ex-ante real estate value at risk calculation method

Noname manuscript No.(will be inserted by the editor)

Ex-ante real estate Value at Risk calculation method

Charles-Olivier Amédée-Manesme ·Fabrice Barthélémy

Published in Annals of Operations Research, https://link.springer.com/article/10.1007/s10479-015-2046-7

Abstract The computation of Value at Risk (VaR) has long been a problematic is-sue in commercial real estate. Difficulties mainly arise from the lack of appropriatedata, the lack of transactions, the non-normality of returns, and the inapplicabilityof many of the traditional methodologies. In addition, specific risks remain latentin investors’ portfolios and thus risk measurements based on market index do notrepresent the risks of a specific portfolio. Following a spate of new regulations suchas Basel II, Basel III, NAIC and Solvency II, financial institutions have increas-ingly been required to estimate and control their exposure to market risk. Hence,financial institutions now commonly use “internal” VaR (or Expected Shortfall)models in order to assess their market risk exposure. This paper proposes the firstmodel designed especially for static real estate VaR computation. The proposalaccounts for specific real estate characteristics such that the lease structures orthe vacancies. The paper contributes to the real estate risk management literatureby proposing for the first time a model that incorporates characteristics of realestate investments. It allows more precise real estate risk measurements and isderived from a regulators’ approach.

Keywords: Risk Management, Value at Risk, Real Estate Finance, Vacancy,Lease.

C-O. Amédée-ManesmeUniversité Laval, Department of Finance, Insurance and Real EstatePavillon Palasis Prince, G1V 0A6, Québéc, QC, CanadaTel.: +(1) 418-656-2131(3890)E-mail: [email protected]

F. BarthélémyCEMOTEV, Université de Versailles Saint-Quentin-en-Yvelines78047 Guyancourt Cedex 33, France

2 Charles-Olivier Amédée-Manesme, Fabrice Barthélémy

1 Introduction

The stock market crash of 1987 triggered the development of new risk measures.This was the first major financial crisis in which practitioners and academics be-came concerned with the possibility of global bankruptcy. The crash was so im-probable given the standard statistical models that quantitative analysts beganto question the appropriateness of their techniques. Numerous academics, pro-claiming that the crisis could easily reoccur, called for reconsideration of all suchrisk models. It became obvious that the possibility of extreme events occurringrequired much greater examination. Limitations of traditional risk measures wereacknowledged, with improved measurement of risks of a major decline in assetvalue having become an urgent task. There was a recognized need for greater re-liance on a risk measure considering the entire return distribution of a portfolio.During the 1990s, a comprehensive new risk measure did emerge: Value at Risk,commonly known by the acronym VaR.1 Practitioners and regulators developedand then increasingly adopted the VaR measure in their subsequent risk analyses.2

Informally, Value at Risk is the largest percentage loss with a given probability(confidence level) likely to be suffered on a portfolio position over a given holdingperiod. In other words, for a given portfolio and time horizon, and having selected aconfidence level3, α ∈]0, 1[, VaR is defined to be that threshold value, assuming nofurther trade, such that the probability of the mark-to-market loss in the portfolioexceeding this VaR level is exactly the preset probability of loss α.4 Thus, VaR isthe quantile of the projected distribution of losses over the target horizon, in thatif α is taken to be the confidence level, VaR then corresponds to the α quantile.By convention, this worst loss is always expressed as a positive percentage in themanner indicated. In formal terms, then, if we take L to be the loss, measured asa positive number, and α to be the confidence level, then VaR can be defined asthe smallest loss - in absolute value - such that:

P (L > V aR) ≤ α. (1)

A more detailed definition of VaR can be found in Jorion (2007) or Bertrand &Prigent (2012).5

The crucial step in the worldwide adoption of VaR was the financial regula-tion Basel II of 1999, which has resulted in virtually complete adoption of thatmeasure (Basel III must be applied by 2019 and rely on VaR but also on anotherpossible risk metric close to VaR: Expected Shortfall. More recently, Solvency IIregulations (for insurers in Europe) have proposed VaR as a reference measurein determining required capital. All these regulations require financial institutions

1 The notion of VaR and more precisely of quantile criterion goes back to the literature about safety-first criteria (see Roy, 1952)

2 The other traditional risk measurement, Expected Shortfall, is not discussed in this paper.3 In practice, VaR is calculated for threshold less than 1%.4 Note that VaR does not give any information about the likely severity of loss by which its level will

be exceeded.5 In terms of gains rather than losses, the VaR at confidence level α for a market rate of return X

whose distribution function is denoted FX(x) ≡ P [X ≤ x] and whose quantile at level α isdenoted qα(X) is:

−V aRα(X) = sup {x : FX(x) < α} ≡ qα(X). (2)

Ex-ante real estate Value at Risk calculation method 3

to compute VaR periodically and to always maintain sufficient capital to coverthose possible losses projected by VaR. The authorities propose a standardizedapproach but leave open the possibility of building further in-house (“internal”)models. Most financial institutions have developed their own internal market riskvaluation model. The present paper enters in this line by proposing a model ac-counting for the specificities of real estate and usable in internal model (or forbacktesting purposes). Our methodology estimates VaR of direct commercial realestate by replicating traditional indices construction.

VaR’s appropriateness as a risk estimator or its adequacy for risk budgetingpurposes have long been discussed in the literature (see Pflug, 2000; Rodríguez-Mancilla, 2010; Krokhmal et al., 2011). In this paper, we do not address thispoint. It suffices that regulators have seen fit to choose the VaR measure forrequired economic capital calculations, and that its computation is mandatoryfor all regulated practitioners. VaR is thus an essential research subject and ofconsiderable interest to a broad spectrum of academics.

In real estate investment, no VaR model dominates. Even worse, to the best ofour knowledge, no model has been specifically developed for real estate. Propertyinvestments harbor many characteristics and discrepancies that must be consid-ered when assessing risk. Among them are heterogeneity (each property is unique),illiquidity (buying and selling may span months), location (property is immov-able), obsolescence (property does not preserve efficiency), lease structures, breakclauses and vacancies (property may be leased long or short-term, possibly withbreak clauses). In addition, idiosyncratic risk is latent: real estate portfolios usuallyrequire larger number of assets than other asset class (more than fifty) in order tobegin diversifying efficiently (see Byrne & Lee, 2001; Callender et al., 2007). Allthese features make real estate a specific asset class that requires a specific riskmanagement tool.

The limited research focusing on direct commercial real estate risk, in spiteof the increasing interest in the topic, is likely due to the lack of data from thereal estate sector. Limited data for this sector is one of the primary obstacles toreliable VaR computation. Either the investment is in listed real estate and itis quoted daily with sufficient data available to compute VaR, or the investmentis in direct real estate and deals with small datasets. The latter point, which isproblematic in practice, is particularly true in commercial real estate (e.g., offices,shopping centers, etc.), where institutional investors invest largely. The real estatemarket is thus comparable to the private equity or hedge fund markets, whereindices are created from small numbers of transactions. Every real estate propertyindex attempts to aggregate real estate market information in order to providea representation of underlying real estate performance. However, observation isgenerally conducted monthly in the best of cases, else quarterly, semi-annually, orsometimes even annually. It has largely to do with the sector under consideration.The residential sector, where many transactions are observable, frequently featuresa monthly index. Commercial real estate faces greater difficulty in regularly de-livering data, and the indices are consequently of longer publication periodicity.To determine VaR of a real estate portfolio at threshold 0.5% (as requested bythe Solvency II framework) using the historic approach, a minimum of 200 valuesare needed, which represents 17 years even for a monthly index. With that num-ber of needed observations, VaR considerations are frequently irrelevant, since thisrequirement typically exceeds the recorded history of the index. Hence, it is nec-


essary to rely on other methods in order to determine VaR for direct commercialreal estate.

Recent regulations urge more reliable methods of computing VaR. This is oneof the primary motivations of this work. Traditional VaR methods are difficultto apply in real estate because of the poverty of the dataset due to their dataconsuming characteristics. Moreover, all these methods suffer at least from onelimitation: they do not consider the characteristics of the real estate portfoliosanalyzed. Computing VaR using market data (indices) gives the market VaR, notthe portfolio VaR. Real estate portfolio characteristics are so different that it isfundamental to consider them in any risk calculation. Investors who argue thatthey replicate the real estate market as a whole are rare. They have preferencesand invest in one type of risk according to their required return. Some invest onlyin core assets (recent properties with a long-term lease); others only in opportunis-tic assets (old vacant properties). Given the various type of investments strategyin real estate, the few indices available and the difficulties of diversifying idiosyn-cratic risk in real estate, VaR models may incorporate portfolios’ characteristicsand intricacies. This point is particularly obvious for property investments sincetwo investors who invest in the same area but under disparate strategies shouldnot exhibit the same VaR (nor the same economic capital). Two portfolios thatcompute VaR using the same index will however obtain the same value if they donot account for the specificities of the investment. To the best of our knowledge,the development of a specific model to determine VaR in real estate has not beenthe subject of extensive research. The present effort is meant to be an advancementon regulatory analysis.

