The Time Value of Housing: Historical Evidence on Discount Rates

Most London housing transactions involve trading long leases of varying lengths. We exploit this to estimate the time value of housing – the relationship between the price of a property and the term of ownership – over a 100 years and derive implied discount rates. For our empirical analysis, we compile a unique historical data set (1987–92) to abstract from the right to extend leases currently enjoyed by tenants. Across a variety of specifications and samples, we find that leasehold prices are consistent with a time declining schedule and low long‐term discount rates in housing markets.

The shape of discount rate functionsor the term structure of discount ratesprovokes considerable research interest across a number of fields. In this article, we use sales of leasehold dwellings to investigate discount rates in housing markets, complementing a recent literature that exploits features of property tenure to estimate market discount rates over long horizons (Wong et al., 2008;Gautier and van Vuuren, 2014;Giglio et al., 2015a, b;Fesselmeyer et al., 2016). Under a leasehold arrangement, a property is owned only for a fixed term so the intuition for why leasehold prices may contain information on discount rates is straightforward. Consider two identical properties, one sold with a fixed term 99-year lease and the other with a 999-year lease. 1 In the absence of any other contractual differences, the gap between the two sale prices must reflect the value of an ownership claim for 900 years, discounted 99 years from now.
As with Giglio et al. (2015a) (henceforth GMS), our empirical analysis centres on the English housing market. 2 Our contribution can be distinguished by two main differences relative to that paper. First, we compile and refine a unique historical data set of property sales from 1987 to 1992 (before the start of the GMS sample), taking advantage of a geographical setting -Prime Central London, the highly urbanised core of London covering Mayfair, Chelsea and Kensingtonin which leaseholds account for four-fifths of sales. In England and Wales, reforms in 1993 gave many leaseholders rights to extend their leases or purchase them outright, at a premium agreed with the landlord or decided by a tribunal (if the two parties fail to reach agreement). This option is regarded as valuable, especially for short lease properties, and is exercised for most leases well before the term runs down. 3 The historical data set allows us to abstract from these rights. Second, we concentrate on the shorter end of the term structure and estimate discount rates for leases in the 1 to 99 year range. (With no extension rights before 1993, we find a greater proportion of short leases in the historical data.) This range is likely to be important for public policies that have medium to long-term consequencesfor example, infrastructure investments, pension savings, mortgages and related financial productsand given the similarities between very short leases and rentals, our findings also relate to research on rent-price ratios (Smith and Smith, 2006;Gallin, 2008;Campbell et al., 2009;Bracke, 2015).
The principal finding from analysis of our historical sales data set is that the housing market discount rate schedule over 100 years is declining, with rates net of rental growth around 3% at 100 years. To bridge our article directly with the GMS study, we next replicate our historical analysis using a sample of sales from the same period . Crucially, we continue to find a declining term structure of discount rates in this later setting. In terms of discount rate levels, our estimate of 2% at 100 years from this exercise sits comfortably with the GMS finding of 1.9% for the same sample period. By comparing the two samples (1987-92 and 2004-13), we can also shed new light on whether the housing market term structure changes over time. We find that on average housing market discount rates are 1.6 percentage points lower in the later setting, showing for the first time that decisions about the long-run may be context specific.
Our regressions use street fixed effects and a large number of property characteristics extracted from sales brochures to disentangle lease length from other neighbourhood and property features. We control for the condition of the property to reflect that a rental externality (Henderson and Ioannides, 1983) may reduce incentives to maintain properties held on short leases. By only comparing leaseholds with other leaseholds, we rule out unobserved differences between (and selection into) leasehold and freehold properties, and in restricting attention to hard-to-redevelop flats we control for potential differences in the value of a redevelopment option (Capozza and Sick, 1991). We also take account of residual contractual differences between leases, carefully separating out those sold with a share in the freehold and controlling for rents paid to the freeholder (the so-called ground rents) where these are significant. Our setting is one in which very few buyers require mortgage finance so this is also unlikely to be driving results. Additionally, we undertake a number of auxiliary regressions that demonstrate that: (i) conditional on our controls there is no relationship between rental value and lease length for properties in our sample; and (ii) that our findings are largely insensitive to changes in sample and specifications, including those that: (a) use minimal controls, 3 A further complication arises in this setting because following the 1993 legislation, a number of real estate companies began to publish and promote graphs purporting to show the relationship between lease length and sales price. These graphs have subsequently become the received wisdom for valuers (and tribunals) in determining the premium for lease extension. Surveyors and agents have used this estimated premium to value and price leasehold properties.
(b) use within building variation, (c) use different time periods or geographies, or (d) rely on different estimation methods.
These results lead us to conclude that omitted variables, for example, omitted structural building characteristics or contractual features, are unlikely to be behind our main results.
Our results are directly relevant to a number of housing settings, including to individuals in England and Wales who own or are contemplating buying a lease. 4 In addition, housing market discount rates may be useful in situations that require estimates of real estate values in the far-off future, for example, long duration mortgages or housing equity release products (reverse mortgages). In some policy settings outside housing, such as pension financing, infrastructure investments and environmental regulation, benefits also accrue only in the far-off future. Debates following the Stern review (Stern, 2006;Weitzman, 2007;Nordhaus, 2007) demonstrate that in such cases, assumptions about discount rates can be paramount in deciding the optimal policy response. Some authorities, notably the Office of Budgetary Management in the United States, guide policy-makers to use a constant discount rate across all time horizons, while others including the UK, France, Norway and Denmark have adopted time-declining policy rates (Cropper et al., 2014). Whether our findings can be usefully deployed to these questions depends on the risk characteristics of residential real estate as compared to the ones of the policy application. For instance, infrastructure investments share with real estate some important features such as indivisibility, low liquidity and location specificity; the discount rates patterns uncovered in this article may therefore provide some additional guidance in cases where housing and infrastructure values are closely related.
The article proceeds as follows. In the next Section, we describe the institutional setting of the leasehold market in the UK. The following Section sets out the data sources that we use for our empirical analysis. We explain our approach in Section 3 before reporting results in Section 4. In Section 5, we discuss threats to identification, paying close attention to findings in the recent literature and outlining techniques and several auxiliary regressions we undertake to resolve them. Section 6 concludes.

