# Modeling wealth over time in a home [closed]

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.
• What is the question you need answered? Oct 23 '21 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 Oct 23 '21 at 3:20