The approach we propose employs a method not based solely on past data. Wepropose an ex-ante approach where a model is used to predict future returns asopposed to the opposite ex-post approach, purely based on observed past returns(see Daripa & Varotto, 2005). We replicated property indexes methodology andbuilt up a model to compute forward looking indices of the portfolio. In doing so,we use the same methodology as the regulators (the so-called “standard model”relies on market indices), but improve it by taking into account the futures returnsof the considered portfolio.

Our model accounts for characteristics of real estate investments (especiallylease structure, vacancies and tenant decisions).6 Here lies the innovation of ourproposal: as far as we know, no paper has considered real estate specific charac-teristics. This proposed model accounts for the specificities of the portfolio and indoing so is able to discriminate strategies: core, value added or opportunistic. Inparticular, the leasing strategies are taken into account. Our proposal thus takesspecific (or idiosyncratic) risk into account. Risky and non-risky investments willthus exhibit different VaR. Strength of our model is its flexibility. Indeed, thegeneric model already integrates several realistic real estate constraints, and themethodology provides many degrees of freedom in order to consistently instantiateadditional real market features or requirements.

In this paper, we contribute to extant literature on VaR and on real estaterisk management.7 To the best of our knowledge, this paper is the first that pro-

6 We explicitly put aside the leverage as it is traditionally not considered by indices.7 Moreover, the methodology developed here can be easily extended to the computation of Expected

Shortfall measurement for Basel III regulation.


poses a VaR model especially designed for real estate and that takes into accountthe specificities of this asset class. Using Monte Carlo simulations, knowing thestochastic processes, we are able to simulate numerous time series of the indices(which can be done on an annual basis, as well as on a monthly or on a quarterlybasis). It allows us to estimate the VaR from the estimated distribution functionof the returns.

The model we propose for VaR implementations is based on the assumptionthat the portfolio mix will not change before the VaR horizon. This hypothesismay be unrealistic, especially when the horizon is large for many asset classes.Nevertheless, in real estate this hypothesis may be acceptable as adjustments donot occur frequently, since properties are illiquid assets and are costly. The oppositewhere VaR is measured dynamically, i.e., taking into consideration portfolio mixadjustments over time, can be the subject of future research (see Fusai & Luciano,2001).

This article is the first to propose a model that is both able to differentiatebetween strategies, able to account for real estate specificities and able to computeex-ante “internal” VaR. Given the current regulatory environment and the need foran “internal” model for practitioners, the current proposal is particularly relevantand opportune.

The remainder of the paper is organized as follows. Section 2 presents therelated literature. Section 3 incorporates a review of index performance measuresand introduces the proposed model while Section 4 implements and illustrates themodel. Section 5 then concludes the paper.

2 Literature review

Methods of computing VaR have already been the subject of considerable re-search, following VaR’s introduction into current banking practice. We note somefundamental articles on VaR assessment and methods of its determination, amongthem generalities on VaR: Duffie & Pan (1997); Monte Carlo simulations: Pritsker(1997); Johnson transformations: Zangari (1996); Cornish-Fisher expansions: Fal-lon (1996) and extreme value theory: Longin (2000), among others.

A lot of research focuses either on determining the best methods to computeVaR or on improving existing approaches. Pichler & Selitsch (1999) compare fiveVaR methods in the context of portfolios and options: Johnson transformations,Variance-Covariance, and the three Cornish-Fisher-approximations of the second,fourth and sixth order, respectively. They conclude that a sixth-order Cornish-Fisher approximation is best among the approaches treated. Mina & Ulmer (1999)compare Johnson transformations, Fourier inversion, Cornish-Fisher approxima-tions, and Monte Carlo simulation, concluding that Johnson transformations donot constitute a robust technique. Monte Carlo and Fourier inversion, instead,are robust, while the Cornish-Fisher approach, though fast, is a bit less robust,particularly when the distribution is far from being normal. Chun et al. (2012)propose the use of quantile regression to estimate conditional risk measures bydeveloping a generalized regression framework. They particularly focus on VaRmeasures. Fuh et al. (2011) discuss the use of small probabilities simulations inthe context of VaR determination. They develop a VaR computation model relyingon heavy-tailed risk factors and multivariate t distributions.


VaR has been the subject of numerous papers in real estate, but they haveprimarily focused on listed (i.e. securitized) real estate and not on direct com-mercial real estate.8 VaR for securitized real estate relies on the same methods asthose used for ordinary stocks and bonds. Indeed, data frequency and typologyare similar and thus approaches developed for traditional assets can be applied forlisted real estate. Zhou & Anderson (2012) concentrate on extreme risks and thebehavior of REITs in abnormal market conditions. They find that estimation ofthe risks requires different methods for different stocks and REITs. Cotter & Roll(2010) study REIT behavior over the past 40 years, highlighting the non-normalityof REIT returns. They compute VaR (called risk of loss in their paper) and com-pare the results for the Case & Shiller index, following three methods that do notrely on Gaussian assumptions, these being the Efficient Maximization algorithm,the Generalized Pareto Distribution method, and the GARCH model. Liow (2008)uses extreme value theory to assess VaR dynamics of ten major securitized realestate markets, allowing one to evaluate the risk of rare market events better thanwould be possible by using traditional standard deviation measures.

Literature focusing on VaR in the context of direct real estate investment (orfunds) is sparse. Nonetheless, some studies do concentrate on risk managementand assessment in real estate. Booth et al. (2002) examine risk measurement andmanagement of real estate portfolios, suggesting that practical issues force realestate investors to treat real estate differently from other asset classes. The reportfocuses on the difference between symmetric measures, such as standard deviation,and downside risk measures, such as VaR. Their work concentrates on all risk mea-sures used in real estate, thus constituting a survey of the current real estate riskmeasures. Gordon & Tse (2003) consider VaR as a tool to measure leveraged riskin the case of a real estate portfolio. Their paper demonstrates that VaR allowsbetter assessment of such risk. In particular, traditional risk-adjusted measures(e.g., the Sharpe or Treynor ratio as well as Jensen’s alpha) suffer from a lever-age paradox. Leverage adds risk along with potential for higher returns per unit ofgreater risk. Therefore, the risk/return ratio does not change noticeably and there-fore does not constitute an accurate tool by which to measure the risk inherentin debt. Contrarily, VaR is quite a good tool for studying leveraged risk. Brown& Young (2011) focus on a new way to measure real estate investment risk usingspectral measures. They begin by refuting the assumption of normally distributedreturns, whose adoption serves to flaw forecasts and decisions. Interestingly, VaRis not their selected measure; instead, an Expected Shortfall technique is adopted.Amédée-Manesme et al. (2014) propose an approach combining the use of CornishFisher expansion and rearrangement methodology in order to compute VaR fordirect real estate. The method presents the advantage of accounting for momentsof order 3 and 4 and does not rely on any distribution assumptions.

The literature focusing on real estate risk management and moreover on VaRis thus pretty thin. To the best of our knowledge, no models have been especiallydeveloped for real estate VaR computation. This paper fills that gap.

8 For instance, listed real estate encompasses REITs.


3 The model

The model we propose replicates the methodology used to compute total returnindices. As opposed to the standard approach of computing VaR on past data (ex-post VaR), our proposal simulates future returns taking into account the specifici-ties of the considered portfolio (we therefore compute ex-ante VaR). The primarychallenge in tracking direct real estate market returns is the difficulty of measuringthe asset price or capital return component (it is often easier and more straight-forward to accurately estimate income returns in real estate investment due totheir predictable characteristics). To abstain from too many assumptions9, we willcompute the portfolio value as the discounted sum of all the future cash flows. Thecash flows are determined using a stochastic approach incorporating both system-atic (through simulation of market rental values) and idiosyncratic risk (leases).This approach is able to account for the tenant’s behaviour concerning break op-tions included in a lease10, and can be tailored for real-world portfolio managersmanaging portfolio risk. Then, using Monte Carlo simulations as proposed in realestate by French & Gabrielli (2005), Hoesli et al. (2006) and Baroni et al. (2007),we estimated the model that leads to estimated indices and estimated VaRs aswell. This way, we determine how taking these specific risks into account altersthe VaR of a real estate portfolio.

