## Sunday, March 20, 2011

### The use and reasonably foreseeable misuse of price to rent ratios

financialinsights has been posting recently on the so-called price-rent ratio indicator of housing valuations. After reading these posts and associated comments by Ben’s readers, I thought I should give an understanding, using textbook finance theories, how “typical” price-rent ratios are derived and how they relate to conventional investment measures such as price-earnings ratios. In addition I will highlight an important difference in price-rent ratios that I believe is in play today that is unusual in other conventional investments.

Pardon if this is elementary for you and if this information is not absolutely correct. I have no degree in finance, but the crux of my assertions I believe to be correct. The post is a bit long but if you’re interested in understanding what price-rent ratios really mean it hopefully provides some "insight".

Net present value

To understand the price-rent ratio, we can first go back to basic finance math and look at the “net present value” (NPV) of a stream of future cash flows. Cash in the future is normally worth less than today and such is discounted at the “discount rate”, sometimes referred to as the “cost of capital”. For example if I promise to pay you \$1000 today or \$1000 a year from now, you will put less value on the future \$1000 because you can invest today’s \$1000 and receive, say, 2% interest risk-free, so my future \$1000 is only worth around \$980 (1000/1.02). If you are skeptical I am capable of paying you back on time (or at all), you will further discount my \$1000 by some amount. If I’m the Canadian government you probably won’t discount much beyond expected inflation, if I’m the Greek government or some Joe of the street you are probably discounting a lot. If you can recoup the money by liquidating collateral put up by the debtor, the discount rate decreases. We can sum multiple discounted cash flows from different payment periods together and get its NPV. NPV is the price an investor would be willing to pay for an investment given the risks involved.

For a property, this series of discounted cash flows is simply revenue (rents) minus expenses (maintenance, taxes, management fees, and capital replenishment) from future payment periods. The revenue and expenses have some variability and risk inherent in property – there is no real way of avoiding it. As a result property will always have a higher discount rate than risk-free. However with property, as a bonus of sorts, revenue and expenses tend to increase roughly with inflation. (Well not quite; as a building ages it depreciates and has higher ongoing maintenance expenses, while the rent will slightly lag inflation – older buildings rent for less than newer ones.) As with any investment we sum the cash flows to get NPV:

NPV = (R-E) + (R-E)(1+i)/(1+d) + (R-E)(1+i)^2/(1+d)^2 …. + (R-E)(1+i)^N/(1+d)^N

where R is the rent, E is expenses, i is the rental inflation rate (which normally but not necessarily tracks CPI), d is the discount rate, and N is the final payment period. Assuming N is large, using simple math this reduces to:

NPV = (R-E)/(d-i)

Notice the denominator d-i for a moment. Remember d is the discount rate, which includes future inflation expectations and risk. In other words, d-i is what we can call an inflation-independent measure of risk. (Again this is a bit of a lie but not much of one.) This number is also referred to as the “cap rate”.

(You may also note that E does not include financing expenses. For the purposes of this analysis, assume we use our own money. Besides, anyone we borrow from or invests with us should be doing the same calculations we are.)

The analog for cap rate in other investments is the price-earnings ratio and the principle is the same (P/E => 1/caprate). With so-called growth investments, the expenses are front-loaded and a potential boon from revenues is pushed out and often heavily discounted, meaning their price-earnings are higher. Property investment will tend to more closely match the price-earnings of utilities like gas, electricity, or water. (Property is, after all, a utility in its own right.)

Relating to price-rent

Price- rent ratio is used because it is a quick mental calculation that is easy to understand and track. It is reasonably simple to see how it relates to the NPV calculation above. Since rents and expenses do not tend to deviate much over time, they can be lumped into a single number: a rule of thumb is to assign 20% of rent to expenses.

A typical residential investment firm would demand, say, a cap rate of 8%. (Wow that seems high… Well, historically, it isn’t.) This means we get an investor’s “desired” price-rent ratio:

desired P/R = 0.8/(cap rate) = 0.8/0.08 = 10

which is a gross yield of 10%. We have a derivation of the price-rent ratio based on an investor’s cursory DCF analysis.

