1. Monthly Mortgage Payment
The fixed monthly payment for a fully amortizing loan is calculated using the annuity formula:
$$M = P \times \frac{r(1+r)^{n}}{(1+r)^{n} - 1}$$
M Fixed monthly mortgage payment ($)
P Principal loan amount ($) = Home Price Γ (1 β Down Payment %)
r Monthly interest rate = Annual Rate Γ· 12
n Total number of payments = 360 (for a 30-year mortgage)
2. Amortization Schedule
Each month, the payment is split between interest and principal. Early payments are mostly interest; later payments are mostly principal.
$$I_t = B_{t-1} \times r$$
$$P_t = M - I_t$$
$$B_t = B_{t-1} - P_t$$
I_t Interest portion of payment in month t ($)
P_t Principal portion of payment in month t ($)
B_t Remaining loan balance after month t ($)
B_0 Initial loan balance = P (the principal)
3. Home Value and Equity
The home appreciates (or depreciates) at a constant annual rate. Equity is the difference between market value and remaining mortgage balance.
$$V_y = V_0 \times (1 + a)^{y}$$
$$E_y = V_y - B_y$$
V_y Home value at end of year y ($)
V_0 Original purchase price ($)
a Annual home appreciation rate (decimal)
E_y Home equity at end of year y ($)
B_y Mortgage balance at end of year y ($)
4. Annual Homeownership Costs
Beyond the mortgage, homeowners pay recurring costs that often increase with inflation or home value:
$$T_y = V_y \times \tau$$
$$H_y = H_0 \times (1 + i)^{y-1}$$
$$R_y = V_y \times \rho$$
$$A_y = A_0 \times 12 \times (1 + i)^{y-1}$$
T_y Property taxes in year y ($)
Ο Property tax rate (decimal)
H_y Home insurance in year y ($)
H_0 Initial annual home insurance ($)
i Annual inflation rate (decimal)
R_y Maintenance/repairs in year y ($)
Ο Maintenance rate as % of home value (decimal)
A_y HOA fees in year y ($)
A_0 Initial monthly HOA fee ($)
5. Tax Benefit from Mortgage Interest
Homeowners who itemize can deduct mortgage interest, reducing taxable income:
$$S_y = I_y^{\text{annual}} \times \tau_m$$
S_y Tax savings in year y ($)
I_y^{annual} Total mortgage interest paid in year y ($)
Ο_m Marginal income tax rate (decimal)
Note: This assumes itemized deductions exceed the standard deduction. Many taxpayers do not benefit from this deduction.
6. Total Net Cost of Buying
The net cost sums all cash outflows, subtracts tax savings, and subtracts final equity (net of selling costs):
$$C_{\text{buy}} = D + K + \sum_{y=1}^{Y}\left(12M + T_y + H_y + R_y + A_y - S_y\right) + V_Y \cdot \sigma - E_Y$$
C_buy Net cost of buying over Y years ($)
D Down payment ($)
K Closing costs ($) = Home Price Γ Closing Cost %
Y Time horizon in years
Ο Selling cost rate (decimal, e.g., 0.06 for 6%)
E_Y Final home equity ($)
7. Renter's Annual Costs
Renters pay monthly rent (which increases annually) plus renter's insurance:
$$L_y = L_0 \times 12 \times (1 + g)^{y-1}$$
$$J_y = J_0 \times (1 + i)^{y-1}$$
L_y Total rent paid in year y ($)
L_0 Initial monthly rent ($)
g Annual rent growth rate (decimal)
J_y Renter's insurance in year y ($)
J_0 Initial annual renter's insurance ($)
8. Renter's Investment Portfolio
The renter invests the money that would have gone to a down payment and closing costs. Each year, any savings from lower housing costs are also invested:
$$W_0 = D + K$$
$$\Delta_y = \max\left(0,\; C_y^{\text{owner}} - C_y^{\text{renter}}\right)$$
$$W_y = W_{y-1} \times (1 + r_i) + \Delta_y$$
W_0 Initial investment ($) = down payment + closing costs
Ξ_y Additional savings invested in year y ($)
C_y^{owner} Homeowner's annual housing cost in year y ($)
C_y^{renter} Renter's annual housing cost in year y ($)
W_y Investment portfolio value at end of year y ($)
r_i Annual investment return rate (decimal)
9. Total Net Cost of Renting
The net cost sums all rent and insurance paid, then subtracts the final investment portfolio value:
$$C_{\text{rent}} = \sum_{y=1}^{Y}\left(L_y + J_y\right) - W_Y$$
C_rent Net cost of renting over Y years ($)
W_Y Final investment portfolio value ($)
10. Final Comparison
The financially superior option has the lower net cost. A negative net cost means you end up wealthier than you started.
$$\text{Savings} = \left| C_{\text{rent}} - C_{\text{buy}} \right|$$
Savings How much better the winning option is ($)
If C_buy < C_rent, buying wins. If C_rent < C_buy, renting wins. If the difference is small (< $5,000), the options are roughly equivalent.