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Rent vs Buy Formulas and Calculations

Learn the rent vs buy formulas for mortgage payments, amortization, ownership costs, sale-adjusted equity, invested savings, and break-even timing.

How do the Rent vs Buy Calculator formulas work?

This formula page defines the variables used by the Rent vs Buy Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Rent vs Buy Formula Overview

The calculator runs two monthly paths over the same period. Buying produces a mortgage balance, changing home value, recurring ownership costs and sale-adjusted equity. Renting begins with the buyer's unused upfront cash invested. Each month, the lower-cost option invests the positive housing-cost difference.

Dt=(Vt(1−s)−Bt+Ib,t)−Ir,tD_t=(V_t(1-s)-B_t+I_{b,t})-I_{r,t}

Dt is the modeled buyer advantage at month t. A positive result favors buying under the assumptions; a negative result favors renting.

Variable Definitions

SymbolMeaningUnit or convention
HPurchase price at month 0Currency
dDown paymentCurrency
PInitial mortgage principal, H - dCurrency
MScheduled mortgage principal and interestCurrency per month
rAnnual mortgage note ratePercentage
iMonthly mortgage rater / 1,200
nScheduled mortgage paymentsMortgage years x 12
BtMortgage balance after month tCurrency
VtModeled home value after month tCurrency
sEntered selling-cost rateDecimal
EtHome equity after selling costsCurrency
Cb,tTotal buyer housing cost in month tCurrency
Cr,tTotal renter housing cost in month tCurrency
Ib,tBuyer investment balance after month tCurrency
Ir,tRenter investment balance after month tCurrency
jEquivalent monthly investment returnDecimal
DtBuyer net worth minus renter net worthCurrency

Monthly Payment

Mortgage principal is the entered home price minus the down payment:

P=H−dP=H-d
i= rac{r}{1200}
M=Pi1−(1+i)−nM=P\frac{i}{1-(1+i)^{-n}}

The annual note rate is entered as a percentage, so 6.50% becomes a monthly rate of 6.5 / 1,200. When the rate is exactly 0%, the engine uses M = P / n. When the balance is smaller than a scheduled payment plus interest, the final payment is capped at the amount required to repay the balance.

Amortization

At each month, interest is calculated from the opening balance. The remainder of the scheduled payment reduces principal:

Jt=Bt−1iJ_t=B_{t-1}i
Qt=max⁡(0,Mt−Jt)Q_t=\max(0,M_t-J_t)
Bt=max⁡(0,Bt−1−Qt)B_t=\max(0,B_{t-1}-Q_t)

Jt is monthly mortgage interest and Qt is principal. The model preserves full calculation precision and rounds only displayed currency.

Annual rates converted to monthly rates

Home appreciation, rent growth, owner-cost growth and investment return use equivalent monthly rates rather than simply dividing by 12:

gm=(1+ga)1/12−1g_m=(1+g_a)^{1/12}-1

This makes 12 monthly growth steps compound back to the annual assumption. For example, home value advances as Vt = Vt-1(1 + ghome,m).

Invested cash-flow differences

Ib,t=Ib,t−1(1+j)+max⁡(0,Cr,t−Cb,t)I_{b,t}=I_{b,t-1}(1+j)+\max(0,C_{r,t}-C_{b,t})
Ir,t=Ir,t−1(1+j)+max⁡(0,Cb,t−Cr,t)I_{r,t}=I_{r,t-1}(1+j)+\max(0,C_{b,t}-C_{r,t})

The renter starts with the buyer's down payment and purchase closing costs:

Ir,0=d+HcI_{r,0}=d+Hc
Ib,0=0I_{b,0}=0

c is the entered buying-closing-cost rate. Existing investments grow first each month, then that month's positive cost difference is contributed.

Extra Payments

The engine does not include extra mortgage principal. The amortization path uses scheduled payments so the rent and buy comparison has one reproducible baseline.

If a buyer plans extra principal, first calculate the scheduled scenario. Then use the Extra Payment Calculator to obtain the revised balance and interest path. Any extra cash assigned to principal must also be removed from money otherwise available for investing; counting it in both places would overstate net worth.

Ownership Costs

Monthly buyer housing cost is:

Cb,t=Mt+Tt+Nt+Rt+At+PMItC_{b,t}=M_t+T_t+N_t+R_t+A_t+PMI_t

Property tax Tt, homeowners insurance Nt and maintenance Rt are annual entered percentages of the current modeled home value divided by 12. HOA dues At begin at the entered monthly amount and grow with the owner-cost assumption.

