The Equity Crowdfunding Platform of
Unison Home Ownership Investors
Student name: Rogier Arends Student number: 10645136
MSc FIN: Dual Track – Real Estate Finance & Corporate Finance Document: Master Thesis
Thesis supervisor: Dr. M. Constantinescu Month and Year: August, 2018
Abstract
This thesis investigates the sustainability of the real estate platform of Unison Home Ownership Investors. The results are achieved by comparing the investments by Unison Home Ownership Investors to real estate derivatives and applying a methodology proposed by Van Bragt et al. (2015) to value options. The research shows that the final call option values indicate a positive value, which means that the equity crowdfunding platform of Unison is profitable. In addition, the findings suggest that each state has a different option value, although all are positive.
Statement of Originality
This document is written by Rogier Arends, who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Table of Contents
1. Introduction 4
2. Theoretical Framework 6
2.1 Potential Diversification Benefits and Drawbacks 6
2.2 The effects of an increase in homeownership 7
2.3 Unison HomeBuyer Transaction 9
2.4 Derivatives 10
2.5 The Product of Unison: HomeBuyer Agreement 11
2.6 Real Estate Derivatives 11
2.7 Real Estate Derivative (Option) Pricing 13
3. Thesis Objectives 15
3.1 Hypotheses 15
3.2 Data 16
3.2.1 Explanation of Data Sources 16
3.2.2 Descriptive Statistics 17
3.2.3 Correlation matrices 19
4. Methodology 21
4.1 Binomial Trees 21
4.2 Methodology to Value European Options 22
4.2.1 Adjusted Volatility 22
4.2.2 Asian options 23
4.3.3 Pricing Formulas 23
5. Results 24
5.1 Binomial Trees 24
5.2 Results Methodology to Value European Options 26
5.2.1 Outcomes Adjusted Volatility 26
5.2.2 Outcomes Asian Options and the Pricing Formulas 27
6. Robustness 30
7. Conclusion and Discussion 32
Reference List 34
1. Introduction
Tech companies are becoming more interested in the world of real estate and they have introduced all kinds of new initiatives (Financieel Dagblad, 10 January 2018). These are called ‘PropTech’, which is a contraction between property and technology. One of those initiatives is, Unison Home Ownership Investors (commonly known as Unison), and it will be explained shortly below.
In the United States a buyer of a house normally only gets 80% of the value of the house financed by debt. This means that when a house costs $500,000, the buyer has to have $100,000 up front. This implies that a large number of people will not be able to buy their own house or the house they want. A company, called Unison, has started in the United States and tries to solve this problem by pairing investors with prospective buyers. Unison provides 50% of the down payment and then will share in the profit or loss of the house. The buyers of the house still need to get a traditional mortgage of 80%, which in this example means a mortgage loan of $400,000. The share of change in value of the house will be split into 65% for the buyer and 35% for Unison (the investors). There is also a possibility to share a down payment of 25% and in that case Unison (investors) share in 43.75% of the change in value. The money Unison provides is an investment, not a loan, so there are no interest charges and you don’t have to make monthly payments to Unison. Instead, Unison hopes to earn a return on its investment from a portion of the appreciation when you sell at a profit. When you sell, Unison receives a single payment equal to its original investment plus or minus a share of the change in value of your home.
The concept of this online real estate platform allows investment- and pension funds and insurance companies access to one of the biggest market is the world: houses. In the US, alone, this market has a value of 21,000 billion dollars, of which approximately half are owned houses and half of them have a mortgage. Unison Home Ownership Investors has announced in February 2017 that they raised over $300 million in total capital (PR Newswire, 2017). This means that investors see a lot of opportunity in the platform, and this raises the question: Are equity crowdfunding platforms for real
estate, such as the platform of Unison Home Ownership Investors, profitable?
There have been various studies on the effect of adding real estate to the portfolios of heterogeneous investors (Hudson-Wilson et al., 2003; Ennis and Burik, 1991; Ziobrowski and Ziobrowski, 1997; Chaudhry, Myer and Webb, 1999; Hoesli et al., 2004). On the other hand, potential drawbacks of including real estate are also mentioned (Hoesli et al., 2003; Solnik and McLeavy, 2013). Others have investigated the effects of an increase in homeownership and Coulson and Li (2013) and Andrews and Sánchez (2011) both give an overview of these studies. Furthermore, the use of real estate derivatives has been researched (Fabozzi et al., 2010). Methodologies to price the real estate derivatives are provided by a number of different studies (Ciurla and Gheno, 2009; Van Bragt et al., 2015; Cao and Wei, 2010; Fabozzi et al, 2011).
In order to answer the research question, I will argue that the investments can be seen as options and thereafter I will analyse the option values. These values will be determined first by calculating the binomial trees for each state that Unison operates in. In addition, the methodology proposed by Van Bragt et al. (2015) will be used to calculate the option values more precisely. There will be three hypotheses investigated in this research. The first and main hypothesis is about the profitability of the equity crowdfunding platform of Unison Home Ownership Investors. This means there will be taken a look at whether the final call option values are positive of negative. The second hypothesis deals with the question whether each state has the same option value or if they differ. The third and final hypothesis examines whether each state is worth investing in or if some states are profitable to invest in and other states are not.
This study adds to the existing literature in multiple ways. First, it applies option pricing to a real estate platform in a way that has not been done before. Second, it tries to investigate whether real estate platforms such as the Unison platform could lead to potential diversification benefits for investors. In addition, it might be a viable way to increase homeownership, which overall has a positive impact on society.
This thesis has the following structure. Paragraph 2 will describe the theoretical background of the topic and elaborate on existing literature on the topic. Paragraph 3 explains the data, the fourth paragraph will propose the methodology and paragraph 5 will clarify the results. In the sixth paragraph, the 3 robustness checks will be explained and finally, the conclusion will follow in paragraph 7.
2. Theoretical Framework
2.1 Potential Diversification Benefits and Drawbacks
As mentioned above, the product offered by Unison Home Ownership Investors allows investors access to the housing market of the states the company operates in. This therefore means that investors are able to invest in another asset class, i.e. they will be able to diversify. Geltner et al. (2014) mention the importance of a bigger quantity, variety and range of alternative assets that can be offered to heterogeneous investors. This is achieved because real estate, and specifically houses, represents a different type of underlying physical assets. In the bigger picture, four major asset classes can be identified: stocks, long-term bonds, cash (T-bills), and real estate (Geltner et al., 2014).
A study by Hudson-Wilson et al. (2003) proposed five reasons for the inclusion of real estate in a properly-managed institutional investment portfolio. The first one is that the overall risk of the portfolio will decrease since asset classes that respond differently to expected and unexpected events are combined. Secondly, an absolute return that is competitive with other asset classes can be achieved. Next, it is a possible hedge against unexpected inflation. A fourth reason may be to manage a portfolio that is closest to an indexed, or market-neutral portfolio. This signals a reasonable reflection of the overall possible investments. A final reason is to achieve an income return, which equals a strong cash flow (Hudson-Wilson et al., 2003).
There have been multiple studies (Ennis and Burik, 1991; Ziobrowski and Ziobrowski, 1997) which concluded that approximately 15 to 30 percent of a mixed-asset portfolio should be allocated to real estate. They argue that real estate returns have a low correlation with the returns of stocks and bonds and in addition, Chaudhry, Myer and Webb (1999) find that stocks are inversely related to real estate on the long term and furthermore they note that the overall impact of stocks on the real estate market is less severe than its impact on bonds and T-bills. Hoesli et al. (2004) determine in their paper that the optimal allocation to real estate using hedged returns is 15 to 25 percent.