3.1 The index methodology

Measuring real estate returns has long been a challenge for the real estate in-dustry. In the 70s, the industry realized how important is was to give a reliablepicture of the direct real estate market at least for credibility of the asset class.Nowadays, many commercial real estate indices producers publish and dissemi-nate series of periodic returns (the two major are NCREIF Property index andInvestment Property Database, IPD). These indices report generally total returns,income returns and capital returns regularly (monthly, quarterly, semi-annually orannually). These indices are now widely accepted and used by industry, academicsand regulators. Many types of indices exist (appraisal-based, transaction-basedand stock-market based). They are all subject to criticisms, advantages and weak-nesses that are not discussed here (extended discussion can be found in Geltner etal., 2007). It suffices that regulators have accepted this type of indices for capital

9 In this model, we assume rational investors and players that make decisions based on discountedcash flows and returns. For clarity, we suppose an optimal world without taxes and do notconsider arbitrage or investment during a simulation (furthermore, it is required by regulators).The model is thus performed for a static portfolio. These restrictions can easily be releasedand dynamic VaR can be the subject of further research. We assume these to keep this articleclear and to a reasonable length.

10 Leases and break options vary significantly with local practice. Lease is nevertheless an essentialcomponent of commercial property appraisal and cash flow. Lease specifies rent, break-options(option to leave in favor of the tenant), possible vacancies, indexation, etc. Rents charged totenants rarely follow market rental values (MRV ). Rents are usually contracted at a valueclose to the MRV at the initiation of the lease. Years after, rents have usually been indexedand do not necessarily represent the current market value (that may collapse in bear marketor raise in bull market). The difference between rent paid and rents available on the marketcan represent a risk for property investors (see Amédée-Manesme et al., 2015).


budgeting purpose to rely on them (for instance, Solvency II framework rely onIPD UK).

The most widely recognised published indices are total return measurements.They are the most important measure of overall investment performance becausethey are used to compare different assets or asset class across time. It incorporatesboth capital and income components, and is calculated as the percentage valuechange plus net income accrual, relative to the capital employed. It is recognised bythe Global Investment Performance Standard set out by the Chartered FinancialAnalyst Institute as the standard composite measure of investment performance.

With respect to a single period t, total return is defined as:11

TRt =CVt − CVt−1 − CExpt + CRptt +NIt

CVt−1 − CExpt(3)

where:

- TRt is the total return in period t,- CVt is the capital value at the end of period t,- CExpt is the total capital expenditure in period t (in real estate, this includespurchases, developments, hard works, refurbishments...),

- CRptt is the total capital receipts in period t (mainly sales),- NIt is the day-dated rent receivable during period t, net of property manage-ment costs, ground rent and other irrecoverable expenditure.

Total return is calculated as the change in capital value, less any capital expen-diture incurred, plus net income, expressed as a percentage of capital employed(value of the asset at the beginning of period plus all capital expenditure) overthe period concerned. Calculated this way, total return for a single period canthen be compounded in order for multi-period total returns. As far as we know,all commercial valuation based real estate indices producers rely basically on thismeasurement method with some possible adjustments. Due to the low number oftransactions, the capital value is generally an appraised value.

3.2 The index replication model

We simulate the receivable rents (NI) taking into account a portfolio’s specifici-ties and deduce the capital value from these cash flows. In fact, the expectationsin terms of returns and risk are modelled using trends and volatilities and aresupposed to represent the market and its systematic risk. We also incorporate id-iosyncratic risks by accounting for leases, and embedded break clauses in futurecash flows. The price path is deducted from the NIs. Because the proposed VaRcomputation supposes a static portfolio, we do not consider arbitrage possibilitiesduring our simulation: this is why CRpt is not discussed here. In addition, CExptis not addressed in this section (a recurrent percentage will be used instead).

11 Notations rely on IPD indices construction guideline Investment Property Database (June 2014).


3.2.1 Notations

We consider a multi-period model for a portfolio where many assets (spaces) invarious submarkets can be considered (the spatial dimension) themselves encom-passing many leases across time (the temporal dimension). First, we develop thetime dimension where just one lease is modeled. Second, we extend this modelingto the submarkets and to the portfolio levels (which corresponds to an aggregationin the spatial approach). We define the following parameters and variables of themodel:

Time and periods:

- T̄ : the number of periods used to estimate the capital values of a single asset(one space),

- h: the number of time periods used to compute the return indices,- rt: the discounted rate at time t.

Portfolio dimension:

- J : the number of submarkets j,- nj : the number of spaces in the submarket j,- n: the total number of spaces in the whole portfolio.

Portfolio, submarkets and spaces characteristics (deterministic values):

- α: the decision-making criterion to exercise the break option,- γ: the vacancy cost,12

- λj : the average duration of vacancy of the submarket j,- µj : the return of the MRV index of the submarket j,- σj : the volatility of the MRV index of the submarket j,- space (i, j): represents the space i located in the submarket j,- δji : the proportional coefficient between the net income (rent) of the space (i, j)and the market rental value index of the corresponding submarket j.

Submarkets and spaces specifications (random variables):

- τ ji,b: the time period where the bth lease contract is signed,- vji,b: the vacancy length before the beginning of the bth lease,- gji,t: the indexation value at time t for the space (i, j).

Net Incomes stochastic processes (random variables):

- NIjt (i): the net income of the space (i, j) at time t,- NIjt : the sum of the net incomes at time t of all the spaces located in thesubmarket j,

- NIt: the net income of the the whole portfolio at time t.

Market Rental Values stochastic processes (random variables):

- MRV jt (i): the market rental value of the space (i, j) at time t,- MRV jt : the market rental value at time t of the submarket j.

Capital Values stochastic processes (random variables):

12 α and γ can vary with the submarket j (αj and γj).


- CV jt (i): the capital value at time t of the space (i, j),13

- CV jt : the sum of the capital values at time t of all the spaces located in thesubmarket j,

- CVt: the capital value at time t of the whole portfolio.

3.2.2 The Net Incomes (NI)

The portfolio contains n spaces split in J different submarkets. Let nj be thenumber of spaces in the submarket j. We define the space (i, j) as the ith spaceof the jth submarket. Let t = 0, 1, . . . , T, . . ., be the time periods corresponding toan infinite time horizon starting at time t = 0. Let τ ji be the set of time periodswhere a new lease contract for the space (i, j) is signed:

∀j, i, τ ji = {τ ji,1, . . . , τji,b, . . .} (4)

Hence, τ ji,b represents the time period where the bth contract of the space (i, j)

begins while τ ji,b+1 represents the beginning of the next lease contract, the (b+1)th.Note that the time intervals [τ ji,b, τ

ji,b+1[ are equivalent to the intervals [τ ji,b, τ

ji,b+1−

1] as the time periods are integer numbers (see Figure 1).

-

τ ji,1 . . .

bth lease contact︷︸︸︷τ ji,b τ ji,b+1−1

(b+1)th lease contact︷︸︸︷τ ji,b+1

0 1 time t. . . . . . t t+1 . . .

Fig. 1: The time intervals for the space (i, j) lease contracts

NIjt (i) represents the cash flow at time t of the space i located in the submarketj. NIjt (i) is equal to 0 if the space (i, j) is vacant. NIt is the cash flow at time tgenerated by all the spaces (i, j) of the portfolio. By convention, NI0 = 0 as weset that the first rent is paid at time t = 1. Therefore:

∀t = 1, 2, . . . , NIt =J∑j=1

NIjt =J∑j=1

nj∑i=1

NIjt (i),

The NI variations are independent of the MRV dynamics. Indeed NI evolution isdriven by indexation and not by market evolution (indexation can be positive evenwhen the MRV is declining). Meanwhile, we need to model the MRV ′s evolution

13 In this model, we do not determine the price of a property but only the price of each space separately.


as it is the base of all the new lease contracts (MRV index is based only on newlease contracts and not on undergoing passing rents). Let MRV jt be the value ofthe MRV index of submarket j at time t. We suppose that the MRV ′s dynamicfollows a diffusion process:

∀j, t, dMRVjt

MRV jt= µjdt+ σjdWt (5)

with µj and σj being the return and the volatility of theMRV index of submarketj, respectively. The correlation between the MRV js of the various submarkets jis also taken into account.

The first positive net income of a given space (i, j), NIjτji,b

(i), is somehow equal

to its MRV (small differences may occur due to market frictions or bargainingpower). The initial rent of a new lease contract is therefore proportional to theMRV index to which it belongs. We can thus derive the initial level of the netincome when a lease is concluded NIj

τji,b(i) as a function of MRV j

τji,b, the index

value at time t for the submarket j:

∀j, i, b, NIjτji,b

(i) = MRV jτji,b× δji (6)

where δji is a coefficient accounting for the specificities of the space (i, j) (size,disposition, floor, etc.).

