Answer to Question #208116 in Financial Math for Beauty Magadlela

Question

Thulisile bought a house and managed to secure a home loan for R790 000 with monthly payments of R9 680,70 at a fixed interest rate of 13,75% per year, compounded monthly, over a period of 20 years. If an average yearly inflation rate of 9,2% is expected, then the real cost of the loan (the difference between the total value of the loan and the actual principal borrowed) is

[1] R201 642.

[2] R270 749.

[3] R588 358.

[4] R1 060 749.

[5] R87 126.

Answer 1

Solution:

We need to calculate total value of loan by using inflation rate, so there will be no use of fixed interest rate of 13,75% per year, compounded monthly, thus it is redundant in the following calculation.

We will use this formula:

Total value of loan $=\frac{\mathrm{PMT}}{\mathrm{r}} \times\left[1-\frac{1}{(1+\mathrm{r})^{\mathrm{n}}}\right]$

Where, $\mathrm{PMT}=$ Monthly payment of loan

r= Periodic rate of interest

n= numbers of periods

Calculation of total value of loan:

Monthly payment PMT=9680.70 Monthly interest rate (r) $=\frac{9.2 \%}{12} =0.007666666$

Number of period $=20 \times 12=240$

Total value of loan:

$=\frac{\mathrm{PMT}}{\mathrm{r}} \times\left[1-\frac{1}{(1+\mathrm{r})^{\mathrm{n}}}\right] \\=\left(\frac{9680.70}{0.00766666}\right) \times\left[1-\frac{1}{(1+0.007666666)^{240}}\right] \\=1262700 \times\left[1-\frac{1}{6.2525070977}\right] \\=1262700 \times[1-0.159935844] \\=1262700 \times 0.840064156 \\=1060749$

Total value of loan is R1060749

Calculation of the real cost of the loan

Real cost of the loan = Total cost of loan -Actual principal borrowed $=\mathrm{R} 1060749-\mathrm{R} 790000 =\mathrm{R} 270749$

Hence, Option 2 is the correct answer.