Despite the optimal allocation to real estate, Chun and Shilling (1998) and Geltner et al. (2014) show that institutional investors allocate a substantially lower weight to real estate. There have been several studies investigating the cause of this phenomenon. Chun et al. (2000) and Craft (2001) state that using an asset-liability framework, the optimal weight assigned to real estate is closer to the weight that institutional investors normally have. This is in contrast with the asset-only framework that is used in most studies regarding this topic. Kallberg et al (1996) argue that the imperfections of the real estate market are often not taken into consideration.
Nonetheless, there are also some drawbacks to investing in the physical real estate market. Hoesli et al. (2003) and Solnik and McLeavy (2013) mention that this market has been characterized by a relative lack in liquidity, high transaction and high maintenance costs, heterogeneity in the assets,
and this market has been subject to negative externalities. However, Garay and ter Horst (2009) state that these costs of investing in real estate might have gone down due to an enhancement in initiatives to decrease illiquidity and increase transparency in the property derivatives market.
2.2 The effects of an increase in homeownership
Moreover, as stated on the website of Unison, ‘the focus of our company is on expanding opportunities available to consumers and giving them the financial flexibility to purchase the home they really want and finance important needs during their life as a homeowner. Unison is dedicated to providing consumers with financing based on partnership that will enrich their lives.’ From the first part of the statement can be taken that the goal of Unison is to allow more people to buy the house they want. This ambition is in reality needed as well, since home prices have appreciated with 8% or higher in many metro areas during the period 2011-2016 (Unison, 2017). Due to their job growth and opportunities, these same metro areas attract demographic groups who are most likely to buy a home, such as Millennials and new families. Since monthly rents paid by renters in most cities across the United States have also gone up significantly (Wall Street Journal, 2015), there is more financial pressure on those who are trying to save for a down payment. Here is where the platform of Unison comes into play and allows more people to buy a house. This would in turn lead to an increase in homeownership rates and there have been multiple studies investigating the effect of homeownership on economic performance as well as broader social benefits. Coulson and Li (2013) identify three different sets of effects in the literature, with contrasting findings in all three of them.
The first one is classified as maintenance and appearance. According to Rossi-Hansberg et al. (2010) there are significant spill-over effects for the neighbourhood when houses are properly maintained and renovated. The literature mentions that owner-occupiers are more likely to perform this neighbourhood maintenance. The reason being that they benefit directly from it, whereas tenants do not. As a consequence, tenants have little incentive to undertake maintenance directly. Henderson and Ioannides (1982) argue that the landlords cannot commit to a compensation for proper maintenance which thus results in less maintenance. This is partly because the costs are higher for absent landlords. In addition, Galster (1983) and Harding et al (2000) both find that rental properties have a worse maintenance than owner-occupied properties. These findings are in confirmation with a paper by DiPasquale and Glaeser (1999). On the other hand, Gatzlaff et al. (1998) report that the differences between appreciation rates of owner-occupied properties and properties with tenants are negligible.
Secondly, the effect on children will be discussed. As stated in papers by Green and White (1997) and Haurin et al. (2002) there is evidence that implies that children growing up in
owner-occupied properties perform better at high school and receive higher cognitive test scores. Aaronson (2000) provides the argument that home-ownership increases residential stability as potential reason. However, it is important to note that there is also literature that suggests that the link between childhood outcomes and ownership is not that straightforward. According to Holupka and Newman (2012) there is little evidence for beneficial homeownership effects and they suggest that previous literature might have falsely interpreted selection differences for the effect of homeownership itself.
Finally, Coulson and Li present their third effect of homeownership, namely on citizenship. DiPasquale and Glaeser (1999) report that homeownership creates barriers to mobility and as a result homeownership stimulates investment in local amenities as well as social capital. Another reason, relating to the constraints to mobility, is that homeowners have more incentive to improve their community. They use a German Socio-Economic Panel to determine a connection between homeownership and citizenship, where they control for individual fixed effects. In addition, they find that homeowners are more informed about their local elected official and also vote more frequently. Another effect of homeownership on citizenship, albeit less productive, is that homeowners are more active adherents of NIMBYs (i.e. not in my backyard), as mentioned by Fischel (2001). In contrast, Engelhardt et al. (2010) did not find large evidence for an increase in neighbourhood involvement by new owners. They also mention that there is an endogeneity bias in the positive impact of homeownership on political engagement, which makes it likely to be over-stated. The reason being that people who are actively participating in community activities are more probable to be homeowners.
Adding to this, Andrews and Sánchez (2011) also give an overview of economic benefits and costs of homeownership. They mention four effects, of which a better outcome for children and community engagement and voting behaviour are in line with the effects Coulson and Li (2013) identify. However, two different effects are also discussed.
Firstly, homeownership could be a vehicle for asset or wealth accumulation. For myopic households it will create an orientation towards the future and thus may result in a more wealth accumulation than otherwise (Sherraden, 1991; OECD, 2003). By getting a mortgage debt, the household spending behaviour may also change since breaking the payment scheme is costly. The effectiveness of this last point has weakened in recent years given the increased possibilities of mortgage refinancing and withdrawal of housing equity (Li & Yang, 2010). On the other hand, there are higher transactions costs involved with buying a house than with renting, making it an illiquid investment (Haurin & Gill, 2002). In addition, Ferreira et al. (2008) and Caldera Sánchez and Andrews (2011) find that the timing of the purchase also matters because of the volatility of housing prices. Recent policies in the United States to promote homeownership have provided evidence that it is adversely related to mobility and can even lead to a rise in negative equity.
Secondly, homeownership is shown to adversely affect labour mobility. Oswald (1996) and Caldera Sánchez & Andrews (2011) show that labour mobility is lower because of high transaction costs of moving. Consequently, unemployment among owner-occupants is higher than among renters. Munch et al. (2006) find that the overall hazard rate into employment is higher for homeowners, which signals a negative correlation between homeownership and unemployment duration.
2.3 Unison HomeBuyer Transaction
With this example, the product of Unison Home Ownership Investors will be explained in more detail. A purchase price of $500,000 will be assumed.
An 80 percent mortgage means that $400,000 will be borrowed and that the down payment will equal $100,000 (20 percent of the purchase price). Unison will provide 50 percent of the down payment, and the remaining 50 percent will be provided by the home buyer. This means that a total of 10 percent of the purchase price will be provided by Unison. In this case, Unison provides $50,000 and the home buyer also provides $50,000. In exchange for their share in the down payment, Unison gets a return on its investment equal to 35 percent of the change in value of the house when it is sold. There are three possible outcomes: the home has gained value, the home has lost value, and home value has stayed the same. With a sale price of $600,000, there is an increase of $100,000 compared to the purchase price. The share of change in value for Unison is 35 percent, which leads to a payment of $85,000 to Unison. This amount consists out of the share in profit and the initial investment of $50,000. With a loss of $100,000, so a sale price of $400,000, Unison will share in the loss for 35 percent as well. Therefore, the payment to Unison after selling will only be $15,000. This is the initial investment of $50,000 minus the share in loss of $35,000. When the home value stays the same, there is no share in profit or loss for Unison and the payment from the home buyer to Unison after selling their house is equal to the initial investment of Unison.
