The Valuation of Houses in an Uncertain World with Substantial Transaction Costs
Margaret Hwang Smith
Department of Economics
Pomona College
Claremont CA 91711
mhwang@pomona.edu
Gary Smith
Department of Economics
Pomona College
Claremont CA 91711
gsmith@pomona.edu
The Valuation of Houses in an Uncertain World with Substantial Transaction Costs
Abstract
This paper presents a dynamic model of residential real estate valuation that
takes into account the uncertain time paths of rents and prices and the substantial
transaction costs. By temporarily postponing decisions, buyers and sellers obtain
additional information about future rents and prices and may avoid transactions
that are costly to reverse. One implication is that renting maintains an implicit
“flexibility” option that may be quite valuable. Another implication
is that the combination of substantial uncertainty and large transaction costs
can create a large wedge between the prices buyers are willing to pay and sellers
are willing to accept.
The Valuation of Houses in an Uncertain World with Substantial Transaction Costs
Some financial advisers view residential housing as a foolproof investment that does not require financial analysis. For example, Barbara Alexander says “there is no bad time to buy” (Alexander 2003); Gary Eldred is slightly more restrained: “Ninety to ninety-five percent of people [who buy a house] will say it is the best investment I ever made” (quoted in Rumbler 2003). It is clearly a mistake to think something is a good investment no matter what the price. Yet people who think price doesn’t matter (or the higher the better!) traded farms for tulips, bought Polaroid stock for 91 times earnings in 1972, paid $10,000 for a Humphrey the Camel Beanie Baby in the 1990s, and paid $2 million for a 2000-square-foot house in Palo Alto in 2001.
The best way to determine whether a house is a good investment is to apply the same intrinsic-value principles that are used to value stocks. The intrinsic value of a stock is the present value of its cash flow. The same is true of a house. The intrinsic value V is the present value of the cash flow Xt, discounted by the required rate of return R:
The implicit income from a house is the rent you would otherwise have to pay to live in this house minus the expenses associated with home ownership. If you would pay $30,000 a year to rent a house, home ownership implicitly gives you $30,000 that you otherwise would pay to someone else. On the other hand, as a homeowner, you must pay property taxes, insurance, maintenance, and some utilities that you would not have to pay if you were a renter. If these expenses are $10,000 a year, your implicit net annual income is X1 = $30,000 - $10,000 = $20,000.
To determine the present value, we must project this implicit income into the future. As with stocks, the constant growth model provides a simple and insightful starting point. If the cash flow grows as a rate g < R, so that Xt+1 = (1 + g)Xt, then the present value formula simplifies to
For example, if the cash flow is $20,000, growing at 5% a year, and the required rate of return is 10%, this house’s intrinsic value is $400,000:
One attractive feature of this model is its internal consistency. The current $20,000 cash flow on a house valued at $400,000 provides a 5% return: $20,000/$400,000 = 0.05. If the cash flow grows, as projected, by 5% a year, then the present value will also grow by 5% a year. Together, the 5% current income and the 5% growth in value provide the requisite 10% return.
A homebuyer can use the projected cash flow and a required rate of return to determine if a house’s intrinsic value is above or below its market price. If the price is less than or equal to the intrinsic value, the house is indeed worth what it costs. If the market price is above intrinsic value, renting is more financially attractive.
Some advisors who consider the possibility that renting is sometimes better than buying list the pluses and minuses, with no numerical calculations at all (Goodman 2001, Miller 2002, Orman 2001); others compare the cost of renting and buying, but ignore uncertainty about future rents and prices (Quinn 1997, Keown 1998, Ramaglia and MacDonald 1999, Winger and Fresca 2000, Kapoor, Dlabay, and Hughes 1999).
As with stocks, uncertainty is key fact of life. One enormous difference between stocks and real estate is that the transaction costs are trivial for the former and substantial for the latter. This combination of substantial uncertainty and high transaction costs has interesting implications for the valuation of houses. We begin with two simple examples to illustrate how renting implicitly involves a “flexibility option” that allows a potential buyer to obtain additional information. These examples also help inform our intuition about why uncertainty and transaction costs can be powerful arguments for inertia. These examples only have uncertainty for a single period and assume that housing purchases are irreversible. We then analyze a more general model.
