On the day his son was born, a father deposited to a trust company a sufficient amount of money so that his son could receive money in the future.

Question

On the day his son was born, a father deposited to a trust company a sufficient amount of money so that his son could receive money in the future. There was a provision that the son can withdraw an amount of P10,000 each year for 3 years starting on his 18th birthday and P15,000 eight months after the last withdrawal. The interest rate is 5% compounded monthly.

1. What is the effective interest rate?

2. How much did the father deposit?

3. How much money can he receive on his 18th birthday, if he decided to withdraw all the money at once?

Answer 1

Let's break down the problem step by step:

1. Effective Interest Rate: The effective interest rate is the rate at which an investment grows when compounding occurs more frequently than annually. In this case, the interest is compounded monthly at a rate of 5% per year.

To find the effective interest rate (reffreff​), we'll use the formula for compound interest:

reff=(1+rn)n−1,reff​=(1+nr​)n−1,

where rr is the nominal interest rate (5% or 0.05) and nn is the number of times compounding occurs per year (12 for monthly compounding). Plugging in the values:

reff=(1+0.0512)12−1≈0.0512 or 5.12%.reff​=(1+120.05​)12−1≈0.0512 or 5.12%.

1. Initial Deposit: The problem states that the father deposited an amount of money that would allow his son to withdraw specified amounts over a period. We'll use the future value of an annuity formula to calculate the initial deposit (PP):

P=A(1−1(1+reff)nt)reff,P=reff​A(1−(1+reff​)nt1​)​,

where AA is the annual withdrawal amount ($10,000), reffreff​ is the effective interest rate, nn is the compounding frequency per year (12 for monthly), and tt is the number of years (3 for the withdrawals and 3 years and 8 months for the final withdrawal). Plugging in the values:

P=10000(1−1(1+0.0512)12⋅3.667)0.0512≈$31,171.12.P=0.051210000(1−(1+0.0512)12⋅3.6671​)​≈$31,171.12.

1. Withdrawal on 18th Birthday: If the son decided to withdraw all the money at once on his 18th birthday, we need to find the future value of the deposit after 18 years using the compound interest formula:

FV=P×(1+reff)18⋅12.FV=P×(1+reff​)18⋅12.

Plugging in the values:

FV=$31,171.12×(1+0.0512)216≈$82,416.05.FV=$31,171.12×(1+0.0512)216≈$82,416.05.

So, if the son decided to withdraw all the money on his 18th birthday, he could receive approximately $82,416.05.

To summarize:

1. Effective interest rate: 5.12%
2. Initial deposit: $31,171.12
3. Withdrawal on 18th birthday: $82,416.05.