The agreement between the home buyer and Unison has a few conditions. The first one is that the home owner/buyer will pay the expenses when the home is sold. This means that they are responsible for paying of their mortgage and any other amounts secured by their home. The home buyer is also responsible for paying all costs that are associated with selling a house, such as the brokerage commission. Another condition is that home buyers can terminate the agreement without selling their house only after the first three years. This is called a Special Termination and in order to do this, an independent and Unison-approved third-party appraisal management firm will determine the market value of the house at that moment. There will be a payment to Unison that is equal to the initial down payment Unison originally provided, as well as their share in profit, if any, Unison would have made if the home buyer would sell their house at that given moment. This means that with a loss
in value, Unison still receives its initial investment back. A possible reason for applying for Special Termination could for instance be that one has information that the house price will increase in the nearby future and also has the resources to pay the initial investment plus the profit, if any. In that way, the home owner can profit optimally from the increased house price without sharing this profit with Unison. A third condition is that in case of remodeling or improvements one can apply for a Remodeling Adjustment. When this is approved, the Remodeling Adjustment will be equal to the portion of the house value that is attributable to the remodeling or the improvement. So, the adjustment is not equal to the cost of the remodeling or improvements and this is because remodeling projects often do not increase the price of a property with a 1:1 ratio. Next, is the application of the Deferred Maintenance Adjustment. This adjustment is applied when the house owner does not ensure that the property only has normal wear-and-tear due to improper maintenance. A final condition is that the Unison HomeBuyer Agreement is not designed as a source of short-term financing. Therefore, the right to request a special termination is after the third-year anniversary of the agreement. This also counts for the remodeling adjustment and at last, if the home buyer sells the property within the first three years of the agreement for a price lower than the purchase price, Unison will not share in the loss (Unison Home Ownership Investors, 2018).
In case the home owner gets behind on their mortgage payments or decides to strategically default because the home value is lower than the value of the debt, Unison has the right to protect its investment with a foreclosure on the property. Moreover, if the home owner faces foreclosure by its lender, Unison will help to sell the home in a normal fashion in order to maximize the sale price. This is because as an investor, they share the desire to protect the equity in your home.
2.4 Derivatives
According to Bodie et al. (2014) derivative securities, also simply called derivatives, are ‘securities whose prices are determined by, or “derive from”, the prices of other securities.’ Options, futures contracts and swaps are all derivative securities since their payoff is based on the value of other securities. There are two distinct kinds of option contracts: a call and a put option. A call option gives its holder the right but not the obligation to purchase an asset at a specified price, the exercise or strike price, on or before a specified expiration date. In contrast, the right to sell an asset for a specified exercise or strike price on or before an expiration date is called a put option. An option can be bought for a premium, which is essentially the purchase price of the option (Bodie et al., 2014). Berk & DeMarzo (2014) state there are two kinds of options; American options allow their holders to exercise the option at any time up to and including the expiration date. These American options are more common than European options. The latter only allow their holders to exercise the option on the
expiration date.
Similar to options, futures and forward contracts specify a purchase or sale of an underlying security at a future date. The key difference, however, is that a futures or forward contract carries the obligation to go through with the transaction. Whereas the holder of an option can decide not to exercise the option if the trade would be unprofitable. Swaps are multi-period extensions of forward contracts (Bodie et al., 2014).
2.5 The Product of Unison: HomeBuyer Agreement
The investment made by the investors of Unison can be seen as a derivative, and specifically a real estate derivative, since the value of the investment depends on the underlying value of the house for which a down payment is provided. Van Bragt et al. (2015) define a real estate derivative in the following way: ‘a contract where the future payoff depends on the realization of a real estate price index or a real estate total return index.’
The down payment or investment that is provided by Unison (and its investors) has similarities with a call option. Similar to normal options, the payoff is completely dependent on the underlying product. The price of the houses on which a down payment is provided can be compared with the stock price on which normal options are written. The average house price of the investments from Unison may be represented by a median or average house price in the states that the product is introduced in. Furthermore, the down payment can be seen as the option premium.
Nevertheless, there are also differences with normal call options. Firstly, the holder of the option does not have the choice whether they want to exercise their option since this right is in hands of the home owner. Secondly, there is no pre-specified exercise price or strike price. A last major difference is the fact that, corresponding with the right for home owners to choose when to sell their house, there is no specified expiration date.
Even so, it will still be possible to value the investment as a call option. This is because the first difference, regarding that the choice to exercise lies with the homebuyer and not the investors, will probably only have a minor negative effect and not a major impact on the valuation of the option. The reason for this is that the home buyer will also face a loss and house owners are less likely to sell below the purchase price (Genosove & Mayer, 2001). Next, the pre-specified exercise price or strike price can be thought of as the house sale price at the moment of selling, which will be estimated with a Monte-Carlo analysis. Finally, the problem of no expiration date can be solved by looking at the average home tenure in the United States, and ideally the specific states. In addition, it could be possible to make a sensitivity analysis by using different holding periods of houses. Because of the possibility for special
termination after the third year, there is a put option embedded in the contract and this implies that the values that are found are a maximum value. This value is limited by a homeowner selling early.
2.6 Real Estate Derivatives
Case and Shiller (1996) recommend the use of index-based futures contacts and options for hedging mortgage risk, default risk, and real-estate price risk. However, the introduction of these derivatives in the real estate market is not straightforward since liquidity is likely to be low when returns are predictable. Carlton (1984) argues that the perception of changes in prices cannot be risky since these changes are predictable. As a result, the market sentiment is unidirectional which makes it difficult to find counterparties. Nonetheless, in their paper, Fabozzi et al. (2010) argue that when a counterparty in a property derivatives contract is taking a position against the market trend, this counterparty does not necessarily need to be a speculator. The reason being that trades may be executed on the futures curve of the real estate index. Consequently, market participants such as hedge funds, pension funds or private equity funds may be providing the desired liquidity.
This liquidity of the property derivatives is crucial to the utilization of these derivatives by investors. Fabozzi et al. (2010) propose that by using property derivatives on some relevant real-estate indices, the real estate risk could be mitigated. Another benefit is that real estate derivatives could potentially lower the price volatility of properties. In a study published in 2008, Hinkelman and Swidler investigate the possibility of hedging housing price risk with futures on financial indices and other commodities. They find that a systematic relationship between national housing prices and prevailing traded futures contracts is not supported by significant evidence. Therefore, completion of the real estate markets would come closer by establishing property futures markets.
In addition, Fabozzi et al. (2010) mention several other advantages of advances in the futures market on real estate indices. Firstly, they may improve the efficiency in the spot markets as well as improving price discovery. Another benefit would be that futures markets may give an indication of the level of spot prices for future and current spot prices. Furthermore, real estate derivatives give investors means of entry to a difficult to access, but important asset class. Finally, due to low correlation of housing prices with other asset classes, including real estate derivatives in diversified portfolios is likely to be beneficial.
Fabozzi et al. also give an overview of categories of end users for whom real estate derivatives are advantageous. The first category includes property owners and private investors who specialize in real estate. This category could potentially be very large, but a majority of these do not make use of derivatives due to a lack of knowledge and transaction costs barriers. A second category consists of asset managers. They may use real estate derivatives to hedge their price risk exposure in domestic
real estate as well as foreign properties. The third category is closely related to the second one, since they are also seeking to hedge their positions. It consists of dealers and portfolio managers in structured products. A final category includes structurers of newly designed structured products for which they can make use of real estate derivatives. Real estate derivatives can be used to hedge against price risk, which is for what the first category uses them. The users of real estate derivatives in the other categories might also employ them for hedging against interest rate risk and currency risk, or exchange rate risk (Fabozzi et al, 2010).
