# The financial stability contour map

Version 2012.04.17,^{∗}
by **Gavin
R. Putland**

#### Abstract

The **financial stability contour map**
(above / hi-res version) shows how the tax system influences the
property market so as to cause or prevent financial crises. It is a
contour graph of the equilibrium rental yield (*y*) as a function
of the holding tax rate (*τ*) and the
product *g**k*, where *g* is the appreciation rate
(treated as exogenous) and *k* is the “capital-gain
preference”, i.e. the factor by which the tax system magnifies
capital gains relative to current income. The graph is calibrated in
terms of the interest rate (*i*). If *y* falls to zero,
“equilibrium” property prices become infinite. But
obviously the financial system cannot support infinite prices. Hence,
if the tax system is such that *y* is zero or negative, financial
instability is guaranteed.

### 1. The yield formula

Suppose that a property

- has a gross annual rental yield
*y*,

- appreciates at an annual rate
*g*(for “*g*ain” or “*g*rowth”),

- can be mortgaged at an effective annual interest
rate
*i*, and

- is subject to a public
**holding charge**or “land tax” at an annual rate*τ*,

where all four variables are expressed as decimals;
e.g. if *y*=0.04, the yield is 4% per annum, and so on.

The “effective” interest rate is the interest paid on the debt-funded part of the purchase price plus the interest forgone on the remainder, all divided by the price.

In the case of an **improved** property (e.g. a property
including a building), *g* accounts for depreciation of the
improvements, and *τ*, as defined here, is expressed in terms
of the improved value (even if the rate defined by legislation is
levied on the land value alone — as it should be, to avoid
penalizing construction). Any maintenance costs can be notionally
included in *τ*.

The applicable appreciation rate is that of a fixed address —
not to be confused with that of the average property or the median
property. As cities grow, average and median properties move further
from city centres, so that their prices do not grow as fast as those
of *particular* properties.

When the market reaches **equilibrium** — not in the sense
that prices are constant, but rather in the sense that *g* is
constant — *buying must be competitive with renting*. Hence
the **total return** (that is, the rent saved or earned, plus the
appreciation) must balance the total holding cost (the interest paid
or forgone, plus the holding “tax”). On a per-unit-price
basis, this is written
\begin{equation}
y+g=i+\tau\,,
\end{equation}
whence
\begin{equation}
\frac{1}{y}=\frac{1}{i+\tau-g}.
\end{equation}

(N.B.: Enable JavaScript™ to see the equations.) Of course
1/*y* is the *P*/*E* (price/earnings) ratio, which in
practice must be positive, so that the denominator on the right-hand
side must also be positive. As that denominator approaches zero,
the *P*/*E* ratio “approaches infinity” —
i.e. increases without limit.

In practice, of course, the *P*/*E* ratio must be finite,
because borrowers have a limited capacity to service loans. Even if
they plan to pay interest out of capital gains, the economy has a
limited capacity to realize capital gains, which in turn limits
borrowers' capacity to service loans. If that capacity is exceeded,
there will be a financial crisis. Hence, if crisis is to be avoided,
the *P*/*E* ratio set by the market must not exceed the
capacity to service loans; that is, *y* must not be too low.

If the denominator on the right side of Eq.(2) is zero, any reduction in the interest rate or the holding
charge or any *increase* in the appreciation rate will cause the
denominator, hence the *P*/*E* ratio, to go negative. But
any one of these changes should make one willing to pay *more*
for the site, not less — as is clear if we
substitute *P*/*E* for 1/*y* and rearrange as
follows:
\begin{equation}
P = \frac{E+gP}{i+\tau}.
\end{equation}
It is as if negative prices were not less than zero, but greater
than infinity!

Not surprisingly, Eq.(3) indicates that if *τ*=0 (that is,
if there is no “land tax” or other holding charge),
the price is the annual accrual (rent plus capital gain) divided by
the interest rate, and that the “land tax” affects
the price like additional interest.

The latter conclusion could have been reached from Eq.(1): because interest and “land tax” appear as a sum, and only as a sum, “land tax” affects the price like additional interest. Of course it would be an equally valid interpretation to say that interest affects the price like another “land tax” — except that (a) interest is paid only by the lower class of property owners, namely those with mortgages, and (b) whereas a rise in the “land tax” rate requires legislative change and is regarded as politically impossible, a rise in interest rates requires nothing but the executive decision of a central bank that is not answerable to the voters.

### 2. Effect of taxes

If financial instability is to be avoided, the rental
yield *y* as set by the market must be *high* enough (i.e.,
the *P*/*E* ratio must be *low* enough) to enable
buyers to service loans. But the “market” is not oblivious
to taxation.

Recurrent **property taxes** have already been taken into
account (through *τ*).

