Loans and Mortgages (Amortization) — Hard Practice Quiz

A Financial Math cheat sheet for Loans and Mortgages (Amortization) — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

The level periodic payment that fully repays a loan or mortgage of principal \(P\). \(i = r/n\) is the periodic rate and \(N = nt\) the total number of payments.

$$PMT = \dfrac{P \times i}{1 - (1 + i)^{-N}} \quad\text{or}\quad PMT = \dfrac{P \times \frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}$$

The loan amount that corresponds to a known periodic payment \(PMT\), where \(i = r/n\) is the periodic rate and \(N = nt\) the total number of payments.

$$P = PMT \times \dfrac{1 - (1 + i)^{-N}}{i} \quad\text{or}\quad P = PMT \times \dfrac{1 - \left(1 + \frac{r}{n}\right)^{-nt}}{\frac{r}{n}}$$

The outstanding balance after \(k\) payments have been made, with \(N - k\) payments still remaining. \(i = r/n\) is the periodic rate and \(N = nt\) the total number of payments.

$$B_{k} = PMT \times \dfrac{1 - (1 + i)^{-(N - k)}}{i} \quad\text{or}\quad B_{k} = PMT \times \dfrac{1 - \left(1 + \frac{r}{n}\right)^{-(nt - k)}}{\frac{r}{n}}$$

Practice quiz

  1. A borrower takes out a $30$-year mortgage for $P = \$300,000$ at an annual interest rate of $r = 6\%$ compounded monthly. After $5$ years, they decide to refinance the remaining balance at a new annual interest rate of $r_{new} = 4\%$ compounded monthly for the remaining term of the original loan. What is the new monthly payment?

    • Approximately $\$1474.30$
    • Approximately $\$1525.80$
    • Approximately $\$1610.50$
    • Approximately $\$1798.65$

    Answer: Approximately $\$1474.30$

  2. Consider two identical loans, Loan A and Loan B, both for $P = \$200,000$ over $N = 240$ months. Loan A has an annual interest rate of $r_A = 5\%$ compounded monthly. Loan B has an annual interest rate of $r_B = 5.5\%$ compounded monthly. What is the approximate difference in the total interest paid over the life of the loans?

    • Approximately $\$13,220$
    • Approximately $\$10,500$
    • Approximately $\$15,800$
    • Approximately $\$18,100$

    Answer: Approximately $\$13,220$

  3. A loan of $P = \$150,000$ is taken out at an annual interest rate of $r = 4.5\%$ compounded monthly for $N = 180$ months. If the borrower decides to pay off the entire remaining balance after $k = 60$ payments, how much less total money (principal + interest) will they pay compared to making all $180$ scheduled payments?

    • Approximately $\$26,750$
    • Approximately $\$22,100$
    • Approximately $\$30,500$
    • Approximately $\$18,900$

    Answer: Approximately $\$26,750$

  4. A person wants to borrow $P = \$250,000$. Option 1 is a $15$-year loan at $r = 4\%$ annual interest, compounded monthly. Option 2 is a $30$-year loan at $r = 4.5\%$ annual interest, compounded monthly. What is the difference in the monthly payment between Option 1 and Option 2, and which option results in a lower total interest paid?

    • Monthly payment difference is approximately $\$582.51$, Option 1 has lower total interest.
    • Monthly payment difference is approximately $\$582.51$, Option 2 has lower total interest.
    • Monthly payment difference is approximately $\$500.00$, Option 1 has lower total interest.
    • Monthly payment difference is approximately $\$500.00$, Option 2 has lower total interest.

    Answer: Monthly payment difference is approximately $\$582.51$, Option 1 has lower total interest.

  5. A loan has a principal $P$, periodic payment $PMT$, periodic interest rate $i$, and total payments $N$. If the borrower decides to double their periodic payment to $2 \times PMT$, how does the new total number of payments $N_{new}$ compare to the original $N$? Assume $i$ remains constant.

    • $N_{new}$ will be less than $N/2$.
    • $N_{new}$ will be exactly $N/2$.
    • $N_{new}$ will be greater than $N/2$ but less than $N$.
    • $N_{new}$ will remain approximately $N$ as the principal is fixed.

    Answer: $N_{new}$ will be less than $N/2$.

  6. A loan of $P = \$10,000$ is taken for $t = 5$ years at an annual interest rate of $r = 6\%$. Scenario A involves monthly compounding, while Scenario B involves semi-annual compounding. What is the approximate difference in the total amount paid over the life of the loan between Scenario A and Scenario B?

    • Approximately $\$123$
    • Approximately $\$85$
    • Approximately $\$150$
    • Approximately $\$200$

    Answer: Approximately $\$123$

  7. A $20$-year loan for $P = \$200,000$ is taken at an annual interest rate of $r = 5\%$ compounded monthly. After $10$ years, what percentage of the original principal has been paid off?

    • Approximately $37.7\%$
    • Approximately $50.0\%$
    • Approximately $25.0\%$
    • Approximately $62.3\%$

    Answer: Approximately $37.7\%$

  8. Given the formula for loan principal $P = PMT \times \frac{1 - (1 + i)^{-N}}{i}$, if a borrower wants to reduce their loan term $N$ by half, assuming $i$ and $P$ remain constant, how must their periodic payment $PMT$ change?

    • $PMT_{new} = PMT \times (1 + (1 + i)^{-N/2})$
    • $PMT_{new} = 2 \times PMT$
    • $PMT_{new} = PMT \times (1 - (1 + i)^{-N/2})$
    • $PMT_{new} = PMT / 2$

    Answer: $PMT_{new} = PMT \times (1 + (1 + i)^{-N/2})$

  9. A loan of $P = \$400,000$ is offered at an annual interest rate of $r = 3.5\%$ compounded monthly. Scenario X is a $15$-year term, and Scenario Y is a $30$-year term. What is the approximate difference in the total interest paid between Scenario Y and Scenario X?

    • Approximately $\$131,760$
    • Approximately $\$100,000$
    • Approximately $\$150,000$
    • Approximately $\$180,000$

    Answer: Approximately $\$131,760$

  10. A loan of $P = \$100,000$ is taken for $N = 120$ months at an annual interest rate of $r = 6\%$ compounded monthly. What is the approximate amount of principal paid in the $61^{st}$ payment?

    • Approximately $\$823$
    • Approximately $\$574$
    • Approximately $\$287$
    • Approximately $\$1110$

    Answer: Approximately $\$823$

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