Mortgage Rates are at Historic Lows

posted Nov 15, 2011, 11:04 AM by Bert Bingley

It is time to refinance. 

Mortgage Comparison with Spreadsheets

posted Dec 1, 2009, 7:38 AM by Bert Bingley

Calculating Mortgage Costs (w/ Equations!)

posted Nov 11, 2009, 6:00 AM by Bert Bingley   [ updated Nov 11, 2009, 4:17 PM ]

   There are many variables that contribute to calculating the future cost of owning a home.  In this article, I outline the variables involved and write out the equations used to calculate the total cost.  

    For most home purchases, there are two periods that need to be considered.  In the first period, the home owner has to make monthly mortgage payments.  Once the mortgage is completely paid off, the home owner enters the second period where mortgage payments no longer have to be made.

 Buying Variables    Assumptions    Selling Variables  
Mortgage Term (years) T Home Appreciation r_{ha}
Cost of Sale (%) cs
Points p Investment Appreciation r_{ia}
Years Owned Y
Interest Rate  r_i Tax Rate r_{tr}
Down Payment (%) {dp} Rent Appreciation r_{ra}
Purchase Price A
Home Insurance r_{hi}
Closing Costs cc
Annual Rental income R

    Next, let's enumerate the nominal cost associated with owning a home.  By nominal, I mean the actual cost paid out of pocket.  To generate the opportunity cost adjusted total cost, or, the real total cost, we will have to incorporate the rate of return on alternative investments, r_{ia}.  For now, let's just look at the nominal costs:

Buying Costs C_B
Annual Costs C_A
Selling Costs C_S
Mortgage Points pA
Mortgage Payment C_{mp}=\\(A-A{dp})\frac{r_i}{1-(1+r_i)^T}
Cost of Sale A cs (r_{ha}+1)^Y
Closing Costs ccA
Taxes A r_{tr} (r_{ha}+1)^Y
Principle Due (A-A{dp})\cdot\\\left(1-\frac{1-(1+r_i)^Y}{1-(1+r_i)^T}\right)
Down Payment A{dp}
Insurance A r_{hi} (r_{ha}+1)^Y Cash from Sale -A (r_{ha}+1)^Y

Rent -R(r_{ra}+1)^Y

    In the table above, the variables C_BC_A, and C_S represent the sum of the buying costs, annual costs, and selling costs, respectively.  Finally, to determine the real cost, we have to adjust each of these cost to account for the opportunity cost of money.  

    Putting it all together, the equation for total mortgage/home cost when Y\le T is

C_{total|Y\le T} = C_B(1+r_{ia})^Y+C_A\frac{(1+r_{ia})\left((1+r_{ia})^Y-1\right)}{r_{ia}}+C_S

When the Y> T, the mortage will have been totally paid off.  To recomupte, we must subtract the mortgage payment for the period after T.  So for this situation, we have the following mortgage/home cost equation,

C_{total|Y> T} = \\C_B(1+r_{ia})^Y+\frac{1+r_{ia}}{r_{ia}}\left[C_{mp} \left((1+r_{ia})^T-1\right)+(C_A-C_{mp})\left((1+r_{ia})^Y-1\right)\right]+C_S

In forthcoming posts, I will analyze these equations to show which terms dominate the costs in the long term.  Another plan I have is to show the break even trade space in terms of home appreciation, investment return and rental income.  Stay tuned....

Mortgage Costs

posted Sep 16, 2009, 6:56 AM by Bert Bingley   [ updated Sep 19, 2009, 8:40 AM ]

    When shopping for mortgages, the obvious question is: which set of mortgage options will cost me the least?  This turns out to be a fairly complicated question to answer.  The difficultly in giving a concrete answers rests in the fact the certain assumptions have to be made in order to calculate long-run mortgage/housing cost. Below, I list the important variables involved in calculating mortgage/housing costs.  The variables in bold are not known a priori when one buys a house, and the parenthetical values are the values assumed for our analysis.
    • Known Variables
      • Closing Cost (4% of purchase price)
      • Home Insurance Rate (0.46% of home value)
      • Tax Rate (1.35% of home value)
      • Cost of Sale (6% of selling price)
      • Mortgage Term
      • Points (0% of mortgage loan)
      • Interest Rate (4.8%)
      • Down Payment
      • Purchase Price ($100,000)
    • Unknown Variables
      • Home Appreciation
      • Investment Appreciation

    So, the process for calculating future value cost of owning a home (financed with a mortgage), is simply a function of all of these variables. Fortunately many of the variables are known (to a large degree). However, two variables must be estimated in order to calculate costs: the rate at which the purchase property will appreciate and the rate at which any other investment assets will appreciate (return on investment; ROI).

    As an estimate for ROI consider that the average ROI on US stocks over long periods of time varies from 8%-12%. On the other hand, house appreciation has only sustained an approximately 3% return over long periods of time. There are notable exceptions like high-demand urban areas where home appreciation approaches 8%. Interestingly, as we will show in a subsequent post, the choice of mortgage is completely independent of the house appreciation rate, but is highly dependent to the non-home ROI.

    Below are several plots for various ROIs and mortgage options. Two general conclusions can be drawn from these plots:
      1. When the ROI (i.e. the opportunuity cost of money sunk in a house) is high, You want to keep as much money as possible and put as little into paying off your house as possible.
      2. When the ROI on alternative investments is low, it also makes sense to pay off your home as quickly as possible.
    By high ROI, we mean and ROI that exceeds the mortgage rate. The plots show, that for high ROI, even after the short-term mortgage has been paid off, the opportunity cost of having all that capital sunk in a house far out paces the cost of additional payment.

Mortgage Considerations

posted Aug 22, 2009, 3:14 PM by Bert Bingley   [ updated Aug 22, 2009, 3:53 PM ]

When shopping for a mortgage it is important to consider your objectives.  The most important factors are:
  1. Your income
  2. How long you plan to own the mortgage/home
  3. The rate of return you expect to earn on other investments
  4. Your level of risk aversion
Your income is important as it determines how large of a mortgage a bank will be willing to grant. The time horizon for selling your home is important because this will determine whether you should explore non-fixed mortgage options like adjustable rate mortgages (ARMs) and interest only (IO) mortgages.  A long time horizon will probably mean that a fixed-rate mortgage is most appropriate.  However, if you only plan to hold the mortgage for a short period of time, it is possible that ARMs and IO mortgages will be more attractive.

In any of these considerations, we must take into account how much your money could be earning in an alternative investment.  If investment alternatives have very low risk normalized yields relative to your mortgage rate, then it would be smart to pay off the mortgage as quickly as possible.  Risk aversion also pays into the payoff time calculation because it is unlikely that a low-risk investment opportunity will yield nearly as much as your mortgage rate.  So, even if a hypothetical investment has a higher risk-adjusted yeild, you will need to be willing to accept a highly level of risk to take advantage of it.  

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