Shiller (2008) gives an overview of two theories that try to explain why the real estate derivative market has trouble to develop. First of all, there is often cited that home owners are reluctant to realize a loss on their home, so they avoid selling until the market has gone up and provides them with a profit. Regret theory models this behavior (Loomis & Sugden, 1982). By using derivatives to hedge, home owners might lose less if house prices go down. However, if house prices are stagnant, home owners will still lose. Thus, regret theory explains why home owners are not likely to hedge using real estate derivatives. Another theory is that owner occupants are already self-hedged and thus do not need to hedge their risks. What this means is that the price of a house is irrelevant to them as people generally expect to live in their property forever. However, Shiller (2008) does not believe that these theories are the primary reason for the slowness of the growth of the real estate derivative markets. He proposes that the principal problem is the lack of liquidity, which as explained above is crucial.
Some special features of the housing market are, according to Case and Shiller (1989), the fact that residential property market is mainly untouched by institutional investors because of the nature of the transactions, the type of contracts, and a lack of knowledge of pricing models (Gemill, 1990). Moreover, there is no single market-price, no frequent trading of housing properties, and the housing properties, i.e. the underlying asset, are heterogeneous. In addition, there is information asymmetry and there is positive serial correlation between house prices and the excess returns.
2.7 Real Estate Derivative (Option) Pricing
There are a large number of studies that provide us with methodologies in order to price real estate derivatives, and specifically options since this is the main goal of this research. For instance, Ciurlia and Gheno (2009) followed a no-arbitrage approach and applied a two-factor model that takes into account the sensitivity to interest rate term structure of the real estate market. Here, they modelled both the real estate asset value as well as the dynamics of the spot rate in order to find the prices of hypothetical European and American options.
real estate derivatives contingent on autocorrelated indices. They believe that the observed index randomly deviates from an unobserved underlying index, which reflects the fundamental value of the property market. By updating the method of Blundell and Ward (1987), they introduce analytical pricing formulas for numerous real estate derivatives.
Cao and Wei (2010) develop an equilibrium model to price housing index derivatives by including the housing market to the financial and goods market of Lucas’ equilibrium model. In their paper, they suggest closed formulas for equilibrium forwards and European vanilla call options. Fabozzi et al. (2011) suggest a pricing framework where the strategy is to find out the market price of risk using their proposed model and a futures or forwards market. Thereafter, it is possible to determine a risk-neutral pricing measure for other real estate derivatives such as options. In this thesis, the methodology proposed by Van Bragt et al. (2015) will be used since this methodology is most easily adapted to the specific case of Unison. Their analytical pricing formulas can be filled in using data that is applicable to the houses for which Unison provides a down payment. For instance, the house price index of the state that the house is located in can be used to calculate a standard deviation. All other data that is needed for the formulas is easily accessible as well, such as the homeownership tenure that is represented by the time (T) in all the formulas, the autocorrelation that can be determined from the house price index, or the drift parameter (π) that is calculated from the short interest rate.
3. Thesis Objectives
3.1 Hypotheses
In order to be able to find the relevant data for the research, it is important to determine what will be researched. There will be investigated, whether, from the perspective of the investor (Unison), the platform is sustainable. As explained above, an analogy can be made with respect to real estate derivatives, and specifically a call option. The inputs to calculate these can be found at the Federal Housing Finance Agency (FHFA) House Price Index or the Standard and Poor Case-Shiller Home Price Indices. The following hypotheses were formed:
Hypothesis 1:
H0: The equity crowdfunding platform of Unison Home Ownership Investors is not profitable. H1: The equity crowdfunding platform of Unison Home Ownership Investors is profitable. It might also be interesting to look at whether the option values vary between the states that Unison operates in. The product is at the moment of writing this thesis available in 13 different states. Since these states have different economic bases, they will have a different economic growth and thus other developments in house prices. Therefore, the following hypotheses were formed.
Hypothesis 2:
H0: The option values do not vary between the states that Unison operates in. H1: The option values do vary between the states that Unison operates in.
By looking at whether the option values are negative in one state and positive in another, it will be interesting for investors to see which states they should invest in and in which states they should not.
Hypothesis 3:
H0: The option values are all negative or all positive in the states that Unison operates in. H1: The option values does vary between the states that Unison operates in.
As will be shown in section 3.2.2, the house price indices of the states are correlated with each other. This means that changes in the house price index in one state are likely to have an effect on the house price index of another state. However, the indices are not perfectly correlated and as a result, the expectation is that option values will vary between the states the Unison operates in.
3.2 Data
Data in this thesis include the Federal Housing Finance Agency (FHFA) House Price Index for the 13 different states that Unison operates in, and the Standard & Poor (S&P) Case-Shiller Home Price Indices for several big cities in the United States. The latter is used to perform a robustness check with respect to the data used to calculate the option values. Furthermore, the 3-month Treasury Bill secondary rate has been retrieved from the Federal Reserve Economic Data. These data will be explained in more detail below.
3.2.1 Explanation of Data Sources
The FHFA House Price Indices give quarterly index values a particular state in the United States. It is available for all 50 states. The index values are estimated using sales prices and appraisal data and the base of the indices is the first quarter of 1980. The datasets contain the years 1975 to 2017 and are not seasonally adjusted. This means that if the time series exhibits a seasonal pattern, this seasonal component is still apparent in the data. However, since real house prices also experience seasonal components, it is better to use the non-seasonally adjusted datasets. Unison Home Ownership Investors is, at the moment of writing this thesis, active in 13 states, namely: Arizona (AZ), California (CA), Connecticut (CT), District of Columbia (Washington, DC), Illinois (IL), Maryland (MD), Massachusetts (MA), New Jersey (NJ), New York (NY), Oregon (OR), Pennsylvania (PA), Virginia (VA), and Washington (WA). Therefore, the house price indices of these states are used in the research. The S&P Case-Shiller Home Price Indices contain monthly index values for the cities: Boston (MA), Chicago (IL), Los Angeles (CA), New York (NJ/NY/PA), Phoenix (AZ), Portland (OR), San Diego (CA), San Francisco (CA), Seattle (WA), and Washington (DC/MD/VA). The S&P Case-Shiller Home Price Indices start from the year 1987 and contain data up to 2017. For Phoenix and Seattle, the index starts respectively from 1989 and 1989. The index values are not seasonally adjusted and contain monthly data. For New Jersey and Pennsylvania, there will be looked at the S&P Case-Shiller Home Price Index from New York. Furthermore, Maryland and Virginia also do not have a specific index for one of their cities and therefore these are included in the index for Washington (DC).
In order to determine the Vasicek Model1, the short-term interest rate is needed. Therefore, the 3-month Treasury Bill, secondary market rate is retrieved from the Federal Reserve Economic Data. It contains monthly observations that are non-seasonally adjusted and start from the year 1975.
Other inputs which are needed are the median house price (i.e. S0), which is found at the website of Zillow on the 18th of June 2018. Furthermore, to determine the strike price (i.e. K), we need
the expected inflation for the years up to 2026 and for the years up to 2023 this value is taken from Statista. The last two years are assumed to have an expected inflation of 2.10 percent. Also, the homeownership tenure for quarter 2 of 2017 for the United States has been found as 8.05 years (Attom, 2017) and therefore there will be made use of 8 years as average time before selling a house in the calculations. Finally, the long-term interest rate is taken from OECD data and this is given to be 2.98 percent in May 2018.
