Fair Value and Risk Profile for Presale Contracts of Condominiums

Abstract

Developers usually presell new condominiums, requiring purchasers to make down payments on a contract that allows them to purchase, at a fixed price, the finished condominiums on a later date. This presale contract is akin to a financial call option sold by the builder to the purchaser of the condo. In this paper, we value the presale contract from both the purchaser’s and the developer’s points of view. We examine the influence of various opt-out clauses, different interest rates and other factors on the value of presale contracts. We discuss the extent of risk sharing between the purchasers and the developers according to varying levels of down payments. We conclude that developers enjoy a reduction in risk without a corresponding reduction in expected profits by holding a presale.

Appendices

Appendix A: Analysis of Lai et al. (2004)

This appendix provides a detailed discussion of Lai et al.’s analysis for the purchaser, which is described in page 341 and Appendix A in their paper. We suppose that the purchaser will make a first payment of Q ₁ at t = t ₁. At t = T the purchaser can either make an additional payment Q ₂ and thus obtain ownership of the unit or she can pay a penalty of A in order to get out of the contract. It is clear that she will only make the additional payment if S > Q ₂ − A where S is the the spot price of a similar unit at the open market at time T.

We suppose that S is given by the stochastic differential equation

$$ dS = \mu S dt + \sigma S dW $$

where μ is the expected growth rate of the unit price, σ is the volatility and W is the Wiener process. We will suppose that the purchaser is risk neutral so we can replace μ by the risk free interest rate r _p . Let C(S,t) be the value of the presale option at time t and price S. Hence we can write the Bellman equation in the form

$$ \label{eqn2} {\partial C \over \partial t}+{1 \over 2} \sigma^2 S^2 {\partial^2 C \over \partial S^2} + \mu S {\partial C \over \partial S} -r_p C=0 $$

(9)

The final condition at t = T can be written in the form

$$ \label{Fin1} C = \max(S-Q_2,-A) $$

(10)

Lai et al. claim that they can replace Q ₂ by $M S e^{r_p(T-t)}$ and state that Q ₂ is “the amount of the last payment and is a function of M (a percentage) and $S e^{r_p(T-t)}$ (the expected spot price at the end of the last period)”. It is not clear what they mean by this statement. At the very least M surely must be a function of t. However, their analysis implies that they treat it as a constant. They also replace the penalty A by $\eta S e^{r_p(T-t)}$ where again they state that η is a percentage which they treat as constant. Thus they replace (10) by

$$ \label{Fin2} C = \max\left(S- (M-\eta) S e^{r_p(T-t)},0\right) - \eta S e^{r_p(T-t)} $$

(11)

Since the last term in this condition depends on S, C vanishes at S = 0. Thus they get the two boundary conditions

$$ \label{boun1} C = 0 \textrm{ at } S=0 $$

(12)

$$ \label{boun2} C \approx S\textrm{ as } S\rightarrow \infty $$

(13)

Thus they get the solution (their equation A4)

$$ C(S,t) = S \Big ( N(d+\sigma \sqrt{T-t}) -(M-\eta)N(d)-\eta \Big) $$

where

$$ d = {-ln(M-\eta) - {1 \over 2} \sigma^2(T-t) \over \sigma \sqrt{T-t}} $$

This solution is linear in S, so the second derivative term in (9) is identically zero, so σ cannot be relevant in their solution. Despite this, σ does appear in their solution. It is also odd that the risk free interest rate r _p does not appear in their solution.

Our position is that we cannot replace the final payment Q ₂ by $M S e^{r_p(T-t)}$ and the penalty A by $\eta S e^{r_p(T-t)}$. Instead we must carry out the calculations using these quantities, Q ₂ and A, directly.

Appendix B: Modeling Assumptions

We assume that the price movements of condominiums can be modelled by Geometric Brownian Motion (GBM). As with stock prices, GBM does not completely fit the behaviour of condo price movements, but we choose it for its analytical tractability.

We use the New Housing Price Index (NHPI) and Construction Price Index (CPI) to calibrate our model parameters, which are provided by Statistics Canada (StatCan). These can be found in the Government of Canada’s CANSIM database (StatCan 2009a, b). The NHPI tracks the monthly average housing price across all major metropolitan regions within Canada. The index adjusts for the change of quality in houses.