Residential Leasehold in England and Wales
In England and Wales as many as 1 million houses and 2 million flats are owned under long leases, 40% of recent new build properties are leased and leaseholds account for around a quarter of residential sales. 5 Leaseholds proliferate where populations are most concentratedthey account for around half of the sales in London and over four fifths of sales in prime central London (Figure 1).
Leasehold ownership is an alternative way to hold residential property outside the more widely studied home-ownership and rental forms of tenure. 6 Conceptualising tenure forms as distinct bundles of use, transfer, and contracting rights and obligations (Besley and Ghatak, 2009), the fundamental characteristic of leasehold ownership is that it grants the purchaser of the leasethe lessee or leaseholderuse rights for a long but finite period, commonly 99 or 125 years at origination, known as the term of the lease. As such, it lies between freehold home ownership (indefinite use rights) and renting (use rights for a short fixed period). As with freehold owners, leaseholders can gift or sell the asset (transfer rights) and mortgage or rent the property (contract rights). 7 Existing leasehold interests can then be bought and sold on the open market. When such a trade takes place, the buyer inherits the existing lease agreement in full, including the duration of the remaining use rights of the contract. This is known as the The data set is available at http://www.landregistry.gov.uk/market-trend-data/public-data. The public version of the data set only contains an indicator variable which labels properties as freeholds or leaseholds. For the main analysis of this article, we use proprietary data from real estate agencies in Central London. Colour figure can be viewed at wileyonlinelibrary.com. 5 Department of Communities and Local Government Table FA1221 (S108): Household type by tenure, 2011-2; housing stock estimates from https://www.gov.uk/government/policies/helping-people-to-buy-ahome. 6 A full account of the history of residential leasehold and its evolution lies outside the scope of this work. Interested readers are referred to McDonald (1969) who describes the origins of residential leasehold ownership in the granting of land, or ground, leases in feudal England. Under such arrangements, tenants would develop leased land, often to agreed parameters, and use it for the term of the lease with the land and buildings reverting to the land owner thereafter. McDonald (1969) suggests several reasons why this arrangement may have evolved, for example, to enable management of the large fixed costs of providing services such as drainage, sea-defences, street lighting and road construction. 7 Although technically the leaseholder cannot assign or sublet without the freeholder's approval. unexpired term of the lease and is simply the original term reduced by the elapsed time since the lease was granted. In contrast to freehold ownership, leasehold ownership implies multiple interests in the same real estate asset since the seller of the leaseholdthe lessor or freeholderretains an interest in the asset beyond the initial sale. 8 Land rents, known as ground rents, are typically paid annually in accordance with a payment schedule agreed at the start of the lease and represent an income to lessors rather than a payment for services. 9 Lessors also commonly retain the right to veto redevelopment or alteration to the property by the leaseholder during the term of the lease. If a leaseholder does wish to redevelop, the freeholder will demand a premium which is subject to negotiation between the parties.
Nearly all flats in England and Wales are owned with leasehold contracts. 10 This ownership structure provides a way to share costs for public goods (e.g. a shared staircase, garden, or lift) when a single building contains more than one dwelling. In some cases, the individual leaseholders collectively own the freehold interest while in other cases it is owned by a third party. The former is known as owning leasehold with a share in the freehold. It effectively allows owners to extend their leases indefinitely and is therefore analogous to freehold ownership of houses in terms of the use rights it grants. 11 The institutional framework around the right to extend or purchase a lease outright is an important consideration in our analysis. Prior to 1993, most leaseholders in England and Wales had no rights over leased property assets at the end of the lease term such that the land and all buildings would revert to the lessor. The only option open to leaseholders that wished to retain ownership was to negotiate a new lease with the lessor, either before an existing lease expired or at the end of the lease term. Major institutional changes in 1993 -described in detail in online Appendix Agranted widespread rights for leaseholds to extend their leases or to purchase them outright, at a price agreed with the landlord or decided by a tribunal. For the reasons set out in the introduction, leasehold sales after 1993 could be less informative about discount rates. Compiling the historical data set, we describe in the next Section permits us to sidestep issues relating to enfranchisement that could confound discount rate interpretations based on later sales. 8 This interestusually thought of as corresponding to the ownership of the ground beneath the real estate asset which has been leasedis known as the freehold interest and can also be traded in secondary markets. Note the distinction between a freehold interest in a real estate asset and freehold ownership of an asset. The former implies that there is a lease over the property and there being two interests. The latter implies a single interest. 9 In some cases ground rents are of a nominal amount, known as a peppercorn ground rent, or a fixed rent with no review. More often, ground rent payments are subject to review in intervals of 20, 25, 30 or 33 years. The lease sets out how the ground rent is reviewed at the review date but according to Savills (2012) it is common for ground rents to either double, to increase by a fixed amount, to be rebased against the retail price index (RPI), or to be rebased against a percentage of the capital value of the underlying property at such times. 10 A few flats are in fact held freehold, rather than share of freehold. These freehold flats will usually be the flat where the freeholder lives. They could have the right to receive ground rents from other leases in the building and, as described below, a stake in the residual interest as with other freehold interests. 11 The owner of the freehold interest for flats usually provides management and maintenance services to the building on behalf of the leaseholder(s), recovering costs through a fee known as a service charge. This applies regardless of whether the block is owned leasehold or share of freehold.

