Question: Question 3 (7 marks)\nA financial advisor is assisting the Khumalo Family Trust with three separate financial decisions. They need to present time-value calculations to the trustees at their upcoming meeting.\nREQUIRED:\n3.1\nThe Trust received a donation of R150 000 today. If the funds are invested at an annual interest rate of 9%, compounded quarterly, calculate the future value of this investment after five years. Show all workings and state whether you are using a financial table factor or a formula. (3)\n3.2\nThe Trust expects to receive an ordinary annuity of R28 000 per year for six years, with the first payment due one year from today. The applicable discount rate is 11% per annum. Calculate the present value of this annuity. (2)\n3.3 A local municipality has offered the Trust a lump-sum payment of R90 000 today, or an alternative stream of R18 500 per year for six years (ordinary annuity) at a discount rate of 11%.\nUsing the annuity factor you calculated in Question 3.2, determine the present value of the annuity option. Compare this to the lump sum and recommend which option the Trust should accept. Justify your answer. (2)

1. **Problem Statement:**\nCalculate the future value of an investment with quarterly compounding, the present value of an ordinary annuity, and compare a lump sum to an annuity option for the Khumalo Family Trust.\n\n2. **Formulas and Rules:**\n- Future Value (FV) with compound interest: $$FV = PV \times \left(1 + \frac{r}{m}\right)^{mt}$$ where $PV$ is present value, $r$ is annual interest rate, $m$ is compounding periods per year, and $t$ is years.\n- Present Value (PV) of an ordinary annuity: $$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$ where $PMT$ is payment per period, $r$ is discount rate per period, and $n$ is number of periods.\n\n---\n\n### 3.1 Future Value of Investment\n- Given: $PV = 150000$, $r = 0.09$, $m = 4$, $t = 5$ years.\n- Calculate: $$FV = 150000 \times \left(1 + \frac{0.09}{4}\right)^{4 \times 5} = 150000 \times (1.0225)^{20}$$\n- Calculate $(1.0225)^{20}$:\n$$ (1.0225)^{20} = e^{20 \times \ln(1.0225)} \approx e^{20 \times 0.02227} = e^{0.4454} \approx 1.5610 $$\n- So, $$FV = 150000 \times 1.5610 = 234150$$\n- **Answer:** The future value after 5 years is approximately $234150$.\n\n### 3.2 Present Value of Ordinary Annuity\n- Given: $PMT = 28000$, $r = 0.11$, $n = 6$.\n- Calculate annuity factor: $$AF = \frac{1 - (1 + 0.11)^{-6}}{0.11}$$\n- Calculate $(1 + 0.11)^{-6} = (1.11)^{-6} = \frac{1}{(1.11)^6}$\n- Calculate $(1.11)^6$:\n$$ (1.11)^6 = e^{6 \times \ln(1.11)} \approx e^{6 \times 0.10436} = e^{0.6262} \approx 1.8704 $$\n- So, $(1.11)^{-6} = \frac{1}{1.8704} = 0.5349$\n- Annuity factor: $$AF = \frac{1 - 0.5349}{0.11} = \frac{0.4651}{0.11} = 4.228$$\n- Present value: $$PV = 28000 \times 4.228 = 118384$$\n- **Answer:** The present value of the annuity is approximately $118384$.\n\n### 3.3 Present Value of Annuity Option and Comparison\n- Given: $PMT = 18500$, $AF = 4.228$ (from 3.2).\n- Calculate present value of annuity option: $$PV = 18500 \times 4.228 = 78218$$\n- Compare to lump sum $90000$.\n- Since $90000 > 78218$, the lump sum is financially better.\n- **Recommendation:** The Trust should accept the lump sum payment of $90000$ as it has a higher present value than the annuity option.\n\n---\n\n**Summary:**\n- Future value of R150000 at 9% compounded quarterly for 5 years is approximately $234150$.\n- Present value of R28000 per year for 6 years at 11% is approximately $118384$.\n- Present value of R18500 per year for 6 years at 11% is approximately $78218$, less than lump sum $90000$.\n- Accept lump sum payment.\n