The CPI is compiled by measuring quarterly changes in building contractors’ quoted prices. It excludes land, design, development charges and real estate fees. Development charges are fees payable to the government. It would be reasonable to include the charge as part of our construction cost, but the size of the charge is immaterial relative to the total cost e.g. order of $10,000 in the City of Vaughan (2009), and so contributes little to the growth of the total construction cost. We therefore do not take it into account.

The graph of the NHPI is shown in Fig. 7, along with the construction price index. The histogram of the log return of the index is shown in Fig. 8. We see that the returns exhibit a distribution with heavier tails than the normal distribution. However, we think it is close enough to a normal distribution that we can gain insight into the market by approximating the returns with a normal distribution. The distribution of the CPI behaves similarly to the NHPI, and so the we also model it using the lognormal process. This is similar to the treatment of construction cost proposed by Wang and Zhou (2006), who however modeled construction cost as a series of cash flows, whereas we model it as a lump sum.

Fig. 7

Quarterly New Housing Price Index and Construction Price Index, as provided in the CANSIM database by Statistics Canada(1997 price = 100)

Fig. 8

Histogram of the of the monthly New Housing Price Inde. The histogram shows that, while the normal distribution is not perfect as a model for monthly logarithmic returns, it is reasonable for purposes of our analysis

There is some evidence of autocorrelation in the housing and construction index returns in Fig. 9. However, this would only significantly impact our model if the purchasers and developers could trade the partly constructed condo. In such a scenario, the use of GBM would paint a flawed picture about the profitability for both the purchaser and the developer. However, the underlying cannot be traded while the contract is in effect, and only the initial and the final prices of the condominium matter. The returns on condominium prices four years from the starting period do not seem to bear any significant autocorrelation, and we can treat the condo price as making one geometric Brownian “leap” from period t = 0 to t = T, with T being four years.

Fig. 9

Scatter plot of log returns of monthly New Housing Price Index. Plotted returns of period k vs k + p, where p is 3 months for the left graph, and 4 years for the right graph. The left graph shows some evidence of pricing momentum in the short term—upward price movement is likely followed by another upward price movement, and vice versa. Over 4 years, the relationship between price movements is weaker. Expectations of condo prices at the time of completion is not affected by the movement in condo prices at the time of the presale agreement since those dates are 4 years apart

However, the change in expectations of final condo price due to the autoregressive nature of the price series cannot be ignored. Rising prices in the last few months may lead a purchaser to expect higher final prices than she would if prices had been falling. We can keep our assumption of GBM for condo prices while incorporating the change in price expectations by adjusting μ. To find a μ that appropriately reconciles the change in expectations, we need to analyze the autocorrelation in the NHPI data.

The Partial Autocorrelation Function of the log returns of the NHPI is shown in Fig. 10. The graph indicates that it would be a good idea to use either an AR(2) or an AR(5) model. The Akaike Information Criterion (AIC) using the Ordinary Least Squares method supports the use of AR(2), while AIC using the Maximum Likelihood Estimation method supports the use of AR(5). We choose to employ AR(5) because the AIC ranking in support of AR(5) is marginally more decisive. This yields the following

$$ \begin{array}{lll} E[\ln(S(t)/S(t-\textrm{1 month}))] = \lambda_t & = & 4.76\times10^{-4}+5.506\lambda_{t-1}+2.192\lambda_{t-2}\\ & & +1.041\lambda_{t-3}+0.883\lambda_{t-4}+0.871\lambda_{t-5} \end{array} $$

Fig. 10

The Partial Autocorrelation Function of NHPI log returns. The results confirm the existence of short term pricing momentum in condo prices

In order to take the autoregression into account, we can set the expected annual rate of appreciation μ to be $\sum_{i=1}^{48}\lambda_i$ which equals the following

$$ (0.118+5.506\lambda_{-1}+2.192\lambda_{-2}+1.041\lambda_{-3}+0.883\lambda_{-4}+0.871\lambda_{-5})/4+\sigma_S^2/2\\ \label{armu} $$

(14)

where λ ₀ thorugh λ _− 4 are the monthly returns on housing for the previous 5 months. By equating μ to Eq. 14, we are able to match our expected appreciation using GBM model over four years to be the same as the expected appreciation implied by the AR(5) model. However, this is not a hard and fast rule. We are free to forecast the value of μ using different methods. Equation 14 was merely derived to show that if we choose to, we can incorporate the expectation of the final condo price as implied by AR(5) model in the GBM model.

