property’s life as well as the probability of lease renewal and downtime between leases.

II. Rent Market Equity: A rent change module which directly correlates changes in rents to changes in factor returns adjusted by the structural profile of the local economy.

III. Mortgage Fixed Income Security: If leverage/gearing is present, a financing module is built as a short on the building’s cash flows. It is also subject to prepayment options and can be cross-collateralized with other buildings or at the fund-level financing. Financing can be modeled differently depending on whether it is fixed rate or floating rate.

It is a key feature of this real estate risk model that all three components are integrated into a coherent aggregate risk estimate by expressing their risk characteristics using the same factor set. For example, since the projected net

11 See: Everything Everywhere: The Multi-Asset Class Risk Model, Northfield Information Services White Paper, http://www.northinfo.com/documents/71.pdf

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operating stream associated with leases is, in essence, a fixed income security, it readily relates to the risk model’s term structure factors, in the same fashion as we would consider each separate bond in a fixed income securities portfolio. The credit risk of tenants is captured similarly to that to the credit risk of bonds. An interesting by-product of the model is that the factor exposure to the “shift” potential of the term structure is comparable to a bond effective duration. The same factor relation is sought in the case of mortgage related fixed income securities, in which case, however, a short “bond” position is assumed.

To the same end, rent risk is expressed using the “equity” factors in the risk model. This is done using bespoke econometric market analysis for each property type and location around the globe.

Ultimately, each of these risks is expressed as functions of factors that are observable in financial markets or in the general economy. The approach is congruent with methodologies used by risk security risk managers and allows for the seamless integration of risk assessment in multi-asset class portfolios and explicit estimation of the risk- reward impact of adding new investments, as well as selling existing ones.

For example, the model allows an investor to consider whether buying a shopping mall in either San Jose or Seattle would be more diversifying, given that their stock portfolio is typically concentrated in high tech companies. Similarly it allows making an informed decision whether to use fixed rate or variable rate financing in any new deal, given the nature of prepayment options, and effective duration, related to the investor’s bond holdings.

The next step in the proposed deal flow process is to calculate the expected loss impact using the Contingency Claim Approach as described earlier. After having calculated the factor risk exposures of any new deal, we have cleared the way to calculate the incremental volatility which the new deal introduces to the portfolio. Provided a value for the imputed (incremental) underlying asset volatility, asset value, and a “strike” price, the value of the real option equal to the investor expected loss can be calculated. In this case, the “strike” price is the threshold for the asset value under which we start accounting for a loss, which is, essentially, the purchase offer price. A typical option pricing model, like Black-Scholes12, uses an asset value distribution based on the mean of a continuously compounded return equal to the risk free rate. From that perspective, the present value of the expected real estate investment operational cash flows, using the risk free rate, is a good approximation of the underlying asset value used in the chosen option pricing model. The importance and relevance of this value will be made clear in the following section.

It should be noted that the use of the risk-free rate fits in the same math employed by the Black-Scholes model or other option models (e.g. Ross-Cox), but the argument behind its use is entirely different. Neither a hedging portfolio or arbitrage-free arguments are invoked as in the Black-Scholes model, nor martingale property rationale is employed as in the Ross-Cox13 model in our framework. It is the utility of the investor that only discounts strictly cash flows of a single sign (i.e. positive) bound by zero, that makes the use of the risk-free rate pertinent, as explained previously.

12 See: Black, Fischer, Myron Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, Vol. 81, Issue 3 (May – June 1973), 637-654 13 See: Cox, J. and S. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics 3, 1976, 145-166.