Context and Data Sources
To undertake the empirical analysis, we first create a data set of transactions in the prime central London (PCL) area for the period 1987-92. 12 We use a definition of PCL provided by real estate agents operating in the London market, including properties that belong to the following postcode districts: SW1, SW3, SW5, SW7, SW10, W1, W2, W8, W11, W14. 13 A large proportion of the PCL housing stock has for 300 years been owned by a small number of private land ownersincluding the Grosvenor, Cadogan, de Walden, Portman, Crown, Ilchester and Phillimore estates. These estates historically made extensive use of the leasehold tenure system to develop land in this area, maintaining some degree of control over the built environment.
Our primary source of data is Lonres.com, a subscription service for real estate agents and surveyors working in the PCL area. Sales information in the Lonres sample is provided by individual agents connected to the Lonres network and collated into a database. Many of the major agencies operating in the PCL market are in the Lonres database, including Savills, Knight Frank and John D Wood & Co. Because the database provides only a limited number of data fields, we extract and merge in additional property attributes from the original PDF sales brochures. In addition to the Lonres.com historical archives, we obtained access to the internal records of John D Wood & Co. (JDW), a real estate agency operating in the PCL area. Sale prices in the JDW sample, which also starts in 1987, have been verified by agents. 14 To obtain a clean data set, we drop suspected duplicate sales where the address is the same and a second sale occurs within 90 days, and data points where street or leasehold information is missing. Because we use a street-fixed effect strategy, all transactions on streets with just one property in the data set are also dropped.
We abstract from the right to extend leases by excluding sales that occurred after the Leasehold Reform Act of 1993 and those occurring in 1992, since this was an election year and both main parties were proposing leasehold reform. By doing so, we minimise concerns that leasehold prices in our data are influenced by the expectation of a reform. 15 Following the earlier 1967 Act, some low-value leasehold houses had already become enfranchisable, i.e. the leaseholder had the right to purchase the freehold of the property in exchange for a premium. Whether a house was enfranchisable or not depended on its rateable value. This is unobserved in our data so we obtained this information from the relevant local authorities, identifying a list of houses which were enfranchisable at the time and further exclude them from our sample. 16 Taken together these restrictions give us confidence we can avoid potentially confounding effects of rights to extend on the leasehold prices in our data. 12 Individual sale data before 1987 are extremely sparse in our data and therefore of little use for econometric analysis. 13 Postcode districts correspond to the first half of British postcodes and, in London, they typically include 10,000-20,000 separate addresses.
14 These prices are likely to be correctly measured because they are used to calculate agents' commissions. 15 As a robustness check, we also ran the analysis including 1992 sales. Results were materially unchanged. 16 Rather than dropping enfranchisable houses, as a robustness check, we also run the analysis including them but assigning them a dummy. This had no material effect on results.