3.2.2 Descriptive Statistics
In this section, the descriptive statistics of the yearly returns will be given. These are computed from the quarterly data. The first column shows the number of observations and the second column gives the average of the returns. In the third column, the median can be found and in column 4 the standard deviation, which represents the volatility, is given. The fifth column gives the standard error of the mean, which is calculated by dividing the standard deviation by the square root of the number of observations. The last two columns show the minimum and maximum values of the quarterly returns. Table 1: Descriptive Statistics FHFA Indices
Note: This table shows the descriptive statistics of the Federal Housing Finance Agency House Price Indices of the 13 states of the United States that Unison operates in.
In table 1, the highest means of the quarterly return of the house price indices can be found in the District of Columbia and in California. These are respectively 7.47 percent and 7.30 percent. The lowest average of the quarterly returns is in Illinois with 4.03 percent, followed by Pennsylvania with 4.32 percent. So, the difference between the highest and lowest mean yearly return is substantial. The highest averages are accompanied by high standard deviations, the volatility in the District of Columbia is 8.43 percent and 10.09 percent in California. The low means are also characterized by low volatilities. Interesting to note is that Arizona has the second highest volatility, but the mean of the yearly returns is only lower in 4 other states. In addition, just 3 states have a lower average yearly return than Connecticut, whereas this state has the third highest volatility. Furthermore, Washington is the state States Number of Observations Mean Median Standard Deviation SE mean Minimum Maximum
Arizona 42 4.82% 4.98% 9.17% 1.42% -17.01% 32.71% California 42 7.00% 7.30% 10.09% 1.56% -18.77% 26.81% Connecticut 42 4.79% 4.03% 8.49% 1.31% -7.16% 30.31% District of Columbia 42 7.47% 6.21% 8.43% 1.30% -5.80% 29.82% Illinois 42 4.03% 3.85% 4.79% 0.74% -8.37% 18.94% Maryland 42 5.04% 4.40% 6.58% 1.02% -9.64% 21.40% Massachusetts 42 6.12% 5.43% 7.79% 1.20% -7.71% 26.83% New Jersey 42 5.28% 3.28% 7.80% 1.20% -6.98% 25.13% New York 42 5.32% 3.98% 7.00% 1.08% -6.26% 19.00% Oregon 42 5.88% 6.11% 7.07% 1.09% -10.51% 20.23% Pennsylvania 42 4.32% 3.44% 4.52% 0.70% -4.38% 15.25% Virginia 42 4.59% 4.24% 5.45% 0.84% -6.76% 18.84% Washington 42 6.31% 5.25% 7.39% 1.14% -10.36% 24.69%
with third highest average yearly return, but there are 6 other states with a higher volatility. As can be expected, columns 6 and 7 show that the states with high standard deviations generally also have smaller minimum values and higher maximum yearly returns.
Table 2: Descriptive Statistics S&P Case-Shiller Indices
Note: This table gives the descriptive statistics of the Standard & Poor Case-Shiller Home Price Indices.
From column 2 of Table 2, it becomes apparent that the mean of the yearly returns of the S&P Case-Shiller Home Price Indices is highest in the cities San Francisco and Portland. The mean is 6.37 percent in San Francisco and 5.86 percent in Portland. The lowest averages of yearly returns can be found in New York (3.41 percent) and Chicago (3.31 percent). Again, the difference between the highest and lowest average yearly return is substantial. The high mean return of San Francisco is accompanied by the second highest volatility of 12.34 percent and the lowest average (Chicago) comes with the lowest volatility of 6.32 percent. On the other hand, Portland has the third lowest standard deviation of the 10 cities, whereas it has the second highest average return. Furthermore, the highest volatility can be found in Phoenix, although the average return is higher in 6 out of the 10 cities. In line with table 1, the indices with the high volatilities have the lowest minimum and maximum yearly returns. For instance, Phoenix has the highest standard deviation and also achieved the lowest minimum and highest maximum of yearly returns. Similarly, San Francisco has the second highest volatility and also the second smallest minimum and second highest maximum. Moreover, Chicago has the lowest volatility and the minimum and maximum of yearly returns are closer to zero.
Cities Number of Observations Mean Median Standard Deviation SE mean Minimum Maximum
Boston (MA) 31 3.72% 3.61% 6.37% 1.14% -10.71% 17.67%
Chicago (IL) 31 3.31% 3.39% 6.32% 1.14% -16.35% 14.29%
Los Angeles (CA) 31 5.76% 6.75% 12.21% 2.19% -25.79% 27.14%
New York (NJ/NY/PA) 31 3.41% 1.85% 7.06% 1.27% -9.73% 17.06%
Phoenix (AZ) 29 4.15% 5.12% 13.02% 2.42% -34.95% 42.65%
Portland (OR) 31 5.86% 7.17% 6.93% 1.24% -13.99% 22.05%
San Diego (CA) 31 5.61% 5.86% 11.21% 2.01% -24.91% 25.46%
San Francisco (CA) 31 6.37% 9.03% 12.34% 2.22% -32.32% 31.16%
Seattle (WA) 28 5.35% 6.75% 7.30% 1.38% -15.00% 18.05%
3.2.3 Correlation Matrices
Table 3: Correlation Matrix, States
Note: This table gives the correlation between the yearly returns of each state.
In table 3, the correlation matrix between the yearly returns of each state is presented. A low correlation implies that the potential for diversification is bigger. This table shows that slightly more than half of the correlations lie between 0.4 and 0.7, which means that the house prices in these states move quite similarly. This signals that it might be smart for investors to diversify their investments between different states. The highest correlation found in table 3 is 0.920, and this is between the yearly returns of Maryland and Virginia. Other high correlations are present between Connecticut and New Jersey (0.894) and between New Jersey and New York (0.890). A likely reason for this is that these states border each other. The lowest correlation is found for Oregon and Massachusetts, with -0.021. This implicates that the house prices in these states move completely independent from each other and thus give great diversification benefits. Other low correlations are between Oregon and New York (0.077) and between Massachusetts and Washington (0.119).
Table 4: Correlation Matrix, Cities
Note: This table gives the correlation between the yearly returns of each city.
The correlation matrix between the yearly returns of each major city that lies in states that Unison operates in can be observed in table 4. What becomes apparent is that the correlations between the returns of the cities are generally higher than the correlations between the returns of states. A potential reason for this is that these cities are all major cities and demand for houses in large cities is generally more stable. The lowest correlation is found between Portland and Boston with 0.130. This is in line with the correlation matrix in table 3, since these cities are located in Oregon and
AZ CA CT DC IL MD MA NJ NY OR PA VA WA AZ 1.000 CA 0.665 1.000 CT 0.474 0.480 1.000 DC 0.622 0.804 0.671 1.000 IL 0.526 0.721 0.492 0.661 1.000 MD 0.679 0.853 0.655 0.788 0.748 1.000 MA 0.398 0.442 0.808 0.475 0.452 0.519 1.000 NJ 0.551 0.571 0.894 0.692 0.602 0.750 0.849 1.000 NY 0.426 0.455 0.838 0.500 0.550 0.663 0.890 0.891 1.000 OR 0.684 0.578 0.172 0.519 0.674 0.541 -0.021 0.198 0.077 1.000 PA 0.519 0.681 0.813 0.761 0.684 0.860 0.569 0.840 0.703 0.375 1.000 VA 0.735 0.780 0.687 0.800 0.692 0.920 0.560 0.797 0.681 0.438 0.874 1.000 WA 0.647 0.719 0.280 0.585 0.722 0.636 0.119 0.328 0.213 0.888 0.519 0.557 1.000
Boston Chicago Los Angeles New York Phoenix Portland San Diego San Francisco Seattle Washington D.C.
