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Appendix A. Why We Do Not Use a Regression Model
We would like to explain in more detail why we seek to provide an alternative analytical approach to a large part of the literature about house prices that relies on time-series or panel-data econometrics.
Our main reason for not using regression analysis is the desire to have a more structural model, with a few clearly interpretable parameters and inputs, like household income, mortgage rates, loan-to-value ratio, without entering the full complexity of general-equilibrium mod-eling.
Other reasons for not relying on regression analysis is that the regression models are often only vaguely motivated by economic theory in terms of their structure and the choice of variables. Variables often included in the regression model are income, multiple interest rates (long, short), stock of credit, stock market indicators, demographic variables, etc. But if both (new) credit and income are included, implicitly the loan-to-value ratio is lurking behind without the analyst being able to use it explicitly for scenario analysis, for instance. Moreover, explanatory variables usually capture both fundamental and transitory factors making it difficult to distinguish between the two in the analysis.
Econometric models also require a significant amount of historical data, preferably over multiple business cycles or financial cycles to avoid the parameter instability. With short data sample, the coefficients can be severely biased, without using theory restrictions. Also, given the long-run trends in nominal and real interest rates, it is not trivial to get theory-consistent values and signs of key coefficients without tight priors. Further, regression models are static in their nature and intrinsically backward-looking, unlike the dynamic borrowing capacity or the present-value computations we try to propose. We believe that the value of an asset should be mainly determined by the current and future conditions not by the past.
We provide a simple illustration of these problems using the data for Prague. We estimate a simple co-integrating relationship using the same variables we use in the borrowing capacity and the intrinsic value approach, i.e. household income and the mortgage rates. We estimate the following relationship:19 in a sample running from 2002:Q1 to 2017:Q4.
The estimated relationship does not have stable coefficients throughout the sample and the fitted value of house prices for an average apartment changes as the coefficients get re-estimated. Further, under no transformation of the interest rate term do the unconstrained coefficients follow a sign implied by the theory. Coefficient instability severely limits the scope of sensitivity and counterfactual scenarios.
Fig. 11 illustrates the magnitude of over- and under-valuation as implied by the estimated set of three regression coefficients. Clearly, as the sample accrues, the ex-post views differ from real-time views. Specifically, the most recent period, the period of most interest, clearly is affected by coefficient instability to a great extent.
Another interesting aspect of standard regression-based estimates is that a linear regression model with an intercept (the ‘constant’) has by construction residuals averaging to zero. The model thus by construction flags periods of under-valuation as well as periods of over-valuation, in contrast to both approaches in this paper which suggest that in Prague the periods of overvaluation and ‘fair’ valuation are most common, without house prices getting undervalued much. Further, in models specified in growth-rates only, the intercept itself is problematic, as it introduces an autonomous drift in house prices that exceeds the dynamics of fundamentals.
It is true that our estimated model is rudimentary. However, the used variables provided enough insight with the borrowing capacity and intrinsic value approaches and adding more variables to the model, given the sample size, would likely over-parameterize the model and thus require more advanced shrinkage estimation with careful cross-validation to avoid over-fitting. In addition, our aim is not to come up with a forecasting model for house prices but to find an ‘equilibrium’ towards which house prices will revert over time.20
Appendix B. Building Prudential Scenarios and Counterfactuals with the SBC Approach
As alluded to in the main body of the paper, for certain policy simulations or counterfactual simulations care must be taken when using the static borrowing capacity approach. One important consideration is whether the counterfactual scenario should capture the short-run, medium-run, or the long-run effects. For each of the horizons we can assume some variables as fixed or at least slow-moving. Another important consideration is what prudential constraints are binding or what is their distance from the bound.21 See Andrle and Plašil (2019) for details how to use the approach proposed for building policy scenarios to assess prudential policies on credit and house prices.
For instance, after a decline in mortgage interest rates, which relaxes the borrowing constraint and for unchanged DSTI ratio, α, the amount of loan available for households increases. The attainable prices may increase thanks to lower interest rates and potentially higher loan amounts (without the need of higher income) but the size of the increase in the short run depends on the behavior of the down payment and the distance from the loan-to-value ratio maximum, φmax. If the down payment stays constant and φ < φmax, households can borrow more and–ceteris paribus–the warranted house prices are higher. The attainable prices correspond to PH1 = L1 +D0, with L1 ≥ L0 and higher leverage until the constraint of the LTVmax becomes binding. If the loan-to-value ratio is at its regulatory maximum, φ = φmax the lower mortgage interest rates won’t allow households to borrow more unless the down payment in-creases, assuming that flexible house prices would immediately absorb the potential increase in the loan. Larger increase in house prices due to a decline in mortgage interest rates would occur if the down payment could increase and accommodate the higher mortgage loan while stabilizing the leverage.
It is thus clear that in the short term, the dynamics of the funds available for the down payment can become an important factor for assessing the effects of prudential policies. One way to see that is to recall the fundamental relationship
where Lt is the loan attainable by the household and Dt is the available down payment.
In the main body of the paper, the assumption of binding debt-service-to-income ratio, α and binding loan-to-value ratio, φ, was used to derive the baseline borrowing-capacity formula, with the implied down payment. This is realistic and the formula is valid for the observed average values α and φ. Yet for some prudential scenarios the fundamental borrowing capacity relationship may be rewritten using explicit expression for the down payment:
for easier short-term analysis of the down payment and income dynamics.
We will illustrate that for the credit-constrained households, the down payment can be linked to their current income, Dt = κtYt, where κt links the existing funds for down payment to household income. The attainable house pricing condition can then be always formulated as
At any point in time, the parameters α, κ,, or LTV can be set or computed, depending on the scope of the analysis. However, they cannot be set independently. These quantities are closely related as
where we have used the abbreviation, f(zt ) for the interest rate term in (4).
