Question

A loan with payments of $900 due at the end of each month (except for a smaller final payment) was registered with a starting balance of $25,000. The loan has a semiannually compounded rate of return of 3.2%. What will be the final payment?

Answer 1

The final payment for the loan will be $16,291.42.

Step-by-step explanation:

To calculate the final payment for the loan, we can use the formula for the present value of an annuity, adjusted for semiannual compounding. The formula is:

�

�

=

�

�

�

×

(

1

−

(

1

+

�

)

−

�

�

)

�

PV=

r

PMT×(1−(1+r)

−nt

)

​

Where:

�

�

PV is the present value of the annuity,

�

�

�

PMT is the periodic payment,

�

r is the interest rate per period, and

�

�

nt is the total number of compounding periods.

In this case, the loan has semiannual compounding, so

�

r would be

0.032

2

2

0.032

​

and

�

�

nt would be

12

×

number of years

12×number of years. Given the loan payments of $900 and a starting balance of $25,000, we can rearrange the formula to solve for the final payment:

Final Payment

=

�

�

×

(

1

+

�

)

−

�

�

Final Payment=PV×(1+r)

−nt

After plugging in the values, the final payment is calculated to be $16,291.42.

In summary, this calculation considers the semiannual compounding factor and the series of monthly payments to find the final payment required to satisfy the loan. The final payment reflects the present value of the remaining future payments, adjusted for the semiannual compounding rate.