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Suppose a person decides to buy a house and finance it by taking out a mortgage. I am attempting to model the wealth of this person over time. I need some help deciding whether this system of recursions is a suitable description for the time-varying quantities.

home price

Home prices randomly appreciate or depreciate.

$$ P_t = P_{t-1}(1+R_t) \tag{1} $$

  • $P_t$ is the price of the home at time $t$
  • $R_t$ is the random arithmetic annualized rate of return

loan value:

The loan value compounds, but more slowly if you pay larger periodic payments.

$$ l_t = (l_{t-1} - p_{t-1})(1+r_t) \tag{2} $$

  • $l_t$ is the amount outstanding at time $t$ of a mortgage
  • $p_t$ is the periodic loan payment at the end of time $t$
  • $r_t$ is the quoted mortgage rate

cumulative money paid

Every period you're paying mortgage, property taxes and insurance.

$$ t_t = t_{t-1} + p_t + c_{t-1} P_{t-1} + i_t $$

  • $t_t$ is the total amount of cash paid towards the house
  • $c_{t-1}$ is the property tax rate
  • $i_t$ is the periodic insurance cost
  • $t_1$ is closing costs and other one-time payments.
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    What is the question you need answered? Oct 23 at 3:00
  • Is this system of recursions a suitable description for the time-varying quantities? @DilipSarwate Are these the pertinent quantities if I’m mostly interested in the evolution of P_t - l_t - t_t
    – Taylor
    Oct 23 at 3:20
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The key to this type of analysis is that the value of the home next year is unpredictable.

Most people in March 2020 as the United States was shutting down due to COVID expected that as inflation hit, many business shuttered, millions lost their job; that home sales would plummet, and home prices would drop in order to allow people to sell that had to sell. Instead the real estate market was hot, and home values went through the roof.

That means that knowing what the housing values will be X years from now is impossible. You can pick any rate you want, and any amount of variability. Just rerun your model every year with the updated change since last year.

In addition to the unpredictability of the value of the house, you have no idea what the property tax rate will be in the future. Sometimes there is pressure to adjust the rate to either bring in more money or to mitigate the impact of raising home values. In amny places that decision is an annual event.

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  • Awesome, thanks. So I’m not missing any important part, it’s just that the dynamics for two of them are difficult?
    – Taylor
    Oct 23 at 14:46

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