Thembi takes out a home loan of R500 000. the interest rate is 9,38% p.a compounded annually. She has two options to repay the loan. In both cases, the loan is repaid 25 years after the loan was drawn by equal monthly instalments. In the first option, she starts the repayments one month after receiving the loan and in the second option, she starts repayments six months after receiving the loan.
Calculate the monthly repayments in each case.

[tex]\boxed{\begin{array}{l}\underline{\textsf{Monthly Payment Formula}}\\\\M=P\cdot \dfrac{r\left(1+r\right)^{n}}{\left(1+r\right)^{n}-1}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$M$ is the monthly payment.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount (loan amount).}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate per month (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the term of the loan (in months).}\\\end{array}}[/tex]

In this case:

P = 500 000
r = 9.38% / 12 = 0.0938 / 12
n = 25 years = 300 months

Substitute the values into the equation and solve for M:

[tex]M=500000\cdot \dfrac{\frac{0.0938}{12}\left(1+\frac{0.0938}{12}\right)^{300}}{\left(1+\frac{0.0938}{12}\right)^{300}-1}\\\\\\M=4326.84810....\\\\\\M=4326.85[/tex]

Therefore, the monthly repayment for option one is R4326.85.

[tex]\hrulefill[/tex]

Option 2

In option 2, Thembi starts the repayments six months after receiving the loan, which means that the loan is accruing interest during the initial 6 months without any payments being made.

To calculate the monthly payment in this case, we need to determine the balance of the loan after the initial 6 months, and then use that new balance to calculate the monthly payment.

Use the compound interest formula to determine the balance of the loan after the first 6 months.

[tex]\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+\dfrac{r}{n}\right)^{nt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}[/tex]

In this case:

P = 500 000
r = 9.38% = 0.0938
n = 1
t = 0.5 years (6 months)

Substitute the values into the equation and solve for A:

[tex]A=500000\left(1+\dfrac{0.0938}{1}\right)^{1 \cdot 0.5}\\\\\\ A=500000\left(1.0938}\right)^{0.5}\\\\\\A=522\;924.47[/tex]

Given that the total number of monthly payments (n) is reduced by 6 months, the values to substitute into the monthly payment formula are:

P = 522 924.47
r = 9.38% / 12 = 0.0938 / 12
n = 300 - 6 = 294

Therefore:

[tex]M=522 924.47\cdot \dfrac{\frac{0.0938}{12}\left(1+\frac{0.0938}{12}\right)^{294}}{\left(1+\frac{0.0938}{12}\right)^{294}-1}\\\\\\M=4548.52408...\\\\\\M=4548.52[/tex]

So, the monthly repayment for option two is R4548.52.