<h2>Analysing Investment and Loan Options</h2>
<p>In the next worked example we consider the effects of the duration of the repayment period on the total amount repaid (the amount borrowed plus the accrued interest) for a loan.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>David and Julie take out a home loan of R \(\text{2.6}\) million with an interest rate of \(\text{10}\%\) per annum compounded monthly.</p>
<ol>
<li>Calculate the monthly repayments for a repayment period of \(\text{30}\) years.</li>
<li>Calculate the interest paid on the loan at the end of the \(\text{30}\) year period.</li>
<li>Determine the monthly repayments for a repayment period of \(\text{20}\) years.</li>
<li>Determine the interest paid on the loan at the end of the \(\text{20}\) year period.</li>
<li>What is the difference in the monthly repayment amounts?</li>
<li>Comment on the difference in the interest paid for the two different time periods.</li>
</ol>
</div>
<div>
<h3>Consider a \(\text{30}\) year repayment period on the loan</h3>

\[x = \cfrac{P \times i}{[1 - (1 + i)^{-n}]}\]\begin{align*} P &amp;= \text{R}\,\text{2,600,000} \\ i &amp;= \cfrac{\text{0.1}}{12} \\ n &amp;= 30 \times 12 = \text{360} \end{align*}\begin{align*} x &amp;= \dfrac{\text{2 600 000} \times \cfrac{\text{0.1}}{12}}{[1 - (1 + \cfrac{\text{0.1}}{12})^{-360}]} \\ &amp;= \text{R}\,\text{22,816.86} \end{align*}

<p>Therefore, the monthly repayment is \(\text{R}\,\text{22,816.86}\) for a \(\text{30}\) year period.</p>
<p>At the end of the \(\text{30}\) years, David and Julie will have paid a total amount of:</p>

\begin{align*} &amp;= 30 \times 12 \times \text{R}\,\text{22,816.86} \\ &amp;= \text{R}\,\text{8,214,069.60} \end{align*}

<p>The total amount of interest on the loan:</p>

\begin{align*} \text{Interest} &amp;= \text{total amount paid} - \text{loan amount} \\ &amp;= \text{R}\,\text{8,214,069.60} - \text{R}\,\text{2,600,000} \\ &amp;= \text{R}\,\text{5,614,069.60} \end{align*}

<p>We notice that the interest on the loan is more than double the amount borrowed.</p>
</div>
<div>
<h3>Consider a \(\text{20}\) year repayment period on the loan</h3>

\[x = \cfrac{P \times i}{[1 - (1 + i)^{-n}]}\]\begin{align*} P &amp;= \text{R}\,\text{2,600,000} \\ i &amp;= \cfrac{\text{0.1}}{12} \\ n &amp;= 20 \times 12 = \text{240} \end{align*}\begin{align*} x &amp;= \dfrac{\text{2 600 000} \times \cfrac{\text{0.1}}{12}}{[1 - (1 + \cfrac{\text{0.1}}{12})^{-240}]} \\ &amp;= \text{R}\,\text{25,090.56} \end{align*}

<p>Therefore, the monthly repayment is \(\text{R}\,\text{25,090.56}\) for a \(\text{20}\) year period.</p>
<p>At the end of the \(\text{20}\) years, David and Julie will have paid a total amount of:</p>

\begin{align*} &amp;= 20 \times 12 \times \text{R}\,\text{25,090.56} \\ &amp;= \text{R}\,\text{6,021,734.40} \end{align*}

<p>The total amount of interest on the loan:</p>

\begin{align*} \text{Interest} &amp;= \text{total amount paid} - \text{loan amount} \\ &amp;= \text{R}\,\text{6,021,734.40} - \text{R}\,\text{2,600,000} \\ &amp;= \text{R}\,\text{3,421,734.40} \end{align*}

<p>We notice that the interest on the loan is about \(\text{1.3}\) times the borrowed amount.</p>
</div>
<div>
<h3>Consider the difference in the repayment and interest amounts</h3>

\begin{align*} \text{Difference in repayments } &amp;= \text{R}\,\text{25,090.56} - \text{R}\,\text{22,816.86} \\ &amp;= \text{R}\,\text{2,273.70} \end{align*}

<p>It is also very interesting to look at the difference in the total interest paid:</p>

\begin{align*} \text{Difference in interest } &amp;= \text{R}\,\text{5,614,069.60} - \text{R}\,\text{3,421,734.40} \\ &amp;= \text{R}\,\text{2,192,335.20} \end{align*}

<p>Therefore, by paying an extra \(\text{R}\,\text{2,273.70}\) each month over a shorter repayment period, David and Julie could save more than \(\text{R}\,\text{2}\) million on the repayment of their home loan.</p>
</div>
</div>
</div>
<p>When considering taking out a loan, it is advisable to investigate and compare a few options offered by financial institutions. It is very important to make informed decisions regarding personal finances and to make sure that the monthly repayment amount is serviceable (payable). A credit rating is an estimate of a person's ability to fulfill financial commitments based on their previous payment history. Defaulting on a loan can affect a person's credit rating and their chances of taking out another loan in the future.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Marlene wants to start saving for a deposit on a house. She can afford to invest between \(\text{R}\,\text{400}\) and \(\text{R}\,\text{600}\) each month and gets information from four different investment firms. Each firm quotes a different interest rate and a prescribed monthly installment amount. She plans to buy a house in \(\text{7}\) years' time. Calculate which company offers the best investment opportunity for Marlene.</p>
<table>
<tbody>
<tr>
<td> </td>
<td>Interest rate (compounded monthly)</td>
<td>Monthly payment</td>
</tr>
<tr>
<td>TBS Investments</td>
<td>\(\text{13.5}\%\) p.a.</td>
<td>\(\text{R}\,\text{450}\)</td>
</tr>
<tr>
<td>Taylor Anderson</td>
<td>\(\text{13}\%\) p.a.</td>
<td>\(\text{R}\,\text{555}\)</td>
</tr>
<tr>
<td>PHK</td>
<td>\(\text{12.5}\%\) p.a.</td>
<td>\(\text{R}\,\text{575}\)</td>
</tr>
<tr>
<td>Simfords Consulting</td>
<td>\(\text{11}\%\) p.a.</td>
<td>\(\text{R}\,\text{600}\)</td>
</tr>
</tbody>
</table>
</div>
<div>
<h3>Consider the different investment options</h3>
<p>To compare the different investment options, we need to calculate the following for each option at the end of the seven year period:</p>
<ul>
<li>The future value of the monthly payments.</li>
<li>The total amount paid into the investment fund.</li>
<li>The total interest earned.</li>
</ul>

\[F = \cfrac{x[(1 + i)^{n}-1]}{i}\]

<h3><strong>TBS Investments:</strong></h3>

\begin{align*} F &amp;= \cfrac{\text{450}[(1 + \cfrac{\text{0.135}}{12})^{84}-1]}{\cfrac{\text{0.135}}{12}} \\ &amp;= \text{R}\,\text{62,370.99} \\ \text{Total amount } (T): \enspace &amp;= 7 \times 12 \times \text{R}\,\text{450} \\ &amp;= \text{R}\,\text{37,800} \\ \text{Total interest } (I): \enspace &amp;= \text{R}\,\text{62,370.99} - \text{R}\,\text{37,800} \\ &amp;= \text{R}\,\text{24,570.99} \end{align*}

<h3><strong>Taylor Anderson:</strong></h3>

\begin{align*} F &amp;= \cfrac{\text{555}[(1 + \cfrac{\text{0.13}}{12})^{84}-1]}{\cfrac{\text{0.13}}{12}} \\ &amp;= \text{R}\,\text{75,421.65} \\ \text{Total amount } (T): \enspace &amp;= 7 \times 12 \times \text{R}\,\text{555} \\ &amp;= \text{R}\,\text{46,620} \\ \text{Total interest } (I): \enspace &amp;= \text{R}\,\text{75,421.65} - \text{R}\,\text{46,620} \\ &amp;= \text{R}\,\text{28,801.65} \end{align*}

<h3><strong>PHK:</strong></h3>

\begin{align*} F &amp;= \cfrac{\text{575}[(1 + \cfrac{\text{0.125}}{12})^{84}-1]}{\cfrac{\text{0.125}}{12}} \\ &amp;= \text{R}\,\text{76,619.96} \\ \text{Total amount } (T): \enspace &amp;= 7 \times 12 \times \text{R}\,\text{575} \\ &amp;= \text{R}\,\text{48,300} \\ \text{Total interest } (I): \enspace &amp;= \text{R}\,\text{76,619.96} - \text{R}\,\text{48,300} \\ &amp;= \text{R}\,\text{28,319.96} \end{align*}

<h3><strong>Simfords Consulting:</strong></h3>

\begin{align*} F &amp;= \cfrac{\text{600}[(1 + \cfrac{\text{0.11}}{12})^{84}-1]}{\cfrac{\text{0.11}}{12}} \\ &amp;= \text{R}\,\text{75,416.96} \\ \text{Total amount } (T): \enspace &amp;= 7 \times 12 \times \text{R}\,\text{600} \\ &amp;= \text{R}\,\text{50,400} \\ \text{Total interest } (I): \enspace &amp;= \text{R}\,\text{75,416.96} - \text{R}\,\text{50,400} \\ &amp;= \text{R}\,\text{25,016.96} \end{align*}</div>
<div>
<h3>Draw a table of the results to compare the answers</h3>
<table>
<tbody>
<tr>
<td> </td>
<td>\(F\)</td>
<td>\(T\)</td>
<td>\(I\)</td>
</tr>
<tr>
<td>TBS Investments</td>
<td>\(\text{R}\,\text{62,370.99}\)</td>
<td>\(\text{R}\,\text{37,800}\)</td>
<td>\(\text{R}\,\text{24,570.99}\)</td>
</tr>
<tr>
<td>Taylor Anderson</td>
<td>\(\text{R}\,\text{75,421.65}\)</td>
<td>\(\text{R}\,\text{46,620}\)</td>
<td>\(\text{R}\,\text{28,801.65}\)</td>
</tr>
<tr>
<td>PHK</td>
<td>\(\text{R}\,\text{76,619.96}\)</td>
<td>\(\text{R}\,\text{48,300}\)</td>
<td>\(\text{R}\,\text{28,319.96}\)</td>
</tr>
<tr>
<td>Simfords Consulting</td>
<td>\(\text{R}\,\text{75,416.96}\)</td>
<td>\(\text{R}\,\text{50,400}\)</td>
<td>\(\text{R}\,\text{25,016.96}\)</td>
</tr>
</tbody>
</table>
</div>
<div>
<h3>Make a conclusion</h3>
<p>An investment with PHK would provide Marlene with the highest deposit (\(\text{R}\,\text{76,619.96}\)) for her house at the end of the \(\text{7}\) year period. However, we notice that an investment with Taylor Anderson would earn the highest amount of interest (\(\text{R}\,\text{28,801.65}\)) and is therefore the better investment option.</p>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>William wants to take out a loan of \(\text{R}\,\text{750,000}\), so he approaches three different banks. He plans to start repaying the loan immediately and he calculates that he can afford a monthly repayment amount between \(\text{R}\,\text{5,500}\) and \(\text{R}\,\text{7,000}\).</p>
<p>Calculate which of the three options would be best for William.</p>
<ul>
<li>West Bank offers a repayment period of \(\text{30}\) years and an interest rate of prime compounded monthly.</li>
<li>AcuBank offers a repayment period of \(\text{20}\) years and an interest rate of prime \(+ \text{0.5}\%\) compounded monthly.</li>
<li>FinTrust Bank offers a repayment period of \(\text{15}\) years and an interest rate of prime \(+ \text{2}\%\) compounded monthly.</li>
</ul>
</div>
<div>
<h3>Consider the different loan options</h3>
<p>To compare the different loan options, we need to calculate the following for each option:</p>
<ul>
<li>The monthly payment amount.</li>
<li>The total amount paid to repay the loan.</li>
<li>The amount of interest on the loan.</li>
</ul>

\[x = \cfrac{P \times i}{[1 - (1 + i)^{-n}]}\]

<h3><strong>West Bank:</strong></h3>

\begin{align*} x &amp; = \cfrac{ \text{750 000} \times \cfrac{\text{0.085}}{12}}{[1 - (1 + \cfrac{\text{0.085}}{12})^{-\text{360}}]} \\ &amp;= \text{R}\,\text{5,766.85} \\ \text{Total amount } (T): \enspace &amp;= 30 \times 12 \times \text{R}\,\text{5,766.85} \\ &amp;= \text{R}\,\text{2,076,066} \\ \text{Total interest } (I): \enspace &amp;= \text{R}\,\text{2,076,066} - \text{R}\,\text{750,000} \\ &amp;= \text{R}\,\text{1,326,066} \end{align*}

<h3><strong>AcuBank:</strong></h3>

\begin{align*} x &amp;= \cfrac{ \text{750 000} \times \cfrac{\text{0.09}}{12}}{[1 - (1 + \cfrac{\text{0.09}}{12})^{-\text{240}}]} \\ &amp;= \text{R}\,\text{6,747.94} \\ \text{Total amount } (T): \enspace &amp;= 20 \times 12 \times \text{R}\,\text{6,747.94} \\ &amp;= \text{R}\,\text{1,619,505.60} \\ \text{Total interest } (I): \enspace &amp;= \text{R}\,\text{1,619,505.60} - \text{R}\,\text{750,000} \\ &amp;= \text{R}\,\text{869,505.60} \end{align*}

<h3><strong>FinTrust Bank:</strong></h3>

\begin{align*} x &amp;= \cfrac{ \text{750 000} \times \cfrac{\text{0.105}}{12}}{[1 - (1 + \cfrac{\text{0.105}}{12})^{-\text{180}}]} \\ &amp;= \text{R}\,\text{8,290.49} \\ \text{Total amount } (T): \enspace &amp;= 15 \times 12 \times \text{R}\,\text{8,290.49} \\ &amp;= \text{R}\,\text{1,492,288.20} \\ \text{Total interest } (I): \enspace &amp;= \text{R}\,\text{1,492,288.20} - \text{R}\,\text{750,000} \\ &amp;= \text{R}\,\text{742,288.20} \end{align*}</div>
<div>
<h3>Draw a table of the results to compare the answers</h3>
<table>
<tbody>
<tr>
<td> </td>
<td>\(x\)</td>
<td>\(T\)</td>
<td>\(I\)</td>
</tr>
<tr>
<td>West Bank</td>
<td>\(\text{R}\,\text{5,766.85}\)</td>
<td>\(\text{R}\,\text{2,076,066.00}\)</td>
<td>\(\text{R}\,\text{1,326,066.00}\)</td>
</tr>
<tr>
<td>AcuBank</td>
<td>\(\text{R}\,\text{6,747.94}\)</td>
<td>\(\text{R}\,\text{1,619,505.60}\)</td>
<td>\(\text{R}\,\text{869,505.60}\)</td>
</tr>
<tr>
<td>FinTrust Bank</td>
<td>\(\text{R}\,\text{8,290.49}\)</td>
<td>\(\text{R}\,\text{1,492,288.20}\)</td>
<td>\(\text{R}\,\text{742,288.20}\)</td>
</tr>
</tbody>
</table>
</div>
<div>
<h3>Make a conclusion</h3>
<p>A loan from FinTrust Bank would accumulate the lowest amount of interest but the monthly repayment amounts are not within William's budget. Although West Bank offers the lowest interest rate and monthly repayment amount, the interest earned on the loan is very high as a result of the longer repayment period. If we assume that William must repay the loan over the given time periods, then AcuBank offers the best option.</p>
<p>However, we know that William can afford to pay more than \(\text{R}\,\text{5,766.85}\) per month, and if the bank allows him to pay back the loan earlier, he should consider taking out a loan with West Bank and take advantage of the lower interest rate.</p>
</div>
</div>
</div>

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Analysing Investment and Loan Options

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Discover the fundamentals of finance and the impact of growth by exploring simple and compound interest, hire purchase, inflation, and foreign exchange rates, and learn how to calculate annuities, sinking funds, and present value annuities while analyzing investment and loan options.


Analysing Investment and Loan Options

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