Pádraig is 25 years old and is planning for his pension - Leaving Cert Mathematics - Question 8 - 2014

Using the same formula:

PV = rac{F}{(1 + i)^n}

we substitute:

• $F = €20,000$
• $i = 0.03$
• $n = t$

Thus, the present value can be expressed as:

PV = rac{20,000}{(1 + 0.03)^t}

The required fund would give payments of €20,000 at the start of each year for 25 years. This can be modeled as a geometric series:

• First payment is €20,000.
• The ratio $r$ of growth per year is $1.03$ (due to the 3% interest rate).

The sum of a geometric series is given by:

A = a rac{1 - r^n}{1 - r}

Where:

• $a = €20,000$
• $r = (1 + 0.03) = 1.03$
• $n = 25$ (number of payments)

Thus the value on the date of retirement is:

A = 20,000 rac{1 - (1.03)^{25}}{1 - 1.03} = 20,000 rac{(1.03^{25} - 1)}{0.03} \\ A ≈ €358,711 \\ ext{(rounded to the nearest euro)}

To find the monthly interest rate $i$, we solve:

$$(1 + i)^{12} = 1.03$$

Taking the twelfth root:

$$1 + i = 1.03^{1/12}\\ i = 1.03^{1/12} - 1 \\ i ≈ 0.002466 \\ ext{(correct to four significant figures)}$$

Thus, the monthly interest rate is approximately 0.2466%.

Using the future value formula for a single payment:

$$$$

Where:

• $P = payment made$
• $i = 0.002466$ (monthly interest rate)
• $n = number of months before retirement$

Thus, the value on the retirement date is: $ext{Future Value} = P(1.002466)^n$

To find the value of P, we express the total fund required as a sum of a geometric series:

$$P imes (1.002466 + (1.002466)^2 + ... + (1.002466)^{480}) = 358711$$

The sum of the series can be calculated using:

$$$$

Solving for P we get:

$$$$

If Pádraig delays his investment for 10 years, the number of months until retirement becomes:

$$$$

Using similar calculations:

$$$$

Thus,

$$$$

From this, we solve for P to find the new monthly payment amount.