Rent Uncertainty
Consider a house that currently costs $20,000 a year to rent (payable immediately) and $460,000 to buy. After 1 year, the annual rent is equally likely to be $20,000 or $25,000 and the price will be $470,000. Thereafter the rent and price will increase by 5% a year forever.
The expected value of next year’s rent is $20,000(0.5) + $25,000(0.5) = $22,500. If the required rate of return is 10%, then a purchase made today has an expected NPV of $10,000:
Suppose, however, that the potential buyer rents for one year before deciding on the purchase. If the rent is $20,000 next year, then a purchase next year has a negative NPV at that time:
If the rent increases to $25,000, then a purchase next year has a positive NPV at that time:
The proverbial bottom line is that if the rent turns out to be $25,000, the house will be purchased; otherwise, it won’t. If the house is purchased, the current NPV is $80,000/1.10 = $72,727. The expected value of the current NPV is
In comparison to an immediate purchase (which has an NPV of $10,000), renting provides a flexibility option that is worth $36,364 - $10,000 = $26,364.
Let’s consider for a moment why it makes sense to pay $20,000 to rent a house whose purchase has an expected NPV of $10,000, knowing that the price will be $10,000 higher next year. In present value terms, a $470,000 price next year is only $470,000/1.10 = $427,273. This saving of $460,000 - $427,273 = $32,727 more than offsets the $20,000 rent. In addition, the potential buyer will be able to learn whether or not the rent increases to $25,000. If it does, then purchase has a positive NPV; if it doesn’t, then purchase is a mistake that the buyer would like to avoid.
Price Uncertainty
A similar analysis applies if the future cost of buying the house is uncertain. Assume now that the annual rent is currently $20,000 and is certain to increase by 5% a year. The price of the house is currently $435,000 and is equally likely to be $440,000 or $470,000 a year from now. A purchase made today has a net present value of $5,000:
Suppose, however, that the potential buyer waits a year. If the price is $440,000 next year, a purchase has a positive NPV at that time:
If the price is $470,000, a purchase has a negative NPV at that time:
If the price is $440,000, the house will be purchased; otherwise, it won’t. If the house is purchased, the current NPV is $22,000/1.10 = $20,000. The expected value of the current NPV with renting is
The option is worth $10,000 - $5,000 = $5,000.
Consider now why it pays to rent a house whose purchase has a $5,000 NPV, in the hopes of buying later at a price that is $5,000 higher than the current price. If the purchase is delayed a year, the buyer will have the exact same implicit cash flow from the house for all years after the delay. The only difference is that the buyer has to pay the current year’s $20,000 rent and postpones the purchase in the hopes of buying at a price that is only 1.1% higher ($5,000/$435,000 = 0.011). If the price does turn out to be $440,000, then, at a 10% required return, delaying the purchase a year is worth, in present value terms,
This confirms the above calculations: if the price rises to $440,000 the NPV increases by $15,000, from $5,000 to $20,000. The expected value of a 50% chance of increasing the NPV by $15,000 and a 50% chance of missing out on the current $5,000 NPV is $5,000.
A General Model
In either of these situations, renting has an option value because it allows the purchase decision to be postponed until valuable information is obtained. We expect renting to be most attractive when the current price-rent ratio is high, there is substantial uncertainty about future rents or prices, and transactions are difficult to reverse. We can generalize this analysis by using dynamic programming to include uncertainty over several years, noninfinite transaction costs, and a variety of parametric assumptions.
A household considering the purchase of a home faces uncertainty regarding both the potential future rent savings from home ownership and changes in the cost of buying a house. It should weigh the benefits from buying immediately against the costs of doing so, including the cost of exercising the purchase option and thereby forfeiting the benefits from waiting for more information about the evolution of rent and prices.