If no vacancy is assumed, there is no time period between two consecutiveleases. Hence, τ ji,b+1 − 1, the time period preceding τ ji,b+1, corresponds to the endof the bth lease contract.

As previously mentioned, the rents evolution is driven by an indexation: attime t for the space (i, j), we denote it gji,t ((g

ji,t)t≥1 can be a stochastic process).

The NI of the space (i, j) at time t is thus:

∀t ∈ [τ ji,b, τji,b+1[, NIjt (i) = (1 + gji,t)

t−τji,b ×NIjτji,b

(i)

From (6) we get:

∀t ∈ [τ ji,b, τji,b+1[, NIjt (i) = (1 + gji,t)

t−τji,b ×MRV jτji,b× δji (7)

Hence, the NI dynamic can be written as:

∀t, i, NIjt (i) = δji∑b≥1

1(t∈[τji,b,τji,b+1[) (1 + gji,t)

t−τji,b ×MRV jτji,b

(8)

Finally, the total net income of the portfolio for each period of time when novacancy is considered is:

∀t, NIt =J∑j=1

nj∑i=1

δji∑b≥1

1(t∈[τji,b,τji,b+1[) (1 + gji,t)

t−τji,b ×MRV jτji,b


If vacancy is made possible. We have to check if at time t the space isvacant or not.

Let us define the random vector vji composed with the vacancy lengths of space(i, j) for each period defined by the vector τ ji set in (4). In the same way:

∀j, b, i, vji = {vji,1, . . . , vji,b, . . .} (9)

The vacancy length vji,b corresponds to the vacancy preceding the beginning of thebth lease. It corresponds to a number of periods and not to a time period. Hence,the random vacancy length vji,1 is preceding τ ji,1 the time period where the firstlease contract is signed. As τ ji,2 corresponds to the time period where the secondlease contract begins, it is preceded by the random vacancy length vji,2. Finally,the first lease contract starts at t = τ ji,1 and ends at t = τ ji,2−v

ji,2. More generally,

the bth lease contract starts at t = τ ji,b and ends at t = τ ji,b+1 − vji,b+1. The time

periods corresponding to the bth lease contract are

[τ ji,b+1, τji,b+1 − v

ji,b+1[= [τ ji,b+1, τ

ji,b+1 − v

ji,b+1 − 1] (10)

which is illustrated on Figure 2.

-

vacancylength

vji,1︷︸︸︷τ ji,1

vji,b︷︸︸︷

vacancylength

vji,b+1︷︸︸︷

vacancylength

bth︷︸︸︷τ ji,b τ ji,b+1−v

ji,b−1

leasecontract

(b+1)th︷︸︸︷τ ji,b+1

leasecontract

0 vji,1+1 time t

. . .

. . . . . . t t+vji,b+1+1 . . .

. . .

Fig. 2: The time periods for the space (i, j) lease contracts with vacancy

These vacancy durations are modelled using Poisson’s law distribution:14

∀j, i, b, vji,b ∼ P(λji ) (11)

where λj is a positive real number equal to the expected number of occurrencesduring a given interval (here, it is equal to the average duration of vacancy ofsubmarket j). Note that the vacancy length can be equal to 0 (if for instance anew tenant arrives as soon as the previous tenant vacates).

14 The Poisson’s law is a discrete probability distribution that expresses the probability of a givennumber of events occurring in a fixed interval of time if these events occur with a knownaverage rate and independently of the time since the last event. The probability function isgiven by:

X ∼ P(λ), P(X = k) =λk

k!e−λ.


As soon as a tenant vacates a unit at time t, the landlord may face a void period(NIjt+1(i) = 0). In order to determine if the space (i, j) is vacant or not, we followAmédée-Manesme et al. (2013) who develop a model that considers lease structure(and therefore for embedded break options that give flexibility to tenants)15. Theirmodel accounts for the differences that may arise at the time of a break optionbetween the market rental value and the net income in place. Note that for everylease τ ji,b+1−v

ji,b+1 corresponds to a break-option exercise date. It is assumed that

a rational tenant exercises its break option as soon as the rent currently paid istoo high in comparison to the market rental values available for similar lettableunits. This is written as:

∀j, i, b,NIjτji,b+1−v

ji,b+1

(i)

MRV jτji,b+1−v

ji,b+1

(i)> 1 + α,

then NIjτji,b+1−v

ji,b+1+1

(i) =

0 if vji,b > 0

MRV jτji,b× δji if vji,b = 0

(12)

where α is a decision-making criterion (α ≥ 0 if the tenant is rational and in-cludes possible moving or transaction costs, for instance). The rationale behindthis model that as with any commodity or product traded on a free market, supplyand demand jointly determine price (here the rent). If a rental property is pricedabove the current market rental value, competitively priced properties are takenup quickly, while overpriced ones remain or become vacant.

As there is no direct link between time t and the different subperiods, a logicaltest is made for all the subperiods. Moreover, the subperiods [τ ji,b, τ

ji,b+1[ have to

be decomposed into [τ ji,b, τji,b+1 − vi,t[∪[τ ji,b+1 − v

ji,t, τ

ji,b+1[. The NI of the space

(i, j) can thus be computed as:

∀j, i, t, NIjt (i) =

δji ×MRVj

τji,b×

(1 + gji,t)t−τji,1 if t ∈ [τ ji,1, τ

ji,2 − v

ji,1[,

− γ if t ∈ [τ ji,2 − vji,1, τ

ji,2[,

...(1 + gji,t)

t−τji,b if t ∈ [τ ji,1, τji,b+1 − v

ji,b[,

− γ if t ∈ [τ ji,b+1 − vji,b, τ

ji,b+1[,

...

...

(13)

where γ corresponds to the cost of being vacant. Indeed a property does not retainefficiency without a minimum level of ongoing capital expenditure. A vacant prop-erty often requires substantial investment in addition to recurrent costs (e.g., local

15 On lease options, see also Al Sharif & Qin (2015).


taxes, security, technical control, etc.). Vacancy costs can therefore be significant.This can finally be rewritten as

∀j, i, t, NIjt (i) = δji∑b≥1

MRV jτji,b

(1(t∈[τji,b,τ

ji,b+1−v

ji,b[)

(1 + gji,t)t−τji,b

× 1(t∈[τji,b+1−vji,b,τ

ji,b+1[) (−γ)

)(14)

We have then defined the stochastic process of the rent:(NIt

)t≥0

={NI0, NI1, . . . , NIt, . . .

}(15)

3.2.3 Capital Value at time 0 (CV0)

The capital value at time t = 0 of a given space is the sum of all the future netincomes - NIjt (i) - discounted at time t = 0. CV j0,+∞(i) will thus denote the capitalvalue of space (i, j) at time t = 0, evaluated using all the net incomes from t = 0 toinfinity (by assumption, NIjt (i) = 0). CV j0,+∞(i) is a random variable that modelsthe theoretical price of space (i, j) at time t = 0. This definition implies that:

CV j0,+∞(i) =

+∞∑t=0

NIjt (i)

(1 + rt)t(16)

where rt is the discounted rate at time t ((rt)t≥1 could be a stochastic process).The capital value of space (i, j) evaluated at time t = 0 can be computed using

(14) in (16):

CV j0,+∞(i) =∑b≥1

τji,b+1−v

ji,b+1−1∑

ν=τji,b

(1 + gji,ν)ν−τji,b

(1 + rν)ν× δji ×MRV

j

τji,b

+

τji,b+1−1∑ν=τji,b+1−v

ji,b+1

−γ(1 + rν)ν

× δji ×MRVj

τji,b

CV j0,+∞(i) = δji∑b≥1

MRV jτji,b

τji,b+1−v

ji,b+1−1∑

ν=τji,b


(1 + rν)ν

+


ji,b+1

−γ(1 + rν)ν

(17)


Remark:In the case where no vacancy is assumed, we get

CV0,+∞(i) = δji∑b≥1

MRV jτji,b

τji,b+1−τ

ji,b−1∑

ν=τji,b


(1 + rν)τν

and in the very special case where the lease contract is just a one-year contractwithout vacancy with a decision-making criterion fixed at 0, as ∀j, i, the set τ jicontains the entire time period from 0 to infinity: τ ji = {1, . . . , t, . . .}.16 We get

CV j0,+∞(i) = δji

+∞∑t=1

MRV jt (1 + rt)t = δji

+∞∑t=1

MRV jt1

(1 + rt)t

The net income dynamics is the one of the MRV . This underlines that the differ-ence in the dynamics arises from the lease structure (the length of the lease, theindexation, the break options and the vacancies).

