Buy vs. Rent Net Worth Comparison Over 30 Years
This simple chart illustrates the difference in the net worth of an individual who buys a home versus one who rents over a 30-year period. This simple examination assumes that the renter saves the additional savings not spent on the mortgage in a high-yield savings account. Adjust the parameters to see how different scenarios affect the outcomes.
Buy vs. Rent Net Worth Comparison Over 30 Years
Homeowner Variables
Renter Variables
Mathematical Formulas Used in the Buy vs. Rent Comparison
1. Homeowner Calculations
1.1 Monthly Mortgage Payment
The monthly mortgage payment \( M \) is calculated using the standard mortgage payment formula:
$$ M = P \times \left( \frac{ r(1 + r)^n }{ (1 + r)^n – 1 } \right) $$Where:
- \( P \): Principal loan amount (home purchase price minus down payment).
- \( r \): Monthly mortgage interest rate (annual mortgage rate divided by 12, expressed as a decimal).
- \( n \): Total number of payments (loan term in years multiplied by 12).
Explanation:
This formula calculates the fixed monthly payment required to fully amortize the loan over its term at a constant interest rate.
1.2 Mortgage Amortization: Principal and Interest Breakdown
For each month \( t \), the interest payment \( I_t \) and principal payment \( PR_t \) are calculated as:
Interest Payment:
$$ I_t = B_{t-1} \times r $$Principal Payment:
$$ PR_t = M – I_t $$Updated Mortgage Balance:
$$ B_t = B_{t-1} – PR_t $$Where:
- \( B_{t-1} \): Remaining mortgage balance from the previous month.
Explanation:
Each monthly payment is split between interest and principal. The interest portion is based on the outstanding balance, while the principal portion reduces the mortgage balance.
1.3 Home Value Appreciation
The home value appreciates annually based on the home appreciation rate. At the end of each year \( y \), the home value \( V_y \) is updated:
$$ V_y = V_{y-1} \times (1 + a) $$Where:
- \( V_{y-1} \): Home value at the end of the previous year.
- \( a \): Annual home appreciation rate (expressed as a decimal).
Explanation:
This formula calculates the increase in home value due to appreciation, compounding annually.
1.4 Home Equity Calculation
The homeowner’s equity \( E_t \) at month \( t \) is calculated as:
$$ E_t = V_t – B_t $$Where:
- \( V_t \): Current home value at month \( t \).
- \( B_t \): Remaining mortgage balance at month \( t \).
Explanation:
Equity represents the portion of the home value that the homeowner owns outright.
2. Renter Calculations
2.1 Monthly Savings Available for Investment
The renter’s monthly savings \( S_t \) available for investment at month \( t \) is:
$$ S_t = \left( M + E \right) – R_t $$Where:
- \( M \): Homeowner’s monthly mortgage payment.
- \( E \): Estimated additional monthly expenses for the homeowner.
- \( R_t \): Renter’s monthly rent at month \( t \).
Note: If \( S_t \) is negative, it is set to zero, since the renter cannot invest negative savings.
2.2 Rent Growth
The renter’s monthly rent increases annually based on the rent growth rate. At the beginning of each year \( y \), the rent is updated:
$$ R_y = R_{y-1} \times (1 + g) $$Where:
- \( R_{y-1} \): Rent at the end of the previous year.
- \( g \): Annual rent growth rate (expressed as a decimal).
Explanation:
This formula accounts for annual increases in rent due to inflation or market conditions.
2.3 Investment Growth
The renter invests their initial deposit (equivalent to the homeowner’s down payment) and monthly savings. The investment balance \( F_t \) grows over time based on the investment return rate.
Initial Investment Balance:
$$ F_0 = D $$Monthly Investment Balance Update:
$$ F_t = F_{t-1} \times (1 + i) + S_t $$Where:
- \( D \): Initial deposit (down payment amount).
- \( F_{t-1} \): Investment balance from the previous month.
- \( i \): Monthly investment return rate (annual rate divided by 12).
Explanation:
This formula calculates the future value of the renter’s investments, considering both the initial deposit and ongoing monthly contributions, compounded monthly.
3. Net Worth Calculations
3.1 Homeowner Net Worth
The homeowner’s net worth \( NW_{\text{homeowner}, t} \) at month \( t \) is:
$$ NW_{\text{homeowner}, t} = E_t $$Explanation:
The homeowner’s net worth is the equity in their home.
3.2 Renter Net Worth
The renter’s net worth \( NW_{\text{renter}, t} \) at month \( t \) is:
$$ NW_{\text{renter}, t} = F_t $$Explanation:
The renter’s net worth is the balance of their investment account.
4. Summary of Variables
- \( P \): Principal loan amount. $$ P = \text{Home Purchase Price} – D $$
- \( D \): Down payment amount. $$ D = \text{Home Purchase Price} \times \left( \frac{\text{Down Payment (\%)}}{100} \right) $$
- \( M \): Monthly mortgage payment.
- \( r \): Monthly mortgage interest rate. $$ r = \frac{\text{Annual Mortgage Rate (\%)}}{12 \times 100} $$
- \( n \): Total number of payments. $$ n = \text{Loan Term (years)} \times 12 $$
- \( B_t \): Mortgage balance at month \( t \).
- \( V_t \): Home value at month \( t \).
- \( a \): Annual home appreciation rate. $$ a = \frac{\text{Home Appreciation Rate (\%)}}{100} $$
- \( E_t \): Home equity at month \( t \).
- \( R_t \): Renter’s monthly rent at month \( t \).
- \( g \): Annual rent growth rate. $$ g = \frac{\text{Rent Growth Rate (\%)}}{100} $$
- \( S_t \): Renter’s monthly savings available for investment at month \( t \).
- \( F_t \): Renter’s investment balance at month \( t \).
- \( i \): Monthly investment return rate. $$ i = \frac{\text{Annual Investment Return Rate (\%)}}{12 \times 100} $$
- \( E \): Estimated additional monthly expenses for the homeowner.
5. Important Notes
- Compounding Periods: Interest rates for both the mortgage and investments are compounded monthly.
- Home Value Updates: The home value is updated annually.
- Rent Updates: Rent is increased annually based on the rent growth rate.
- Savings Investment: The renter invests any positive difference between the homeowner’s total monthly expenses and their rent.
- Expenses Excluded from Net Worth: Regular expenses such as mortgage interest, property taxes, maintenance costs, and rent payments are not included in the net worth calculations since they are expenditures, not assets.
6. Conclusion
These mathematical formulas provide a comprehensive framework for comparing the net worth of a homeowner versus a renter over time. By understanding and applying these calculations, individuals can make informed decisions about whether buying or renting is more advantageous for their financial situation.