Stackelberg Follower notes that housing prices in Canada continue to not crash. Should we expect them to? Nick Rowe revisits some recent evidence and passes this along:

[I thank Tsur Somerville for emailed comments on a previous draft of this Note]

An August 2008 paper by Tsur Somerville and Kitson Swann of the Sauder School of Business UBC presents estimates of the extent to which houses are overpriced in major Canadian cities. The paper can be found on the website of the Centre for Urban Economics and Real Estate here. It’s certainly a very timely paper, on an important topic, and based on a lot of data and careful research. But the conclusions nevertheless struck me as strange. They concluded for example that: Toronto houses were not overpriced; Vancouver houses were 11% overpriced; and Ottawa houses were 25% overpriced (see their Table 7). They reach these conclusions despite Vancouver having the lowest ratio of rents to prices (3.6%), Toronto having a higher ratio (5.2%), and Ottawa having the highest ratio of all nine cities (6.5%).

Reading through their paper, I found one assumption in particular that I didn’t like, and which seems to be important in driving their results. It’s a nice clear paper, which makes it easy to see how changing the assumptions will affect the conclusions. What I do here is revise their assumptions to get revised estimates of the extent to which houses are over-priced in nine Canadian cities.

Ignoring property taxes, insurance, structural depreciation and maintenance for the moment, an annual rent/price ratio of 3.6% means that the rate of return on owning a house (as opposed to renting a similar house) is 3.6%, provided rents and house prices stay constant. We can compare this own rate of return on housing to the market interest rate to decide if houses are over-priced. If market interest rates are 7.2%, for example, then house prices seem to be double what buyers should rationally pay. But if there is inflation, so that rents and house prices cannot be expected to stay constant, we need to make some sort of adjustment to the rate of return on houses. Somerville and Swann adjust by adding the expected rate of nominal house price appreciation to the rent/price ratio (actually, they subtract it from the market rate of interest to get a revised cost of capital, but that amounts to the same thing). They base this estimate of expected house price appreciation on an average of past appreciation in each city, peak-to-peak, and trough-to-trough.

That’s the assumption of the paper which really worries me. Based on past appreciation, they assume future appreciation of 5.4% for Vancouver, which gives a rate of return of 3.6%+5.4%=9% before costs (taxes, insurance, structural depreciation and maintenance). The same 5.4% assumed future appreciation for Toronto pushes the total return before costs on Toronto houses to 5.2%+5.4%=10.6%, which is what makes Toronto houses seem to be so reasonably priced.

I don’t buy this assumption about future house price appreciation. If we could assume there were no bubbles in house prices, so that house prices had always reflected fundamental values, it might be reasonable to extrapolate from past to future appreciation. But the question of bubbles is exactly what is at issue. Their use of peak-to-peak and trough-to-trough averages (rather than a simple trend) is an attempt to avoid this problem, but I’m not sure it works. Also, since overall inflation is likely to be lower in the future than it has been on average over the last 30 years, their approach might overstate future house price appreciation even if there were no bubble. I replace their assumption with an alternative.

The fundamental value of a house is defined as the present value of the future stream of rents net of costs (taxes, insurance, structural depreciation and maintenance). If we ignore costs, and assume that rents grow at a constant rate g, the fundamental value can be calculated as F=Rent/(r-g) where r is the rate of interest (we can either use a nominal r and g or a real (inflation-adjusted) r and g, it makes no difference). To compare the current price to the fundamental value F, we can then compare (Rent/Price)+g to r. So I am going to revise Somerville and Swann’s estimates by replacing their future price appreciation estimates with my own rent growth estimates. I will add the expected growth rate of rents to the rent/price ratio to get my own estimate of the gross rate of return on housing.