3.2.4 Capital Value at time t (CVt)

We are interested in the capital value of space i evaluated at time t. It is thesum of all the future net incomes NIjt (i) from time t to infinity, each value beingdiscounted at time t. This is denoted at time t for the space (i, j), CV jt,+∞(i).

This modelling can be inferred from the one of CV j0,+∞(i). Indeed, for a givensimulation, the path deriving from a given space remains the same over time.Therefore, not all the τ ji,b values have to be taken into account. There is a subsetof τ ji which will be in the past at time t, t being now the origin of the analyzedperiod. Hence, only the τ ji,b’s posterior or equal to t are of interest. We can thustruncate the CV j0,+∞(i) dynamics from time t. The discounted factors are thereforetranslated from t periods in order to have the portfolio value expressed at time t:(1 + rν)ν−t instead of (1 + rν)ν in (17).

This analysis can be conducted as well taking into account the set vji definedin (9). We can write:

∀j, i, t, CV jt,+∞(i) = δji∑b≥1

1(τji,b≥t)×MRV j

τji,b

×

τji,b+1−v

ji,b+1−1∑

ν=τji,b


(1 + rν)ν−t+


ji,b+1

−γ(1 + rν)ν−t

(18)

Taking the weighted sums on the different spaces and the different submarkets,we get the whole portfolio value at time t

CVt,+∞ =

J∑j=1

CV jt,+∞ =J∑j=1

nj∑i=1

CV jt,+∞(i) (19)

16 As ∀b, we get τ ji,b+1 − vji,b+1 − τ

ji,b − 1 = 0 (because vji,b+1 = 0 and τ ji,b+1 − τ

ji,b = 1).


Then we get for the portfolio value dynamic(CVt,+∞

)t≥0

={CV0,+∞, CV1,+∞, . . . , CVt,+∞, . . .

}(20)

This approach differs from the one usually used in literature where the price ofthe asset is simulated as a diffusion process. However, we argue that the diffusionprocess does not sufficiently account for property specificities (and, in particular,of possible voids in cash flows). This is particularly true in commercial real estatewhose values are mainly driven by leases contracts. For this reason, we have con-sidered a traditional discounted cash flows at infinity. In addition, taking infinityinto account allows avoiding an assumption on the terminal value of the portfolio(and thus on infinite growth rate, correlation between prices processes and marketrental values, etc.).

3.2.5 Total Return dynamics (TRt)

We can build the one period total return as follows:

TRt,1 =CVt,+∞ − CVt−1,+∞ − CExpt + CRptt +NIt

CVt−1,+∞ − CExpt(21)

We are thus finally able to compute the stochastic process of the one year totalreturn above: (

TRt,1)t≥1

={TR1,1, TR2,1, . . . , TRt,1, . . .

}Using Monte Carlo simulations, we estimate the indices and derive from all thescenarios the VaR of the portfolio. By expanding previous models, we are able totake into account many factors such as rent in place, indexation, rental value dy-namics and volatilities, vacancy duration and the leases’ structure(break options).The proposed approach is thus an accurate means of simulating the main risks inreal estate cash flows, combining systematic risks (on the MRV ’s as well as theircorrelation over time) and specific risks (the tenant’s behavior facing a possibilityto leave). Additionally, our model is based on forward looking indices (instead ofpast data) and therefore the estimated VaR is an ex-ante VaR.

4 Estimation

The issues that emerge when working with general model is their application. In-deed, it is impossible to simulate an infinite number of steps. For this reason, weneed to estimate the functions of interest. For instance, the price of the assets area function of the NIs and of the simulation time period. In this section, we specifyhow each defined function is estimated. In particular, we estimate the stochasticprocess of the capital values on the time horizon h that will be used to estimatethe returns (h returns by simulation) and then the VaR.

The length of the time period. To estimate the price at time t of space (i, j)we have to estimate its capital value at time t defined in (18). To do so, we truncatethe infinite time period [0,+∞[ to a finite time period [0, T̄ ]. This specification has


one main advantage: it considerably reduces the time of computation, especiallywhen the dimensions of the model are large. Thus when T̄ tends to infinity

∀t ≤ h, CV j0,+∞(i)− CV j0,T̄

(i) =

+∞∑t=T̄+1

NIjt (i)

(1 + rt)t−→T̄→∞

0 (22)

which leads to

CV j0,+∞(i) =

+∞∑t=0

NIjt (i)

(1 + rt)t'

T̄∑t=0

NIjt (i)

(1 + rt)t= CV j

t,T̄(i) (23)

As a consequence, and with the same approximation order:

∀t ≥ 0, CV jt,+∞(i) 'T̄+t−1∑tt=t

NIjtt(i)

(1 + rtt)tt(24)

The value of T̄ depends on the parameter values of the rt’s and on NIj0(i) (thefirst net income, which is a function of MRV j0 , δ

ji and τ ji,1).

The indices. The total returns (and the capital returns) are computed fromthe corresponding estimated index over h periods (for instance, practitioners oftenuse 10 years). Hence, we need h+ 1 values of CV jt (i) from t = 0, . . . , h. This caninterpreted as the index on h periods for the space (i, j):

∀j, i, k,(CV jt (i)

)(0≤t≤h)

={CV j

(k)

0 (i), CV j(k)

1 (i), . . . , CV j(k)

h (i)}

(25)

where CV j(k)

t (i) is the kth simulated value of the capital value index of space (i, j)at time t.

The net incomes. The stochastic process (MRV jt )(t≥0) is the fundamentalunderlying process to explain the random variables NIjt (i) as presented in (6). LetK be the number of paths of this index. We get

∀j, k,(MRV j

(k)

t

)(0≤t≤T̄+h)

={MRV j

(k)

0 ,MRV j(k)

1 , . . . ,MRV j(k)

T̄+h

}(26)

where MRV j(k)

t is the kth simulated value of the MRV j index at time t.

The first values in (26), from 0 to T̄ , makes it possible to estimate CV j0 (i),the first value of the index defined in (25). The last ones, from h to T̄ + h, leadto the estimation of its last value CV jh (i). In order to compute the capital values,we have to compute the net incomes path associated with the simulated MRV j

defined in (26).For each MRV j path we get a realization of the NIj(i) stochastic process

according to its modeling defined in (14)(NIj

(k)

t (i))(0≤t≤T̄+h)

={NIj

(k)

0 (i), NIj(k)

1 (i), . . . , NIj(k)

T̄+h(i)}

(27)


The capital values. We compute the K paths of the index stochastic processdefined in (25):(

CV j(k)

t (i))(0≤t≤h)

={CV j

(k)

0 (i), CV j(k)

1 (i), . . . , CV j(k)

h (i)}

(28)

which can be expressed as functions of realized net incomes as defined in (27)

(CV j

(k)

t (i))(0≤t≤h)

=

T̄∑t=0

NIj(k)

t (i)

(1 + rt)t, . . . ,

T̄+h∑t=h

NIj(k)

t (i)

(1 + rt)t

(29)

At the level of one of the MRV ’s

(CV j

(k)

t

)(0≤t≤h)

=

nj∑i=1

δji

T̄∑t=0

NIj(k)

t (i)

(1 + rt)t, . . . ,

nj∑i=1

δji

T̄+h∑t=h

NIj(k)

t (i)

(1 + rt)t

(30)

And at the portfolio level(CV

(k)

t

)(0≤t≤h)

=J∑j=1

nj∑i=1

δji

T̄∑t=0

NIj(k)

t (i)

(1 + rt)t, . . . ,

J∑j=1

nj∑i=1

δji

T̄+h∑t=h

NIj(k)

t (i)

(1 + rt)t

(31)

which given in a more compact form(CV

(k)

t

)(0≤t≤h)

={CV

(k)

0 , CV(k)

1 , . . . , CV(k)

h

}(32)

The capital returns. It is straightforward to derive the capital returns from(37) (

CVR(k)

t

)(1≤t≤h)

=

{CV

(k)

1 − CV(k)

0

CV(k)

0

, . . . ,CV

(k)

h − CV(k)

h−1

CV(k)

h−1

}(33)

This leads to (CVR

(k)

t

)(1≤t≤h)

={CVR

(k)

1 , . . . , CVR(k)

h

}(34)

The K × h one-period returns (h returns on K paths) leads to an estimationof c.d.f. of the one-period capital return as follows

∀x ∈ CVRt, F̂CVR(x) =1

K × h

K∑k=1

h∑t=1

1(CVR

(k)t ≤x

) (35)

and the estimation of the quantile α, denoted q̂α is such that

F̂CVR(q̂α) =1

K × h

K∑k=1

h∑t=1

1(CVR

(k)t ≤q̂α

) = α (36)

which leads automatically to the estimation of VaR at level α.