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In the last part of the Results Section, we compare our findings with a similar sample of properties sold in PCL between 2004 and 2013. The information on these property transactions is again taken from Lonres.com. While the next subsection describes the 1987-1 sample, statistics on the 2004-13 sample are presented in online Appendix C. Table 1 describes the complete data set of 8,184 records, splitting the data into categories based on the type of dwelling (flats and houses) and data source (Lonres and JDW records). More than half the data points are leaseholds with less than 100 years unexpired term. Figure 2 shows the distribution of lease lengths in the sample: there are many data points for leases with 55-65 years left, for 85-100 years left, and between 120 and 125 years; there is a group of sales with unexpired leases between 950 and 999 years. The third column of Table 1 includes freehold houses and share of freehold flats, which are displayed for illustrative purposes since they are not part of the empirical analysis. Although share of freehold flats have a lease term, it is critical to put them together with freehold properties since their purchase includes a share of the freehold value of the building and, with it, the right to extend one's lease indefinitely. 17 Figure 3 shows the location of sales in the data set and Figure C2 in online Appendix C shows how observations in the data set are spread across the different quarters between 1987 and 1991.

Descriptive Statistics
All sale prices reported in the John D Wood & Co. archive are verified exchange prices. By contrast, only around 15% of Lonres data points have been verified against other data sources. When the price is non-verified, the figure may coincide with the  Table shows the number of sales in our data set which belong to the following categories: leaseholds with unexpired term below 100 years, leaseholds with unexpired term above 100 years and freeholds (including flats sold with a share of freehold). Our empirical analysis is restricted to leasehold properties; we only include freehold properties in this Table for illustrative purposes. The average unexpired term for leasehold flats with more than 100 years to expiry is 307, whereas the median unexpired term is 124. 17 Leasehold term for these properties tends to be long. In our data set, more than a third of freehold flats have a lease term longer than 945 years. A failure to account for these shares of freehold properties could result in spurious conclusions about the implied value of lease term. This is even more critical for recent data: Table C1 in online Appendix C shows that more than one third of PCL flats were sold as share of freehold in 2004-13. original asking price. Non-verified properties are equally found, in roughly the same proportion, across leasehold of different lengths and our hedonic regression contains a variable that flags non-verified properties. 18 Table 2 refers to the estimation sample (leasehold properties) and contains the descriptive statistics for all variables. Those that were immediately available from the original data tables include: HOUSE (whether the property is a house, as opposed to a flat), BEDROOMS (entered as a categorical variable), SALE QUARTER, STREET (entered as fixed effect), FLOOR LEVEL, VERIFIED (for sales in the Lonres data set, this variable indicates whether the sale price has been verified), MAISONETTE (indicates multi-level apartments), and ONEROUS GROUND RENT (we define the ground rent as onerous when it is above 0.1% of the sale price) (Figure 4). 19 All other variables shown were extracted from pdf brochures. Most are self-explanatory; INNEED indicates the presence, in the property advert, of the key phrase 'in need', which is often followed by expressions such as 'of improvements', 'of refurbishment' and so on.

Methodology
This Section develops a simple approach to estimate discount rates from sales prices, relying on the intuition that the gap between the sale prices of the property held forever and a property leased only for t years reflects the value of full ownership discounted t years from now. We call this the present value of use rights, i.e. the present value of consumption and/or investment returns that flow from the asset. We proceed in two steps: first estimating the discounts associated with leaseholds of a given length, then retrieving the implied discount rates.
Our identifying assumption is that conditional on controls, the only source of discounts are differences in the present value of use rights. Potential confounders include any unobserved factors which drive price differences between properties that are related to the term of the lease but do not arise because of discounting.