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The way to impute the incremental volatility relevant to the calculation of the CCA expected investment loss is as follows: + 2 + 2 2 2 2 휔퐶푢푟푟푒푛푡 푃표푟푡푓표푙푖표 퐶표푛푡푒푛푡 ∗ 휎퐶푢푟푟푒푛푡 푃표푟푡푓표푙푖표 퐶표푛푡푒푛푡 휔푁푒푤 퐼푛푣푒푠푡푚푒푛푡 ∗ 휎 푁푒푤 퐼푛푣푒푠푡푚푒푛푡, . = + 휔퐶푢푟푟푒푛푡 푃표푟푡푓표푙푖표 퐶표푛푡푒푛푡푠휔푁푒푤 퐼푛푣푒푠푡푚푒푛푡 ∗ 퐶푂푉푁푒푤 퐼푛푣푒푠푡푚푒푛푡 퐶푢푟푟푒푛푡 푃표푟푡 2 2 ퟐ 휔( 퐶푢푟푟푒푛푡 푃표푟푡푓표푙푖표 퐶표푛푡푒푛푡 ∗ 휎 퐶푢푟푟푒푛푡 푃표푟푡푓표푙푖표 퐶표푛푡푒푛푡) 흎푵풆풘 푰풏풗풆풔풕풎풆풏풕 ퟐ 푰풎풑풖풕풆풅 푰풏풄풓풆풎풆풏풕풂풍 푽풂풓풊풂풏풄풆 푫풖풆 풕풐푵풆풘 푰풏풗풆풔풕풎풆풏풕 The weights in this expression∗ are흈 all calculated relative to the sum of the value of the existing portfolio and additional investable cash. The risk model, as explained before, is essential in the calculation of all variance and covariance terms, and as such imperative that it covers a broad based of asset classes and market that can be present in the current portfolio. The expression then reduces to:

( ) ퟐ + 2 =흈 푰풎풑풖풕풆풅 푰풏풄풓풆풎풆풏풕풂풍 푽풂풓풊풂풏풄풆 푫풖풆 풕풐 푵풆풘 푰풏풗풆풔풕풎풆풏풕 , . 2 2 휔푁푒푤 퐼푛푣푒푠푡푚푒푛푡 ∗ 휎푁푒푤 퐼푛푣푒푠푡푚푒푛푡 휔퐶푢푟푟푒푛푡 푃표푟푡푓표푙푖표 퐶표푛푡푒푛푡푠휔푁푒푤 퐼푛푣푒푠푡푚푒푛푡 ∗ 퐶푂푉푁푒푤 퐼푛푣푒푠푡푚푒푛푡 퐶푢푟푟푒푛푡 푃표푟푡 2 Or: 휔푁푒푤 퐼푛푣푒푠푡푚푒푛푡

. , . = 2 2 (1) �휔푁푒푤 퐼푛푣푒푠푡푚푒푛푡∗휎푁푒푤 퐼푛푣푒푠푡푚푒푛푡+2휔퐶푢푟푟푒푛푡 푃표푟푡 휔푁푒푤 퐼푛푣푒푠푡푚푒푛푡∗퐶푂푉푁푒푤 퐼푛푣푒푠푡푚푒푛푡 퐶푢푟푟푒푛푡 푃표푟푡 흈푰풎풑풖풕풆풅 휔푁푒푤 퐼푛푣푒푠푡푚푒푛푡 The expression on the right hand side will yield the incremental volatility that should serve as the input of an option pricing model. The other inputs of the CCA valuation will be a “strike” price equal to the offer/purchase price and an underlying price equal to the present value of the new investment that is derived by discounting future cash inflows at the risk free rate.

Once we have calculated the NPV and the expected loss we simply take the difference in the two to arrive at the risk-adjusted NPV of any particular investment deal. The final step is to rank order all the investments that are under consideration for any given portfolio for any given period of time and to select that ones whose total cost is equal to or less than the (possibly) leveraged investable cash available to the investor over the investment period. An illustration of this process is provided in Table 1 and Figure 2. We can effectively include the effect of a variable budget constraint due to the possibility of borrowing (Figure 3), by incorporating cash flows at increasing loan rates that reduce NPV, (loans will be also discounted at the risk free rate as ruthless default is not considered an possibility) and in that case the cutoff point of the invested set will be the investment before which the marginal contribution to NPV turns negative due to high capital costs. While these graphs may seem well familiar from existing finance practice, the information that enters in them is categorically new and richer in terms of the risk- adjusted choices investors should make.

