Calculate Mortgage Payments - Nationwide Multistate Licensing System NMLS

1. State the monthly principal portion formula given payment $P$ and interest portion $I$.

   $\text{Principal}=P-I$. Subtracts the interest portion from the total payment to isolate the amount applied to reducing the principal.

2. State the formula for total payments made over the full term for monthly payment $P$ and $n$ payments.

   $\text{Total paid}=Pn$. Aggregates the fixed monthly payment over all $n$ periods to find the total amount disbursed by the borrower.

3. Find the new balance after paying $200 of principal on a $\$200{,}000$ balance.

   $200{,}000-200=\$199{,}800$. Deducts the principal paid from the existing balance to update the remaining loan amount.

4. State the monthly interest portion formula when the current balance is $B$ at monthly rate $i$.

   $I=Bi$. Computes the interest accrued over one month on the outstanding loan balance at the given monthly rate.

5. Identify the payment structure where each payment includes both principal and interest and ends at $0$ balance.

   Fully amortizing payment. Payments include both interest and principal components, systematically reducing the balance to zero by term end.

6. Identify whether amortization is positive, zero, or negative if $P=\$900$ and monthly interest due is $\$1{,}000$.

   Negative amortization. Occurs when the payment is less than interest due, causing the unpaid interest to increase the balance.

7. What is the interest-only payment on $L=\$300{,}000$ at $6\%$ annually (monthly rate $i=0.005$)?

   $P=300{,}000\times 0.005=\$1{,}500$. Calculates the payment as interest on the full loan amount, with no principal reduction in this structure.

8. Find and correct the formula error: $P=L\frac{i(1+i)^n}{(1+i)^n+1}$ for a fully amortizing payment.

   Correct: $P=L\frac{i(1+i)^n}{(1+i)^n-1}$. Replaces the erroneous +1 in the denominator with -1 to correctly apply the annuity formula for amortization.

9. State the formula for total interest paid over the full term for loan amount $L$, payment $P$, term $n$.

   $\text{Total interest}=Pn-L$. Subtracts the original loan principal from total payments to determine the cumulative interest expense over the term.

10. State the interest-only monthly payment formula $P$ for balance $L$ and monthly rate $i$.

    $P=Li$. Multiplies the constant loan balance by the monthly interest rate since no principal is repaid in this structure.

11. State the fully amortizing monthly payment formula $P$ for loan $L$, rate $i$, term $n$.

    $P=L\frac{i(1+i)^n}{(1+i)^n-1}$. Derives from the present value of an annuity formula to calculate the fixed payment that fully repays the loan over $n$ periods at rate $i$.

12. Find the annual rate as a decimal $r$ when $APR=5.5\%$.

    $r=\frac{5.5}{100}=0.055$. Divides the APR percentage by 100 to convert it to a decimal form for rate calculations.

13. Find total interest if $L=\$300{,}000$, $P=\$1{,}500$, and $n=360$.

    $Pn-L=540{,}000-300{,}000=\$240{,}000$. Subtracts the initial loan amount from total payments to isolate the total interest cost.

14. Find total paid over $n=360$ payments if $P=\$1{,}500$ per month.

    $Pn=1{,}500\times 360=\$540{,}000$. Multiplies the monthly payment by the total number of payments to sum the overall amount paid.

15. State the formula for total number of monthly payments $n$ for a $t$-year loan.

    $n=12t$. Multiplies the loan term in years by 12 to determine the total number of monthly payments.

16. Identify the payment structure where payments are less than interest due, causing the balance to increase.

    Negative amortization. Payments fall short of covering accrued interest, resulting in unpaid interest being added to the principal balance.

17. What is the number of monthly payments $n$ for a $30$-year mortgage?

    $n=12\times 30=360$. Multiplies the years by 12 months to compute the total number of payments in a monthly mortgage.

18. Identify the payment structure where payments cover only interest and the principal balance does not decline.

    Interest-only payment. Payments equal only the accrued interest, maintaining the principal balance unchanged throughout the period.

19. State the new balance formula after one payment when starting balance is $B$ and principal paid is $p$.

    $B_{new}=B-p$. Reduces the prior balance by the principal portion paid to reflect the updated outstanding amount.

20. Find the monthly interest due $I$ on a balance $B=\$200{,}000$ at $i=0.005$.

    $I=200{,}000\times 0.005=\$1{,}000$. Multiplies the current balance by the monthly interest rate to determine the interest accrued for that period.

21. State the formula to convert an annual percentage rate $APR\%$ to a decimal annual rate $r$.

    $r=\frac{APR\%}{100}$. Divides the percentage figure by 100 to express the annual rate as a decimal for use in calculations.

22. Find the principal paid if the monthly payment is $P=\$1{,}200$ and interest is $I=\$1{,}000$.

    $P-I=1{,}200-1{,}000=\$200$. Subtracts the interest due from the total payment to find the portion reducing the principal.

23. State the formula for monthly interest rate $i$ from annual rate $r$ (as a decimal).

    $i=\frac{r}{12}$. Divides the annual decimal rate by 12 to convert to the equivalent monthly interest rate.

24. State the remaining balance formula after $k$ payments for loan $L$, rate $i$, payment $P$.

    $B_k=L(1+i)^k-P\frac{(1+i)^k-1}{i}$. Grows the initial loan by compound interest for $k$ periods and subtracts the accumulated value of payments made.

25. What is the monthly rate $i$ when the annual rate is $6\%$ (as a decimal)?

    $i=\frac{0.06}{12}=0.005$. Converts the annual decimal rate to monthly by dividing by 12 for periodic interest calculations.

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