Measuring Leasehold Discounts
We model the logarithm of the price of a leasehold property, held for t years, as: where p(∞) is the log price of a property held on an infinite lease. The function f (t) represents the discount associated with a given lease length as opposed to a property Notes. The Table does not contain information on sale dates (described in Figure C2), sale locations (mapped in Figure 3), and lease length (see Figure 2) held forever (but still on a leasehold arrangement to avoid potential biases deriving from price differences between leaseholds and freeholds that do not depend on lease length).
To model the price of a property held forever, we follow the literature on hedonic regressions (Hill, 2013): where a j are street fixed effects, X are property attributes, and k s are quarterly dummies denoting the time of the sale (s). 20 Our baseline specifications include the full set of property attributes listed in Table 2, with the exception of square footage which is available for a subset of data points.
To estimate ln f (t), we employ three methods: (i) leasehold buckets; (ii) leasehold dummies; and (iii) a semiparametric approach based on Yatchew (1997  The pattern in sales well reflects the experience of market practitioners in that period and is consistent with national and local price indices. 1988 was a boom year, with real estate agents enjoying 'high volumes, high prices, and high commissions'. After that came a fall in the market in 1989, and the number of sales stabilised in 1990-1. Colour figure can be viewed at wileyonlinelibrary.com 20 Repeat sales regressions are an alternative to hedonic models but they require a sample with a sufficient number of properties which have been sold twice. Since the main sample includes only the years from 1987 to 1991, the repeat sales regression would only include properties that sold twice within five years. The resulting sample would be small and potentially affected by selection bias, as property that sold often could have particular (observed or unobserved) characteristics.
The bucket method divides leasehold properties into several large groups according to their lease length so that price effects can be estimated for each bucket. The dummy method pushes this further such that each integer value of lease length up to 999 years (the highest in our data) takes a categorical variable: 21 The semiparametric estimation approach described in Yatchew (1997) is reported in online Appendix B as a robustness check. By sorting all the observations in ascending order with respect to t and differencing them, we take advantage of the fact that ln f ðt 0 Þ À ln f ðtÞ tends towards zero. We can then use simple OLS to estimate a version of (1) that does not contain f (t). In a second step, we can apply common nonparametric estimation techniques to retrieve ln f (t) frompðtÞ ¼ p t Àpð1Þ, where the predicted price of a property held forever (pð1Þ) is derived from the first step described above.

Estimating Discount Rates
Taking Gordon's (1959) simple constant discount rate model to (1) implies that: 22 ln f ðtÞ ¼ lnð1 À e ÀR t T Þ: ( Our aim is to explore whether R t is constant, i.e. R t 1 ¼ R t 2 ¼ R, or varies over the time horizon in question. Prior expectations are that f (t) should satisfy f (0) = 0, f 0 ðtÞ [ 0 and lim x!1 f ðtÞ ¼ 1, indicating that a zero year lease should have no market value, that all else equal more years on a lease should make the property more valuable, and that at some point very long but finite leases should be equivalent to infinite leases. In practice, since we estimate the c t s in an unconstrained way, these conditions do not always hold and in Figure 5 the points are scattered and some estimates lie above the long-lease line. Before attempting inferences about discount rates, we therefore fit a local polynomial through the estimated points, fine tuning the bandwidth of the polynomial within reasonable limits. We then use the predicted values of the polynomial curves to compute the discount rates at each point in the term range by solving for each R t that corresponds to a pair fR t ; tg in (3). 21 To retrieve the true price discounts in each category the c coefficients must be exponentiated. Jensen's inequality could cause the estimated discount to be larger than the actual discount, because an average of logarithms is not the same as the logarithm of an average. In practice, the consequences of Jensen's inequality are likely to be limited. We confirmed this by running our baseline regression on simulated data. The impact of Jensen's inequality on estimates was apparent only at the third or fourth decimal point. 22 If the price of owning the property for one period is P(1), then the price of owning the property forever is: where R 1 is the net discount rate applied with an infinite horizon. In turn, R 1 ¼ r 1 À g 1 , where r 1 represents the gross discount rate and g 1 the growth rate of P(1) over time. For a property held for t years, we have that:

Leasehold Discounts
Our first results focus on leasehold discounts in the historical setting (1987-91). Table 3 shows the output of the hedonic regressions. The first specification uses the bucket approach and includes both leasehold houses and leasehold flats. The second specification is the baseline specification which focuses only on leasehold flats and adopts the more granular dummy approach where each lease term integer has its own categorical variable. Table B1 of Online Appendix B contains the first stage of the semiparametric approach alongside other robustness regressions. Coefficients across the two models are generally in line with intuition. Houses command a premium of 20% over flats controlling for other attributes such as bedrooms and street. 23 The coefficient on INNEED of 15-20% implies a discount for properties advertised as 'in need of improvement', an important control if poor maintenance is correlated with lease term. The (unreported) coefficients on  Table 3. The chart also plots the 95% confidence bands associated with each coefficient. The dashed horizontal line represents the value of long leases and in this case represents the value of a 999-year lease. Colour figure can be viewed at wileyonlinelibrary.com.

