If you buy a house, the risk is minus the covariance between the price of the house and the present value of the rents of the house you would desire to live in.
I disagree with Felix Salmon. Strangely, Felix seems to have misplaced the significance of an idea he had earlier: we are born with a short position in housing.
So if you know for sure that you (and your heirs) would always want to live in the house you buy, there is a perfect positive correlation between the price of your house and the PV of the rents on the house you would want to live in, and you eliminate risk by buying the house.
But if there's a chance you would want to live in another house, you have to think about how the price of the house you buy now will correlate with the present value of the rents of the house you would want to live in in future years.
If there's a positive correlation between the price and your future desired rents, you reduce risk by buying the house, but do not eliminate it. Only if there's a negative correlation do you increase risk. If you buy a house in the country, and sell it and move into the city at the same times as everyone else moves into the city, the price of the house you sell will fall just as city rents rise. You increase risk by buying that house.
Really, it's just an application of CAPM, except you have a short position equal to the present value of the rents on the house(s) in which you will be living.
P.s. if anyone reading this knows CAPM better than me, and can figure out more precisely what I'm trying to say, and should be saying, go for it!
Worthwhile Canadian Initiative 
Just one point: “risk” in neither generic nor neutral. There are many different kinds of risk, and everyone has different tolerances for different kinds of risk. I may have a high tolerance for real-estate-based financial risk, but a low tolerance for real-estate-based location risk (i.e., I may care much more about living in a cool neighborhood than making a financially optimal housing purchase). CAPM recognizes only aggregated, neutered risk and aggregated, neutered returns–useful, perhaps, for managing an investment portfolio that includes a variety of asset classes, but useless for individual home buyers.
So, the sentence, “You increase risk by buying that house,” sounds suspiciously like this whole question might be a misapplication of CAPM. It’s fundamental that the kinds of things that can be true or false for large populations of people and decisions cannot be true or false for individual persons and decisions. While Salmon’s insight be useful for some highly restricted purposes in policy formulation, I cannot see how it sheds any light on decisions related to personal finance.
So, how do you see this insight bearing on housing policy or real-estate portfolio management?
[Disclaimer: I have no formal training in economics or finance, just experience in housing finance.]
I rent – I’m short 1 house.
I own – I’m “flat” housing.
I own 2 houses – I’m long 1 house.
What’s my position if I’m homeless? Consistency, please.
Your question above is about systemic risk and specific risk, before its about covariance or correlation risk. CAPM type math is essentially what did in LTCM 10 years ago, and CDO traders 2 years ago – an absurd and foolish fetish with correlation risk statistics. Quants taking over the asylum.
Don’t forget the tax shelter effects of tax free capital gains and mortgage interest deductibility in the case of the US.
Nick, another way of saying what you said, using CAPM analysis, is that the higher the likelihood that you will want to change house in the future,the higher your risk premium in investing in a house now. So given the same future rent expense expectations, your higher risk premium lowers your NPV for making the house investment right now.
But if you do not ever plan to change house, you probably see no risk premium, and hence your NPV will be almost 100% of the nominal sum of all future rental payments (discounted only by the risk-free rate).
J Powers: I think I’m talking just about risk to one’s wealth: i.e. monetary risk. Since you can always sell a house if you don’t like what has happened to the neighbourhood, the risk that matters is measured by the price, compared to the cost of living where you want to live. I am ignoring the transactions costs of selling and buying, and moving costs, it is true.
anonTDH: if homeless, because you can’t afford to rent anywhere, then you are once again “flat” housing (and flat broke).
Rogue: thinking about it some more (I wrote that post in too much of a hurry): you have a short position in housing, but that short position keeps switching from one house to another, depending on where you would want to live at the time. It’s like you were always short one stock, but that short position kept changing in a way that you cannot predict perfectly. So you want to take a long position in a stock whose value will be most closely correlated with your short position.
Here is my interpretation:
NPV of housing = P – PV
where P=price of your house, PV=PV of the rents on the house you would want to live in
Var(NPV)= Var(P) – 2*Cov(P,PV) + Var(PV)
where Var represents Variance, and Cov represents Covariance.
If P is always identical to PV, then NPV is always 0, and Var(NPV) is also always 0.
If Cov(P,PV) is positive, your risk is smaller than the sum of each risk.
If Cov(P,PV) is negative, your risk is larger than the sum of each risk.
But I think this is still in the realm of Markowitz, not CAPM.
