Homeownership, Part III-B

T writes:


Comparing WR(T) and WHO(T)

I think the best place to start is by looking at how the two “worths” compare at the outset, when T = 0.

Comparing the two worths at T = 0 really just means comparing WR(0) and WHO(0):

WR(0) = Do; and
WHO(0) = Do – BuyCosts1; so

WR(0) – WHO(0) = BuyCosts1

What this means is that the act of buying a home actually reduces our immediate worth. This isn’t really unexpected if you think about it: the process of buying a home (which may involve inspections, realtor and lawyer fees, etc.) costs money, and so when we buy our first home our worth will be immediately reduced by the size of these BuyCosts1. People are immediately made financially worse off by buying a home, but people still buy homes so there must be some incentive to do so. An example of a good incentive would be if our worth while we own WHO(T) actually becomes greater than what our worth while we rent WR(T) would be at some time T. And in order for our WHO(T) to become greater than our WR(T), it is necessary that dWHO(T)/dT > dWR(T)/dT for at least some range of T. It turns out that dWHO(T)/dT > dWR(T)/dT when:

In the above inequality, I have used square brackets [ ] to group three sets of terms that are of particular relevance in this comparison. These are:

I have already shown that WHO(0) < WR(0), so the only way to ever have WHO(T) > WR(T) at some time T (e.g., at T = F) is for dWHO(T)/dT > dWR(T)/dT for some range of T. This can only occur if [A] + [B] + [C] > 0. It follows that if we are to buy a home, we would like to have [A] > 0, we would like to have [B] > 0, and we would like to have [C] > 0. We don’t actually need ALL of them to be > 0, but we need their sum to be > 0 so any positive contribution is preferred to a negative contribution.


I’m going to look at [C] first, because its sign is the most definite. [C] invokes a comparison between the rate of the ROI (in $/month) on our free cash savings that we can earn while we own and that which we can earn while we rent. The $/month that we earn on our investments depends on two things: the size of our monthly savings contributions and the return rate r as a %. There is no reason why the way in which we invest our homeownership savings (I – HO) should be different from the way in which we invest our rental savings (I – R), so we should expect the same r whether we own or we rent. However, our monthly savings contributions are not the same when we own as they are we rent. I’m only considering situations where HO > R, and therefore (I – HO) < (I – R). This means that the $/month rate of return that we earn on our homeownership savings is always less than the rate of return that we earn on our rental savings. So, [C] < 0.


Now on to [A], which embodies a comparison between the rate at which the value of our first home appreciates and the rate of the ROI that we could otherwise earn on our first down payment Do (both in $/month). Similar to our ROIs, the average $/month that we can earn (once we sell) from the appreciation of our first home depends on two things: the initial value
(Vo = Do – BuyCosts1 + L) of our home and the appreciation rate v as a %. Now, v is highly case-specific, so to make things a bit simpler I’m going to initially treat it as the same as r (e.g., something around 2% – 5%). This probably isn’t too far off in many real-world scenarios.

With v = r, the sign of [A] only depends on the relative magnitudes of Do, BuyCosts1, and L. In [A], I’ve broken down the rate of appreciation into two components: the rate of appreciation of the (Do – BuyCosts1) component and the rate of appreciation of the L component (again, I point out that our mortgage principal appreciates and actually makes us money – an incentive for a bigger mortgage!). Obviously, Do – BuyCosts1 < Do, so the $/month rate of appreciation of the (Do – BuyCosts1) component is definitely less than the $/month rate of the ROI on Do (given v = r). However, the positive rate of appreciation of the L component should more than compensate for this, because it is pretty certain that L > BuyCosts1 (otherwise we probably don’t even really need a mortgage in the first place). This means that, for v ? r, [A] > 0. Things are a little bit more complicated if v < r, but in general Vo is probably sufficiently larger than Do that [A] will be greater than 0 for 0 ? T ? F even when v is a little bit less than r.

While [A] may be generally > 0, it is important to consider that [A] is the most volatile and unpredictable term of the three. We may have some idea of how the value of our home will appreciate over time, but a drastic swing in the housing market (in an economic crisis, for example) could very suddenly bring our [A] close to or below zero.


Lastly, on to [B], which reinforces that familiar comparative monthly savings result from Part I: [B] is only greater than 0 if P > HO – R.

The Sum, [A] + [B] + [C]

I have determined that [C] is generally < 0 and [A] is probably > 0, so it is quite feasible that the sign of [B] could be the crucial factor in determining whether or not [A] + [B] + [C] > 0. This confirms the importance of choosing the right L < Q*R so that P > HO – R. It is true that our WHO(T) can still become greater than our WR(T) even if P ? HO – R (as long as [A] + [C] > |[B]|), but [A] is variable and cannot be forecast with 100% certainty at the time of buying, whereas the relative sizes of P, HO and R are all pretty much definitely known for all T in 0 ? T ? F. To sum up, [C] is always < 0 for HO > R, we don’t have 100% control over the sign of [A] in the long term, but we can reliably choose our L to ensure that [B] > 0.

Option A and Option B

Remember Option A and Option B? This whole analysis is intended to help us decide between Option A: keep renting for now and buy our first home at T = F, and Option B: buy our first home now at T = 0 and sell it to upgrade to a second home at T = F. I have developed expressions for WR(T) which describe our worth if we follow Option A, and I have developed expressions for WHO(T) which describe our worth if we follow Option B. I have shown that WR(0) > WHO(0), and I have shown how this means that WHO(F) can only become greater than WR(F) if dWHO(T)/dT > dWR(T)/dT in the range 0 ? T ? F. I then went on to show that having P > HO – R is a very important factor in getting dWHO(T)/dT to be greater than dWR(T)/dT. The simple question remains, though: if P > HO – R such that [A] + [B] + [C] > 0, does this automatically mean that Option B is better than Option A?

Option B is only better than Option A if F is greater than the amount of time needed for us to make up for all the costs associated with buying our first home AND selling our first home AND buying our second home. That is, Option B is better than Option A if:

The actual time at which the above becomes true is case-specific, but experimenting with various parameters that are representative of our situation I find that it is very realistic to satisfy the above relation with F < ~30 months.

Key Result # 3: Establishing P > HO – R is one of the most controllable things we can do to ensure that our worth while we own grows to exceed our worth while we rent. However, such growth invariably takes some time. Therefore, we are only better off pursuing Option B instead of Option A if we are willing to commit a time of at least F in our first home, so that
WHO(F) – WR(F) ? BuyCosts1 + SellCosts1 + BuyCosts2.


This concludes the now epic homeownership series. The results, for Alexis and me at least, confirm that we are actually better off renting while we fulfill our short-term West coast lifestyle. We should, however, consider buying when we move on to our next phase of life if we plan to remain fixed in that next phase for long enough to cover all of our buying and selling costs.

It’s always nice to confirm by lengthy analysis that decisions which you felt were intuitively correct were indeed so. More importantly, though, I have developed a pretty manageable set of criteria to consider when we actually do go shopping for our first home.

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