Tt=Vt−1ptax1200,Nt=Vt−1pins1200,Rt=Vt−1pmaint1200T_t=V_{t-1}\frac{p_{tax}}{1200},\quad N_t=V_{t-1}\frac{p_{ins}}{1200},\quad R_t=V_{t-1}\frac{p_{maint}}{1200}

Modeled PMI applies only while the opening mortgage balance divided by opening home value exceeds 80%:

PMIt={Bt−1pPMI1200,Bt−1/Vt−1>0.800,otherwisePMI_t=\begin{cases}B_{t-1}\frac{p_{PMI}}{1200},&B_{t-1}/V_{t-1}>0.80\\0,&\text{otherwise}\end{cases}

Monthly renter cost is current rent plus renter insurance:

Cr,t=Rentt+Insurancer,tC_{r,t}=Rent_t+Insurance_{r,t}

Sale-adjusted equity and break-even

Et=Vt(1−s)−BtE_t=V_t(1-s)-B_t
NWbuy,t=Et+Ib,tNW_{buy,t}=E_t+I_{b,t}
NWrent,t=Ir,tNW_{rent,t}=I_{r,t}
Dt=NWbuy,t−NWrent,tD_t=NW_{buy,t}-NW_{rent,t}

Break-even is the first annual checkpoint with Dt greater than or equal to 0. The calculator evaluates the selected endpoint as approximately even when the absolute difference is no more than 0.50% of the larger ending net-worth value, with a minimum tolerance of $100.

Worked Calculation

Use the default 10-year scenario: $500,000 home, $100,000 down, 6.50% rate, 30-year term, $2,500 rent, 3.00% purchase costs, 6.00% selling costs, 3.00% appreciation and rent growth, and 5.00% investment return. The remaining ownership inputs use the defaults shown in the calculator.

  1. Mortgage principal: $500,000 - $100,000 = $400,000.
  2. Monthly principal and interest: $2,528.27.
  3. First-month property tax, insurance and maintenance: $500.00 + $145.83 + $416.67.
  4. Initial buying cost: $3,590.77; initial renting cost: $2,520.00.
  5. Renter starting investment: $100,000 + $15,000 = $115,000.00.
  6. Modeled home value after 10 years: $671,958.19.
  7. Remaining mortgage balance: $339,104.51.
  8. Sale-adjusted home equity: $671,958.19 x 94% - $339,104.51 = $292,536.19.
  9. Renter investment balance: $319,265.66.
  10. Net-worth difference: $292,536.19 - $319,265.66 = -$26,729.47, so renting leads at year 10.

The tested engine calculates every month before reporting annual checkpoints. See the three rent vs buy examples for the complete baseline, short-stay and long-stay comparisons.

Assumptions

  • The buyer sells at the end of the selected period and pays the entered selling-cost percentage.
  • The mortgage uses a fixed note rate and scheduled monthly amortization.
  • Home value, rent, HOA dues, renter insurance and investments change at constant entered rates.
  • Positive monthly savings are invested at month-end after existing investments grow.
  • Property tax, homeowners insurance and maintenance remain constant percentages of modeled home value.
  • PMI follows the simplified entered-rate and 80% loan-to-value rule in the engine, not a specific policy.
  • Tax deductions, capital-gains taxes, utilities, deposits, moving, renovations, refinancing, investment taxes and fees, and unentered costs are excluded.
  • The result measures modeled financial net worth only; it does not assign a value to stability, flexibility, location or repair responsibility.

Formula FAQ

Why is mortgage principal not treated as a cost?

Principal reduces the loan balance and therefore increases home equity. The monthly cash leaves the buyer's budget, but it is not lost in the same way as interest, tax or maintenance. Review the Rent vs Buy Calculator formula assumptions before relying on this result for decisions.

Why does the renter invest the down payment?

The comparison assumes the renter retains cash the buyer uses upfront. Investing it exposes the opportunity cost of moving that cash into home equity. Use realistic inputs and review the Rent vs Buy Calculator formula assumptions before relying on this result for decisions.

Can break-even disappear after it is reached?

The displayed break-even is the first annual checkpoint when buying reaches renting. Later values still depend on all growth and cost assumptions, so users should review the entire annual series rather than assuming the lead is permanent. Review the Rent vs Buy Calculator formula assumptions before relying on this result for decisions.

Why does the model deduct selling costs?

The buyer's home value is not immediately spendable net worth. The model assumes a sale at the selected endpoint and subtracts the entered percentage before removing the mortgage balance. Review the Rent vs Buy Calculator formula assumptions before relying on this result for decisions.

Does the formula include a mortgage-interest tax deduction?

No. Eligibility and value depend on current tax law and the household's facts, so the model does not assume a tax benefit. Use realistic inputs and review the Rent vs Buy Calculator formula assumptions before relying on this result for decisions.

References

Use the Rent vs Buy Calculator