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SALEQUARTER together imply a mix-adjusted index of house prices in prime central London. This is increasing in 1987-9 and decreasing thereafter, a pattern consistent with other historical indices such as the Nationwide regional house price index for London (see Figure B1 in online Appendix B). The R 2 indicates that these models are able to explain approximately 78-82% of the variation in house prices. The first model of the Table, which excludes freehold properties, is designed to test for price differences between long leasehold properties of different lease lengths. We group leaseholds into four buckets: below 80 years, between 80 and 99 years, between 100 and 124 years, between 125 and 900 years and above 900 years (the baseline group). 24 The coefficients for other leasehold categories are not significant except for the coefficient on leaseholds with less than 80 years, which is significant at the 10% level. These results suggest that in this historical setting, very long maturity leaseholds Notes. Standard errors in parentheses clustered at the street level. *p < 0.10, **p < 0.05, ***p < 0.01. The baseline categories are flats, 1-bedroom properties, 1st floor. Both regressions include leasehold properties only. The second model is run on leasehold flats. All models have street and quarter fixed effects.
cannot be easily statistically distinguished from other very long maturity leaseholds of a different length. The second model of Table 3 is our baseline specification in which we drop houses to focus purely on leasehold flats and adopt the leasehold dummy estimation approach. Our main object of interest, the dummy coefficients, are not tabulated but are displayedexponentiatedin Figure 5. These estimates indicate the discount associated with all leasehold flats of a specific lease length with respect to leasehold flats with 999 years remaining. The point estimates are shown as dots with the 95% confidence intervals represented by the bars appearing to vary in line with the histogram of Figure 2, with the smallest errors corresponding to leases groups computed from more observations.

Discount Rates
The exponentiated c t coefficients in Figure 5 define the shape of f (t) in (3) above. 25 As expected, the estimates are slightly scattered, so in Figure 6 we fit a local polynomial to these points, weighting by the number of sales at that specific lease length. The curve is a second-degree local polynomial with a bandwidth of 15 years on both sides and an Epanechnikov weighting scheme. Confidence bands are represented by the grey areas  Figure 5. The chart focuses on the 1-99 year range. The dummy estimates are plotted as circles where the size of the circle is proportional to the number of observations for that specific lease length. The grey bands around them represent 95% confidence bands, computed using bootstrapping with 1,000 iterations. Colour figure can be viewed at wileyonlinelibrary.com. around the curve. Since we implement a two-step procedure, we compute the confidence bands using bootstrapping, reshuffling the original data set one thousand times. Although the curve is fitted across the whole lease range, we focus on leases of 1 to 99 years given our findings in the previous Section.
The polynomial fulfils the conditions described above: it is increasing and remains below the line representing the value of 999-year leases (the horizontal line at 100). We next use the predicted values of the polynomial curve to compute the discount rate at each point in the term range by solving for each R t that corresponds to a pair fR t ; tg in (3). The result is shown by the line in Figure 7, with circles representing the discounts derived from the original dummy estimates. Overall, these results indicate that leasehold prices in our setting appear to be consistent with a declining discount rate schedule. Very short leases imply discount rates of around 5-6%, whereas long leases, close to 100 years left, imply discount rates close to 3%.
These net discount rates can be used to estimate the gross discount rates prevailing at the time of our analysis (1987-92). One way of doing so is to add the long-run rate of real rent growth, as in GMS who take a real rent growth of 0.62% using the CPI component 'actual rents for housing' (series D7CE) from the UK Office of National Statistics for the period 1996-2012 . This would imply a 0.62% upward shift of the dots in Figure 7.

