Loans and Mortgages (Amortization) — Practice Quiz

A Financial Math cheat sheet for Loans and Mortgages (Amortization) — every key formula with its symbols defined — plus a medium-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 loan of \$200,000 at an annual interest rate of $5\%$ compounded monthly for $30$ years. What is the monthly payment? Round to the nearest cent.

    • \$1,073.64
    • \$10,000.00
    • \$7,105.87
    • \$680.92

    Answer: \$1,073.64

  2. You can afford a monthly loan payment of \$1,200. If the annual interest rate is $4.5\%$ compounded monthly and the loan term is $15$ years, what is the maximum loan amount you can afford? Round to the nearest cent.

    • \$156,864.12
    • \$26,657.01
    • \$17,471.28
    • \$216,000.00

    Answer: \$156,864.12

  3. A loan of \$300,000 is taken out at an annual interest rate of $4\%$ compounded monthly for $20$ years. What is the outstanding balance after $5$ years (i.e., after $60$ payments)? Round to the nearest cent.

    • \$245,771.35
    • \$300,000.00
    • \$190,923.54
    • \$98,712.50

    Answer: \$245,771.35

  4. Which of the following correctly defines the periodic interest rate $i$ and the total number of payments $N$ for a loan with an annual interest rate $r$, compounded $n$ times per year, over a term of $t$ years?

    • $i = r/n$ and $N = nt$
    • $i = r$ and $N = t$
    • $i = r/t$ and $N = n$
    • $i = r \times n$ and $N = t/n$

    Answer: $i = r/n$ and $N = nt$

  5. A loan of \$150,000 has an annual interest rate of $6\%$ compounded quarterly for $10$ years. What is the quarterly payment? Round to the nearest cent.

    • \$5,014.07
    • \$1,665.31
    • \$9,969.23
    • \$16,265.13

    Answer: \$5,014.07

  6. If you make semi-annual payments of \$2,500 on a loan with an annual interest rate of $5.5\%$ compounded semi-annually for $7$ years, what was the original principal amount of the loan? Round to the nearest cent.

    • \$28,727.52
    • \$173,973.04
    • \$23,974.12
    • \$15,723.52

    Answer: \$28,727.52

  7. A loan of \$100,000 is taken out at an annual interest rate of $3.5\%$ compounded monthly for $15$ years. What is the outstanding balance after $36$ payments? Round to the nearest cent.

    • \$83,960.23
    • \$100,000.00
    • \$74,264.23
    • \$24,396.99

    Answer: \$83,960.23

  8. A loan of \$250,000 is taken out at an annual interest rate of $4.2\%$ compounded monthly for $25$ years. What is the total interest paid over the life of the loan? Round to the nearest cent.

    • \$154,206.74
    • \$404,206.74
    • \$262,500.00
    • \$3,150,000.00

    Answer: \$154,206.74

  9. You are comparing two loan options for \$180,000, both compounded monthly. Option A has an annual interest rate of $4\%$ over $20$ years. Option B has an annual interest rate of $4.5\%$ over $15$ years. How much lower is the monthly payment for Option A compared to Option B? Round to the nearest cent.

    • \$286.22
    • \$-286.22
    • \$-13,925.68
    • \$902.35

    Answer: \$286.22

  10. You can afford a maximum monthly payment of \$1,800. If the annual interest rate is $3.8\%$ compounded monthly and you want a $25$-year loan, what is the maximum principal amount you can borrow? Round to the nearest cent.

    • \$348,259.10
    • \$47,367.77
    • \$43,199.15
    • \$540,000.00

    Answer: \$348,259.10

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