A broad-based **consumption tax**, in so far as it simply
devalues the currency in which all values are measured, has no effect
on the above analysis. In *theory*, because
“investment” in land delays the opportunity to consume, a
consumption tax may affect land prices if it is known to discriminate
between current and future consumption, whether because the tax rate
will change or because the items that will be consumed later (after
selling the land) will be taxed more or less severely than the items
that could be consumed now (instead of buying the land). But
in *practice* such effects are unlikely to be known or even
guessed at, let alone acted upon. (The word “investment”
is in quotation marks because “investment” in land does
not of itself produce a new net asset.)

To account for **income tax**, all quantities in
Eq.(1) — or all quantities
except *P* in Eq.(3) — must be
replaced by after-tax equivalents.

For convenience, let us define a **neutral** income tax as
having a flat marginal rate, no discrimination between current income
and capital gains, and full deductibility of interest and property
taxes (that is, no quarantining of “negative gearing”).
Under these conditions, the affected quantities are converted to their
after-tax equivalents by multiplying by the **scale factor**
(1-*r*), where *r* is the income-tax rate (e.g. *r*=0.3
for a 30% marginal rate). When this is done in
Eq.(1) or
(3), the factor (1-*r*) cancels out
and the equation is left unchanged. So a neutral income tax does not
affect *P* or the pre-tax *P*/*E* ratio.

The requirement that a “neutral” income tax has
“no discrimination between current income and capital
gains” does not apply to current income that is outside the
present analysis, e.g. labour income. Discounting of income from
assets relative to income from labour does not violate neutrality
provided that rents, capital gains, interest and holding costs are all
discounted by the *same factor*.

In Australia, the treatment of *owner-occupied housing* is an
extreme case of uniform discounting: imputed rents and capital gains
are not taxable, while interest and council rates are not deductible;
so income tax is neutral. But for commercial and investment property,
neutrality is violated in that capital gains alone are discounted for
tax purposes. Hence, when we substitute after-tax equivalents in
Eq.(1), the scale factor for *g*
becomes (1-*r*′), where *r*′ is the effective
rate of capital gains tax (**CGT**); and when we divide through by
(1-*r*), the scale factors don't all cancel out, but *g*
ends up being scaled by the factor
\[\frac{1-r'}{1-r}\,,\]
which is *greater* than 1 (if capital gains are
taxed *less* than current income). For brevity, let's call this
factor *k*. So Eq.(1) gets modified
as follows:
\begin{equation}
y+gk=i+\tau.
\end{equation}

For example, taxing capital gains at 15% and current income at 30%
gives *k*=(1-0.15)/(1-0.3)=85/70. Exempting capital gains while
taxing current income at 50% would give *k*=2. Taxing capital
gains and current income at the same rate would
give *k*=1. Taxing capital gains at 50% while exempting current
income would give *k*=0.5. If *k*=0, capital gains are
confiscated. In each case, it is assumed that interest and holding
taxes are deductible at the same marginal tax rate at which rental
income is taxable.

(This simple scaling of the appreciation rate is valid for short-term or non-compounding appreciation. The treatment of longer-term appreciation is beyond the scope of this article. A more comprehensive paper dealing with that subject is in preparation.)

If the ability to deduct current losses on property against other income (“negative gearing”) is restricted in any way, the effect is equivalent to that of increasing the interest rate for the affected owners.

What about **conveyancing stamp duty**? From the viewpoint of
someone who buys a property and re-sells it, the stamp duty is
equivalent to a holding tax at a rate inversely proportional to the
time for which the asset is held. In a rising market, it is
alternatively equivalent to a capital gains tax (which
reduces *k*) at a rate inversely related to the time for which
the asset is held. Either way, it tends to impose a lower limit on the
holding time, but has little effect on buyers who intend to hold for
long periods.

Eq.(4) can be rearranged as
\[\frac{1}{y}=\frac{1}{i+\tau-gk}.\]
Of course 1/*y* is the *P*/*E* (price/earnings)
ratio. Because the denominator on the right-hand side is a difference,
it can approach zero. As it does so, a small increase in *τ*
or a small reduction in *k* can produce an arbitrarily large
reduction in the price. And conveyancing stamp duty can be represented
by an increase in *τ* or a reduction in *k*. So this
simple theory is good enough to explain the following
counter-intuitive observation on p.16 of
a paper by Andrew
Leigh:

Across all neighbourhoods, the short-term impact of a 10 percent increase in the tax rate is to lower house prices by 1–2 percent.... Since stamp duty averages only 2–4 percent of the value of the property, these results imply that the economic incidence of the tax is entirely on the seller... Indeed, the house price results are in some sense “too large”, in that they imply a larger reduction in sale prices than the value of the tax.

