$
\underline{Amortization} \\[3ex]
PV = R1500000 \\[3ex]
t = 20\:years \\[3ex]
r = 10.5\% = \dfrac{10.5}{100} = 0.105 \\[5ex]
Compounded\:\:monthly \rightarrow m = 12 \\[3ex]
PMT = ? \\[3ex]
(9.1) \\[3ex]
PMT = \dfrac{PV}{m} * \left[\dfrac{r}{1 - \left(1 + \dfrac{r}{m}\right)^{-mt}}\right] \\[10ex]
PMT = \dfrac{1500000}{12} * \left[\dfrac{0.105}{1 - \left(1 + \dfrac{0.105}{12}\right)^{-1 * 12 * 20}}\right] \\[10ex]
= 125000 * \left[\dfrac{0.105}{1 - \left(1 + 0.00875\right)^{-240}}\right] \\[7ex]
= 125000 * \left[\dfrac{0.105}{1 - \left(1.00875\right)^{-240}}\right] \\[7ex]
= 125000 * \left[\dfrac{0.105}{1 - 0.123580101}\right] \\[5ex]
= 125000 * \left[\dfrac{0.105}{0.876419899}\right] \\[5ex]
= \dfrac{125000 * 0.105}{0.876419899} \\[5ex]
= \dfrac{13125}{0.876419899} \\[5ex]
= 14975.6983 \\[3ex]
\approx R14,975.70 \\[3ex]
(9.2) \\[3ex]
12\:months = 1\:year \\[3ex]
144th\:\:payment \rightarrow \dfrac{144}{12} = 12th\:year\:\:payment \\[5ex]
Remaining = 20 - 12 = 8\:years \\[3ex]
$
Ask your students how they will calculate the outstanding balance after the $144^{th}$ payment.
Note their responses.
Some of the responses may include:
Find the total payment...for 20 years ...which is 240 months
Find the payment for 144 months
Subtract that payment from the total payment
...among other responses.
Remind them that these are Mortgage payments...Fixed Rate Mortgage
Show them how it works...using the example on my website
You can also use the Loan Amortization Calculator to confirm the calculations
Any monthly payment made is used to offset the interest earned during that period...first step
Then, any remaining balance is applied toward the principal.
The same principle applies to car payments and other amortized payments.
We can find the outstanding balance at the end of the $144th$ payemnt (at the end of $12$ years) in at least two ways.
Use any method you prefer.
First Method: Find the present value of the ordinary annuity for the remaining $8$ years.
OR
Second Method:
Compare to the previous response of "probably" one of your students.
But, explain the concept well for your students to understand.
Use the Compound Interest formula to calculate the future value of the loan amount for $12$ years.
Remember that the monthly payment of $14975.70$ was found using the Amortization formula for the
entire term of the loan.
So, use the Future Value of an Ordinary Annuity formula and calculate the future value of that payment for $12$ years.
Then, subtract the future value of that payment from the future value of the loan amount.
$
\underline{First\:\:Method} \\[3ex]
\underline{Present\:\:Value\:\:of\:\:Ordinary\:\:Annuity} \\[3ex]
t = 8\:years \\[3ex]
PMT = R14975.6983 \\[3ex]
r = 0.105 \\[3ex]
m = 12 \\[3ex]
PV = m * PMT * \left[\dfrac{1 - \left(1 + \dfrac{r}{m}\right)^{-mt}}{r}\right] \\[10ex]
PV = 12 * 14975.6983 * \left[\dfrac{1 - \left(1 + \dfrac{0.105}{12}\right)^{-1 * 12 * 8}}{0.105}\right] \\[10ex]
= 179708.38 * \left[\dfrac{1 - \left(1 + 0.00875\right)^{-96}}{0.105}\right] \\[7ex]
= 179708.38 * \left[\dfrac{1 - \left(1.00875\right)^{-96}}{0.105}\right] \\[7ex]
= 179708.38 * \left[\dfrac{1 - 0.43329075}{0.105}\right] \\[5ex]
= 179708.38 * \left[\dfrac{0.56670925}{0.105}\right] \\[5ex]
= \dfrac{179708.38 * 0.56670925}{0.105} \\[5ex]
= \dfrac{101842.401}{0.105} \\[5ex]
= 969927.629 \\[3ex]
\approx R969,927.63 \\[3ex]
\underline{Second\:\:Method} \\[3ex]
\underline{Compound\:\:Interest\:\:and\:\:Future\:\:Value\:\:of\:\:Ordinary\:\:Annuity} \\[3ex]
\underline{Compound\:\:Interest} \\[3ex]
P = R1500000 \\[3ex]
r = 0.105 \\[3ex]
t = 12\:years \\[3ex]
m = 12 \\[3ex]
A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[7ex]
A = 1500000\left(1 + \dfrac{0.105}{12}\right)^{12 * 12} \\[7ex]
= 1500000\left(1 + 0.00875\right)^{144} \\[7ex]
= 1500000\left(1.00875\right)^{144} \\[7ex]
= 1500000(3.50615308) \\[3ex]
= R5259229.62 \\[3ex]
\underline{Future\:\:Value\:\:of\:\:Ordinary\:\:Annuity} \\[3ex]
m = 12 \\[3ex]
PMT = R14975.6983 \\[3ex]
r = 0.105 \\[3ex]
t = 12\:years \\[3ex]
FV = m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] \\[10ex]
FV = 12 * 14975.6983 * \left[\dfrac{\left(1 + \dfrac{0.105}{12}\right)^{12 * 12} - 1}{0.105}\right] \\[10ex]
= 179708.38 * \left[\dfrac{\left(1 + 0.00875\right)^{144} - 1}{0.105}\right] \\[7ex]
= 179708.38 * \left[\dfrac{\left(1.00875\right)^{144} - 1}{0.105}\right] \\[7ex]
= 179708.38 * \left[\dfrac{3.50615308 - 1}{0.105}\right] \\[5ex]
= 179708.38 * \left[\dfrac{2.50615308}{0.105}\right] \\[5ex]
= \dfrac{179708.38 * 2.50615308}{0.105} \\[5ex]
= \dfrac{450376.71}{0.105} \\[5ex]
= R4289302 \\[3ex]
\underline{Balance\:\:after\:\:144th\:\:payment} \\[3ex]
Balance = 5259229.62 - 4289302 = R969,927.62 \\[3ex]
$
Formulas Used: Mathematics of Finance