Misuse of price-rent

Price-rent ratio metrics are an approximation. There are some deviations that can and do exist, some substantial. We hear stories of properties with 1000:1 price-(monthly) rent in parts of Greater Vancouver. When using price-rent to value a property it’s important to take into account its highest and best use. For example, a small house surrounded by larger more affluent houses (or condominimums) is likely going to be re-developed. Its value will be based on comparing various DCF scenarios and choosing the highest one. An underdeveloped piece of land is akin to a call option. In this scenario there are definite grounds for a higher price-rent (though maybe not 1000:1!) because an owner can sell today for today’s best use.

A piece of land expected to be redeveloped (or simply to have its revenue outpace inflation due to significant income growth) at some point in the future, but is not occurring today, is akin to a growth investment, where the market is assuming future cash flows will increase significantly some point in the future. This may or may not be rational and the premiums placed on these properties are speculative.

An important consideration for residential real estate, unlike other investments, is that it is not just investors who are active in the market. Owner-occupiers who prefer to own will be willing and able to pay a premium over rental value for a property, a so-called consumer surplus. In the case where owner-occupiers are heavily active in a market an investor either tags along for the ride, hoping to sell at a higher price in lieu of a low cap rate, or simply sits on the sidelines or invests in better returns elsewhere. Canada’s ownership rate has increased from 64% at the turn of the century to close to 70% today. If one believes the newly-minted Canadian owners place a premium on ownership, this will tend to have driven marginal prices higher. At some point, however, there are no more marginal owner-occupiers buying and we’re left with the “sidelines” investors requiring rental value without the premium.

In some markets there are few comparables that are rented. In this case, price-rent ratios are not useful due to poor data quality. A proxy of price-income ratios is used to infer “imputed rent” of a property. Country-wide this makes sense since, well, 70% of occupied properties are not rented! Nonetheless it is a proxy for what an investor would demand.

A final important point is how the role of financing would adjust price-rent ratios. I will not pass any opinion on this, only to state that, obviously, lower costs of capital have effect at the margins but it is far from clear to me permanently lower costs of capital would mean a secular shift to permanently higher price-rent ratios.

In summary the price-rent ratio can be construed as being derived from simple financial calculations of discounted cash flows. Since rational investors will eventually act as marginal buyers, the price-rent ratio serves as an indicator of where prices will revert before such investors will support prices. In the right circumstances it’s the purest indication we have of what gives housing value in the long run.

takloo said...

illuminating stuff... keep it coming :)

interesting that underdeveloped land is akin to a call option. I'm curious to find out how cities/towns determine land value for new development.

Etienne de Cochon said...

Thanks Jesse. Great post.

mohican said...

Yes, lots of great food for thought. Thanks Jesse.

David said...

There is something missing here, which is taking account of inflation! Profit expectations used to be 8% because there was 6% annual inflation, so that is the same as about 4% profit now.

If you believe property prices will rise, for example due to tight zoning restrictions and the constraints of geography, such as for downtown Vancouver, then it is perfectly rational to invest in property with low rents. This has made thousands of millionaires in Vancouver who would not have invested based on your formulas. However, I do agree that prices are even beyond what can be supported by these arguments, so we are due for a fall. Its the timing and size of a correction that is completely unpredictable.

By the way, this is certainly the best real estate blog for Vancouver! Your postings make us think.

jesse said...

@David, The argument is the one I avoided: financing rates are very low in part due to low inflation and people "expect" lower returns. I would argue for an asset whose earnings track this same inflation, that is a mistake in the long run. The effect, however, can be delayed by years.

patriotz said...

If you believe property prices will rise, for example due to tight zoning restrictions and the constraints of geography, such as for downtown Vancouver, then it is perfectly rational to invest in property with low rents.

It's not rational at all, that's a speculative bubble by definition. It's rational to buy at a high price/rent if you think rents will rise, but real rents have in fact been falling for decades, which means that those "tight" restrictions don't exist in objective terms. You really don't have to look at the stats to see this, just look out the window.

This has made thousands of millionaires in Vancouver who would not have invested based on your formulas.

Greater fool theory. And how many of those "millionaires" have actually cashed out?

Thanks for the post Jesse. I will add that to escape the issues of rental market transparancy and future densification one can simply look at condo price/rents. A massive bubble, pure and simple, and detached house prices cannot escape a condo bust.

jesse said...

"one can simply look at condo price/rents"

Agreed. Looking at detached dwellings suffers from two major problems: future use changes and lack of statistically significant rental equivalence data. Condos and apartments are a better gauge on both fronts.

vreaa said...

Good analysis, thanks jesse.

Charles Peake said...
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