Because the present model is an *equilibrium* model, it
doesn't predict the transient effects of *changes* in the tax
system. For example, there is empirical evidence that a new stamp-duty
concession (or a new grant!) for a particular class of buyers will
bring forward demand from that class, and that the counterparties will
“lever up” their capital gains through the financial
system in order to “trade up”, and so on, causing a
temporary speculative spiral. The aim of the present model is not to
predict those dynamics, but rather to determine whether the tax system
is compatible with financial stability. As the word
“stability” suggests, an equilibrium model is satisfactory
for that purpose.

### 3. Note on inflation

In the above analysis, capital gains and interest have been taken
as nominal. This is appropriate for Australia, where the tax system
assesses nominal capital gains and allows deductions for nominal
interest. If the tax system assessed *real* capital gains (all
else being unchanged), that would be represented by a higher value
of *k*. If only *real* interest were deductible (all else
being unchanged), the effect would be equivalent to that of a higher
interest rate.

### 4. The contour graph

Eq.(4) can be written in the form
\begin{equation}
gk=\tau+i-y\,,
\end{equation}
which shows that the graph of *g**k* vs. *τ* is
a straight line, with unit slope and an intercept of *i*-*y*
on the “*g**k* axis”. Each value
of *y* gives a different line, so that each line can be
understood as a contour in a graph of *y* vs. *g**k*
and *τ*. Because the intercept (*i*-*y*) is a
linear function of *y*, equally spaced values of *y* give
equally spaced contours.

The most interesting contours are *y*=0, for which the
intercept is *i*, and *y*=*i*, for which the intercept
is 0. From these we may deduce the regions for
which *y*>*i* (positive gearing at 100% LVR),
0<*y*<*i* (negative gearing), and *y*<0
(guaranteed financial instability), as shown in the graph above
(reproduced below).

Because equally spaced values of *y* give equally spaced
contours, we can easily add contours for other values of *y*. For
example, the contour for *y*=*i*/2, for which the equilibrium
rental yield is *half* the interest rate, is in the middle of the
“negative gearing” band. Empirically, a rental yield of
less than half the interest rate should make one fear an imminent
crash. Hence, if the tax system is such that *y*<*i*/2 —
that is, if it places us closer to the red region of the contour map
than the green region — it invites a strong suspicion that the
tax system is incompatible with long-term financial stability. If the
tax system places us *in* the red region, suspicion gives way to
certainty.

If the tax system causes the “equilibrium” rental yield to be unsustainably low, prices will rise until the financial system collapses, then fall until the bad debts are somehow worked out, then rise again, and so on. At any stage of the cycle, the price of a property will be determined by what one can borrow against it.

### 5. Where are we?

In Australia, the long-term appreciation rate is similar to the
long-term interest rate. For residential owner-occupants, *k*=1,
so *g**k* is roughly *i*; and the property tax rate is
a small fraction of *i*. That places us close to the red region
— too close for financial stability. For other classes of
property owners, *k* is higher, and the total property tax rate
is also higher, but probably by an insufficient margin to compensate
for the higher *k*, in which case the destabilizing tendency is
even greater than that from ordinary home owners.

Under these conditions, arguments about population growth and the
unresponsiveness of housing supply are relevant to the direction of
rents, but *not* to the direction of prices or price/rent ratios,
which are limited by the financial system.

### 6. Implications

If the tax system places us in the red region, raising interest
rates might shift us into the amber region. But because monetary
policy affects prices of goods and services, it is not necessarily
available for the purpose of taming asset prices. Moreover, the above
analysis deals with *constant* interest rates, not changes in
interest rates. In reality, of course, if high property prices have
endangered the financial system, raising interest rates will tend to
precipitate the collapse.

As explained in the preceding article in this series, financial regulations aimed at limiting credit on the supply side are not politically robust.

So we must look to the tax system. In the long term, financial
stability is improved by higher “land tax” and/or
higher taxation of capital gains relative to current income (including
at least income from assets). At present, capital gains are taxed less
than income from assets. If it were the other way around, financial
crises would be less likely. Any of these reforms, by making it more
attractive to generate income from land and less attractive to hold
idle land in pursuit of capital gains, would improve the
responsiveness of housing supply to population growth,
reducing *g* and improving financial stability. Any of these
reforms involves a change in the tax mix. None of them requires an
overall increase in taxation.

Implications for housing affordability are considered in the next post.

__________

^{∗} First posted Nov.16,
2011. On Apr.17, 2012, the text was amended to suggest that
maintenance be included in *τ* instead of *g*; the
reference to Leigh was added; and the discussion of discounting of
future rent was deleted because it was based on the incorrect (if
common) practice of applying the pre-tax discounting rate to pre-tax
cash flows. The correct treatment of the discounting rate must await
the “more comprehensive paper” mentioned in the
text. Equations were redisplayed in `MathJax` on Sep.1,
2013.