The total returns. For computing the kth path of the total returns index twodistinct paths are simulated. First, one of the net incomes linked to the J differentMRV paths (see (38) for the NIjt (i) path). Second, one of the potential sellingprices, CV , the capital value (see (32)). These two processes are deeply linked asthey depend on the same lease structure. The risk associated solely to the sellingprice is embedded into the capital return.17 Taking into account these two risksleads to:(

TR(k)

t

)(1≤t≤h)

={CV

(k′)

1 +NI(k)

1 − CV(k′)

0

CV(k′)0

, . . . ,CV

(k′)

h +NI(k)

h − CV(k′)

h−1

CV(k′)h−1

}(37)

where

- the NI(k)

t are defined using (27) and summing on j and k (with the weight δji ).(NI

(k)

t

)(0≤t≤T̄+h)

={NI

(k)

0 , NI(k)

1 , . . . , NI(k)

T̄+h

}(38)

- the CV(k′)

t are one path of the whole portfolio capital value as defined in (32)(CV

(k′)

t

)(0≤t≤h)

=

{CV

(k′)

0 , CV(k′)

1 , . . . , CV(k′)

h

}(39)

The quantiles are estimated in the same way as in (36).

5 Application

The theoretical model presented above has been tested on the most standard leaseand real estate portfolios. The objectives of this experiment are twofold. From afinancial perspective, we want to examine the nature and features of lease structureon real estate cash flows and management. From a computational perspective, wewant to investigate how VaR behaves when the specificities of the portfolio areconsidered.

In what follows, we illustrate numerically how the model accounts for specificand market risk factors. We compute VaR at 0.5% threshold (as requested inSolvency II regulation). As a first step, we illustrate our model. In particular, weconcentrate on the way the model accounts for real estate specificities (ex. leases,among other things). Then we compute and analyse VaR for a portfolio composedof one lease in order to isolate the lease’s effects and finally, we compute VaRfor more leases and analyse the results and highlight the portfolio diversificationeffect.

In the base case, we consider the following parameters: the indexation of theleases contracts (g) and the trend of the market rental value index (µ) are fixed at

17 We can compute solely the risk due to the net incomes by taking the expected value of the cap-ital value. However this would be a biased estimator of the whole risk as the prices are notdeterministic.


2%, the volatility of market rental value (σ) equals 20%, the discounted factor (rt)is set at 6.5%, and the standard lease structure is 5-10 years with a possibility ofbreak at year 5 in favour of the tenant. The average vacancy length (λ, in years)is set at 3 years, the cost of vacancy (γ) is fixed to 0, and the tenant’s decision-making criterion (α) is equal to 0 (a tenant vacates at the time of a break-optionas soon as the MRV is lower than the rent). Each space being initially vacant,we set the probability of being vacant at 30% (70% of the simulation will exhibitrents at the 1st period). Finally, for clarity of the presentation, at the end of eachlease, the rental value will be set at the market rental value without vacancy (thetenant renews the lease). This assumption can be questionable because in practice,landlords often try to dissuade a tenant from leaving at the dates determined inthe lease, but they do not succeed each time. Nevertheless, this assumption wastaken in order to more clearly distinguish the lease effect on our model. It caneasily be removed. For each scenario, we generate 10,000 paths over 100 years(T̄ = 100, see Appendix B) of a hypothetical portfolio (see appendix A).In sum, the parameters of the base case are: g = µ = 2%, σ = 20%, rt = 6.5%,λ = 3 years, lease: 5|10, γ = 0, α = 0.

5.1 Methodology illustration

Real estate cash flows rely on rent and capital growth. As such, rent is the mostfundamental points for models because they are the basis of the net incomes aswell as of the capital values. The proposed model accounts for the lease structurein the cash flows. Instead of the classical methodology, which suggests consideringa certain percentage of vacancy over time, we consider the leases of the portfolio.We thus take into account the possibility of void periods in the cash-flows.

In order to illustrate the model, we consider the rent generated by a lease overthe simulation. We thus simulate - on the basis of the base case - three NI pathsof a single space (i, j) in Figure 3 in line with equation 13. The drops to zero valuecorrespond to a vacancy period after the exercise of a break option (γ = 0 in thebase case, null net income in case of vacancy).

The most fundamental information of the net income dynamics are the firstdate of each lease contract, the end of the lease (by contract or by break-option)and the length of vacancy, respectively defined in equation 10, 4 and 9. All theseparameters are specific parameters (vs market parameters). For instance, the re-alization of these random vectors for the bold path of Figure 3 are:

- {0, 10, 22, 31, 39, 49, 59, 69, 79, 85, 95} for the beginning of the new leases for therandom vector τ ji ,

- {0, 0, 7, 4, 3, 0, 5, 0, 0, 1, 0} for the vacancy lengths for the random vector vji- {15, 27, 36, 54, 84} for the time periods where a break option is exercised.

The analysis of this path has to be conducted relative to the correspondingMRV dynamics, defined in equation 26. Figure 4 underlines the link between thenet incomes evolution and the observed market rental values dynamic. We set δjiequal to one for the comparison. The first net income of every new lease is equalto the MRV (see equation 5). The first 5|10 lease runs from t = 0 to t = 10, witha break option at time t = 5. As at this time the net income is lower than theMRV , therefore the option is not exercised. Hence, the first lease runs until its


Fig. 3: Three paths NIj(k)

t (i) for a 5|10 lease

ending period, t = 10, time where the new (the second) 5|10 lease starts (witha net income equal to the MRV by assumption). This new 5|10 lease runs fromt = 10 to t = 20, which leads to a break option at time t = 15. At this point, asthe net income exceeds the MRV , the option is exercised. The range of this leaseis then [10, 15[. The (random) vacancy length, simulated to be equal to 7 years,implies that the following (the third) lease contract begins at time t = 22. And soon for the whole period.

Fig. 4: One NIj(k)

t (i) path and its corresponding MRV j(k)

t (5|10 lease)

Figure 3 highlights the importance of taking into account the difference betweenthe market rental value and the rent. Indeed, the rent is indexed (here, g = 2%) andcan exceed the market rental value (in a bear market for example, when indexation


remains positive). The proposed model therefore considers the differences thatarise between the dynamics of MRV s and the rent in place (usually contractedinto years before a break option).

From Figure 3 we can compute the corresponding 10-year capital value indexdefined in equation 25. The parameter h is equal to 10 for the 10-year index whichleads to 11 values. The value at time t = 0 corresponds to the initial value ofthe portfolio, composed here with only one space, i, in a given submarket, j. Thisvalue is the sum of the first 91 discounted net incomes (see equation 24) withT̄ = 90 and t = 0. We get CV j

(k)

0 (i) =∑90tt=0[NIj

(k)

tt (i)/(1 + rtt)tt], the first value

of equation 25.

In the same way, we get the different values of the index defined in equation25.

- the 2d value, CV j(k)

1 (i) =∑91tt=1[NIj

(k)

tt (i)/(1 + rtt)tt],

- the hth value, CV j(k)

h (i) =∑90+htt=h [NIj

(k)

tt (i)/(1 + rtt)tt],

- the 11th value, CV j(k)

10 (i) =∑100tt=10[NIj

(k)

tt (i)/(1 + rtt)tt].

Following equation 33 we get 10 annual returns for each path.18

Let us still consider only one MRV but 10 spaces with the lease structure5|10, each having a probability to be vacant initially equal to 30% (as stated). Theillustration of the stochastic processes of interest is still built with 3 observed paths.The capital return and total return pdfs are estimated with 10,000 replications of11-year indices, which leads to 100,000 values.

18 Note that from a theoretical point of view, constant net income growth without vacancy leads to aconstant annual return equal to the discount rate (here rt = 6.5%), because the initial valueis determined by discounting all the future cash flows at this discount rate.