Boston 1.000 Chicago 0.606 1.000 Los Angeles 0.582 0.771 1.000 New York 0.836 0.784 0.711 1.000 Phoenix 0.469 0.783 0.757 0.678 1.000 Portland 0.130 0.653 0.481 0.349 0.753 1.000 San Diego 0.684 0.740 0.940 0.692 0.623 0.415 1.000 San Francisco 0.658 0.731 0.859 0.617 0.699 0.483 0.899 1.000 Seattle 0.379 0.786 0.692 0.578 0.760 0.838 0.610 0.670 1.000 Washington D.C. 0.621 0.803 0.919 0.804 0.779 0.434 0.879 0.789 0.569 1.000
Massachusetts. The highest correlation is given between the returns of San Diego and Los Angeles, which is logical since these are both located in California.
Table 5 on the next page shows that the house prices in major cities move in similar directions as the house prices of the whole state that they are located in. This also implicates that looking at these major cities presents a viable robustness check2.
Table 5: Correlation Cities with States
Note: This table gives the correlation between the yearly returns of the city and the state that it represents.
2 See paragraph 6 for the robustness section
ρ (City, State)
Boston (AZ) 0.909
Chicago (IL) 0.847
Los Angeles (LA) 0.941
New York (NJ) 0.930
New York (NY) 0.891
New York (PA) 0.697
Phoenix (AZ) 0.911
Portland (OR) 0.936
San Diego (CA) 0.914
San Francisco (CA) 0.835
Seattle (WA) 0.908
Washington D.C. (DC) 0.881
Washington D.C. (MD) 0.852
4. Methodology 4.1 Binomial Trees
The value of an option can be calculated by using the Binomial Option Pricing Model, which is a model that prices options by assuming that at the end of the next period, the price can only have 2 values (Berk & DeMarzo, 2014). Cox et al. (1979) propose a method, commonly called the Cox-Ross-Rubinstein binomial option pricing model, in their paper to determine the likelihood of an up movement and the likelihood of a down movement. These are given by the following formulas3:
p: up = 𝑒𝜎∗√Δ𝑡 (1)
1 – p : down = 𝑒−𝜎∗√Δ𝑡 (2)
Figure 1: Binomial Tree, where S0: Median house price in state, using e.g. Zillow Home Value Index.
Thereafter, a new binomial tree can be calculated by working backwards from the final values found in the first binomial tree to reconstruct the net present value. This net present value, which is equal to the call option, can be calculated using the following formulas:
𝑞 =𝑒(𝑅𝑓∗∆𝑡)−𝑑
𝑢−𝑑 & 1 – q (3)
However, the underlying security on which the binomial tree will be made is input from a real estate house price index. Geltner et al. (2003) give an overview of literature that states autocorrelation can occur in appraisal-based indices. The reason for this is that the index values are slowly updated with new market information since appraisers do not appraise properties continuously. In addition, transaction-based indices also display positive autocorrelation as the private real estate market is less informationally efficient than the market of public securities. Consequently, the effectiveness of price discovery and information aggregation is lower for the private real estate market. As a result, calculating net present value via binomial trees will not give accurate values for the call option since binomial trees assume that the underlying asset can be modeled by a geometric Brownian motion.
4.2 Pricing Formulas
4.2.1 Adjusted Volatility
Van Bragt et al. (2015) therefore propose an alternative methodology to value European options. The option is valued at time t and the European option expires at time T > t 0 with no possibility for early exercise. They state that if K = 1 an exact pricing formula exists, and this is a modification of the Black (1976) equation. In order to take the effect of stochastic interest rates into account, an adjustment is made to the implied volatility parameter. Brigo and Mercurio (2006, p. 888) mention that this adjusted volatility can be calculated in the following way:
𝜎
2∗2(𝑡, 𝑇) = 𝑉(𝑡, 𝑇) + 𝜎
22(𝑇 − 𝑡) + 2𝜌
𝜎1𝜅𝜎2(𝑇 − 𝑡 −
1𝜅(1 − 𝑒𝑥𝑝(−𝜅(𝑇 − 𝑡)))
(4) Where,𝑉(𝑡, 𝑇) =
𝜎12 𝜅2(𝑇 − 𝑡 +
2 𝜅𝑒𝑥𝑝(−𝜅(𝑇 − 𝑡)) −
1 2𝜅𝑒𝑥𝑝(−2𝜅(𝑇 − 𝑡)) −
3 2𝜅)
(5) and is the correlation between the Wiener processes for the short interest rate (see equation 6) and the efficient market process (see equation 7). The correlation can be calculated between the residuals of both.𝑑𝑟
𝑁(𝜏) = 𝜅 (
θ(τ)𝜅− 𝑟
𝑁(𝜏)) 𝑑𝜏 + 𝜎
1𝑑𝑍
1(𝜏)
(6)𝑑ln𝑦(𝜏) = (𝑟
𝑁(𝜏) − 𝑞 − 𝜎
22/2)𝑑𝜏 + 𝜎
2𝑑𝑍
2(𝜏)
(7) Equation (6) is the Vasicek model written in other form. In this model, the risk-neutral process for r is:𝑑𝑟 = 𝑎(𝑏 − 𝑟)𝑑𝑡 +
𝑑𝑧
(8)where a, b, and represent nonnegative constants. Mean reversion is included in the Vasicek model. This is the assumption that the short rate will move towards it average over time. The short rate will be pulled to a level b at rate a. This “pull” is joined by a normally distributed stochastic term, * dz. Equation (7) represents the efficient market process, from which the annual returns can be figured as:
𝑟𝑐𝑎(𝑡) = 𝐾∗𝜋 + (1 − 𝐾∗)𝑟
𝑐𝑎(𝑡 − 1) + 𝐾∗𝜀(𝑡) (9)
where, K* equals 1 minus the first-order autocorrelation of the index returns, π is the drift parameter that is assumed to be constant, and (t) represents the residuals of the efficient market process.4
4.2.2 Asian Options
If K < 1, an approximating pricing formula can be formulated. The index value at time T equals the weighted sum of T – t lognormal distributions, which can be seen from equation (10).
