An investment occount pays interest at the rate
of per year, compounded semi-annually What
is the amount that should be deposited now so
that R250 can be withdrawn at the end of every
six months for the next ten years?

Question

An investment occount pays interest at the rate
of

per year, compounded semi-annually What
is the amount that should be deposited now so
that R250 can be withdrawn at the end of every
six months for the next ten years?

UpStudy AI · Accepted Answer

1. We first determine the interest rate per semi‐annual period. Since the annual rate is $$5\%$$ compounded semi-annually, the rate per period is
   $$
   i = \frac{5\%}{2} = 0.025.
   $$

2. The withdrawals of $$R250$$ occur at the end of each six‐month period for 10 years. The total number of periods is
   $$
   n = 10 \text{ years} \times 2 = 20 \text{ periods}.
   $$

3. We use the present value formula for an annuity immediate, which is
   $$
   PV = R \times \frac{1 - (1+i)^{-n}}{i},
   $$
   where $$R = R250$$, $$i = 0.025$$, and $$n = 20$$.

4. Substitute the values into the formula:
   $$
   PV = 250 \times \frac{1 - (1.025)^{-20}}{0.025}.
   $$

5. Calculate $$(1.025)^{-20}$$:
   - First, compute $$(1.025)^{20}$$. Approximately, 
     $$
     (1.025)^{20} \approx 1.6386,
     $$
   - Hence,
     $$
     (1.025)^{-20} \approx \frac{1}{1.6386} \approx 0.6105.
     $$

6. Now, compute the numerator:
   $$
   1 - 0.6105 = 0.3895.
   $$

7. Divide by the interest rate per period:
   $$
   \frac{0.3895}{0.025} \approx 15.58.
   $$

8. Finally, multiply by $$R250$$:
   $$
   PV \approx 250 \times 15.58 \approx 3895.
   $$

Thus, the amount that should be deposited now is approximately $$R3895$$.
You should deposit approximately R3895 now to withdraw R250 every six months for the next ten years with a 5% annual interest rate compounded semi-annually.

An investment occount pays interest at the rate
of per year, compounded semi-annually What
is the amount that should be deposited now so
that R250 can be withdrawn at the end of every
six months for the next ten years?

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An investment occount pays interest at the rate of per year, compounded semi-annually What is the amount that should be deposited now so that R250 can be withdrawn at the end of every six months for the next ten years?

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Latest Economics Questions

An investment occount pays interest at the rate
of per year, compounded semi-annually What
is the amount that should be deposited now so
that R250 can be withdrawn at the end of every
six months for the next ten years?