Final answer:
To decide the most cost-effective payment arrangement for an asset, calculate the present value of each option using a 12% annual interest rate. For a perpetuity starting in 4 years, calculate its present value today using a 5% interest rate.
Step-by-step explanation:
To determine the most cost-effective way to pay for an asset from the given options, we need to compute the present value (PV) of each payment arrangement using the interest rate of 12%. We'll use the formula PV = R × (1 - (1 + i)^(-n)) / i for the annuity options and PV = FV / (1 + i)^n for the lump sum option, where 'R' is the regular payment, 'i' is the interest rate per period, 'n' is the number of periods, and 'FV' is the future value.
- Option 1: PV = R15,000 × (1 - (1 + 0.12)^(-120)) / 0.12
- Option 2: This is an annuity due, so we calculate the PV of an ordinary annuity and then adjust it by multiplying by (1 + i). PV = R14,850 × (1 - (1 + 0.12)^(-120)) / 0.12 × (1 + 0.12)
- Option 3: PV = R3,500,000 / (1 + 0.12)^120
After calculation, the option with the smallest present value will be the most cost-effective.
To find the present value of a R2500 annual perpetuity starting at the end of 4 years with an interest rate of 5%, we use the formula PV = C / r, where 'C' is the annual cash flow and 'r' is the interest rate.
Notice that the cash flows start after 4 years, so we first find the PV at year 4, and then find the present value of that amount as of today:
- PV at year 4 = R2500 / 0.05
- Present Value today = PV at year 4 / (1 + 0.05)^4
Populating this into the formulas provides the present value of the perpetuity.