𝑎(𝑡) = 𝐾 ∑
𝑚(1 − 𝐾)
𝑖−1𝑦
∗(𝑡 − 𝑖 + 1) + (1 − 𝐾)
𝑚𝑎
∗(𝑡 − 𝑚)
𝑖=1 (10) Where 1 m t and𝑎
∗(𝑡 − 𝑚) ≡ 𝑎(𝑡 − 𝑚)𝑒𝑥𝑝 (∫
𝑟
𝑁(𝜏)𝑑𝜏)
𝑡 𝑡−𝑚𝑒𝑥𝑝 (−𝑞𝑚)
𝑦
∗(𝑡 − 𝑖 + 1) ≡ 𝑦(𝑡 − 𝑖 + 1)𝑒𝑥𝑝 (∫
𝑟
𝑁(𝜏)𝑑𝜏)𝑒𝑥 𝑝(−𝑞(𝑖 − 1))
𝑡 𝑡−𝑖+1 (11)This signals that there is an Asian basket option. According to Hull (2018), Asian options are ‘options where the payoff depends on the arithmetic average of the price of the underlying asset during the life of the option.’ Making use of these options is effective because, in this research, the payoff of the asset of the option is determined by the standard deviation of the returns of the underlying asset. In order to determine the value of this option, the first moment M1 and the second moment M2 of the exact probability distribution at time T have to be calculated (Hull, 2009, pp. 578-579):
𝑀
1,0= exp ((𝑟
𝑁,𝑇(𝑡) − 𝑞)(𝑇 − 𝑡)) 𝑎(𝑡)(1 − 𝐾)
(𝑇−𝑡)𝑀
1,𝑖= exp ((𝑟
𝑁,𝑇(𝑡) − 𝑞)(𝑇 − 𝑡)) 𝑦(𝑡)𝐾(1 − 𝐾)
(𝑖−1)𝑀
1= 𝑀
1,0+ ∑
𝑇−𝑡𝑖=1𝑀
1,𝑖(12)
𝑀
2= ∑
𝑇−𝑡𝑖=1𝑀
1,𝑖2exp(𝜎
2∗2(𝑡, 𝑡 + 𝑖)) + ∑
𝑖<𝑗𝑀
1,𝑖𝑀
1,𝑗exp(𝜎
2∗2(𝑡, 𝑡 + 𝑖))
(13)
4.2.3 Pricing Formulas
By using the first moment M1 and the second moment M2, the exact distribution can be fitted with an approximating lognormal distribution, as first proposed by Levy (1992). With the following formula, it is possible to determine the forward price FT (t) and the implied volatility (Hull, 2009):
𝐹
𝑇(𝑡) = 𝑀
1; 𝜎 = √
𝑇−𝑡1𝑙𝑛
𝑀𝑀21
(14)
The price of a European put and call option can then be calculated by the familiar Black (1976) price for an option on a forward contract:
𝑝(𝑡) = exp (−𝑟
𝑁,𝑇(𝑡)(𝑇 − 𝑡)) [𝑋𝑁(−𝑑
2) − 𝐹
𝑇(𝑡)𝑁(−𝑑
1)],
Where,
𝑑
1=
ln(𝐹𝑇(𝑡)𝑋 )+𝜎2(𝑇−𝑡)/2𝜎√𝑇−𝑡
, 𝑑
2= 𝑑
1− 𝜎√𝑇 − 𝑡,
(16)p(t) denotes the price of a put option, c(t) represents the price of a call option, and X is the strike price.5
5. Results
5.1 Binomial Trees
Applying the calculations of section 4.1 to data of the state of Arizona gives us the following binomial trees.
Figure 2: Binomial Tree stage 1
Figure 3: Binomial Tree stage 2
The value at time t = 8 in figure 2 shows the potential profits or losses that the median house in Arizona can obtain. The highest profit is $258,034 and the biggest loss in value that the house can undergo is $123,895. Since the investors of Unison Home Ownership only share for 35 percent in the profit/loss, the value at time t=8 in figure 3 shows the value of time t=8 in figure 2 times 0.35. The NPV of these potential profits/losses in Arizona has a positive value of $23,133. The binomial trees are constructed for each state that Unison operates in and these can be found in Appendix B. Appendix C shows the binomial tree constructions of major cities in the states that Unison operates in.
Table 6: Binomial Tree Outcomes, States
Note: This table gives the net present values (call value) in US dollars, calculated by the binomial trees which are shown in section 5.1 and appendix B.
As can be seen above, all net present values, which represent the value of the call options, are positive. This implies that investing in houses in the states, in which Unison Home Ownership Investors operates, is profitable. The call option values are highest in California and District of Columbia. Both states also have the highest median house price. The lowest median house prices can be found in Illinois and Pennsylvania and this in turn also results into the lowest call options.
Table 7: Binomial Tree Outcomes, Cities
Note: This table gives the net present values (call value) in US dollars, appendix C shows the binomial trees of the cities.
The table above shows the net present values, i.e. the call option value, of the binomial trees of 10 major cities in the states that Unison operates in. Interesting to note is that almost in every case,
State Median House Price Investment by Unison Net Present Value (Call)
Arizona 238,500 23,850 22,234 California 542,900 54,290 50,979 Connecticut 234,800 23,480 21,783 District of Columbia 566,600 56,660 52,543 Illinois 173,500 17,350 15,951 Maryland 282,300 28,230 25,930 Massachusetts 391,700 39,170 36,177 New Jersey 313,100 31,310 28,920 New York 282,600 28,260 25,998 Oregon 334,500 33,450 30,782 Pennsylvania 166,300 16,630 15,315 Virginia 250,400 25,040 22,973 Washington 371,000 37,100 34,192
City Median House Price Investment by Unison Net Present Value (Call)
Boston (MA) 585,100 58,510 53,712
Chicago (IL) 225,900 22,590 20,735
Los Angeles (CA) 674,600 67,460 64,519
New York (NJ/NY/PA) 660,000 66,000 60,732
Phoenix (AZ) 229,000 22,900 22,063
Portland (OR) 585,100 58,510 53,810
San Diego (CA) 625,500 62,550 59,297
San Francisco (CA) 1,334,800 133,480 127,814
Seattle (WA) 767,000 76,700 70,656
the call value is higher in the major city than the call value for the whole state. A likely reason for this is that land is more scarce in major cities and thus has a higher price. In rural parts of the states, the house values are on average lower as the land is less scarce. The only exceptions to the larger value in the major city than in the whole state are Arizona and Phoenix and Washington D.C. and District of Columbia. For the latter is a simple clarification since they represent the same area.
Since there is autocorrelation present in the index values that these binomial trees are based on, no conclusions can be made from the call values. Therefore, the results of the methodology provided by Van Bragt et al. (2015) will be discussed below.
5.2 Results Methodology to Value European Options
5.2.1 Outcomes Adjusted Volatility
In order to calculate the adjusted volatility one needs V(t, T), for which in turn the volatility (12) of the short interest rate and the mean reverting speed ( or a) are needed. These are calculated with the Vasicek model. Furthermore, to determine the adjusted volatility, one also has to determine the volatility of the index values as well as the correlation () between the risk-neutral process and the efficient market process. The tables below show the inputs for the calculation of the adjusted volatility and the outcomes.
Table 8: Adjusted Volatility, States
Note: This table shows the inputs for the calculation of the adjusted volatility and its outcome.
State σ1 V (t,T) σ2 ρ (ε1, ε2) σ22 Arizona 1.63% 0.095 0.027 9.17% 0.080 10.01% California 1.63% 0.095 0.027 10.09% -0.149 9.58% Connecticut 1.63% 0.095 0.027 8.49% 0.021 8.60% District of Columbia 1.63% 0.095 0.027 8.43% -0.207 6.93% Illinois 1.63% 0.095 0.027 4.79% -0.129 4.00% Maryland 1.63% 0.095 0.027 6.58% -0.081 5.71% Massachusetts 1.63% 0.095 0.027 7.79% -0.129 6.71% New Jersey 1.63% 0.095 0.027 7.80% -0.085 7.00% New York 1.63% 0.095 0.027 7.00% 0.061 6.96% Oregon 1.63% 0.095 0.027 7.07% 0.003 6.70% Pennsylvania 1.63% 0.095 0.027 4.52% -0.007 4.29% Virginia 1.63% 0.095 0.027 5.45% -0.157 4.35% Washington 1.63% 0.095 0.027 7.39% 0.195 8.24%
Table 9: Adjusted Volatility, Cities
Note: This table shows the inputs for the calculation of the adjusted volatility and its outcome.
