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A local company is considering investing in a gas project. The project requires an initial investment of three payments, each of $\mathrm{R} 120,000$. The first is due at the start of the project, the second three months later, and the third payment is due one year after the start of the project.
At the beginning of the $15^{\text {th }}$ year, it is assumed that a major refurbishment of the infrastructure will be required, costing R300,000. The project is expected to provide a continuous income stream as follows:
- R 30,000 in the second year
- R 33,000 in the third year
- $\mathrm{R} 36,000$ in the fourth year
- R 39,000 in the fifth year
Thereafter the continuous income stream is expected to increase by $5 \%$ per annum (compound) at the start of each year. The income stream is expected to cease at the end of the $35^{\text {th }}$ year from the start of the project.

Calculate the discounted payback period for the project, assuming an effective rate of interest of $10 \%$ peranm Note: The discounted payback period is after the major refurbishment.
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\begin{aligned}
&-120,000\left(1+v^{0.25}+v\right)-300,000 v^{14}+\bar{a}_{\mathrm{ה}}\left(30,000 v+33,000 v^{2}+36,000 v^{3}+39,000 v^{4}\right) \\
&+39,000 \bar{a}_{ה}\left((1.05) v^{5}+(1.05)^{2} v^{6}+\ldots+(1.05)^{t-5} v^{t}\right)=0 \quad @ i=10 \% \\
&-322,032.1592+39,000 \bar{a}_{\mathrm{i}} v^{4}\left((1.05) v^{1}+(1.05)^{2} v^{2}+\ldots+(1.05)^{t-5} v^{t-5}\right)=0 \quad(a i=10 \%
\end{aligned}

\begin{aligned}
&\text { With } \frac{1.05}{1.1}=\frac{1}{1+j} \Rightarrow j=4.7619048 \%\\
&-322,032.1592+39,000 \bar{a}_{\uparrow 10 \mathrm{os}} v_{105}^{4}\left(v_{j}^{1}+v_{j}^{2}+\ldots+v_{j}^{t-5}\right)=0\\
&-322,032.1592+39,000 \bar{a}_{7100 \mathrm{~s}} v_{103}^{4} a_{t-5, \%_{6}}=0\\
&a_{t-5 / \%}=12.67469016\\
&t-5=19.888690318\\
&\mathrm{DPP}=19.888690318+5=24.888690318
\end{aligned}
by Platinum (91,198 points)

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