# A ‘general formula’ for the price-rent ratio in a rational property market

How pro-speculation tax systems can lead to “sub-intrinsic-value bubbles”

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### The formula

In a rational property market, the gross rental yield of a property, i.e. the reciprocal of the price/rent ratio, is given by \begin{equation}\label{eygen} y = h + \frac{i-g/u}{1-e^{(g-ui)T}} \left\{ (1\!+\!s)\left[1\!-(1\!-\!v)e^{-uiT}\right] - (1\!-\!r)ve^{(g-ui)T\strut} \right\}\,, \end{equation} where

- \(s\) is the
**stamp duty**rate payable by the buyer on the sale price of the property (negative for a grant or subsidy);

- \(r\) is the
**resale cost**payable by the seller, expressed as a fraction of the resale price, and includes any commissions, legal fees and “vendor stamp duty”, but*not*capital-gains tax (which is treated separately);

- \(h\) is the
**holding charge rate**, expressed as a fraction of the current market price per unit time, and includes any property taxes, “rates”, maintenance costs and body-corporate fees;

- \(u\) is the fraction of current
**property income**and related expenses remaining after income tax;

- \(v\) is the fraction of any net
**capital gain**remaining after income tax;

- \(g\) is the continuously compounding rental
**growth rate**; thus the rent at time \(t\) is \begin{equation}\label{eE} E = E_0\,e^{gt}\,, \end{equation} where \(E_0\) is the*initial*rent (at \(t\!=\!0\));

- \(i\) is the continuously-compounding
**grossed-up discount rate**; that is, \(ui\) is the continuously-compounding*after-tax*discount rate, so that a future cash flow at time \(t\) must be multiplied by \begin{equation}\label{edisc} e^{-uit} \end{equation} to find its present value (PV) at time \(0\);

- \(T\) is the time for which the property will be held;

and \(y\), \(s\), \(r\), \(h\), \(u\), \(v\), \(g\) and \(i\) are assumed to be constant through the holding period.

(*N.B.:* Defining \(i\) as the grossed-up discount rate is
convenient for interpreting certain special cases, but is *not*
meant to imply that the grossed-up rate is more fundamental than the
after-tax rate.)

### So what?

The formula explains the empirical observation that a stamp duty on
property buyers can *reduce* prices by more than the value of
the duty, while a subsidy (or “grant”) for buyers
can *raise* prices by more than the value of the subsidy.

If rents are rising, the formula predicts absurdly high price/rent
ratios for reasonable values of the parameters. Thus it predicts the
existence of **sub-intrinsic-value bubbles** — situations in
which prices, although more than justified by net-present-value
calculations, are unsustainable due to unserviceable debt. It also
indicates that tax reform involving greater reliance on **land-value
tax** and/or **capital-gains tax** can prevent
sub-intrinsic-value bubbles by bringing “intrinsic” values
(net present values) within the debt-servicing capacity of
prospective buyers. The capital-gains-tax option is found to be more
effective in reducing the annual cost of home ownership relative to
renting. As these conclusions say nothing about taxation of
consumption or labour income, they have no bearing on the overall
level of taxation or public expenditure.

### Derivation

By the above definitions, if \(P\) is the market price at time
\(t\), then
\begin{equation}\label{ey}
E=yP\,.
\end{equation}
From \((\ref{eE})\) and \((\ref{ey})\) we have
\begin{equation}\label{eP}
P = E_0\,e^{gt}/y\,,
\end{equation}
showing that the rental growth rate \(g\) is also (unsurprisingly)
the *price* growth rate. The initial price (at \(t\!=\!0\)) is
\begin{equation}\label{ePo}
P_0 = E_0/y\,.
\end{equation}

Let the property be bought at \(t\!=\!0\) and sold at \(t\!=\!T\). Let \(P_0^{\rm\,rent}\) denote the PV of the rent received during the holding period, and let \(P_0^{\rm\,resale}\) denote the PV of the resale price, where both the rent and the resale price are net of taxes and other costs (but not interest, which is accounted for in the discount rate). For the initial buyer, the net present value (NPV) is \(P_0^{\rm\,rent}\) plus \(P_0^{\rm\,resale}\).

