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Renting and Homeownership

Evaluate advantages and disadvantages of renting.

Learning Objectives

After completing this section, you should be able to:

  1. Evaluate advantages and disadvantages of renting.
  2. Evaluate advantages and disadvantages of homeownership.
  3. Calculate the monthly payment for a mortgage and related interest cost.
  4. Read and interpret an amortization schedule.
  5. Solve application problems involving affordability of a mortgage.

Advantages and Disadvantages of Renting

When renting, you will likely sign a lease, which is a contract between a renter and a landlord. A landlord is the person or company that owns property that is rented. The lease will detail your responsibilities, restrictions on activities, deposits, fees, maintenance, repairs, and rent during the term of the lease. It also defines what your landlord can, and cannot, do with the property while you occupy the property.

Like leasing a car, there are advantages to renting but also some disadvantages. Some advantages are:

  • Lower cost.
  • Short-term commitment.
  • Little to no maintenance cost. The landlord pays for or performs most maintenance.
  • You need not stay at end of lease. Once the lease term is over (the lease is up), you are not obligated to stay.
  • If renting in an apartment complex, there may be a pool, gym, or community room for renters to use.

Of course, there are disadvantage too:

  • No tax incentives.
  • Housing cost is not fixed. When the lease is up, the rent can change.
  • No equity. When you are done living in a rental, you have built no value.
  • Restrictions on occupants. There may be a limit on how many can live in the apartment.
  • Restrictions on decorating. The property is not yours, so any decorating or improvements need landlord permission.
  • Limits on pets. Permission for pets, and their number and type, will be set forth in the lease.
  • May not be able to remain when lease term is over. The landlord can, at the end of your lease, invite you to leave.
  • The building may be sold, and the new landlord may institute changes to the lease when the previous lease expires.

Renting has fees to be paid at the start of the lease. Typically, when you rent, you will pay first and last months’ rent and a security deposit. A security deposit is a sum of money that the landlord holds until the renter leaves the rental property. The deposit will cover repairs for damage to the apartment during the renter’s stay but may be returned if the apartment is in good condition. If your landlord runs a credit check on you, the landlord may charge you for that.

Advantages of Buying a Home

The advantages to buying a home mirror the disadvantages of renting, and the disadvantages of home ownership mirror the advantages of renting.

Some advantages to buying a home are:

  • There are tax incentives. The interest you pay for your mortgage (more on that later) is deductible on your federal income tax.
  • There are no restrictions on pets or occupants, unless laws in your area specify limits for homes.
  • You can redecorate any way you wish, limited only by the laws in your area.
  • Once your mortgage is set with a fixed-interest rate, your housing cost is fixed.
  • Your home grows equity, that is, the difference between what you owe and what the house is worth grows. You can use the equity to secure loans, and you recover the equity (and more if you’re fortunate) when you sell the house.
  • As long as you pay your mortgage and maintain the home to the standards of your community, you can stay as long as you wish.

Some disadvantages to home ownership are:

  • The cost is higher than renting. Mortgages and associated costs are typically higher than rent for a similar living space.
  • The owner is responsible for upkeep, maintenance, and repairs. These can be extremely costly.
  • The owner cannot walk away from the property. It can be sold, but simply leaving the property, especially if not paid off yet, has serious consequences.

The big question of affordability looms large over the decision to rent or buy. Renting, strictly from an affordability viewpoint, comes with much less initial outlay and smaller commitment. If you do not have sufficient income to regularly save for possibly expensive repairs, or your credit isn’t quite as good as it needs to be, then renting may be the best choice. Of course, even if you can afford to buy a home, you may choose to rent based on the comparative advantages.

Buying a home really involves two buyers. You and the mortgage company. The mortgage company has interest in the home, as they are providing the funds for the home. They want to protect their investment, and many fees are about the bank as much as the buyer. They fund a mortgage based on the value they assign the property. Not you. This means they will want some certainty that the home is sound, and you are a good investment.