CAPM is just a derivative of Markowitz.
himaginary: Thanks. I’m just going to copy your comment, adding some more interpretation:
This is a simple 2-period model. You know where you want to live in the first period, and what the rents will be. But you don’t know where you will want to live in the second period. Suppose you buy a house in the first period. Does that reduce or increase your second period risk?
NPV of owning a house at the beginning of the second period = P – PV
where P=price of your house in period 2, PV=PV of the rents on the house you would want to live in in period 2.
So your risk in the second period is:
Var(NPV)= Var(P) – 2*Cov(P,PV) + Var(PV)
where Var represents Variance, and Cov represents Covariance.
If P is always identical to PV (because you know you will always want to live in the same house in period 2), then NPV is always 0, and Var(NPV) is also always 0.
If Cov(P,PV) is positive (because the house you will want to live in in period 2 will be similar to the house you want to live in in period 1), your risk is smaller than the sum of each risk. This means you reduce risk by buying in period 1.
If Cov(P,PV) is negative (because the house you will want to live in in period 2 will be very different from your period 1 house, so the rents and prices will move in opposite directions) , your risk is larger than the sum of each risk. This means you increase risk by buying in period 1.
I still don’t have the intuition as clear as I want it to be.
But thanks himaginary!
A solution to your quandary of buying now and not knowing whether you’ll want to move in the future is to enter into a future swap agreement with someone who lives in the house you might want to move to in the future and, vice-versa, might also be wanting to live in your house.
You will have an agreed price (strike price) at which you will swap your houses, exercisable over, say, 5 years from now (the maturity). The value of this swap to you will depend on how much is each house’s expected (volatility) of sale prices 5 years from now. Obviously, if your house has more expected volatility, it will trade at a discount to the other house, and you have to pay a bigger fee to your counterparty for him to agree.
CAPM is just a derivative of Markowitz.
Markowitz’s theory suggests how to form one’s portfolio. CAPM describes what happens to the whole economy if everyone followed that suggestion.
What I wanted to point out at the last sentence of my previous comment is that this post’s story is about the housing portfolio of individual person, not the resulting equilibrium state of the housing economy.
Rogue: that makes good theoretical sense, and helps me get my head clearer.
If you knew in advance where you would want to rent in future, you would buy (or lease) your future house in a futures/forward(?) market. (Your swap arrangement requires a double-coincidence of wants).
But suppose you didn’t know where you would want to live in future, because it would depend on contingencies? Presumably you would want to enter into contingent future (Arrow-Debreu) contracts for your future uncertain housing needs.
But I expect this brings us to think about why those contracts are almost never seen. Unlike stocks and bonds, if the owner doesn’t maintain a house, its value will fall. And will the future owner like your taste in wallpaper?
himaginary: “Markowitz’s theory suggests how to form one’s portfolio. CAPM describes what happens to the whole economy if everyone followed that suggestion.” Beautifully clear!
Back to Rogue’s point: and then the question is: if those Arrow Debreu contracts don’t exist, what comes closest? Buying or renting?
There is an ideal bundle of Arrow-Debreu contracts you want to buy. But the market doesn’t provide it. Instead, you have the choice of buying a fixed bundle (called “a house”) that overlaps with the bundle you want to buy. Some parts of the bundle are parts you don’t want. And the bundle excludes some A-D contracts you do want.
So you want to know whether the parts of the bundle you buy but don’t want are positively or negatively correlated with the bundles of Arrow Debreu contracts that you do want but don’t buy.
Nick:”But I expect this brings us to think about why those contracts are almost never seen. Unlike stocks and bonds, if the owner doesn’t maintain a house, its value will fall. And will the future owner like your taste in wallpaper?”
In other words, housing market is far from complete. I suppose that was also a main theme of your previous post.
But Robert Shiller is trying to make it as complete as possible, although Felix Salmon doesn’t like his idea very much. Optimistically speaking, if his attempt succeeds, things might change.
anon: “CAPM is just a derivative of Markowitz.”
himaginary: “Markowitz’s theory suggests how to form one’s portfolio. CAPM describes what happens to the whole economy if everyone followed that suggestion.”
No, it’s not quite that simple (though himaginary is essentially correct).
The CAPM requires more assumptions then simply the applicability of portfolio theory. In particular it requires all assets (including human capital) to be tradeable.
