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 loan of 150,000 is taken out at an annual interest rate of 4.5% compounded monthly. If the loan term is 30 years, what is the approximate monthly payment $PMT$?

    • 763.88
    • 716.20
    • 810.50
    • 695.30

    Answer: 763.88

  2. If a borrower can afford a monthly payment of 1,200 for 15 years on a loan with an annual interest rate of 6% compounded monthly, what is the maximum loan amount $P$ they can afford?

    • 141,624
    • 135,800
    • 150,000
    • 128,950

    Answer: 141,624

  3. A loan has a monthly payment of 800, an annual interest rate of 5% compounded monthly, and a total term of 20 years. What is the outstanding balance $B_k$ after 60 payments have been made?

    • 101,320
    • 115,000
    • 98,750
    • 105,500

    Answer: 101,320

  4. In the provided formulas, what does the variable $i$ represent?

    • Annual interest rate
    • Total number of payments
    • Periodic interest rate
    • Principal loan amount

    Answer: Periodic interest rate

  5. If the total number of payments, $N$, for a loan increases while the principal $P$ and periodic rate $i$ remain constant, what happens to the periodic payment $PMT$?

    • $PMT$ increases
    • $PMT$ decreases
    • $PMT$ remains the same
    • $PMT$ becomes zero

    Answer: $PMT$ decreases

  6. A borrower takes out a 200,000 loan at an annual interest rate of 3.6% compounded monthly. How much lower would the monthly payment be if the loan term is 30 years instead of 15 years?

    • 531.10
    • 480.25
    • 610.75
    • 450.00

    Answer: 531.10

  7. The formula $B_k = PMT \times \dfrac{1 - (1 + i)^{-(N - k)}}{i}$ calculates:

    • The total interest paid over the loan term
    • The principal amount of the loan
    • The outstanding balance after $k$ payments
    • The total amount paid after $k$ payments

    Answer: The outstanding balance after $k$ payments

  8. A loan has a total of 240 monthly payments. If the outstanding balance $B_k$ is calculated after 90 payments, how many payments are still remaining?

    • 150
    • 90
    • 240
    • 330

    Answer: 150

  9. If the annual interest rate $r$ for a loan increases, assuming all other variables ($P$, $n$, $t$) remain constant, how does this affect the periodic payment $PMT$?

    • $PMT$ increases
    • $PMT$ decreases
    • $PMT$ remains the same
    • $PMT$ becomes zero

    Answer: $PMT$ increases

  10. A car loan requires a monthly payment of 450 for 5 years. If the annual interest rate is 2.4% compounded monthly, what was the original principal amount $P$ of the loan?

    • 25,425
    • 24,800
    • 26,150
    • 23,900

    Answer: 25,425

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