Economic Analysis – Should I Buy a House?

Overview

For the past year or so I’ve been thinking about whether or not after I finished paying off student loans I would consider purchasing a house.  I’ve done a little research, but it seems that there are a lot of different opinions out there and none really made the case one way or another.  Opinions ranged from: “Yes, as long as you’re planning on staying there for at least x years” to “Yes, as long as you can put x% down” to “No, you’re better off renting and investing the difference in monthly payment in the market” to “It just depends on preferences – flexibility, mobility etc”.  After reading through their reasons, I’ve decided to do my own analysis on the economic reasons to buy or not to buy.  Note that I am not concerning myself with the intangible elements that come with home ownership, but merely looking at the financial implications, because to me, it’s purely an economic decision – at least at this point in my life.

Summary

The following analysis will be based in basic economic price theory and financial mathematics.  For those of you who are uninterested in this (though I don’t see how someone wouldn’t!), I’ll provide a brief summary of my conclusions. 

Buying a house is tentatively a mediocre financial decision.  Like any investment it has its own risks.  Also, it appears that the optimal time to hold onto your house is about 10 years (this will maximize your investment yield). I’ll need to do some more analysis to be completely comfortable with the risk/reward tradeoff.

Methodology

I start by calculating what my investment cash flows are.  The equation I used for my investment is the following:

Investment(t) = I(t) = Mortgage Payment(t) +Other Monthly Payment(t) - Rent(t) - Tax Savings(t)

Investment(t=0)= I(0) = Down Payment  + Buying Costs

Essentially you start with you total monthly outlay and subtract anything you would have paid if  you continued to rent (sunk costs – in this case rent) and any monthly tax savings (you can write off interest payments on your income taxes every year).  “Other monthly payment” here is for things like property taxes, insurance and condo fee (if I got a place in Cambridge or the surrounding area it’d probably be a condo or duplex of some sort). 

The next step is to calculate what the cash flow you receive when you sell the house is.  You can think of this as what all of those investment cash flows accumulated to, or your net equity.  Here is the equation I used:

Net Equity (T) = E(T) =Home Price(T) – Loan Balance(T) – Selling Costs

The capital “T” here denotes the time at which you sell the house. 

The next step is to solve for the interest rate that would make the string of investment payments accumulate to the net equity you received from the sale of the house.  Here is an illustration

The “I”s on the top represent the investment outflows and the E(T) on the bottom represents the investment inflow.  Without going into too much detail, these cash flows imply a certain interest rate which represents the yield on this investment.  The yield will tell you how good of an investment it is.

Assumptions

A lot of assumptions go into this analysis.  Some of the assumptions I used are probably better than others and they were all based on limited research.  I started with a base set of assumptions and then tested how sensitive they were to the financial outcome by modeling slightly different assumptions.  Here are the base assumptions:

A few notes:

• 500k is about what I would have to pay to buy the apartment I live in now (I know it’s crazy.  Cambridge/Boston real estate is super expensive)
• 500 additional monthly would be for property taxes, insurance, condo fee etc. 
•   Home appreciation rate of 2% seemed conservative (just about inflation expectation)
• My rent is ridiculous
• Rent increase of 2% per year keeps it in line with my home appreciation rate.  It’s more likely that this wouldn’t be smooth though (probably more like a 15% increase every 5 years or something)
• Income tax rate should be your marginal tax rate, not average
• Value Shock = set to 0 for base scenario, I’ll use this in the analysis
• Buyer/seller costs – a friend of mine who recently bought a house in the area gave me these numbers.  These represent lawyer fees, broker fees etc.

Results

Running these numbers through a model I built (just in excel), I got the following yields:

In other words, with these inputs, the implied annual return on my investment in a house would be 9.4%.  Certainly not bad especially in current economic conditions!  However, this is based on static assumptions I came up with on the spot.  In order to understand the sensitivity of these results to the assumptions, I ran several scenarios, shocking the assumptions (both up and down) to see what the yields could be.  Here is tabular view of the results:

Changes in down payment and housing price assumptions

A few notes:

• The less you put down the higher your yield is
• This is because the less you put down the more leveraged you are in the investment.  To illustrate this, consider you have $100 and you want to purchase an investment for $1,000 that matures in one year.  You borrow the $900 more you need at an interest rate of 5%.  Suppose the asset is worth $1,100 at the end of the year implying a 10% gain.  You pay your $900 loan back with interest of $45 – (.05*900 = 45), leaving you with $1,100 - $945 = $155.  Your $100 grew to $155 or a 55% gain!  While this sounds awesome, it is very risky.  Assume your asset was worth only $900 at maturity.  You’d lose over 100% of your investment.  (By the way, this is how financial institutions make a lot of money)
• On down shock scenarios, the implied yield is higher at later durations. This is because it takes time for the loss to be recouped

Other Scenarios without Commentary (I’m getting tired of writing)

Impact of different rent assumptions:

Impact of different loan rates: (by the way these were for 30 year fixed interest loans)

Impact of different home prices:

Conclusion

I feel the jury is still out, however, it does look like it could be a pretty decent investment.  Like I said in the beginning, this analysis didn’t take into account intangible benefits of home ownership (I want to paint my walls black an orange for example—fictional example), or I want to be able to move on a whim to California and don’t want a house to tie me down.  Perhaps the final decision to purchase a house should be based on these intangible criteria as I’m not convinced there is a super strong financial incentive to own (or to not own) a home (at least not for me right now).  I do think that the assumptions in my model could be flushed out a bit better and I would like to introduce some dynamic element to the assumptions.  It’d be interesting to model this with a stochastic home price forecast.  I’d like to see not only what my potential returns could be, but a full distribution of the returns.