If I have non-tradeable assets like human capital Makowitz’s portfolio theory still applies to me (under enough other assumptions that make me want to mean/variance optimize my portfolio). When making a portfolio of traded assets I still assess based on their covariance with the other stuff I’m holding, including the non-tradeable stuff.
However, even if everyone is doing this, if we have non-traded assets then this won’t imply the CAPM. We all end up with different optimal portfolios non of which will be the market portfolio.
The CAPM results when everyone wants to mean/variance optimize (quadratic utility and/or joint normality of asset returns gets us this) and there are no non-traded assets, all wealth is traded. Then, each person’s optimal portfolio of risky assets turns out to be the same, it is the tangency portfolio in the usual effecient frontier picture. Since everyone holds the same portfolio it follows that this portfolio must be the market, (agents differ only in how much of their wealth to invest in that portfolio and how much to have in the risk-free asset).
CAPM requires human capital to be tradeable?
Interesting – but really?
Doesn’t the Consumption-based CAPM (CCAPM) get around that problem? (I’m not sure.) That’s where you multiply the payoffs by the marginal utility of consumption; or look at the covariance wrt consumption, or something, IIRC.
I suppose Adam P is referring to Roll’s critique.
no, absolutely nothing to do with Roll’s critique. where did you get that idea?
anon, read the last paragraph of my comment. The CAPM comes from us all having the same optimal portolio.
This can only happen if human capital is tradeable, we basically insure each other’s income so there is no idiosyncratic consumption risk, only aggregate risk. Thus I hold as much of your human capital as you do, and you hold as much of my human capital as I do (talking portfolio weights here). Thus, any idiosyncratic consumption shock is born equally by everyone in the entire economy which, in the limit as the number of agents gets large, leaves the idisyncratic shock having only neglible effect on everyone and a noticeable effect on nobody.
Thus, since we are all holding the same portfolio it must be the market. But of course, market here means all wealth including all of our respcetive human capital.
If we couldn’t trade out of our human capital exposure then we’d seek to hedge it with the tradeable assets, and since it wouldn’t be perfectly correlated we’d each have differnt hedging demands and thus hold different portfolios.
Nick says: “But suppose you didn’t know where you would want to live in future, because it would depend on contingencies? Presumably you would want to enter into contingent future (Arrow-Debreu) contracts for your future uncertain housing needs…..If those Arrow Debreu contracts don’t exist, what comes closest? Buying or renting?”
The only efficient way to hedge uncertainty in this sense is to rent. Perhaps we could also say that rent is the way for people to buy “occupancy rights” from house owners in any location they so wish in exchange for rental income streams.
The only other ‘theoretical’ arrangement I could think of minimizing your downside in selling a house at severely declined prices is to buy a put. The way this could ‘theoretically’ be done is for you to offer the chance of selling your house at a fixed but discounted price (the strike price) to a potentially interested party at some point in the next 5 years (maturity period). The chance to buy at the discounted price engages interest from the counterparty, but the option to sell is yours. The value of this put option is then dependent on the amount of the discount, the volatility of your house’s price, and the risk-free rate which provides the buyer an alternative of earning your discount. Thus minimizing your selling risk, you can have more courage of going back into the market to buy .
Nick, I have a problem with your analysis: doesn’t it confuse the discount rate with the social return to capital. The risk-adjusted return on capital exceeds the discount rate.
Which is why it almost always superior to invest in capital goods directly and use the resulting yield to expense your housing, provided you invest in those areas where you possess superior information. It is precisely this reason that a demand for loans exists at all and is basic finance: the risk adjusted yield on the investment must exceed the cost of funds.
Adam P:”no, absolutely nothing to do with Roll’s critique. where did you get that idea?”
Well, I thought what you said was similar to Roll’s statement about the unobservability of market portfolio. Here is what Roll says (from the second link of my previous comment):
“No two investigators who disagree on the market’s measured composition can be made to agree on the theory’s test results.”
And he refers to human capital and other non-tradable assets right after this sentence.
Jon:”the risk adjusted yield on the investment must exceed the cost of funds.”
IIRC, all risk-adjusted yields are identical in CAPM world.
In CAPM, individual asset return Ri can be expressed as
Ri = Rf + beta*(Rm – Rf)
where Rf is riskfree rate, and Rm is return on market portfolio.
So, if you adjust for the risk represented by beta and subtract the second term from the above equation, risk-adjusted return always equals to Rf.
And similar thing happens in Arrow-Debreu world by using Risk-neutral measure, IIRC.