In the end, you must weigh your options and carefully consider your priorities in choosing to rent or buy a home.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Mortgages

Some people will purchase a home or condo with cash, but the majority of people will apply for a mortgage. A mortgage is a long-term loan and the property itself is the security. The bank decides the minimum down payment (with your input), the payment schedule, the duration of the loan, whether the loan can be assumed by another party, and the penalty for late payments. The title of the home belongs to the bank.

Since a mortgage is a loan, everything about loans from The Basics of Loans holds true, including the formula for the payments.

Monthly Mortgage Payments

The formula to calculate your monthly payments of principal and interest uses APR as the annual interest rate.

To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of payments.

30-Year Mortgage at 4.8% Interest

Try it.

Evan buys a house. His 30-year mortgage comes to $132,650 with 4.8% interest. Find Evan’s monthly payments.

Solution

Using the information above, \(P\) = $132,650, \(r\) = 0.048 and \(t\) = 30. Substituting those values into the formula \(pmt=\frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1}\) and calculating, we find the payment is \[\begin{array}{lll}pmt & = & \frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1} \\ & = & \frac{\text{\$}132,650\times (0.048/12)\times {(1+0.048/12)}^{12\times 30}}{{(1+0.048/12)}^{12\times 30}-1} \\ & = & \frac{\text{\$}132,650\times (0.004)\times {(1.004)}^{360}}{{(1.004)}^{360}-1} \\ & = & \frac{\text{\$}2,233.07781448}{3.20858992551} \\ & = & \text{\$}695.97\end{array}\]

His mortgage payment is $695.97.

To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of payments. This can be useful information, but not too many people reach the end of their mortgage. They tend to move before the mortgage is paid off.

With the principal of the mortgage and how much total is paid over the life of the mortgage, the cost of financing can be found by subtracting the principal of the mortgage from the total paid over the life of the mortgage.

30-Year Mortgage at 5.35% Interest

Try it.

Cassandra buys a house. Her 30-year mortgage comes to $99,596 with 5.35% interest. What was Cassandra’s cost of financing?

Solution

In , we found that the total Cassandra will pay for the $99,569 mortgage is $200,217.60. Subtracting those we find the cost of financing \(\text{CoF}=T-P=\text{\$}200,217.60-\text{\$}99,596=\text{\$}100,621.60\).

Condensed — the full section is in OpenStax Contemporary Mathematics.

Reading and Interpreting Amortization Tables

Amortization tables were addressed in The Basics of Loans. They are most frequently encountered when analyzing mortgages.

The amortization table for a 30-year mortgage is quite long, containing 360 rows. A full table will not be reproduced here. We can, though, read information from a portion of an amortization table.

Amortization Table for a 30-Year, $165,900 Mortgage

Try it.

shows a portion of an amortization table for a 30-year, $165,900 mortgage. Use that table to answer the following questions.

  1. What is the interest rate?
  2. How much are the payments?
  3. How much of payment 175 goes to principal?
  4. How much of payment 180 goes to interest?
  5. What’s the remaining balance on the mortgage after payment 170?
Solution
  1. Reading at the top of the table, we see the interest rate is 5.61%.
  2. Reading from the top of the table or from the column labeled Payment, we see the payments are $953.44 per month.
  3. In the row for payment 175, we see that the amount that goes to principal is $400.44.
  4. In the row for payment 180, we see that the amount that goes to interest is $543.56.
  5. In the row for payment 170, we see the remaining balance is $119,873.35.

Escrow Payments

The last few examples have looked at mortgage payments, which cover the principal and interest. However, when you take out a mortgage, the payment is sometimes much higher than that. This is because your mortgage company also has you pay into an escrow account, which is a savings account maintained by the mortgage company.

Your insurance payments will be set by your insurer and the mortgage company will pay them on time for you from your escrow account. Your property taxes are set by where you live and are typically a percentage of your property’s assessed value. The assessed value is the estimation of the value of your home and does not necessary reflect the purchase or resale value of the home. Your property taxes will also be paid on time by the mortgage company from your escrow account.