Weighing all these factors and considering the significant analytic simplification, we feel that using geometric Brownian motion is adequate to describe the uncertainties associated with the price of a single condominium. By the same logic, we also model the construction cost of condominiums to follow a geometric Brownian motion. The Q-Q plot and the histogram of the returns of CPI are very similar in shape to that of the returns of NHPI.

Rosenthal (1999) finds that construction cost and housing prices are co-integrated. To capture the connection between the two without sacrificing analytic tractability, we instead assume the two time series are correlated, and we denote the correlation as ρ. Any long-term divergence between construction cost and condo price would lead to increasingly greater or smaller profitability for the developer. This leads to our assumption that condo prices and construction costs appreciate at the same rate μ. Our view is validated by comparing the average returns of the NHPI and CPI. However, this does not preclude a divergence of condo and construction prices on a given realization, particularly in the short term.

Appendix C: Analytic Solution to Purchaser’s Position

The Bellman equation we are trying to solve is identical to the well-known Black Scholes equation, but with the rate of return on an asset allowed not to equal the risk free rate. The fundamental solution of this equation is

$$ K(S',t) = e^{-\left(\ln{S\over S'}+(\mu-{1\over 2}\sigma^2)(T-t)\right)^2/2\sigma^2(T-t)}/S' $$

Given our final condition

$$ F(S',T) = \max(S-(Q_2-A),0)-A $$

and defining B(t) to be the following

$$ B(t) = e^{-r(T-t)}/{\sigma\sqrt{2\pi(T-t)}} $$

Our solution can be obtained by computing the following

$$ \begin{array}{lll} C(S,t) & = & B(t)\left[\int_{-\infty}^{\infty}K(S',t)F(S',T)\frac{dS'}{S'}\right] \\ &=& B(t)\left[\int_{-\infty}^{\infty}K(S',t)(\max(S'-(Q_2-A),0)-A)\frac{dS'}{S'}\right]\\ & = & B(t)\left[\int_{Q_2\!-\!A}^{\infty}K(S',t)(S'\!-\!(Q_2-A)) \frac{dS'}{S'}\right]\!-\!B(t)\left[\int_{-\infty}^{\infty}K(S',t)A\frac{dS'}{S'}\right]\\ \end{array} $$

Our calculations become much more easy to follow if we use the following easily verifiable relationships.

$$ \begin{array}{lll} & & B(t)\int_{\alpha}^{\infty}K(S',t)S'/S'dS' = S N(\delta_{1})e^{(\mu-r)(T-t)}\\ & & B(t)\int_{\alpha}^{\infty}K(S',t)\beta/S'dS' = \beta N(\delta_{2})e^{-r(T-t)}\\ & & \delta_{1} = { \ln\big({S \over \alpha}\big) +\big(\mu+{1 \over 2} \sigma^2\big)(T-t) \over \sigma \sqrt{T-t} }\\ & & \delta_{2} = \delta_1 -\sigma\sqrt{T-t} \end{array} $$

Using the above relationships, the value of a presale contract from the purchaser’s point of view is found to be

$$ C(S,t) = SN(d_{1})e^{(\mu-r)(T-t)}-(Q_2-A)N(d_{2})e^{-r(T-t)}-Ae^{-r(T-t)} $$

(15)

where

$$ \begin{array}{lll} d_{1} & = & { \ln\big({S \over (Q_2-A)}\big) +\big(\mu+{1 \over 2} \sigma^2\big)(T-t) \over \sigma \sqrt{T-t} }\\ d_{2} & = & d_1 -\sigma\sqrt{T-t} \end{array} $$

Appendix D: Variance of Purchaser’s Profitability

We calculate the variance of the profitability for the purchaser when she holds the contract without employing any hedging strategies. We do this by calculating the value of the option at time T, which is equal to the payoff of the option.

