If you’ve read through these “Homeownership” entries in series, then be forewarned: this is the longest Part. In truth, it isn’t as long as it could be or ought to be (I’ve just recently abandoned a substantially longer version), but Alexis tells me that web-posts are supposed to be concise and I’ve conceded that making this final entry too long or overly complicated won’t help to encourage more readership. After several iterations of varying depth and complexity, what follows is my most approachable answer to the main motivational question.

I have shown (Part I) that our monthly savings while we own *S _{HO}* may be greater than our monthly savings while we rent

*S*even if the total monthly cost of homeownership

_{R}*HO*exceeds the total monthly cost of renting

*R*. This occurs when

*P > HO – R*. I have also shown (Part II) that having

*P > HO – R*involves working with a mortgage principal

*L*that satisfies the relation

*L < Q*R*, where

*Q*is given by

*(a + t + z – pa)*. I have since introduced a new concept, our “worth”, which I defined as the total value of money that we can put towards a home at any given time. In the rental scenario, our worth

^{-1}*W*includes only the total amount of savings we have for a down payment

_{R}*D*; while in the homeownership scenario, our worth

_{R}*W*includes both the total amount of savings we have available for a second down payment

_{HO}*D*and the total equity

_{HO}*E*in our first home.

There are many factors to consider in studying homeownership, and many different scenarios that could arise. In an effort to be as concise as possible, I’m going to specifically consider a decision that is most relevant for Alexis and me, and for others in similar circumstances. The decision is this: given that we rent now and we have money *D _{o}* for a down payment now, should we a) keep renting and buy later or b) buy now and sell later to upgrade to a new place. I think this question is common enough to be of interest to a general audience.

To begin, I need to introduce the concept of time *T* into my analysis, where *T* is in units of months. I define today as the beginning of time, so *T* = 0 today. Today we have two options: Option A) we can keep renting and eventually buy a place at some time in the future when*
T = F*, or Option B) we can buy a first place now (at

*T*= 0) and hold onto it for a while, then sell it at

*T = F*and upgrade to a second place. My goal is to assess which option lets us end up in the best (i.e., highest value) place when

*T = F*, and to identify the key influential factors. To keep things general, I need to assume that we won’t be doing any fancy renovations on our first home or attempting to achieve any form of home flipping.

Our total monthly costs of renting are *R*, and our total monthly costs of homeownership would be *HO*. We are pretty much always better off buying rather than renting for *HO < R*, so the situation that I am interested in is *HO > R*. This is because in this situation we have more “free cash”* (I – R)* while we rent than we would *(I – HO)* if we bought (“free cash” is equal to *S _{R}* while we rent and

*S*if we buy), and we can invest that extra free cash to earn extra returns. What I want to examine now (and this addresses the main motivational question from the Preface) is how the evolution of our worth while we rent

_{HO}– P*W*compares to the evolution of our worth after we’ve bought

_{R}*W*.

_{HO}**Our Worth while we Rent, W_{R}**

I’ll begin this analysis with a general description of our worth while we rent *W _{R}(T)*, and then I’ll use this description to model how our worth will evolve if all of our other parameters (

*I, R, S*, etc.) stay the same. This analysis begins now, so

_{R}*T*= 0 today, and we currently have total savings

*D*as our starting worth

_{o}*W*.

_{R}(0)While we rent, the entirety of our rental payments go to someone else and do not contribute to our worth. This is the main deterrent from renting. However, our worth *W _{R}(T)* does still increase on a monthly basis because we have monthly savings

*S*, which I’m treating as constant from month to month because I’ve defined

_{R}= I – R*I*and

*R*to be constant and I’m ignoring consumption (but it’s also constant so that’s okay). For simplicity, I assume that all of our monthly savings

*(I – R)*are intended to be put towards a down payment

*D*when we buy a home, so that each month we add a quantity

_{R}*(I – R)*to our

*D*. We have no equity while we rent, so our worth

_{R}*W*at any given time

_{R}*T*is simply equal to our down payment

*D*. That is:

_{R}*W _{R}(T) = D_{R}(T)*

However, we don’t just hoard our savings under our mattress; we invest them and earn returns on those investments. We may achieve these returns through many different methods, such as savings accounts, GICs, mutual funds, stocks, bonds, etc., each of which may evolve in a different way. However, regardless of how we invest our savings, it is always true that at some time *T* in the future we’ll have earned a respective amount *ROI _{n}(T)* from each of our

*n*investments (each of which may be positive or negative, of course). Herein I am using a respective “

*ROI*” term to represent the actual dollar value of our accumulated returns from each of our

_{n}(T)*n*investments at time

*T*. This completely avoids the manner in which our returns were earned and, in that way, allows for the most general scope (because the equations that are specific to each investment type are generally quite complicated and in some instances unreliable anyway). Believe me, I have learned from experience that ignoring the composition of the

*ROI*terms is actually the most telling way to approach this problem. As an example, if after one year we have contributed $10,000 to our savings but we actually have $10,300, then we have incurred a

_{n}(T)*ROI(12)*of $300.

We may invest our starting worth *D _{o}* and our monthly savings

*(I – R)*separately, so the most general expression for our worth while we rent is:

where *ROI _{Do}(T)* and

*ROI*are the dollar values of the returns we’ve earned as of time

_{(I – R)}(T)*T*by investing our starting worth

*D*and our monthly savings

_{o}*(I – R)*, respectively. Another thing I’d like to know, though, is how our worth evolves, which is given by the derivative of our worth with respect to time,

*dW*:

_{R}(T)/dTThe *dROI _{n}(T)/dT* terms represent the rates at which we earn returns on our investments, in $/month. These units are very important: they are $/month and NOT %. The essential distinction is that a % rate might be the same for any size of investment and/or saving contribution schedule, while a $/month rate specifically depends on the size of the investment and/or saving contributions (i.e., the quantity against which the % is measured). To distinguish between the $/month rate and the % rate, I’m going to use the term

*r*whenever I’m talking about the average % rate. Going back to my previous example of $10,000 with a

*ROI(12)*of $300, that situation describes a

*dROI(12)/dT*of $25/month and an

*r*of 3%.