For example, in Kalamazoo, Michigan, the effective tax rate for property is 1.69% of the assessed value of the home. These escrow payments, which cover bills for the home, can increase the monthly payments for your home well beyond the basic principal and interest payment.

Adding Escrow Payments to Mortgage Payments

Try it.

Jenna decides to purchase a home, with mortgage of $108,450 at 6% interest for 30 years. The assessed value of her home is $75,600. Her property taxes come to 5.7% of her assessed value. Jenna also has to pay her home insurance every 6 months, which is $744 per six months. How much, including escrow, will Jenna pay per month?

Solution

Using the payment function to find her mortgage payments, \(pmt=\frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1}\), with \(P\) = $108,405, \(r\) = 0.06, and \(t\) = 30, her payments are

\[\begin{array}{lll}pmt & = & \frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1} \\ & = & \frac{\text{\$}108,405\times (0.06/12)\times {(1+0.06/12)}^{12\times 30}}{{(1+0.06/12)}^{12\times 30}-1} \\ & = & \text{\$}649.95\end{array}\]

Jenna also pays into escrow 1/12 of her property taxes per month. Her property taxes are 5.7% of the assessed value of $75,600, which comes to \(0.057\times \text{\$}75,600=\text{\$}4309.20\). This is an annual tax, so she pays 1/12 of that each month, or $359.10. Jenna’s home insurance is $744 per 6 months, so each month she pays $124.00 for insurance. Adding these together, her monthly payment is \(\text{\$}649.95+\text{\$}359.10+\text{\$}124.00=\text{\$}1,133.05\). This is quite a bit more than the $649.95 for the principal and interest.

Key Concepts

  • There are many points of comparison between renting and buying a house.
  • Before deciding to buy a house, you should carefully consider all the responsibilities that come with home ownership.
  • Renting comes with more restrictions on the renter, but with fewer costs and is easier to move from.
  • Owning a house has more costs but has more freedom, plus the owner creates equity.
  • Mortgages are loans, and payments are calculated in the same way as any other loan.
  • Amortization tables help a homeowner understand the mortgage and how the payments are applied to the principal and interest.
  • In addition to paying the amount financed for a mortgage, the monthly payment will include an escrow payment, which covers insurance and taxes.

Practice (5)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Evan buys a house. His 30-year mortgage comes to $132,650 with 4.8% interest. Find Evan’s monthly payments.

    បង្ហាញ​ចម្លើយ

    Using the information above, \(P\) = $132,650, \(r\) = 0.048 and \(t\) = 30. Substituting those values into the formula \(pmt=\frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1}\) and calculating, we find the payment is \[\begin{array}{lll}pmt & = & \frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1} \\ & = & \frac{\text{\$}132,650\times (0.048/12)\times {(1+0.048/12)}^{12\times 30}}{{(1+0.048/12)}^{12\times 30}-1} \\ & = & \frac{\text{\$}132,650\times (0.004)\times {(1.004)}^{360}}{{(1.004)}^{360}-1} \\ & = & \frac{\text{\$}2,233.07781448}{3.20858992551} \\ & = & \text{\$}695.97\end{array}\]

    His mortgage payment is $695.97.

  2. Cassandra buys a house. Her 30-year mortgage comes to $99,596 with 5.35% interest. If Cassandra pays off the mortgage over those 30 years, how much will she have paid in total?

    បង្ហាញ​ចម្លើយ

    To find the total paid over the life of the mortgage, use the formula \(T=pmt\times 12\times t\). To calculate this, the payment must be found. Using the information above, \(P\) = $99,596, \(r\) = 0.0535 and \(t\) = 30. Substituting those values into the formula \(pmt=\frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1}\) and calculating, we find the payment is

    \[\begin{array}{lll}pmt & = & \frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1} \\ & = & \frac{\text{\$}99,596\times (0.0535/12)\times {(1+0.0535/12)}^{12\times 30}}{{(1+0.048/12)}^{12\times 30}-1} \\ & = & \frac{\text{\$}99.596\times (0.004458\overset{\bar}{3})\times {(1.004458\overset{\bar}{3})}^{360}}{{(1.004458\overset{\bar}{3})}^{360}-1} \\ & = & \frac{\text{\$}2,202.45911222}{3.96013414693} \\ & = & \text{\$}556.16\end{array}\]

    Using the mortgage payment of $556.16 and \(t\) = 30 years in the formula \(T=pmt\times 12\times t\), the total that Cassandra will pay for the mortgage is $200,217.60.

