6.2 Loans, Mortgages, Credit Cards & Amortization Concepts

6.2 Loans, Mortgages, Credit Cards & Amortization Concepts

Installment loans, residential mortgages, and revolving credit cards are direct practical applications of Present Value of an Ordinary Annuity mathematics. Understanding how financial institutions compute monthly payments, apportion payments between interest and principal, apply revolving finance charges, and price mortgage discount points is vital for scoring well on the CLEP College Mathematics examination.

---

1. Installment Loans & Fixed-Rate Mortgage Payments

An installment loan is a contract where a borrower receives a lump-sum principal ($P$) today and repays it through a sequence of equal monthly payments ($PMT$) over a predetermined term of $t$ years ($N = 12t$ payments).

Because the lender provides principal $P$ today in exchange for a stream of $N$ future payments, the loan principal $P$ is identically equal to the Present Value of an Ordinary Annuity ($PVOA$):

$P = PMT \cdot \left[ \frac{1 - (1 + i)^{-N}}{i} \right]$

Solving algebraically for the monthly payment $PMT$ yields the Standard Loan Amortization Payment Formula:

$PMT = P \cdot \left[ \frac{i}{1 - (1 + i)^{-N}} \right]$

Where:

• $PMT$ = Fixed monthly payment
• $P$ = Loan principal (initial amount borrowed)
• $i = \frac{r}{12}$ = Monthly periodic interest rate (annual nominal rate $r$ divided by 12)
• $N = 12t$ = Total number of monthly payments across $t$ years

Total Amount Repaid & Total Lifetime Interest

• Total Amount Repaid: $\text{Total Repaid} = PMT \times N$
• Total Interest Paid: $\text{Total Interest} = (PMT \times N) - P$

Worked Example 1: Calculating Mortgage Payment and Total Interest

Problem: A family purchases a home with a $$300{,}000$ fixed-rate mortgage for $30$ years at an annual interest rate of $6%$ compounded monthly. Calculate:

1. The monthly mortgage payment ($PMT$).
2. The total amount repaid over the 30-year loan term.
3. The total interest paid to the lender.

Solution:

1. Parameters: $P = 300{,}000$, $r = 0.06$, $i = \frac{0.06}{12} = 0.005$, $N = 30 \times 12 = 360$.
2. Compute discount factor: $(1 + 0.005)^{-360} = (1.005)^{-360} \approx 0.166041$.
3. Compute denominator: $1 - 0.166041 = 0.833959$.
4. Compute monthly payment: $PMT = 300{,}000 \cdot \left[ \frac{0.005}{0.833959} \right] = 300{,}000 \times 0.00599551 \approx \$1{,}798.65$
5. Total Amount Repaid: $\text{Total Repaid} = 1{,}798.65 \times 360 = \$647{,}514.00$
6. Total Interest Paid: $\text{Total Interest} = 647{,}514.00 - 300{,}000.00 = \$347{,}514.00$

Critical Observation: Over 30 years, the total interest paid ($$347{,}514.00$) exceeds the original borrowed principal ($$300{,}000.00$).

---

2. Loan Amortization Mechanics & Schedule Construction

Amortization is the process of gradually extinguishing debt through scheduled periodic payments. Every fixed monthly payment ($PMT$) is partitioned into two distinct components:

$PMT = I_k + PR_k$

Where:

• $I_k$ = Interest portion for payment period $k$
• $PR_k$ = Principal reduction portion for payment period $k$

+-----------------------------------------------------------------------------+
|                   MONTHLY AMORTIZATION STEP-BY-STEP CYCLE                   |
|                                                                             |
|   1. INTEREST DUE:           I_k = Balance_(k-1) * i                        |
|      (Interest is charged only on the remaining unpaid loan balance)        |
|                                                                             |
|   2. PRINCIPAL REDUCTION:    PR_k = PMT - I_k                               |
|      (The remainder of the monthly payment reduces the loan balance)        |
|                                                                             |
|   3. NEW ENDING BALANCE:     Balance_k = Balance_(k-1) - PR_k               |
+-----------------------------------------------------------------------------+

The Amortization Inversion Dynamic

Because the loan balance declines with each payment, the monthly interest charge ($I_k$) steadily decreases over time. Consequently, the principal reduction ($PR_k$) steadily increases over time.

+-----------------------------------------------------------------------------+
|                  30-YEAR MORTGAGE PAYMENT COMPOSITION PROFILE               |
|                                                                             |
|   $1,799 | [ INTEREST PORTION (Declines) ]                                  |
|          | \                                                                |
|          |  \                                                               |
|          |   \                                 / [ PRINCIPAL (Grows) ]      |
|          |    \                               /                             |
|          |     \                             /                              |
|          |      ----------------------------                                |
|       $0 +------------------------------------------------------------------|
|         Month 1                 Month 180 (Year 15)              Month 360  |
+-----------------------------------------------------------------------------+

Sample Amortization Schedule Table (First 3 Months)

For a $$300{,}000$ loan at $6%$ ($i = 0.005$, $PMT = $1{,}798.65$):

• Month 1 Calculation:
  • Interest: $300{,}000.00 \times 0.005 = $1{,}500.00$
  • Principal: $1{,}798.65 - 1{,}500.00 = $298.65$
  • Ending Balance: $300{,}000.00 - 298.65 = $299{,}701.35$
• Month 2 Calculation:
  • Interest: $299{,}701.35 \times 0.005 = $1{,}498.51$
  • Principal: $1{,}798.65 - 1{,}498.51 = $300.14$
  • Ending Balance: $299{,}701.35 - 300.14 = $299{,}401.21$

---

3. Credit Card Finance Charges: Average Daily Balance Method

Unlike fixed installment loans, credit cards represent revolving credit. If the cardholder carries an unpaid balance from month to month (does not pay the statement in full during the grace period), financial institutions compute interest charges using the Average Daily Balance (ADB) method.