That’s it. These two equations are all that I need to analyze our worth while we rent *W _{R}(T)* and compare it to our worth while we own

*W*in order to estimate when we should buy (or should have bought…) our home.

_{HO}(T)**Our Worth while we Own, W_{HO}**

At some point in the future, *T = F*, we will buy our first home. What I’d like to do now, though, is imagine we bought our first home today, while *T* = 0, and that we plan to sell this home and move to a second one at time *T = F*. If we did this, our worth during the period *0 < T < F* would not be described by *W _{R}(T)*, but instead by

*W*. I will assume that, in this situation, we’ll have used all of our current savings

_{HO}(T)*D*on our first down payment so that our future savings

_{R}(0) = D_{o}*D*will be starting from zero

_{HO}(T)*D*= 0.

_{HO}(0)While we own, a portion of each mortgage payment goes to someone else and a portion *P* of each mortgage payment goes towards our mortgage principal *L*. This means that a portion *P* of our monthly homeownership expense *HO* is effectively saved each month. I had already accounted for this in my expression for monthly homeownership savings *S _{HO} = I – HO + P*, but now I want to think of it in terms of equity. Equity

*E*is defined at any time

*T*as the difference between the *current* value

*V(T)*of our first home and the amount of money

*L(T)*that we still owe on our mortgage principal

*L*. By bringing in the concept of equity

*E(T)*, our worth while we own becomes:

*W _{HO}(T) = E(T) + D_{HO}(T)*

Over time, our equity *E(T)* evolves because both the current value *V(T)* of our first home and the amount of money that we owe the bank *L(T)* evolve. So,

*E(T) = V(T) – L(T)*,

and now I need to look at *V(T)* and *L(T)* individually.

The value of our first home at any given time *V(T)* should be evolving. This evolution may be treated in many different ways depending on the specific scenario, so what I’m going to do is adopt the same sort of “*ROI(T)*” approach that I used for our savings investments. That is, at any given time *T* in the future the value of our first home will have appreciated by some dollar amount (could be +, -, or 0), such that:

where *V _{o}* is the value of our first home when we buy (i.e., at

*T*= 0 in this case) and

*App*is the total dollar amount by which the value of our first home has appreciated as of time

_{Vo}(T)*T*.

When we actually do buy our first home, there will be costs associated with the purchase. These *BuyCosts* are incurred in the form of various fees (e.g., lawyer fees, realtor fees, inspector fees, one-time insurance payments, etc.) and serve to effectively lower the value of our down payment. So, *V _{o} = D_{o} – BuyCosts_{1} + L*, which means that the value of our first home may also be written as:

This is really interesting and something that I hadn’t previously considered: since we borrow money *L* to cover the value of our home *V _{o}*, and its the total value of our home that increases with time, then we actually earn a return on our mortgage

*L*(our mortgage “appreciates” as a portion of the value of our home,

*App*). This return will surely not be greater than the interest that we pay, but it will serve to reduce the net cost of the mortgage.

_{L}(T)The amount of money that we owe the bank *L(T)* should also evolve thanks to our regular *P* contributions towards our mortgage principal *L*. In Part II, I briefly stated that the relative size of our *P* contributions may change over time, which means that:

However, to keep things simple I am going to assume that our *P* contributions are constant:

Combining all of this together, our total equity is:

In addition to our equity, our monthly homeownership costs *HO < I* still allow us to save free cash every month. This is represented by the *(I – HO)* part of the *S _{HO}* expression. I’ll again assume that all of this free cash will be put towards a down payment

*D*(this time when we sell our first home at

_{HO}*T = F*in an attempt to upgrade to a second home) and that in the meantime we invest it to earn a return. Thus, we’ll have monthly contributions of

*(I – HO)*and a return at time

*T*of

*ROI*:

_{(I – HO)}(T)However, when we do buy a second place, we’ll incur a fresh round of *BuyCosts _{2}* and we’ll also incur

*SellCosts*to sell our first home. These two

_{1}*Costs*effectively reduce the down payment

*D*that we can pay on our second home at time

_{HO}*T = F*. Also, notice that we have no lump sum savings

*D*in

_{o}*D*because we’ve put all of that money into the equity of our first home.

_{HO}(T)As previously stated, our worth while we own *W _{HO}* at any time

*T*is given by our equity

*E(T)*combined with the money that we have available for a second down payment

*D*. This means:

_{HO}As with the rental case, though, the rate of change of our worth is also pretty important here. At the highest level, our worth while we own *W _{HO}(T)* evolves as both our equity

*E(T)*and the money that we have available for a down payment

*D*evolve:

_{HO}(T)So now looking at the rate of change of our worth while we own, *dW _{HO}(T)/dT*, I get:

Which is very similar in form to the equation that I developed for *dW _{R}(T)/dT*. It’s almost as if that was deliberately set up to facilitate comparisons…

A note about the rate of appreciation terms, *dApp _{n}(T)/dT*: as was the case with the rate of

*ROI*terms, the rate of appreciation that I am using is in units of $/month and not %. To distinguish the two, I will use the term

*v*to denote an average % appreciation rate.

To be continued… oh the suspense!

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