Question: Exercise 3: Old Mind Action Series Textbook 1. Simon starts saving money in a Unit Trust fund. He immediately deposits R900 into the fund. Thereafter, at the end of each month, he deposits R900 into the fund and continues to do this for the five years. Interest is 18% per annum compounded monthly. (a) Calculate the future value of his investment at the end of the five- year period. (b) Simon leaves his money in the fund to grow for two years without making any further payments of R900. The interest rate changes to 10% per annum compounded quarterly. Calculate the value of his investment after the two-year period. 6.1 On 31 January 2020, Tshepo made the first of his monthly deposits of R1 000 into a savings account. He continues to make monthly deposits of R1 000 at the end of each month up until 31 January 2032. The interest rate was fixed at 7,5% p.a., compounded monthly. 6.1.1 What will the investment be worth immediately after the last deposit? (4) 6.1.2 If he makes no further payments but leaves the money in the account, how much money will be in the account on 31 January 2033? (2)

### Problem 1: Simon's Unit Trust Investment 1. **Problem statement:** Simon deposits R900 immediately and then R900 at the end of each month for 5 years. Interest is 18% per annum compounded monthly. Calculate the future value at the end of 5 years. 2. **Formula for future value of an annuity with initial deposit:** The future value $FV$ after $n$ months with monthly interest rate $i$ and monthly payment $P$ plus an initial deposit $P_0$ is: $$FV = P_0(1+i)^n + P \times \frac{(1+i)^n - 1}{i}$$ where: - $P_0 = 900$ - $P = 900$ - Annual interest rate $r = 0.18$ - Monthly interest rate $i = \frac{0.18}{12} = 0.015$ - Number of months $n = 5 \times 12 = 60$ 3. **Calculate each part:** - Calculate $(1+i)^n = (1.015)^{60}$ Using a calculator, $(1.015)^{60} \approx 2.4596$ 4. **Calculate future value:** $$FV = 900 \times 2.4596 + 900 \times \frac{2.4596 - 1}{0.015}$$ $$= 2213.64 + 900 \times \frac{1.4596}{0.015}$$ $$= 2213.64 + 900 \times 97.3067$$ $$= 2213.64 + 87576.03 = 89789.67$$ 5. **Answer for part (a):** The future value after 5 years is approximately **R89789.67**. --- ### Problem 2: Simon's investment grows for 2 more years at 10% p.a. compounded quarterly without further payments 1. **Problem statement:** After 5 years, Simon leaves the money to grow for 2 years at 10% p.a. compounded quarterly with no further deposits. Calculate the value after 2 years. 2. **Formula for compound interest:** $$FV = PV \times (1 + i)^n$$ where: - $PV = 89789.67$ (value from part a) - Annual interest rate $r = 0.10$ - Quarterly interest rate $i = \frac{0.10}{4} = 0.025$ - Number of quarters $n = 2 \times 4 = 8$ 3. **Calculate:** $$FV = 89789.67 \times (1.025)^8$$ Calculate $(1.025)^8 \approx 1.2184$ $$FV = 89789.67 \times 1.2184 = 109374.15$$ 4. **Answer for part (b):** The value after 2 years is approximately **R109374.15**. --- ### Problem 3: Tshepo's monthly deposits of R1000 from 31 Jan 2020 to 31 Jan 2032 at 7.5% p.a. compounded monthly 1. **Problem statement:** Tshepo deposits R1000 monthly for 12 years (from 31 Jan 2020 to 31 Jan 2032). Interest is 7.5% p.a. compounded monthly. Find the investment value immediately after the last deposit. 2. **Parameters:** - Monthly deposit $P = 1000$ - Annual interest rate $r = 0.075$ - Monthly interest rate $i = \frac{0.075}{12} = 0.00625$ - Number of months $n = 12 \times 12 = 144$ 3. **Formula for future value of an annuity:** $$FV = P \times \frac{(1+i)^n - 1}{i}$$ 4. **Calculate:** Calculate $(1.00625)^{144}$: Using a calculator, $(1.00625)^{144} \approx 2.4596$ $$FV = 1000 \times \frac{2.4596 - 1}{0.00625} = 1000 \times \frac{1.4596}{0.00625} = 1000 \times 233.536 = 233536$$ 5. **Answer for 6.1.1:** The investment is worth approximately **R233536** immediately after the last deposit. --- ### Problem 4: Tshepo leaves money for 1 more year without deposits at 7.5% p.a. compounded monthly 1. **Problem statement:** Tshepo makes no further payments after 31 Jan 2032 and leaves the money for 1 year at 7.5% p.a. compounded monthly. Find the value on 31 Jan 2033. 2. **Parameters:** - Principal $PV = 233536$ - Monthly interest rate $i = 0.00625$ - Number of months $n = 12$ 3. **Formula:** $$FV = PV \times (1+i)^n$$ 4. **Calculate:** Calculate $(1.00625)^{12} \approx 1.0774$ $$FV = 233536 \times 1.0774 = 251460.5$$ 5. **Answer for 6.1.2:** The value on 31 Jan 2033 is approximately **R251460.50**.