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It should be noted that there is a special case regarding the covariance term in the calculation of incremental variance in expression (1). This happens when the new investment is strongly diversifying the existing portfolio, the covariance term is negative and the expression under the radical is negative. In this case, it should be taken with a positive sign, and the resulting calculated “expected loss” value should be considered with a positive sign (i.e. a windfall profit in excess of the present value). The practical significance of this different treatment is that the inclusion of the asset actually equates with the so computed decrease in volatility, which is equivalent to an additional “expected benefit”.

Table 1. Potential investments are sorted by Adjusted NPV (per dollar invested) and included in the investment set up to the budget constraint.

Investment PV Cash Offer PV (per NPV (per Time Imputed CCA Adjusted Cumulative Cumulative Inflows Price dollar dollar Horizon Volatility Drawdown NPV (per Investment Budget (mill (mill Invested) Invested) Value per dollar (mill Constraint dollars) dollars) dollar invested) dollars) (mill invested dollars) Property 8 67 43 1.56 0.558 10 28.2 0.131 0.427 43 100 Property 4 31 22 1.41 0.409 10 19.5 0.067 0.342 65 100 Property 7 27 18 1.5 0.500 10 35.1 0.209 0.291 83 100 Property 6 46 31 1.48 0.484 10 36 0.221 0.263 114 100 Property 1 27 20 1.35 0.350 10 22 0.098 0.252 20 100 Property 5 52 40 1.3 0.300 10 24 0.125 0.175 60 100 Property 3 20 16 1.25 0.250 10 19 0.084 0.166 76 100 Property 2 43 38 1.13 0.132 10 18 0.095 0.037 114 100

Figure 2. We sort all investment possibilities by Risk-Adjusted NPV in descending order. The cutoff point will be the acceptance threshold for possibilities

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Figure 3. Increasing cost of capital due to borrowing will contribute to increasing present value of financing cash flows (discounted at the risk-free rate), presenting a dynamic capital constraint.

Comparing ODFRE methodology with Mean-Variance Optimization

Before comparing the ODFRE approach to mainstream techniques used in quantitative portfolio construction, a quick review of its theoretical underpinnings is required. ODFRE’s inherent optimization philosophy is that NPV, using the risk-free time value of money (TVM) discount rate, is the true economic value of any investment in absence of any downside risk other than the initial purchase price cash outflow. A related but separate argument is that the expected PV using a risk-free rate is the correct underlying (spot) value that should be used in the drawdown-CCA-related option valuation procedure (DCC). Formulating this argument in different way, in the absence of any potential loss (no purchase price), the investor is concerned only with the expected cash flows and time value of money when determining his stochastic preferences of one payoff distribution relative to another. Hence the pricing under the “risk-neutral” DCC assumption is justified. Essentially, if an investor is about to be granted for free, and without any consideration in exchange, a set of non-negative cash flows ranging from zero (e.g. failed business venture) to potentially very high levels (a very successful business venture), and he is not to be faced with the possibility of any cash outflow, the only risk characteristic the investor will most likely be interested in will be the average of the present value of those cash flows, as an aggregate measure of getting more over getting less. This is because under any outcome the investor cannot experience a drawdown on their contemporaneous consumption or wealth. For example, a choice of two Christmas gift boxes with an unknown

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amount of known content (e.g. widgets) – one large and one small – would normally prompt the choice of the bigger – it can be empty or full to the top, while the smaller can also be empty or full to the top, but at best with fewer widgets than the larger one. The choice criteria applied is that the maximum outcome with a bounded downside and a distribution of the same type will be ranked synonymously with the first moment of the distribution – the average. In the case of the usual distributions used in finance, the maximum tends to infinity but the same logic holds regarding the mean as a gauge of attractiveness under the same conditions.

It is the fact that the investor needs to pay a consideration for the cash inflows pertaining to the investment that introduces his or her aversion to the dispersion of the cash inflows. Under some of those cash inflow outcomes, the investor would realize a loss, and it is this set of outcomes that would make the investor suffer some level of disutility as to make her averse to the moments of the distribution that describe a higher probability mass in the lower tail (second and higher moments).