We will often refer to the net implicit cash flow C from home ownership—rent you would otherwise have to pay minus the expenses associated with home ownership—simply as rent. Assume that the evolution of rent C can be described by this geometric Brownian motion equation:
 |
(1)
|
where a is the trend rate of growth of rent and dz
is the increment of a standard Wiener process.
The cost P of buying a house also evolves according to geometric Brownian motion:
 |
(2)
|
The correlation coefficient between rent C and price P is r.
The purchase of a house costs P and provides an expected present value V of the cash flow:
 |
(3)
|
where R is the risk-adjusted required rate of return.
We know from the dynamic programming literature that the simple investment rule of buying when V is larger than P and selling when V is smaller than P is inappropriate in an uncertain world where there is an option value to waiting to make investments that are costly to reverse (Jensen, 1982; Pindyck, 1988; Dixit, 1989). Because the purchase of a house is expensive to undo, there is a potential benefit from postponing the purchase until price/rent conditions are decisively favorable. Dynamic programming can be used to determine the value F[C, P] of this purchase option.
Setting the return on the option, rF, equal to the expected capital gain on the option and using Ito's Lemma, we obtain this differential equation for the region in which the household waits to purchase:
A natural assumption that makes the analysis tractable is that the option value
is homogeneous of degree one: F[C, P] = Pf[C/P]. The substitution of the requisite
partial derivatives into our differential equation yields
where f = s2 - 2rsq + q2.
This is a homogeneous second-order linear differential equation whose general solution has the form
where
As the rent-price ratio C/P approaches 0, the value of the buy option does too; therefore, B1 = 0:
 |
(4)
|
For the homeowner, the value G[C, P] of the house includes the cash flow C and
the value of the option to sell if the rent/price ratio falls sufficiently.
Assuming this value to be homogeneous of degree one, G[C, P] = Pg[C/P], the
differential equation in the region where the home is held is:
The general solution has the form
As the rent-price ratio C/P increases, the value of the sell option approaches 0; therefore, A2 = 0:
 |
(5)
|
A household that is waiting to buy will do so when the rent/price ratio rises to the threshold l1; a homeowner waiting to sell will do so when the rent/price ratio falls to the threshold l2. At C/P = l1, the value of the buy option is equal to the value of owning the house net of the purchase price:
Thus
 |
(6)
|
The first derivatives are also equal at C/P = l1:
 |
(7)
|
At C/P = l2, the value of the sell option is equal to the value of the buy option plus the sale price net of the proportional sales cost g,
Thus
 |
(8)
|
and the first derivatives are equal at C/P = l2:
 |
(9)
|
The substitution of the differential equations (4) and (5) into the value-matching and smooth-pasting conditions (6) - (9) gives these four equations,
which can be solved for the thresholds l1 and l2 and the differential-equation parameters A1 and B2.
To illustrate this model, consider the following base case: a = 0.05, s = 0.10, b = 0.05, q = 0.20, r = 0.50, R = 0.10, and g = 0.08. Rent has a 5% trend growth rate and 10% standard deviation; price has a 5% trend growth rate and 20% standard deviation. The correlation between rent and price is 0.50. The required rate of return is 10% and the transaction cost for house sales is 8% (including brokerage commission, legal fees, and fixup costs).
Using these parameter values, the present value of the rent is given by Equation 3:
If there were no transaction costs, the house should be bought when the rent-price ratio is above 0.05 and sold when the rent-price ratio is below 0.05. With an 8% sales cost, the threshold rent-price ratios work out to be l1 = 0.069 and l2 = 0.034. A household that is waiting to buy should do so when the rent-price ratio rises to 0.069; a homeowner should sell when the rent-price ratio falls below 0.034. A buyer requires a rent-price ratio 38% larger than 0.05 because a purchase has the additional cost of extinguishing the possibility of buying at terms that are even more favorable and also less likely to incur future transaction costs. A seller requires a rent-price ratio 32% below 0.05 because a sale incurs transaction costs and extinguishes the possibility of selling at more favorable terms that are less likely to be reversed sufficiently to persuade the seller to buy again. Figure 1 shows how increased transaction costs widen the thresholds.