Above, it becomes clear that when the volatility in the index values is higher, the adjusted volatility is also higher. The fourth column shows that the highest volatility in a house price index was in California (10.09%), whereas in the last column of table 8 the highest adjusted volatility can be found, namely Arizona with 10.01%. The lowest volatility was found in Pennsylvania (4.52%) and for the adjusted volatility this was Illinois (4.00%). In table 9, we can see that the lowest volatility in the house price index was in Chicago (6.32%), while the lowest adjusted volatility is for Boston (4.18%). Furthermore, the highest volatility in a house price index was found in Phoenix (13.02%) which is accompanied with the highest adjusted volatility (13.24%).
5.2.2 Outcomes Asian Options and the Pricing Formulas
With the calculated adjusted volatility of section 5.2.1, the first moment and the second moment can be discovered. Thereafter, the forward price and the implied volatility can be calculated using the formulas (14) mentioned in paragraph 4.2.3. The strike price is calculated by using the projected inflation rates for the United States (Statista, 2018). Next, by using the equations of (15) and (16), the call option can be determined. The outcomes of the equations are shown below in table 10 and table 11. The final column shows the final value of the call option.
The call option in table 10, shown in the 9th column, is the highest for California ($92,599) and for District of Columbia ($79,847). The lowest determined call options are found to be $17,583 and $17,569 for respectively Pennsylvania and Illinois. In order to determine the value in the final column, the original investment by the investors of Unison has to be subtracted from the call option. The investment is assumed to be equal to 10 percent of the median house value in a particular state. All values in the final column are positive and this means that the average investment of Unison is profitable. The states in which the investments are determined to be most profitable are the same states with the highest call option, California and District of Columbia. The values of their investments are respectively $38.309 and $23.187. The investments in Illinois and Pennsylvania are determined to
City σ1 V (t,T) σ2 ρ (ε1, ε2) σ22
Boston (MA) 1.63% 0.095 0.027 6.37% -0.333 4.18% Chicago (IL) 1.63% 0.095 0.027 6.32% -0.230 4.68% Los Angeles (CA) 1.63% 0.095 0.027 12.21% -0.375 10.83% New York (NJ/NY/PA) 1.63% 0.095 0.027 7.06% -0.439 4.11% Phoenix (AZ) 1.63% 0.095 0.027 13.02% -0.281 13.23% Portland (OR) 1.63% 0.095 0.027 6.93% 0.111 7.15% San Diego (CA) 1.63% 0.095 0.027 11.21% -0.303 9.94% San Francisco (CA) 1.63% 0.095 0.027 12.34% -0.285 11.97% Seattle (WA) 1.63% 0.095 0.027 7.30% 0.025 7.10% Washington D.C. (DC/MD/VA) 1.63% 0.095 0.027 8.99% -0.432 5.94%
be the least profitable. For Arizona, California, Connecticut and Washington, the value of the investment is less than 65 percent of the call option. This means that these states are probable to have the highest return on investment. On the other hand, for Virginia, Pennsylvania and Illinois, the investment is over 90% of the call option and therefore the return on investment is most likely to be low.
In table 11, the call values of the cities are presented. It becomes immediately clear that all final values of the call options are positive again, which implicates that investing in houses in these cities is profitable. The highest final value is present within San Francisco, followed by Los Angeles and San Diego. These cities are all located in California, which had the highest final call option value of table 10. There are three cities with a low, but slightly positive final value and these are Boston, Chicago and New York. The final call option values of Phoenix, Portland and Seattle are higher than the value of their respective state.
Table 10: Call Value, States
Note: Inputs for calculating the value of the call option in the whole state and its outcome shown in the last column.
Table 11: Call Value, Cities
Note: Inputs for calculating the value of the call option in a major city and its outcome shown in the last column.
State M1 M2 FT (t) σ K d1 d2 N (d1) N(d2) C (t) I I / C (t) Value
Arizona 269,330 9.81E+10 269,330 19.44% 283,684 0.180 -0.369 0.5716 0.3559 41,733 23,850 57.15% 17,883
California 613,078 5.02E+11 613,078 18.99% 645,754 0.172 -0.365 0.5682 0.3575 92,599 54,290 58.63% 38,309
Connecticut 265,151 9.09E+10 265,151 17.93% 279,283 0.151 -0.356 0.5601 0.3609 37,583 23,480 62.47% 14,103
District of Columbia 639,842 5.03E+11 639,842 16.01% 673,944 0.112 -0.341 0.5445 0.3665 79,847 56,660 70.96% 23,187
Illinois 195,927 4.31E+10 195,927 12.05% 206,370 0.018 -0.323 0.5072 0.3734 17,583 17,350 98.68% 233
Maryland 318,792 1.2E+11 318,792 14.47% 335,782 0.078 -0.332 0.5310 0.3701 35,462 28,230 79.61% 7,232
Massachusetts 442,333 2.39E+11 442,333 15.74% 465,909 0.106 -0.339 0.5422 0.3672 54,154 39,170 72.33% 14,984
New Jersey 353,573 1.54E+11 353,573 16.10% 372,418 0.114 -0.342 0.5452 0.3663 44,412 31,310 70.50% 13,102
New York 319,130 1.25E+11 319,130 16.04% 336,139 0.112 -0.341 0.5448 0.3664 39,927 28,260 70.78% 11,667
Oregon 377,739 1.74E+11 377,739 15.73% 397,872 0.106 -0.339 0.5421 0.3672 46,217 33,450 72.38% 12,767
Pennsylvania 187,797 3.99E+10 187,797 12.48% 197,806 0.029 -0.324 0.5117 0.3731 17,569 16,630 94.65% 939
Virginia 282,768 9.08E+10 282,768 12.58% 297,839 0.032 -0.324 0.5128 0.3730 26,705 25,040 93.77% 1,665
Washington 418,957 2.24E+11 418,957 17.53% 441,287 0.143 -0.353 0.5570 0.3622 57,928 37,100 64.04% 20,828
City M1 M2 FT (t) σ K d1 d2 N (d1) N(d2) C (t) I I / C (t) Value
Boston (MA) 660,733 4.93E+11 660,733 12.32% 695,949 0.025 -0.323 0.5101 0.3733 60,866 58,510 96.13% 2,356
Chicago (IL) 255,101 7.46E+10 255,101 13.06% 268,697 0.044 -0.325 0.5176 0.3725 25,164 22,590 89.77% 2,574
Los Angeles (CA) 761,802 8.06E+11 761,802 20.28% 802,405 0.196 -0.377 0.5778 0.3530 123,630 67,460 54.57% 56,170
New York (NJ/NY/PA) 745,315 6.26E+11 745,315 12.22% 785,039 0.023 -0.323 0.5090 0.3733 67,991 66,000 97.07% 1,991
Phoenix (AZ) 258,602 1.01E+11 258,602 22.58% 272,385 0.238 -0.401 0.5941 0.3444 47,137 22,900 48.58% 24,237
Portland (OR) 660,733 5.4E+11 660,733 16.28% 695,949 0.117 -0.343 0.5468 0.3658 84,053 58,510 69.61% 25,543
San Diego (CA) 706,355 6.73E+11 706,355 19.36% 744,002 0.179 -0.369 0.5710 0.3562 108,981 62,550 57.40% 46,431
San Francisco (CA) 1,507,343 3.28E+12 1,507,343 21.39% 1,587,681 0.217 -0.388 0.5858 0.3489 259,251 133,480 51.49% 125,771
Seattle (WA) 866,146 9.26E+11 866,146 16.21% 912,310 0.116 -0.342 0.5462 0.3660 109,643 76,700 69.95% 32,943
