Phinance for Physicists #1

Thanks to jesse for putting this together and doing some legwork.

I had a brief email conversation with Dr. Somerville on his recent paper regarding proper valuations of residential houses in different Canadian cities. He and I agreed that the life of an economist involves a bit more than plugging numbers into equations. However economics is a science so, having no significant formal finance/economics training but a bit more physics training, I thought I would bridge the divide with some much touted but rarely practiced cross-disciplinary legwork. Scientist to scientist.

A good friend of mine opined about physics education that the number of lies told during one's schooling slowly decreases. In many ways, in my current view, economics is like this. We use simple formulas, supply-demand curves, etc. in an attempt to understand the world around us.

With my limited understanding of finance, here starteth the lies.

Net Present value

The concept of The Net Present Value (NPV for short) is determining fundamental value for an asset, such a car, a boat, a stock, a pair of droids, a sail barge, or a house. The concept is simple: that the purchasing of this asset ties up one's money that cannot be used for other things, that people generally want to maximize their wealth, and that there is risk in tying up money in an asset.

Rule#1: The Net Present Value (NPV) of an asset is the sum of a series of expected discounted future cash flows.

What it means is that rational investors, given all other available choices with what to do with their money (opportunity cost), the risks involved, the expected revenue, expenses, and a little something for their trouble, will look at an asset and determine what the maximum they would be willing to pay to produce a fair return.

We can derive a rough formula for the net present value of, say, a condominium that is rented out. We assume the condo is rented out forever, which is a reasonable approximation (forever and 30 years produce about the same number). Returns and inflation compound so the series of cash flows increases geometrically but are discounted:

(1)

where i is the expected inflation of rent and expenses (say, 2% per year), and r is the so-called discount rate that is a combination of the long term bond rate (that includes expected inflation), a risk premium, and depreciation. The discount rate is a method of combining risk, inflation, and opportunity cost into one number. Rent is gross annual rent, and Expenses are annual maintenance, tax, and insurance costs (et cetera). Financing costs are not included here as the decision to buy is based upon having cash in hand, though more advanced analysis can explicitly include them. (Financing costs are, to a degree, implicit in the above formula) Note depreciation could be abstracted in Expenses or as part of the discount rate r.

Note that some condos will have more problems with tenants than others. How this is handled in the calculation is typically by having a higher risk premium included in "r" though in essence it could also be handled by using the expected value of rent, not nominal rent. The same principle applies for expenses, like the probability and expense of a condo being leaky for example.

This is in fact the formula mohican used to calculate fundamental value. He chose to use r-i (also known as the "cap rate") as the 5 year mortgage rate as a rough approximation. In his words, "The bond market and the banks are particularly efficient at determining the risk premium and adequately price that in to the 5 year mortgage rate. This is why I use it." Mohican also clarified to me that he assumes land appreciation and structure depreciation are roughly equivalent. It's also worth noting that mohican has chosen to use a relatively simple formula to estimate fundamental price and not spend too much time in details. In general there is nothing wrong with this but in certain situations it helps to know what to do if there are significant deviations.

The working paper Dr. Somerville et al published uses a formula similar to this but in a different form. It can be derived from the same basic principle, resulting in:

(2)

Where k is the long term mortgage rate, t is tax rate, m is maintenance, d is depreciation, (the last 3 are as % of property value) and E(DP/P) is expected capital appreciation.

Without going into the details, both mohican's and Dr. Somerville et al's chosen formulas can be construed to have been derived from Equation 1 above, from a strict financial, not economic, perspective.

Example

Let us use mohican's chosen formula in a simple example. A condo rents for $1000 per month, just sold for $220,000 and has expenses that are 20% of rent. The mortgage rate is 6%. NPV = 1000*12*0.8/0.06 = $160,000. This means that a rational investor who is looking only at future cash flows will be willing to pay $160,000 for this condo, according to mohican's formula.

Now, let us use Dr. Somerville et al's chosen formula for the same condo. We will assume that this condo will "conservatively" appreciate at E(DP/P) = 3% per year, k is 6%, t is $1200/year, or 0.5%, m is 0.5%, and d is about 1%. According to this formula a rational investor will pay 12000/(0.06+0.005+0.005+0.01-0.03) = $240,000

Analysis

Why the difference? There are two reasons. The main reason is contained in E(DP/P). It turns out, according to first principle derivations, that E(DP/P) is equal to the expected appreciation of future cash flows. In other words, a value of E(DP/P) increasing faster than cash flows are increasing from the current asset is expecting to either sell the asset to someone in the future for a price inflated by E(DP/P) (selling an asset is a cash flow), or that cash flows can increase more sometime in the future by using the asset more productively. Assuming 3% annual appreciation is more than what was implicitly assumed by mohican.

The second difference is more subtle. Remember in Equation 2 the expense costs were as a percentage of value. If capital appreciation continually outpaces rent inflation, these percentages perpetually drop; in other words the cost of capital perpetually decreases. This works mathematically but one must realize that the capital appreciation inherently includes productivity increases above what the current structure can support. The potential flaw in this is that the expenses must include construction costs for the productivity enhancements and would boost the percentages higher than what was calculated. Furthermore, and more importantly, one must think hard about what value to use in the denominator of the percentages. For this reason it may (must) be preferable to pull m, t, and d and make them nominal, not as a % of an arbitrary pre-calculated value.

Remember Equation 1 assumes an infinite series of geometrically increasing cash flows. If at some point in the future rents increase faster than expected or the land can be further sub-divided and used to produce more rent (say turning a 5 story condo into a 20 story condo 15 years from now), the NPV could be higher than what current cash flows suggest. One can approximate this by adjusting upwards the exponent on the geometric series. Expenses, too, can be nonlinear if there are major renovations or redevelopments in the future. If, however, the cash flows spike up (or down) too far in the future, their present values are diminished so as not to significantly change the expected capital appreciation rate. For example 1000 years from now a giant space tower with 6000 floors is likely to be built in a detached residential neighbourhood. This will have close to no impact on NPV, even if it were a certainty.

We can of course solve for E(DP/P) to figure out, if the condo just sold for $220,000, what the rational investor expects for capital appreciation.

Another measure thrown around local blogs is the 100X-125X monthly rent multiplier for condos. This would put our example condo at $100,000-$125,000. There is historical precedent for this occurring, though just as prices can be above one's calculated NPV, they can drop below as well. It IS reasonably possible to calculate NPV equal to 100X rent in Vancouver; in fact I think there is a good chance of select condo sale prices equaling this someday soon.

Regardless of which formula, mohican's, Dr. Somerville's, or some other fundamental asset formula, you may choose to use, remember that their roots are the same but their assumptions can be different.

The next (lesser) lie to discuss is looking at E(DP/P) -- capital appreciation -- in more detail; naturally, with housing in mind of course.