The Average Daily Balance (ADB) Protocol

$\text{Average Daily Balance (ADB)} = \frac{\sum (\text{Daily Closing Balance} \times \text{Number of Days})}{\text{Total Days in Billing Cycle}}$

$\text{Finance Charge} = \text{ADB} \times \left( \frac{\text{APR}}{365} \right) \times \text{Days in Cycle} = \text{ADB} \times \left( \frac{\text{APR}}{12} \right)$

Worked Example 2: Average Daily Balance & Finance Charge

Problem: A credit card billing cycle contains $30$ days with an APR of $24%$ ($i = \frac{0.24}{12} = 0.02$ monthly). The account activity is as follows:

• Days 1–10 (10 days): Unpaid starting balance is $$800$.
• Day 11: A purchase of $$600$ is made, increasing the balance to $$1{,}400$ for Days 11–20 (10 days).
• Day 21: A payment of $$300$ is credited, reducing the balance to $$1{,}100$ for Days 21–30 (10 days).

Calculate the Average Daily Balance and the monthly finance charge.

Solution:

1. Calculate total dollar-day sum:
   • Days 1–10: $$800 \times 10 = $8{,}000$
   • Days 11–20: $$1{,}400 \times 10 = $14{,}000$
   • Days 21–30: $$1{,}100 \times 10 = $11{,}000$
   • Sum of daily balances $= 8{,}000 + 14{,}000 + 11{,}000 = $33{,}000$
2. Calculate Average Daily Balance: $\text{ADB} = \frac{\$33{,}000}{30} = \$1{,}100.00$
3. Compute Finance Charge using monthly rate $i = 0.02$: $\text{Finance Charge} = 1{,}100.00 \times 0.02 = \$22.00$

Minimum Payment Dynamics

Credit card issuers typically set minimum monthly payments at $2%$ to $3%$ of the outstanding balance (or $$25$, whichever is greater). When a consumer pays only the minimum:

• The vast majority of the payment covers the monthly finance charge.
• Only a negligible fraction reduces the principal.
• Paying off a $$5{,}000$ balance at $20%$ APR via minimum payments can take over 20 years and cost more than $$7{,}000$ in cumulative interest.

---

4. Consumer Financial Decision-Making & Comparisons

15-Year vs. 30-Year Fixed-Rate Mortgages

When choosing a mortgage term, borrowers evaluate the trade-off between monthly cash flow affordability and lifetime interest savings.

Comparative Numerical Case: Borrowing $$250{,}000$:

• 30-Year at 6.0%: $PMT = $1{,}498.88$ | Total Repaid: $$539{,}595$ | Total Interest: $$289{,}595$
• 15-Year at 5.4%: $PMT = $2{,}029.47$ | Total Repaid: $$365{,}304$ | Total Interest: $$115{,}304$

Result: The 15-year loan requires $$530.59$ more per month, but saves $$174{,}292$ in total interest.

Discount Points and Origination Fees

• Discount Point: An upfront fee paid to the lender at closing to permanently reduce the mortgage interest rate. $\text{Cost of 1 Point} = 1\% \text{ of Total Loan Amount} (P)$
  • Typically, buying 1 point ($1%$ of principal) reduces the annual interest rate by $0.25%$ ($25$ basis points).
• Break-Even Analysis: To determine whether paying for discount points is mathematically beneficial, compute the number of months required for cumulative monthly payment savings to recoup the upfront cost: $\text{Break-Even Duration (Months)} = \frac{\text{Total Upfront Cost of Points}}{\text{Monthly Payment Savings}}$

Worked Point Example: On a $$200{,}000$ mortgage, purchasing $2$ discount points costs $2% \times 200{,}000 = $4{,}000$ upfront. If this reduces the monthly payment from $$1{,}264.14$ to $$1{,}231.43$ (saving $$32.71$/month): $\text{Break-Even} = \frac{\$4{,}000}{\$32.71/\text{month}} \approx 122.3 \text{ months} \approx 10.2 \text{ years}$ Decision: If the buyer plans to stay in the home for longer than $10.2$ years, purchasing the points saves money; if relocating or refinancing earlier, it results in a net financial loss.

Lease vs. Purchase Decisions

---

5. Common CLEP Traps & Strategic Checkpoints

• Trap 1: Confusing Simple Interest with Amortized Installment Loans: Never compute total mortgage interest as $I = P \cdot r \cdot t$. In an amortized loan, interest is assessed monthly on the declining balance, not on the original principal.
• Trap 2: Forgetting to Convert Annual Rates to Monthly Rates: In the payment formula $PMT = P\left[\frac{i}{1-(1+i)^{-N}}\right]$, always convert annual rate $r$ to $i = r/12$ and term in years $t$ to $N = 12t$.
• Trap 3: Average Daily Balance Calendar Miscounting: When calculating credit card daily balances, ensure the sum of days in all intervals exactly equals the total days in the billing cycle (e.g., $10 + 10 + 10 = 30$ days).
• Trap 4: Discount Point Basis: Remember that discount points are a percentage of the loan amount borrowed, NOT the purchase price of the home.