Specifically, we can assume that the down payment of the household comes from accumulation of past savings from their income and a portion due to other wealth:
For credit-constrained households, a simple law of motion for the down payment can be specified for the analysis. Assume the household has been saving every year a portion of their income for a down payment, s×Yt, for a R years in total. Assuming zero interest on their liquid saving for simplicity, their funds available for down payment in the current periods are thus given by:
where gt stands for the gross income growth of the household in period, t. For actual computations time-varying growth rates are easy and important to use. But to help with the intuition, we can write down a steady-state version of the down payment to income ratio as
where the crucial dependence on R, the number of years of saving for the down payment, and on the income growth, g, should be noted. This simple relationship should help the intuition on linking the down payment to (past) levels of income.
The scenarios for bounds of the effects of prudential policies can be devised. However, it should be clear that a more comprehensive structural model is needed for more advanced tasks. The important assumption behind the SBC approach is that households are constrained and their perceived value of housing makes them willing buyers up to their debt-service-to-income and loan-to-value potential.
Appendix C. Toolbox
Together with this paper we are developing a user-friendly toolbox for house prices assessment. We have used the power of R and Matlab computing environments. Using other plat-forms would be straightforward.
In addition, we have developed a user-friendly web-based graphic user interface (GUI) to improve the user experience. For instance, the interface developed in Shiny/R environment, see Fig. 12, can be used for flexible and interactive exploration of the borrowing capacity approach.
Simply put, given their income the households can purchase a small but expensive, luxurious property or the opposite.
While there are plausible economic reasons why the share of income devoted on housing can increase with the increase in real income, this would be reflected in the change of parameter α and very likely would be gradual.
Adjustable-Rate Mortgages are not mortgages with flexible floating interest rate. Typically, the mortgage duration is N period, where after each K periods, often two or five years, the mortgage interest rate can be changed.
Finding the maximum loan for the baseline DBC and the offset-account DBC differs only in how amortization schedule is constructed. For the offset-account DBC annuities are further adjusted for the saved deposits and the interest proceeds.
In practical applications, a simple grid search with reasonable scaling coefficients applied to static borrowing capacity can be used. A more computationally-involved alternative is to use traditional optimizers where the violation of constraints is heavily penalized.
The ‘life vest’ parable we borrow from Ashwat Damodaran thoughts on valuation. www.stern.nyu.edu/ damodar/pptfiles/eq/ValClosing.ppt
At retail level, it is unlikely that an increase in leverage also increases the opportunity cost, ie, as would be the case for corporations.
This implies no loss of generality. More frequent changes in interest rates (or ever floating rates) can be considered at the cost of heavier notation. A common value for K is K = 5 years.
We create a fine grid of scaling coefficients for the size of the loan, compute multiple valuations, and choose the one that satisfies all the criteria, including the LTV criterion. For instance, (Damodaran, 2012, pp. 749) points to this issue: “it is worth bearing in mind that the [debt] ratios should be based on the value of the property rather than the funding needs...This, of course, creates circular reasoning since the cost of capital is necessary to estimate the value of the property in the first place.” Our view is that while problem is self-referential it is a standard, solvable fixed-point problem and not circular reasoning.
In the Czech Republic the income measure in the mind of public is the monthly one, traditionally. We make sure that the corresponding annual totals are used where appropriate.
The scaling factor essentially reflects two different phenomena. First, the higher declared income is partly due to a higher average number of applicants included in the documented income. Second, the income level of applicants stands above the nation-wide averages thus positive selection bias can be observed.
For the years 2002–2003 where the mortgage interest rate data are not available, we use the yields of the 5Y government bonds to interpolate the mortgage loans using a simple regression relationship, given the very tight co-movement of the mortgage rates and government bonds yields before government bonds hit the effective zero or negative yield in the aftermath of the great financial crisis.
While the valuation formula accommodate a time-varying opportunity cost, we use a constant value in the application. First, the dynamics of risk-free yields with the risk-premium on equity markets can result in a rather stable profile of the required yield. Second, given the long horizon of the analysis, the medium- and long-run value of the required yield clearly dominate the valuation. This holds for most NPV computations but is further aggravated by the interaction of discounting profile and net adjusted rent. In early years, the rent adjusted for mortgage payments is quite small.
Unlike in the United States, where property taxes can range over 1 percent of the appraised value.
All data are available from the authors, with the exception of the Czech National Bank’s nominal income forecasts. Note that no forecasts are needed to obtain the baseline static borrowing capacity measure.
Effective from October 2018, the Czech National Bank issued a recommendation to commercial banks that the initial debt-service-to-income ratio, α, should not be higher than 45 %.
The key point here is mostly to compare lower expected income growth persistence and higher income growth persistence, the value is mostly illustrative. The half-life of an AR(1) process with ρ = 0.85 is 4.3 years, that is, after 4.3 years the current growth reduces its deviation from the long-run growth by half. The baseline persistence of ρ = 0.25 is much smaller and it should be understood mainly in the context of annual frequency and the need to extend the CNB’s medium-term forecast towards the long-run growth.
Alternative specifications of the interest-rate term do not matter for the stability considerations or fit. Note that in case of growth-rate specifications, the intercept should be restricted to zero to avoid a deterministic trend in house prices in excess of value implied by fundamentals.
A detailed analysis by Hejlová, Hlaváček, and Komárek (2017) of the Czech house prices with much richer model also points to instability of the estimated coefficients.
Alluding to short-term and long-term analysis and the distance to binding constraints may be reminiscent of traditional Keynesian/Neoclassical analysis with fixed wages and economic slack in the short run and flexible prices the economy on its potential output in the long run.