2. TVM Calculations ( 40 marks) a) Calculate the future value of R50,000 invested for 5 years at an annual interest rate of $$ 8 \% $$ compounded annually. ( 5 marks) b) Calculate the present value of R100,000 to be received in 10 years if the discount rate is $$ 6 \% $$ per annum. ( 5 marks) c) A company wants to accumulate R1,000,000 in 8 years. How much should it invest today in a savings account offering $$ 5 \% $$ interest compounded annually? (5 marks) d) A business purchases equipment and finances it through an annuity payment of R5,000 per month for 3 years at an interest rate of $$ 12 \% $$ per annum. Calculate the present value of this annuity. ( 5 marks) e) Calculate the future value of an investment of R25,000 made annually for 10 years at an interest rate of $$ 7 \% $$ compounded annually. ( 5 marks) f) Determine the present value of an annuity that pays R10,000 per year for 15 years at a discount rate of $$ 9 \% $$. ( 5 marks) g) A project will generate cash flows of R50,000 per year for the next 6 years. Calculate its present value if the discount rate is $$ 8 \% $$. ( 5 marks)

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**a) Future Value of R50,000 for 5 years at 8% compounded annually**

Use the formula  
$$
FV = PV \times (1 + r)^n
$$
where  
$$PV = 50\,000$$,  
$$r = 0.08$$, and  
$$n = 5$$.

Thus,  
$$
FV = 50\,000 \times (1 + 0.08)^5 = 50\,000 \times (1.08)^5.
$$

Calculating $$(1.08)^5$$:
$$
(1.08)^5 \approx 1.46933.
$$

So,  
$$
FV \approx 50\,000 \times 1.46933 \approx 73\,466.50.
$$

---

**b) Present Value of R100,000 received in 10 years at 6%**

Use the formula  
$$
PV = \frac{FV}{(1 + r)^n}
$$
with  
$$FV = 100\,000$$,  
$$r = 0.06$$, and  
$$n = 10$$.

Thus,  
$$
PV = \frac{100\,000}{(1.06)^{10}}.
$$

Calculating $$(1.06)^{10}$$:
$$
(1.06)^{10} \approx 1.79085.
$$

So,  
$$
PV \approx \frac{100\,000}{1.79085} \approx 55\,837.00.
$$

---

**c) Amount to Invest Today to Accumulate R1,000,000 in 8 years at 5%**

We again use the present value formula:
$$
PV = \frac{FV}{(1 + r)^n}
$$
with  
$$FV = 1\,000\,000$$,  
$$r = 0.05$$, and  
$$n = 8$$.

Thus,  
$$
PV = \frac{1\,000\,000}{(1.05)^8}.
$$

Calculating $$(1.05)^8$$:
$$
(1.05)^8 \approx 1.47746.
$$

So,  
$$
PV \approx \frac{1\,000\,000}{1.47746} \approx 677\,000.
$$

---

**d) Present Value of an Annuity: R5,000 per Month for 3 Years at 12% per annum**

For monthly payments, first calculate the monthly interest rate and number of payments.  
$$r_{monthly} = \frac{12\%}{12} = 0.01$$,  
$$n = 3 \times 12 = 36$$.

The present value of an annuity is given by  
$$
PV = PMT \times \frac{1 - (1 + i)^{-n}}{i},
$$
where $$PMT = 5\,000$$ and $$i = 0.01$$.

Thus,  
$$
PV = 5\,000 \times \frac{1 - (1.01)^{-36}}{0.01}.
$$

Calculate $$(1.01)^{-36}$$:  
$$(1.01)^{36} \approx 1.431$$ so  
$$(1.01)^{-36} \approx \frac{1}{1.431} \approx 0.699$$.

Then,  
$$
PV \approx 5\,000 \times \frac{1 - 0.699}{0.01} = 5\,000 \times \frac{0.301}{0.01} = 5\,000 \times 30.1 \approx 150\,500.
$$

---

**e) Future Value of an Investment: R25,000 Annually for 10 Years at 7%**

The future value of an annuity (with payments made at the end of each period) is given by  
$$
FV = PMT \times \frac{(1 + r)^n - 1}{r},
$$
with $$PMT = 25\,000$$, $$r = 0.07$$, and $$n = 10$$.

Thus,  
$$
FV = 25\,000 \times \frac{(1.07)^{10} - 1}{0.07}.
$$

Calculate $$(1.07)^{10}$$:
$$
(1.07)^{10} \approx 1.96715.
$$

So,  
$$
FV \approx 25\,000 \times \frac{1.96715 - 1}{0.07} = 25\,000 \times \frac{0.96715}{0.07} \approx 25\,000 \times 13.8164 \approx 345\,410.
$$

---

**f) Present Value of an Annuity: R10,000 per Year for 15 Years at 9%**

Use the annuity present value formula  
$$
PV = PMT \times \frac{1 - (1 + r)^{-n}}{r},
$$
with $$PMT = 10\,000$$, $$r = 0.09$$, and $$n = 15$$.

Thus,  
$$
PV = 10\,000 \times \frac{1 - (1.09)^{-15}}{0.09}.
$$

Calculate $$(1.09)^{15}$$:
$$
(1.09)^{15} \approx 3.643.
$$

So,  
$$
(1.09)^{-15} \approx \frac{1}{3.643} \approx 0.2744.
$$

Then,  
$$
PV \approx 10\,000 \times \frac{1 - 0.2744}{0.09} = 10\,000 \times \frac{0.7256}{0.09} \approx 10\,000 \times 8.0622 \approx 80\,622.
$$

---

**g) Present Value of a Project with Cash Flows of R50,000 per Year for 6 Years at 8%**

Treating the cash flows as an annuity, use  
$$
PV = CF \times \frac{1 - (1 + r)^{-n}}{r},
$$
with $$CF = 50\,000$$, $$r = 0.08$$, and $$n = 6$$.

Thus,  
$$
PV = 50\,000 \times \frac{1 - (1.08)^{-6}}{0.08}.
$$

Calculate $$(1.08)^6$$:
$$
(1.08)^6 \approx 1.58687,
$$
so  
$$
(1.08)^{-6} \approx \frac{1}{1.58687} \approx 0.63017.
$$

Then,  
$$
PV \approx 50\,000 \times \frac{1 - 0.63017}{0.08} = 50\,000 \times \frac{0.36983}{0.08} \approx 50\,000 \times 4.6229 \approx 231\,145.
$$
**a) Future Value:** R73,466.50 **b) Present Value:** R55,837.00 **c) Investment Amount:** R677,000 **d) Present Value of Annuity:** R150,500 **e) Future Value:** R345,410 **f) Present Value of Annuity:** R80,622 **g) Present Value:** R231,145

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