Fortunately for me, the UBC Sauder School’s Centre also has an excellent accessible collection of data, including data on rents in those nine cities. Eyeballing that data, it is hard to reject the conclusion that real rents have very little long term trend over the last 20 years, though Halifax, Montreal, and Winnipeg real rents have been declining at about 1% per year. Even Vancouver real rents show a slight downward trend, and not the upward trend which would warrant a low rent/price ratio based on fundamentals. But since I cannot be sure whether those slight downward trends will continue, for simplicity I will first assume nominal rents will grow at 2% per year (the Bank of Canada’s inflation target) in all cities.

Despite the absence of any big long-term trend in real rents, my scientific eyeball suggests that Calgary and Edmonton real rents are currently about 10% above trend. This makes sense given the recent oil boom, and the fact that house construction cannot immediately increase the supply of housing in response. But I expect these high rents to be short-lived, and accordingly will consider adjusting the rent/price ratios in those two cities downwards by 10% to adjust for this.

I have only one minor quibble about Somerville and Swann’s estimates of the costs of home ownership (property taxes, insurance, structural depreciation and maintenance): they present these costs as an annual percentage of the price of the house. But if house prices (say) halved, many of these costs would stay the same, and so would double as a percentage of the price of the house. If so, this would mean they underestimate the extent to which house prices might be over-valued compared to fundamentals, but it would not affect whether houses are over-valued. Accordingly, I will ignore my minor quibble, and take their cost estimates as they are, and subtract those costs from the rent/price ratio to get the net rate of return on housing.

Starting with the rent/price ratio, adding 2% nominal growth in rents, and subtracting costs, I get the following revised estimates of the net nominal rate of return to buying a house (remember that a low rate of return means house prices are too high):

(Rent/price) + growth of rents – costs = net rate of return (nominal)

Calgary 5.0% +2% -3.1% = 3.9%

Edmonton 6.4% +2% -3.3% = 5.1%

Halifax 6.0% +2% - 4.6% = 3.4%

Montreal 5.8% +2% - 4.3% = 3.5%

Ottawa 6.5% +2% -3.9% = 4.6%

Regina 5.2% +2% - 4.5% = 2.7%

Toronto 5.2% +2% -3.2% = 4.0%

Vancouver 3.6% +2% -2.1% = 3.5%

Winnipeg 6.0% +2% -4.9% = 3.1%

If we adjust Edmonton and Calgary rents down by 10% (to reflect the reversal of the recent increase), we get a revised rate of return of 3.4% for Calgary, and 4.5% for Edmonton. If we assume 1% rather than 2% growth rate of rents for Halifax, Montreal and Winnipeg, their rates of return become 2.4%, 2.5%, and 2.1%. (Since my own house is near Ottawa, I am pleased to see that Ottawa house prices now become the most reasonable, but then everyone thinks that it’s only everyone else’s houses that are over-priced.)

Next step is to compare these rates of return to the market rate of interest. Which market rate of interest? For a buyer financing 100% by borrowing, the relevant interest rate would be a mortgage rate, of (say) 6%. By this metric, all Canadian cities have house prices which are far too high. For a buyer with a 100% down-payment, the after-tax interest rate would be the relevant opportunity cost of capital. With long term bonds paying (say) 4%, and a buyer in a (say) 50% marginal tax bracket, the after-tax rate of 2% makes house prices look low. (This assumes an owner-occupier, who pays no tax on the implicit rental income of the house. A buy-to-let landlord, who will pay income tax on the rents, and can deduct interest payments, should compare the return on housing to the before-tax rate of interest.) For equilibrium house prices, what matters is the source of financing for the marginal buyer, who will typically have a large mortgage, and small down-payment, and whose interest rate is a weighted average of the mortgage rate and the after-tax alternative return. For the marginal buyer, using mostly mortgage financing, Canadian houses look a bit over-priced.

I’m still not sure I believe those estimates. They don’t seem to correspond well across cities to recent price increases. Differences in costs drive much of the differences in return, and tax differences across cities drive much of the differences in costs.

[Tsur Somerville notes in correspondence that when you buy a house you also get the option value of redeveloping the site, to earn higher (implicit or explicit) rents. An interesting point, which may lead house prices to appreciate faster than rents. This effect may be stronger in some cities than others.]