You’re making up the human capital element. It’s not in the original formulation of CAPM.
himaginary, I see what you mean. I wasn’t thinking Roll because he’s talking about observability of the market, I’m basically talking about market incompleteness.
My point was about one of the assumptions of the model, that markets be (at least dynamically) complete.
Roll was talking about a situation where you accept the assumptions of the model and try to test the implication, you can’t because you can’t observe the market portfolio.
himaginary: “all risk-adjusted yields are identical in CAPM world.”
This is true, both in the CAPM world and in every other world in which there are no arbitrage opportunities (that is every world in which there exists a risk-neutral measure).
himaginary, forget my last comment, I see you said the same at the end of your comment.
Let’s just say that I agree:)
Jon: at the margin, in equilibrium, the rate of return on investment should equal the (risk-adjusted) rate of interest.
Buying a house is buying a bundle of A-D contracts: you get the right to live in this house in period 1, and period 2, and 3…in all states of the world.
But the bundle you really want to buy is different: you get the right to live in this house in period 1, the right to live in a bigger house in period 2 if you have 6 kids in period 2, the right to live in a smaller cheaper house in period 2 if you lose your job in period 2, etc.
Renting a house is buying the right to live in this house in period 1. That contract has zero value in period 2.
Does the period 2 value of the buying a house bundle correlate positively or negatively with the period 2 value of the ideal bundle?
If it’s positive, I think you reduce risk by buying a house, as opposed to renting. If it’s negative, I think you increase risk. ?
At first glance, the argument is interesting, but really the real situation is a lot more complicated than it seems. You aren’t really reducing risk, you’re swapping one type of risk (size of a stream of payments) for another (value of an asset). In a world without credit constraints you might be able to consider changes in net present value of that stream to changes in the value of that asset. In the real world, a sudden massive drop in house value below the outstanding balance on the mortgage is significantly worse than a future increases in rent which might be equal in NPV terms.
I’ve been looking at this post over and over and there was something else that kept bugging me. I finally managed to put my finger on it: I don’t really like the assumption of constant covariance between rental prices and home values. If it’s not guaranteed that the covariance will remain positive, ownership is a poor hedge for rental costs.
There’s a few other assumptions that are overly restrictive: unless your preferences are absolutely constant the model of being born short one house doesn’t really make sense. Right now, I’m short one apartment. Perhaps next year I will be short an apartment in a totally different city. Given that uncertainty, it makes more than a little difficulty to hedge rental price risk. Furthermore, buying/selling a house comes with hefty transaction costs. So really, unless you don’t plan on moving any time soon, it makes little sense to use the short one house model. If you aren’t planning on moving anytime soon, it might make sense to buy a house regardless. (At this point this comes down to a non-financial consideration.) If you’re looking to hedge rental prices, the ideal hedge would something you could hold independent of your variable preferences or at least would have a low transaction cost.
Ignoring transaction costs of buying/selling the house doesn’t make sense unless you’re trying very hard to prove that people should buy houses. Those costs are a huge structural argument in favor of renting. Furthermore, contrary to your assertion in comments, it isn’t possible always possible to sell a house if you don’t like it. In fact this is one of the primary risks involved in purchasing a house. In a down market it may take months to sell the house or it may not sell at all unless you are willing to accept a price significantly below market value. Now if you have insufficient assets, you may not be able to afford to accept too low of a price because it would be insufficient to cover the mortgage and you may be unable to take out an unsecured loan to cover the shortfall.
Also, I should have really read the earlier post before commenting on this one, as it seems you cover most of my points.
Victor: “At first glance, the argument is interesting, but really the real situation is a lot more complicated than it seems.”
Agreed. I haven’t got this idea fully worked out. And I’m pleased to see you working your mind around some of the same things I was trying to work my mind around, plus some more.
But all I’m hoping for at this stage is that I might be onto something!
This assumes the house itself will never have any home ownership black swan events in regards to maintenance, losing your job in a down economy/housing market, or change in personal marital status. Far too many people do not factor in maintenance, and the opportunity costs of the time home ownership takes – yard work for the well paid accountant comes to mind.
While equity can be remortgaged as a source of income smoothing, a rational renter can access a presumed surplus of cash.
I know a lot of people presume home ownership is security and very often it provides a last resort pool of emergency capital because it is forced savings, but it isn’t risk free. Losing a home under duress is one of the most stressful events in North American social life, and it is a rare but real and large risk that needs to be accounted for.