(a) Net Incomes paths, NIj(k)

t (b) Capital Value index, CV j(k)

t

(c) Capital Value Return, CVRj(k)

t(d) Capital Return distribution

(e) Total Return, TRj(k)

t (f) Total Return distribution

Fig. 5: Three NIj(k)

t , CV j(k)

t , CVRj(k)

t and TRj(k)

t (chosen randomly) for a 10-spaceportfolio, 1-year Capital Return and Total Return pdfs (10,000 replications)

Figures 5d and 5f are particularly interesting in the VaR context because theyhighlight the differences between total returns and capital returns. Many commentscan be raised. First, capital return is clearly left skewed compared to total return.This is due to the net income returns that are strictly positive (γ = 0 in ourscenarios relating to the base case). Therefore they “smooth” the total returns


(total returns are the sum of the capital return and the net income returns).Second, capital return values are significantly lower than the total returns values.This comes from the more erratic nature of capital returns. Basically, when manyleases are considered, it is unlikely that all the spaces become vacant and thusthe income return would not change that much. This will be highlighted in nextsection.

5.2 VaR analysis on one lease

In this section, we study the VaR at 0.5% for the 1-year capital return (VaRCVR

0.5%,1)

and the 1-year total return (VaRTR

0.5%,1). Table 1 reports the effect on the VaR

according to a variation of the parameters of interest related to theMRV dynamics,µ and σ, the vacancy length in year λ and the deterministic growth rate g. Figure9 illustrates the joint effect of σ and λ on the 1-year total returns VaR.

Table 1: VaR0.5%,1 for a single space according to the model parameters

Capital Total MRV Expected NI MRV LeaseReturn Return Volatility vac. length growth Return StructureVaR

CVR

0.5%,1VaR

TR

0.5%,1σ λ g µ

7.8% 4.5% 10%13.7% 7.9% 15%20.8% 11.9% 20% 3 2% 2% 5|1029.3% 16.8% 25%38.8% 21.8% 30%

12.8% - 015.2% 5.7% 118.0% 9.0% 220.8% 11.9% 20% 3 2% 2% 5|1023.0% 14.3% 425.7% 16.7% 5

20.8% 11.9% 20% 3 2% 2% 5|1021.1% 12.2% 3%21.6% 12.8% 4%

20.8% 11.9% 20% 3 2% 2% 5|1017.5% 9.8% 3%14.2% 7.4% 4%

Table 1 raises many comments. Volatility of the market rental value is clearlythe most sensitive parameter. Volatility changes have implications on exercisingbreak-options and on the level of net incomes (themselves based on MRV ). It canbe interpreted as a volatility transmission dynamic. Volatility is the traditional riskmeasurement in finance and therefore the transmission of the volatility dynamicis not surprising. What is most astonishing, is the level of VaR compared to thelevel of volatility (σ). VaR grows far more quickly than the volatility: when thevolatility increases by 200% (from 10% to 30%), VaR grows by almost 400%. For


comparison purpose, when the volatility of a Gaussian distribution19 increasesby 200% (from 10% to 30%), VaR increases in the same proportion (∼ 200%).This difference in VaR variation can be attributed to the idiosyncratic risk thatis taken into account in our model: when the volatility increases, the number ofexercised break options increase too and finally the volatility of the NI increasesmore quickly than the volatility of the MRV .

The other parameters also have an impact on VaR. The length of vacancy -one of the most important parameter of specific risk with the lease structure - ispositively linked to the level of VaR. An increase of the average vacancy lengthimplies a rise in the VaR level. It comes from the reduction of the cash flowsgenerated by the considered leases. The effect is particularly marked in this casewhere only one lease is considered; in fact no diversification can reduce the impactof the void period. Similar conclusions can be drawn from the net income growthor from the market rental value trend. In the case of the NI growth, an increase ofthis parameter goes hand in hand with an increase of VaR. This comes from thenumber of break-options exercised. Indeed, if the level of rent indexation is higherthan the level of MRV trends, more options will be exercised because the rentalvalue will on average be higher than the rental growth. Conversely, an increaseof the market rental value trend leads to a decrease of the VaR level because onaverage the market rental value will exceed the rent and thus fewer options willbe exercised.

Beyond the analytical discussion, the application raises a comment about thetype of risk: market risk vs idiosyncratic risk. Market risk, represented by volatility,σ, has much more impact on VaR than the idiosyncratic risk. This comes from thefact that σ impacts also the idiosyncratic risk. Indeed, if σ equal 0, there is no riskat all and therefore breaks options will be exercised only when the indexation g isbelow the MRV trend µ. The importance of σ in our model thus comes from thefact that MRV has a strong impact on idiosyncratic risk (it increases the impactsof leases on cash flows).

The combined impact of the average vacancy length and of the volatility on a5|10-year lease can be observed in Figure 9 in Appendix C.

In Table 2, we consider more precisely the impact of the lease structure: thenumber of break options and the lease length. We observe that VaR decreases withthe security level of the lease. Let us specify that a lease is considered as moresecure if it exhibits no break-options. In this sense, from a landlord’s point of viewa 10-year lease contract will be more secure than a 10-year lease contract witha break-option at the end of year 5 (a 5|10-year lease).20 The lease clearly has astrong impact on VaR. The bottom of Table 2 displays VaRs for some long securelease terms. The VaRs in this case are clearly lowered. This comes from the lownumber of possible renegotiation periods (in this case, no breaks are considered

19 Comparing our model to a naïve Gaussian one is tricky. Indeed, our model is based on cash-flowsand not directly on a process that follows a certain distribution. However Gaussian can beused as a reference. For the same level of trend and volatility that the base case (respectively2% and 20%), the VaR attributed to a Gaussian distribution equals 50% (vs 12% in our basecase).

20 In this application, we have chosen to consider the end of the lease like a renegotiation period andnot like a break-option possibility (see introduction of Section 5). Therefore, at the end ofthe lease, the rent returns to the market rental value and no vacancies can be observed. Thisassumption reinforces the cash flows level in the case of fixed 10-year leases because no voidcan be observed.


given the assumption under which the end of the lease is not a break-option). Thisexplains the attractiveness of such assets (or leases) for investors. Indeed, theypromise long-term indexed future cash flows (generally for reliable tenants) andtherefore the required premium is often reduced (which implies a higher price).This kind of lease is common in London (UK) where initial yields are among thelowest (if not the lowest) in the world. Among other things, it explains why theLondon office market is one of the most appealing and active of the world. The casein point, where it is possible to break each year, are noticeably too risky. Exceptin some rare and specific cases, these kind of leases do not exist in commercial realestate. Long-term secured lease are attractive for investors but often they do notoffer enough flexibility for tenants. The 5|10-year lease is a good alternative as itoffers flexibility to tenants and is a good compromise in terms of risk. The VaRlevel is acceptable with this kind of lease.

Table 2: VaR0.5%,1 for a single space according to the lease structure

VaRCVR

0.5%,1VaR

TR

0.5%,1σ λ g µ Lease Structure

13.7% 3.2% 1020.8% 11.9% 20% 3 2% 2% 5|1028.4% 22.6% 1|2|3|4|5|6|7|8|9|10

12.6% 5.4% 520.8% 11.9% 20% 3 2% 2% 5|1022.0% 12.9% 5|10|15|20|2522.5% 13.0% 5|10|15|20|25|30|35|40|45|50

28.4% 22.6% 1|2|3|4|5|6|7|8|9|1028.8% 22.8% 1|2|3. . . 10. . . |14|1528.7% 22.5% 1|2|3. . . |10|. . . |24|2529.1% 23.0% 1|2|3|. . . |9|10|. . . |98|99

3.4% 0.8% 206.5% 1.7% 1513.7% 3.2% 1012.6% 5.4% 511.8% 6.2% 2

Figures 10 of Appendix C further illustrates the combined impact of averagelease length and volatility in the case where the lease structure is 1|2|3|4|5|6|7|8|9|10instead of 5|10.

Table 3 shows how the VaR0.5%,1 is modified according to the lease structureand the expected vacancy length (in year) λ. All parameters are those of the basecase.

As expected, the VaR whether it be for Total Return or Capital Value Returnincreases with both the vacancy length and the risk attached to the lease. The valueof VaR is thus positively linked to the vacancy length, meaning VaR numbersincrease with the average vacancy length. The difference between Total ReturnVaR and Capital Value Return VaR is somehow constant regardless of the averagevacancy length. In the same vein, the VaR level increases with the number ofbreak-options. Indeed, as soon as the security attached to the lease is reduced, the


VaR numbers raise. Interestingly, the difference between Total Return VaR andCapital Value Return VaR exhibits a gap when the security of the lease is minimal.In this case, the Total Return VaR steps up significantly more than the CapitalValue Return VaR (in particular for low value of λ).