Comparison to Existing Estimates and Changes Over Time
In this subsection, we first compare our findings to existing estimates of the effect of leasehold term on sales prices. The most natural starting place is to compare our Notes. The chart shows the discount rates implied by the curve fitted in Figure 6. The discount rates implied by the corresponding individual dummy estimates are also plotted. As in Figure 6, the circle size is proportional to the number of observations for that specific lease length. This Figure represents a leasehold flat v.s. leasehold flat analysis, significantly departing from the conventional basis for establishing relativity used by market practitioners. Colour figure can be viewed at wileyonlinelibrary.com. estimates to heuristics commonly followed by leasehold valuers and tribunals in the UK markets because, like us, these mostly focus on the 1 to 99-year range and also purport to abstract from leaseholder rights. 26 Such a comparison highlights that the shape of the curve we fit diverges substantially from the curves in common use by practitionersas we show in Figure B7 in online Appendix B.
Although our central focus is not on very long-run discount rates implied by leases of maturities longer than 100 years, we can also demonstrate that our results sit comfortably with the recent academic findings in GMS. To illustrate, in the first column of Table 4, we show results from GMS Table III. In the second column, we present results from a similar specification but using our 2004-13 PCL sample. To be consistent with GMS main results, in this regression, we use freehold properties as the baseline. There are no major differences in coefficients derived from the two studies except for the shortest-lease bucket. 27 Following the calibration method adopted by GMS, our results for this sample are consistent with very long-term discount rates of around 2% in 2004-13, very close to GMS's main finding that very long term net discount rates in housing markets are close to 1.9% in this period for England as a whole.
At the other end of the spectrum, the discount rate on very short leases can be compared to the average rent-price ratio in the area, given some degree of substitutability between renting for a few years and buying a short lease. Bracke Table 4 Comparison with Giglio et al. Giglio et al. (2015a) England + Wales Notes. The first column of the Table reports the results from GMS Table III, column (1). The second column reports results from an analogous regression run on PCL properties advertised for sale between 2004 and 2013 (the same sample period used by GMS) and recorded by the portal Lonres. In these regressions, the baseline category is freehold properties. 26 We are sceptical about the validity of these heuristics due to lack of a rigorous statistical approach and the impossibility of replication. See online Appendix A for more details. 27 This likely derives from differences in the spatial scope of the two studies. In a robustness check, Giglio et al. (2015a) report findings for London graphically (with standard errors unreported). These show coefficients that are similar to GMS's baseline results above but with a 13% discount for the shortest-lease bucket. Interestingly, the GMS London analysis also finds a positive coefficient on the 700+ year group. To investigate changes over time in more depth, we next contrast estimates of discount rates across the 1 to 99-year range in our two sample periods. In Figure 8, we show that discount rates in the 2004-13 sample of sales are consistently lower than in the 1987-91 data across the whole range. Both lines decline slowly with regressions confirming that the slope of both lines is non-zero. Importantly, the average level of the two lines is materially different with the average 1987-91 rate (4.1%) being significantly higher than the average rate in 2004-13 (2.5%). There could be many explanations behind these findings, for instance changes in the risk-free rate, the riskiness of housing and changes to the institutional setting (including the influence of the existing relativity graphs). Although unable to distinguish fully between them at this time, we do demonstrate in Figure B4 in online Appendix B that implied discount rates were stable either side of the 4.5% drop in the Bank of England base rate between October 2008 and March 2009. At the same time, Clark's (1988) findings that longterm interest rates in land markets fell from around 10% in Medieval England to around 4% by the start of the nineteenth Century place our results within a much longer historical perspective.