With the benefit of this observation we can look back to the distribution of strictly positive cash flows in the absence of a purchase price. We have already observed that being bound by zero from below, and having a relatively symmetrical distribution, an increase in the mean will imply an increase in any of the quantiles – a clear preferred choice for the investor embedded in the mean value measure. One might ask, but what if the distribution is skewed towards lower or higher values and not relatively symmetrical. It is the cutoff created by the purchase price in the lower region of this distribution that will generate the loss distribution that will measure the impact of this difference in probability mass in the upper or lower region – i.e. the introduction of the purchase price will correctly factor in higher moments as relevant to the investor considered from his entry point – the price paid.

Figure 4. Comparing the expected loss of two investments at the same price, and different volatility and mean value levels

small volatility

large volatility

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The usual portfolio utility theory chooses to use variance as a measure of dispersion between the “gain” and “loss” cash flows realizations.14 Our approach, on the other hand, does not take the face value of return variance, but takes it in relation to the mean and purchase price, as we are only interested in the expectation of the loss as embedded in the draw-down CCA value (DCC).

Assuming a normal distribution of return, which is also assumed by the derivation mean variance utility,15 we can project the comparison of this approach with MVO. It is worth noting that a normal assumption in linear return will also precipitate a normal assumption for the investment cash flows (for simplicity let us assume they occur at a fixed investment horizon). We should also note that while we theorize that the investment cash inflows are bound from below at zero, the normal distribution is not. However, given a very low probability of below zero cash flows in the normal distribution, the assumption will be shown to keep the relationship outline that follows to hold approximately true.

The value of the expected shortfall is:

( ) ( ) = 푥−휇 2 (2) 퐷 1 − 2휎2 µ퐷퐶퐶 and σ ∫denote−∞ 퐷 − respectively푥 휎√2휋 푒 the푑푥 mean and standard deviation of the distribution of investment payoff cash inflows; D denotes the purchase price.

The above expression can then be transformed in the following way:

( ) ( )

= 푥−휇 2 푥−휇 2 퐷 1 − 2휎2 퐷 1 − 2휎2 The퐷퐶퐶 integral퐷 ∫− ∞in휎 the√2휋 푒first term푑푥 −evaluates∫−∞ 푥 휎√ 2to휋 푒a particular푑푥 p-value constant p. In the second term we complete to integrate on the interval.

( ) ( ) ( ) = . + [ 푥−휇 2 푥−휇 2] = 휎 퐷 휇−푥 − 2휎2 휇 − 2휎2 2 2 퐷퐶퐶 퐷 푝 √2(휋 ∫−)∞ 휎 푒 − 휎 푒 ( ) 푑푥 = . + 퐷−휇 2 0 푥−휇 2 = 휎 − 2휎2 퐷 1 − 2휎2 퐷 푝 √2휋 �푒 − ( � −) 휇 ∫−∞ 휎√2휋 푒 푑푥 . . + 1 (using approximation on the exponential term) 2 휎 퐷−휇 2 ≈ 퐷 푝 − 휇 푝 √2휋 � − 2휎 �

14 Markowitz, Harry, “Portfolio Selection”, Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91. 15 Meucci, Attilio, “Risk and Asset Allocation”, Springer, 2007 edition, p. 274

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Now we return to our stated objective (function): = (3)

( ) 푂푏푗푒푐푡푖푣푒= + 휇( − 퐷 −) 퐷퐶퐶 + 2 휎 퐷−휇 To 휇convert− 퐷 to푝 휇mean− 퐷 and− √ 2standard휋 2휎√2 휋deviation of return we need to divide all terms by D

( ) [ + ( ) + ]/ = 2 휎 퐷−휇 휇 − 퐷 푝 휇 − 퐷 − √2휋 2휎√2휋 퐷 ( ) = 1 1 + + (1 + 1) + (1 1 ) 휎푟 퐷−휇 푟 푟 푟 The subscript− 휇 “r”푝 denotes휇 − distribution− √2휋 parameters− − 휇 2that휎√2휋 relate to units of return rather than value. Dividing the denominator and numerator by D in the last terms then yields:

( ) = + ( ) + ( ) 휎푟 1−1−휇푟 푟 푟 푟 푟 which휇 finally푝 휇 yields− √2휋 for −the휇 objective2휎 √2휋 function defined for the Optimal Deal Flow process for Real Estate (ODFRE):

ODFRE Objective = + ( ) + (4) 2 휎푟 휇푟 푟 푟 푟 In comparison (disregarding휇 푝 휇 temporarily− √2휋 2휎 the√2휋 impact of varying risk aversion levels):

MVO Objective = 2 It is obvious that 휇the푟 − ODFRE휎푟 objective just like MVO benefits from the increase in expected return and decrease in volatility. In the ODFRE objective function, however, the impact of the two variables is more involved. Put simply, the reason for the higher complexity of the ODFRE expression is that we are penalizing the shortfall of the distribution under the principal threshold and that shortfall depends on the interaction of the mean as well as the variance of the distribution, and separately on their relation to the principal threshold (i.e. purchase price).

It should be noted that the ODFRE objective, broadly related to MVO, has a convex component (decreasing marginal utility) with increasing wealth. At constant the value of will decrease proportionately more than a potential increase of , making the second term smaller. This is partly (but less than) compensated by the fourth 푟 term which will increase at a lower rate than ( ), unless휎 for very 푝large values of . 푟 휇 푟 푟 To incorporate a risk aversion constant similarly푝 휇 to MVO, namely: 휇

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we have to observe all terms that are related to risk disutility in our ODFRE objective and apply a constant to all of them simultaneously. In equation (3), that would be all but the first term. As a side note, market risk aversion is embedded in the sale price, which is empirical, immediate, and makes no theoretical assumption with respect to a pricing model. Yet it fits seamlessly within the DCC option pricing model’s theoretical framework as it enters as the strike price, or principal protection threshold. Hence if the offer price is accepted by the investor, in essence, it entails that they match the market level of risk aversion, which is already reflected in equation (4).

ODFRE differs from MVO in one important aspect. While MVO is typically applied to a single-period setting, the tool set of the ODFRE does not restrict its application to a single period. Neither the calculation of the net present value, nor the option model required by the methodology needs to be constrained to a single period. This is especially relevant for the typical time horizon for the typical real estate investor.

Another advantage of ODFRE compared to MVO is the choice of asset present value distribution. Even if the previously derived closed form solution is based on a normal distribution, the framework can be easily generalized to any distribution where the expected loss or preference function can be computed either analytically or computationally.

As a final note in the comparative section, we would like to point out why use contingent claim pricing methodologies to estimate the downside risk rather than far more commonly accepted methods such as Value-at- Risk or Conditional VaR. The simple answer is that the ODFRE approach offers all of the benefits of these methods and more. For all practical purposes, the expected loss under the “option” framework is calculated like a CVaR statistic with one major difference; it is that ODFRE is calculates the size of loss considered from a purchase price threshold, rather than a p-value determined threshold as cVaR, which makes the ODFRE approach more relevant from a purchaser’s or seller’s point of view, especially in view of capturing all relevant distribution moments of the payoff.

Conclusion

The approach presented in this paper outlines a number of significant practical benefits while offering the same level of theoretical rigor used in traditional optimization techniques. The main benefit and purpose of the methodology is to find a clear prescription for investors as to how to efficiently allocate their real estate sales and acquisition capital that takes into consideration the uncertainty of both their real estate as well as their total portfolio’s performance. While taking up this task, the approach resorts to intuitive and transparent metrics like present value and expected loss, which, in turn, tie into the well familiar framework of capital budgeting and marginal cost-benefit analysis. Given this innovative methodology’s familiar backdrop, the authors hope that the paper will gain acceptance with practitioners and give them a rigorous tool for making better risk-adjusted investment decisions, similar to the capabilities which their brethren in the public investment space were granted with the introduction of Modern Portfolio Theory.

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