Before income tax, the rent net of holding charges is
\(E\!-\!hP\). *After* income tax, the net rent received
or saved during an infinitesimal interval \(dt\) is
\begin{equation}\label{erdt}
u(E-hP)\,dt\,,
\end{equation}
which we multiply by \(e^{-uit}\) to obtain its present value. Adding
the PVs for all the infinitesimal intervals, we have
\begin{equation}\label{ePrent}
P_0^{\rm\,rent} = \!\int_0^T \!\! u(E-hP)\,e^{-uit}dt\,.
\end{equation}
Substituting from \((\ref{eE})\) and \((\ref{eP})\), we find
\begin{align}
P_0^{\rm\,rent}
&= uE_{0\,}(1-h/y)\!\int_0^T \!\! e^{(g-ui)t}dt\label{ePrentTi}\\
&= \tfrac{E_0(1-h/y)}{i-g/u}\left[1-e^{(g-ui)T}\right]\label{ePrentT}
\end{align}
provided that \(g\!-\!ui \neq 0\).

The acquisition price *including duty* is \(P_0(1\!+\!s)\),
which, by \((\ref{ePo})\), can be written
\begin{equation}\label{ePos}
E_0(1\!+\!s)/y \,.
\end{equation}

The resale price [from \((\ref{eP})\)] is \(E_{0\,}e^{gT}\!/y\). Resale costs reduce this to \(E_0(1\!-\!r)e^{gT}\!/y\). Deducting the acquisition cost (\ref{ePos}), then multiplying by \(v\), we obtain the after-tax capital gain \[ \tfrac{\,vE_0\,}{y}\left[(1\!-\!r)e^{gT}\!-\!(1\!+\!s)\right]\,. \] We add this to the cost base (\ref{ePos}) to find the after-tax resale price, which is then discounted to find its present value, denoted by \(P_0^{\rm\,resale}\). The result is \begin{equation}\label{ePoresale} P_0^{\rm\,resale} = \tfrac{\,E_0\,}{y} \left[(1\!-\!r)ve^{gT} + (1\!+\!s)(1\!-\!v)\right]e^{-uiT}\,. \end{equation} [Of course we get the same result if we subtract the capital-gains tax from the resale price (net of resale costs) and discount the difference.]

If the price of acquisition (\ref{ePos}) is the NPV, we
have
\begin{equation}
E_0(1\!+\!s)/y = P_0^{\rm\,rent} + P_0^{\rm\,resale}\,.
\end{equation}
The use of the price *including duty* on the left-hand side
does *not* amount to an assumption that the price is simply
reduced by the value of the duty. Rather, when we substitute from
\((\ref{ePrentT})\) and \((\ref{ePoresale})\), the yield \(y\) appears
on both sides of the equation, which is solved in order to discover
the effects of the various parameters on \(y\), hence on the
price. Making those substitutions and solving for \(y\), we obtain the
general formula \((\ref{eygen})\).

### Further details

G.R. Putland, “The Price Cannot be Right: Taxation,
Sub-Intrinsic-Value Housing Bubbles, and Financial
Instability”, *World Economic Review: Contemporary Policy
Issues*,
No. 5 (July 2015), pp. 73–86. Journal
version: PDF, 14pp. Author's two-column
version: PDF, 8pp.

Errata in journal version:

- Page 76, after Eq.(2): “If =
*u*” should be “If*v*=*u*”;

- Page 85, 2nd line after Eq.(29):
“contour =
`0`” should be “contour*y*=`0`”.

[This post first appeared on 9 December
2013. It was updated on 11 & 13 July 2015.]