  3. Cassandra buys a house. Her 30-year mortgage comes to $99,596 with 5.35% interest. What was Cassandra’s cost of financing?

    បង្ហាញ​ចម្លើយ

    In , we found that the total Cassandra will pay for the $99,569 mortgage is $200,217.60. Subtracting those we find the cost of financing \(\text{CoF}=T-P=\text{\$}200,217.60-\text{\$}99,596=\text{\$}100,621.60\).

  4. shows a portion of an amortization table for a 30-year, $165,900 mortgage. Use that table to answer the following questions.

    1. What is the interest rate?
    2. How much are the payments?
    3. How much of payment 175 goes to principal?
    4. How much of payment 180 goes to interest?
    5. What’s the remaining balance on the mortgage after payment 170?
    បង្ហាញ​ចម្លើយ
    1. Reading at the top of the table, we see the interest rate is 5.61%.
    2. Reading from the top of the table or from the column labeled Payment, we see the payments are $953.44 per month.
    3. In the row for payment 175, we see that the amount that goes to principal is $400.44.
    4. In the row for payment 180, we see that the amount that goes to interest is $543.56.
    5. In the row for payment 170, we see the remaining balance is $119,873.35.

  5. Jenna decides to purchase a home, with mortgage of $108,450 at 6% interest for 30 years. The assessed value of her home is $75,600. Her property taxes come to 5.7% of her assessed value. Jenna also has to pay her home insurance every 6 months, which is $744 per six months. How much, including escrow, will Jenna pay per month?

    បង្ហាញ​ចម្លើយ

    Using the payment function to find her mortgage payments, \(pmt=\frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1}\), with \(P\) = $108,405, \(r\) = 0.06, and \(t\) = 30, her payments are

    \[\begin{array}{lll}pmt & = & \frac{P\times (r/12)\times {(1+r/12)}^{12\times t}}{{(1+r/12)}^{12\times t}-1} \\ & = & \frac{\text{\$}108,405\times (0.06/12)\times {(1+0.06/12)}^{12\times 30}}{{(1+0.06/12)}^{12\times 30}-1} \\ & = & \text{\$}649.95\end{array}\]

    Jenna also pays into escrow 1/12 of her property taxes per month. Her property taxes are 5.7% of the assessed value of $75,600, which comes to \(0.057\times \text{\$}75,600=\text{\$}4309.20\). This is an annual tax, so she pays 1/12 of that each month, or $359.10. Jenna’s home insurance is $744 per 6 months, so each month she pays $124.00 for insurance. Adding these together, her monthly payment is \(\text{\$}649.95+\text{\$}359.10+\text{\$}124.00=\text{\$}1,133.05\). This is quite a bit more than the $649.95 for the principal and interest.

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\approx
approximately equal
Equal to the precision shown, not exactly.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
a \bmod n
remainder
What is left after dividing a by n.
\%
per cent
Per hundred: 15% = 15/100.
a : b,\ \frac{a}{b}
ratio, fraction
a for every b; a divided by b.

How to: Renting and Homeownership

  1. Evaluate advantages and disadvantages of renting.
  2. Evaluate advantages and disadvantages of homeownership.
  3. Calculate the monthly payment for a mortgage and related interest cost.
  4. Read and interpret an amortization schedule.
  5. Solve application problems involving affordability of a mortgage.
  6. Lower cost.
  7. Short-term commitment.
  8. Little to no maintenance cost. The landlord pays for or performs most maintenance.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

ព្យាយាម​របស់​អ្នក​ផ្ទាល់

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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