Table 3: VaR0.5%,1 for a single space according to the lease structureand the expected vacancy length λ

λ = 1 λ = 3 λ = 5

VaRCVR

0.5%,1VaR

TR

0.5%,1VaR

CVR

0.5%,1VaR

TR

0.5%,1VaR

CVR

0.5%,1VaR

TR

0.5%,1

10 14.1% 1.4% 13.7% 3.2% 13.4% 4.7%5|10 15.2% 5.7% 20.8% 11.9% 25.7% 16.4%

5|10|15|20|25 15.9% 6.3% 22.0% 12.9% 27.8% 18.3%5|10|15|20|25|30|35|40|45|50 16.2% 6.3% 22.5% 13.0% 28.4% 18.3%

1|2|3|4|5|6|7|8|9|10 18.1% 11.6% 28.4% 22.6% 37.4% 30.7%

5.3 V aR analysis of a whole portfolio

The benchmark case is the same as in the previous subsection (5|10 lease, g =µ = 2%, rt = 6.5%, σ = 20%, λ = 3, the probability of being initially vacantequal 30%). The analysis is mainly conducted in order to highlight the impact oftwo parameters: the number of spaces for a given sub-market and the number ofsub-markets.

(a) VaR function of the number of spaces (b) VaR function of the number of MRV s

Fig. 6: Impacts on VaR from the number of spaces (one sub-market) and fromthe number of MRV s (one space in each sub-market)

Figures 6a and 6b underline classical portfolio diversification results. The VaRis vanishing with the number of spaces and with the number ofMRV s. The number


(a) σ = 20%, λ = 3 (b) σ = 30%, λ = 5

Fig. 7: Total return VaR0.5%,1 as a function of J , the number of submarkets(each encompassing five 5|10-year leases)

of spaces and MRV being in this case the number of assets. Note that the conclu-sion remains true whatever the values of σ, λ and ρ. It can be noticed in Figure6b that the VaR for one lease is equal to 11.9%, the value we get as the initialvalue of every curves. The analysis of the sub-market diversification is conductedaccording to the value of the correlation between the J sub-markets. For the illus-tration we set all the correlations equal to ρ. When ρ tends to 1, this is equivalentto having different spaces in the same sub-market, the bold lines in Figures 6aand 6b are thus similar. The diversification effects are far more important in thecase of the MRV ’s diversification than in the case of lease number. Indeed, in thelease diversification case, a decline in the MRV will impact the whole portfolio,however in the MRV ’s diversification case, the correlation is taken into accountand therefore a decline in one lease’s MRV will not necessarily impact the otherlease and thus the diversification effect is much more pronounced. In Figure 6b,VaR reach 0 (when ρ = 50%). In this case, at the 0.5% level, no loss are observedunder our assumptions.

Lastly, we consider a double diversification in Figures 7a and 7b where weanalyse the sub-markets diversification in the case where leases’ diversification ineach sub-market is already conducted. In this figure, we compute VaR for differentnumber of sub-markets (1 to 15), each containing 5 spaces (and thus a maximum of75 leases) and for various level of correlation. VaR levels clearly decrease even if it ismore obvious for lower volatility. In particular, when the number of sub-markets(each containing 5 leases) increases as the correlation simultaneously decreases,VaR vanishes. This has to deal with the real estate diversification. Real estateinvestors can choose to diversify by region or area (sub-markets), by the numberof assets (leases or spaces) or by sector (office, residential, industrial...). It has longbeen a debate in the literature. This last figure underlines how diversification of aportfolio in the number of leases and in the number of sub-markets is efficient (andmore efficient than in only one strategy). The double diversification is therefore astrategy that may be sought.


6 Conclusion

The challenge of real estate risk modelling is to adequately incorporate the speci-ficities of this asset class. Since the under or overestimation of risk can alternativelylead to high losses or to significant missed opportunities, it is important to reliablyestimate it. This paper is the first that proposes a model design for real estate VaRand takes into account the specificities of real estate asset class. Furthermore theproposed model allows computing ex-ante VaR, based on forward looking indicesof the portfolio.

The aim of this paper has been to compute VaR for direct commercial realestate investment. The existence of latent idiosyncratic risk in real estate portfolioinvestment, combined with the lack of data that the real estate industry faces,results in a systematic mis-estimation of risk when using the conventional VaRcomputation approach. To accurately estimate real estate VaR risk measure, tak-ing into account specific property features and future expected returns, we haveconsidered lease structure and their impacts on both price and income. Using themodelled price and income components, we built-up real estate indices. We thenperformed Monte Carlo simulations in order to deduct VaR from the replications.

A key feature of our analysis has thus been to deal with the difficulties forstandard risk modelling posed by the specificities of commercial real estate mar-kets. In light of all the recent regulations (Solvency, Basel) that have followed thesubprime crisis, risk measurements - in particular VaR estimates - are in greatdemand by the real estate industry, as well as by regulation authorities. Yet, todate, little research has concentrated on VaR analysis - or more generally on riskmeasurement - in the case of direct commercial real estate. This paper fills this gapby employing an approach that accounts for specific risk (particularly the leasingrisks). In practice, the model could be used to optimize real estate investmentgiven the required capital for a fixed horizon.

Doing so, the proposed model is able to differentiate between portfolio strate-gies (in terms of undertaken specific risk). In this sense, this model is the firstable to differentiate specific portfolio risk. Our article thus contributes to the ex-tant literature by proposing an original approach to commercial real estate riskassessment.

The approach could be adapted - with a few changes - to nearly all asset classesnot traded regularly such as art, wine, private equity venture funds, hedge funds,etc. Risk managers with a need to develop appropriate models of risk should finda useful approach here, one yielding “internal” models applicable to real estate, aswell as to many other areas. Other real estate specificities - such as obsolescence,leverage or other dynamics (non-normality of returns) - could be easily added tothe current proposal. In this study, we assume static portfolio (as requested byregulators). This assumption naturally derives from the fact that we are consid-ering a fixed horizon (1 year) and that the portfolio is closed at inception of theVaR estimation. An extension of the proposed model to dynamic VaR assessmentincorporating different aspects of real estate portfolio allocation is straightforwardand could be an interesting topic for further study.


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A Expected Shortfall

Expected shortfall (ES) - also called conditional value at risk (CV aR), or expected tail loss(ETL) - is another risk measurement derived from VaR. It is a valuable alternative to VaRas it is more sensitive to the shape of the tail of the loss distribution. This measure presentsthe advantage of being a coherent risk measure as opposed to VaR which is not, in general,a coherent risk measure (See (Artzner et al., 1999) or (Ahmed et al., 2007) for a detaileddescription of coherent risk measure).

Informally, ES is defined as the average of all losses which are greater than or equal toVaR. Thus, it is the average loss in the worst α cases, where α is the confidence level. In otherwords, it gives the expected value of an investment in the worst α% of the cases.21

ESα(X) = E(X|X < V aRα)

21 ES is not the worst case scenario (always 100% loss). ES is simply an average of losses below theselected risk threshold, α.


where VaR is the Value at Risk at the threshold α defined by P (X < V aR) = α (see intro-duction of this paper).

Despite its discrepancies and imperfections, VaR remains the most widely used risk mea-sure adopted as best practice by nearly all banks and regulators. However, over the past years,ES popularity has increased. In particular in the light of Basel III regulation where ES hasbeen proposed as replacement to VaR. The approach presented in this paper remains valid forES computation.

B Length of simulation: the choice of T̄

The choice of T̄ is dependant on the choice of the different parameters values. To set itsvalue for the application, we study the robustness of the VaR estimators according to thisparameter (the proxy of the infinite time horizon). Figure 8 illustrates that i) there is a bias(here upward) when considering values of T̄ no high enough, ii) this bias tends to zero asmentioned in equation 22 when T̄ increases.

T̄ is also strongly linked to the number of simulations. The number of simulation is itselfdependant on the requested precision. After 100 periods (T̄ = 100) the variations of the esti-mation reach an asymptote and VaR is somehow fixed. This suggest that T̄ has to be at leastequal to 100 + h as pointed out by equations 24 and 25.

(a) Capital Value Return (b) Total Return

Fig. 8: VaR0.5% as a function of T̄

In this article, we will use T̄ = 100 as a proxy for the infinite time period.


C Combined impact of vacancy length and volatility on V aR level

(a) VaR function of λ for a given σ (b) VaR function of σ for a given λ

Fig. 9: Total Return VaR0.5%,1 as a function of volatility and vacancyfor one 5|10 lease (g = µ = 2%; rt = 6.5%)

(a) VaR function of σ for a given λ (b) VaR function of λ for a given σ

Fig. 10: Total Return VaR0.5%,1 as a function of volatility and vacancyfor one 1|2|3|4|5|6|7|8|9|10 lease (g = µ = 2%; rt = 6.5%)

ex-ante real estate value at risk calculation method

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