These effects are substantial. A rent-price ratio of 0.05 implies that a house with a $20,000 cash flow is worth $20,000/0.05 = $400,000. The 0.069 threshold implies that a buyer is only willing to pay $20,000/0.069 = $290,000; the 0.034 threshold implies that a homeowner isn’t willing to sell for less than $20,000/0.069 = $588,000.
The other ceteris paribus comparative static multipliers in the neighborhood of the base case are reasonable. A higher trend growth rate of rent or price reduces the thresholds for buying and selling. This is analogous to growth stocks which trade at low earnings/price ratios (high price/earnings ratios). The faster that rent and price are expected to increase in the future, the lower the rent-price ratio at which trades take place. Similarly, a higher required rate of return increases the threshold rent-price ratios: for given growth rates, houses need a high current rent-price ratio to provide households with a high rate of return. If the current rent-price ratio isn’t sufficiently high, households will be less willing to buy and more willing to sell.
Larger standard deviations of rent and price increase the value of the buy and sell options and thereby make it more attractive to wait and not exercise these options. Thus the buy threshold is higher and the sell threshold is lower. Households waiting to buy should consider the possibility that, if they do buy now, the rent-price ratio may subsequently fall, causing them to regret their purchase. They should also consider the possibility that, if they do not buy now, the rent-price ratio may subsequently rise, giving them the opportunity to buy at even more favorable terms that are less likely to be reversed in the future. Thus high standard deviations make waiting more attractive and necessitate a higher rent-price threshold. On the other hand, homeowners who are waiting to sell should consider the possibility that, if they sell now, the rent-price ratio may subsequently rise, causing them to regret their sale. They should also consider the possibility that, if they do not sell now, the rent-price ratio may subsequently fall, giving them the opportunity to sell at even more favorable terms that are less likely to be reversed in the future. Thus high standard deviations make waiting more attractive and necessitate a lower rent-price threshold.
Figure 2 shows the effects of the standard deviations on the thresholds. In these calculations, the standard deviation of price is fixed at twice the standard deviation of rent. As the standard deviations approach 0, the rent-price threshold for buying approaches 0.05 and the rent-price threshold for selling approaches 0.046, which is 0.05 minus the 8% transaction cost.
Figure 3 shows that a higher correlation between rent and price reduces the threshold for buying and raises the threshold for selling because it is less likely that the rent-price ratio will change significantly in the future. Put another way, the smaller the variability of the rent-price ratio, the smaller is the probability that conditions will be even more favorable in the future and the smaller, also, is the probability that the household will someday want to reverse a decision made today.
Conclusion
This paper presents a dynamic model of residential real estate valuation where purchases and sales are affected by uncertain projections of rents and prices. Higher anticipated growth rates of rent and prices and a higher required return increase the rent-price thresholds for both buyers and sellers. Higher transaction costs, higher standard deviations of rents and prices, and a higher correlation between rents and prices increase the rent-price threshold for buyers and reduce the rent-price threshold for sellers.
One implication is that renting maintains an implicit “flexibility” option that may be quite valuable. Another implication is that the combination of substantial uncertainty and large transaction costs can create a large wedge between the prices buyers are willing to pay and sellers are willing to accept. This wedge may explain why—unlike the stock market—so many real estate transactions seem motivated by socio-demographic necessity (marriage, divorce, relocation), rather than purely economic calculations. In the absence of this wedge, we would expect to find households shifting back and forth between renting and buying. Because of this wedge, homeowners typically sell because they have to, not because they consider the sale price high enough to make renting more attractive than homeownership.
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Figure 1 The Effect of Transaction Costs on the Thresholds
Figure 2 The Effect of Uncertainty on the Thresholds
Figure 3 The Effect of the Rent-Price Correlation on the Thresholds