Threats to Identification and Robustness Checks
The baseline specification in column (2) of Table 3 incorporates a number of strategies to isolate the present value of use rights from other sources of variation. The street fixed effects partial out granular location-specific effects and help us control for some unobserved housing attributes, for example, where properties on the same street share the same style and layout. 28 This regression uses the most complete set of structural dwelling attributes that our historical data set allows. We control for the condition of the property to reflect that a rental externality (Henderson and Ioannides, 1983) may reduce incentives to maintain properties held on short leases. 29 By only comparing leaseholds with other leaseholds, we rule out potentially unobserved differences between leasehold and freehold properties and related concerns, for example, endogenous selection of properties into freehold and leasehold tenure, buyer preferences for freeholds, or other factors that drive systematic value differences between the property groups. Remaining observable contractual differences between leases are accounted for by carefully separating out and excluding those leases sold with a share in the freehold and by controlling for rents paid to the freeholder (the so-called ground rents) where these are significant. Auxiliary analysis in Giglio et al. (2015a) Appendix A.1.7 gives us confidence that additional contract featuressuch as restrictive covenantsare unlikely to vary systematically with remaining lease term. Similarly by comparing flats only with other flats, we avoid unobserved differences between flats and houses, including corresponding concerns around market segmentation and endogenous dwelling structure. Since flats cannot usually be redeveloped to a higher density, restricting attention to these dwellings has the additional benefit of controlling for potential differences in the value of a redevelopment option which could be correlated with the term of the lease (Capozza and Sick, 1991). 30 We aim to mitigate omitted variable concerns further in two supplementary regressions. In the first, we test whether lease length has an effect rental value conditional on our set of controls. To do so, we match properties in our main specification to a data set of property rentals in the period 2004-14 which restricts the sample to around 1,000 properties. Figure 9 shows that there is no clear relationship between lease length and rental price. We conclude that if rental values are strongly correlated over time, omitted property characteristics that drive both rents and prices are unlikely to be biasing our results. For the second auxiliary regression, we repeat our baseline analysis but additionally including a building fixed effect for all properties that share the same street name and number. Results, displayed in Figure 10, demonstrate that our main finding of a declining discount rate is robust to this demanding specification which controls for all unobserved variation at the level of the building, including, for example, age of the structure. 31 28 We also ran our analysis using postcode fixed effects instead of street fixed effects and obtained nearly identical estimates. 29 It should also be noted that UK leaseholders have an obligation to maintain a property in good state and that failure to do so might trigger a dilapidation claim from the freeholder. 30 The value of a redevelopment option is likely a function of the up-front costs of redevelopment and the increased rents that will result. With a short lease, the value of the option is low because there are few periods over which to recover capital costs. Our argument is that if flats cannot be redeveloped to a higher density then redevelopment gains will be hard to achieve whatever the term of the lease. 31 In unreported results, we also confirm our main findings our insensitive to two additional variations on our baseline specification. In the first of these, we use postcodes instead of streets as our geographical fixed effects. In the second, we include the dwelling's Council Tax band, which is based on the assessed value of the property used for taxation purposes, as an additional control. In both cases, our sample size is reduced so we prefer the baseline specification above. (i) the dependent variable is price per square foot; 32 (ii) we interact street and quarter dummies to allow for street-quarter intercepts, which amounts to comparing only properties within the same street and sold in the same quarter; 33 (iii) we split the sample into the submarkets of Kensington versus Chelsea; and (iv) we split the sample into the boom period (1987-8) versus the bust period (1990-1).
Finally, in the spirit of Altonji et al. (2005) and Oster (2018), we show in Figure B8 of online Appendix B that a regression specification with only street fixed effects and no other control variable is able to replicate the shape of leasehold prices shown in Figure 6 (delivering a declining schedule of discount rates). Moreover, since the R 2 of  Table 3 but adds the quarter of the rental. As previously, the dots are the point estimates and the whiskers the 95% confidence interval. Colour figure can be viewed at wiley onlinelibrary.com. 32 Square footage is available for around half of our data points. Examining the data set reveals no clear pattern to omission, i.e. expensive and less expensive properties, or big or small properties, are equally likely to have square footage recorded. 33 In practice, this reduces the effective sample size by a third but results remain the same. our full specification is 81%, there is relatively little scope for omitted variables to have a large impact on results. A more general concern may be that our results lack external validity to policy settings if the discount rates we uncover are driven by time preferences as well as horizon-specific features of housing markets, such as the riskiness of housing or financing frictions in mortgage markets specific to some parts of the term range. In this context, we note that a high proportion of buyers in this area were not dependent on mortgage financing. 34

Conclusion
This article describes the association between lease length and sales prices of flats in the London market, using data from two distinct periods: 1987-91 when leaseholders had no rights over leased assets on lease expiry and 2004-13 when they did. We compute housing market discount rates through the application of the simple Gordon model. Results are suggestive of declining discount rates over the 1 to 99 year range in both samples. To the extent that discount rates in housing markets are a useful indicator for social discount rates, these findings could support the use of a declining discount rate function for policy-making, as have already been adopted in the UK and in France. The chart shows discount rates implied by a local polynomial fitted through leasehold estimates derived from a model that mirrors column (3) of Table 3 but additionally includes building fixed effects. Discount rates implied by individual dummies are also plotted, with circle size proportional to the number of observations. Colour figure can be viewed at wileyonline library.com. We investigate whether the average level of housing market discount rates stays constant between two samples of sales: in 1987-91 and 2004-13. We find rates in the later period are significantly lower (by 1.6 percentage points on average) than rates in the earlier setting, albeit slightly converging at very long horizons to between 2% and 3%. To the best of our knowledge, this is the first demonstration that long-run housing market discount rates implied by leaseholds may change over time.
Our findings are also relevant to current and potential leaseholders in England and Wales where the relationship between lease length and property value, assuming no rights to extend a lease, is an important factor in determining the price required to purchase a lease extension or to enfranchise a leasehold property. The results stand in direct contrast to rule-of-thumb approaches to valuing lease term used by market practitioners. These differences suggest that lease extensions could result in transfers between leaseholders and freeholders that are out of kilter with market values (Badarinza and Ramadorai, 2014). Additional Supporting Information may be found in the online version of this article: