A firm is evaluating an investment proposal to install new milling machines. The project requires an initial investment of R$50,000.$ The equipment has a lifespan of $5$ years and no salvage value. The company operates under a tax rate of $55\%$ and utilizes straight$-$line depreciation. The estimated annual profits before depreciation from the investment over the next $5$ years are as follows: Year $1$: R$10,000,$ Year $2$: R$11,000,$ Year $3$: R$14,000,$ Year $4$: R$15,000,$ Year $5$: R$25,000.$ Compute:

a$)$ The payback period, and

b$)$ The Net Present Value $($NPV$)$ at a discount rate of $10\%.$

$($Note that depreciation is a non$-$cash transaction, and tax is applied after depreciation has been accounted for. Straight$-$line depreciation means that the depreciation amount remains constant over time. Please ensure clarity by researching unfamiliar concepts, as this is an assignment.$)$

Key:

PBDT: Profit before interest and tax

PBT: Profit Before Tax

PAT: Profit after Tax

CIF: Cash inflow

PVF: Present Value Factors