$$ \begin{array}{lll} \textrm{Variance}(\textrm{Prof\/it}) & = & E\left[\max(S(T)-Q_2,-A)^2\right] - E[\max(S(T)-Q_2,-A)]^2\\ & = & E\left[\max(S(T)-Q_2,-A)^2\right]-C(S,T)^2 \end{array} $$

$$ \begin{array}{lll} &&E\left[\max(S(T)-Q_2,-A)^2\right] \\ &&{\kern1pc} = \int^{\infty}_{-\infty}\max\left(S(0)e^{\big(\mu-\sigma_S^2 /2\big)T+\sigma_S W(T)}-Q_2,-A\right)^2 P(W(T))dW\\ &&{\kern1pc} = \int^{\infty}_{\alpha}\left(S(0)e^{\big(\mu-\sigma_S^2 /2\big)T+\sigma_S W(T)}-Q_2\right)^2 P(W(T))dW\\ &&{\kern2pc} -A^2\int^{\alpha}_{-\infty}P(W(T))dW \end{array} $$

where

$$ \alpha = \frac{\ln((Q_2 -A)/S_0)-\big(\mu-\sigma^2/2\big)T}{\sigma} $$

To ease notation, denote S(0) = S ₀, W(T) = W and σ _S  = σ. Since W is normally distributed with mean 0 and standard deviation of $\sqrt{T}$,

$$ \begin{array}{rll} &\int^{\infty}_{\alpha}&\left(S_0 e^{(\mu-\sigma^2 /2)T+\sigma_S W}-Q_2\right)^2 P(W)dW -A^2\int^{\alpha}_{-\infty}P(W(T))dW\\ & = & \frac{S_0^2}{\sqrt{2\pi T}}\int^{\infty}_{\alpha}e^{-\frac{1}{2T}((W-2\sigma T)^2-4\mu T^2-2 T^2 \sigma^2)}dW \\ &&-\frac{2 Q_2 S_0}{\sqrt{2\pi T}}\int^{\infty}_{\alpha}e^{-\frac{1}{2T}((W-\sigma T)^2-2\mu T^2)}dW \\ & & + \frac{Q_2^2}{\sqrt{2\pi T}}\int^{\infty}_{\alpha}e^{-\frac{W^2}{2T}}dW - \frac{A^2}{\sqrt{2\pi T}}\int^{\alpha}_{-\infty}e^{-\frac{W^2}{2T}}dW\\ & = & \frac{S_0^2}{\sqrt{\pi}}e^{2\mu T+\sigma^2 T}\int^{\infty}_{\frac{\alpha-2\sigma T}{\sqrt{2T}}}e^{-U^2}dU-\frac{2 Q_2 S_0}{\sqrt{\pi}}e^{2\mu T}\int^{\infty}_{\frac{\alpha-\sigma T}{\sqrt{2T}}}e^{-U^2}dU\\ &&+Q_2^2 N(d_2)-A^2 (1-N(d_2))\\ & = & S_0 e^{2\mu T+\sigma^2 T}N(d_3)-2 Q_2 S_0 e^{\mu T}N(d_1)+Q_2^2 N(d_2)-A^2 (1-N(d_2)) \end{array} $$

where, as for the purchaser’s solutions, we have

$$ \begin{array}{lll} d_{1} & = & {\ln\big({S \over (Q_2-A)}\big) +\big(\mu+{1 \over 2} \sigma^2\big)T \over \sigma \sqrt{T} }\\ d_{2} & = & d_1 -\sigma\sqrt{T}\\ d_3 & = & d_1+\sigma\sqrt{T} \end{array} $$

Substituting our answer for C(S,T) in the full expression for the variance yields

$$ \begin{array}{lll} \textrm{Variance} & = & S_0^2 e^{2\mu T}\left(N(d_3)e^{\sigma^2 T}-N(d_1)^2\right) + 2 A S_0 e^{\mu T} N(d_1)(1-N(d_2))\\ &&+Q_2^2 N(d_2)(1-N(d_2)) + A^2(1-N(d_2))N(d_2)\\ &&-2 A Q_2 N(d_2)(1-N(d_2))-2 Q_2 S_0 e^{\mu T}N(d_1)(1-N(d_2)) \end{array} $$