Question

You have a choice to invest in one of three different college funds to save up for college. You need to save at least $14,000 for your first year. In which of the three accounts do you choose to invest your money given that you have $10,000 to invest and you would like to accrue as much interest as possible?​

Answer 1

Answer: C) You choose account #2

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Explanation:

Let's compute the final balance for account 1.

We'll use the compound interest formula

A = P*(1+r/n)^(n*t)

where,

• P = principal or amount deposited
• r = decimal form of the annual interest rate
• n = number of times we compound the interest per year
• t = number of years

where in this case

• P = 10,000
• r = 0.035
• n = 4
• t = 10

So with all that in mind, we get

A = P*(1+r/n)^(n*t)

A = 10,000*(1+0.035/4)^(4*10)

A = 14,169.0883793113

A = 14,169.09

Account #1 gets us $14,169.09 which is over our goal of $14,000

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Let's repeat this idea for account 3. I'll get back to account 2 in the next section.

For account 3, we have these inputs

• P = 10,000
• r = 0.032
• n = 1
• t = 10

leading to...

A = P*(1+r/n)^(n*t)

A = 10,000*(1+0.032/1)^(1*10)

A = 13,702.4104633564

A = 13,702.41

This isn't larger than 14,000. So we cannot use account 3. The interest rate is too small and needs to be bumped up.

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Now let's compute account 2. We'll use the continuously compounded interest formula which is

A = P*e^(rt)

The P, r and t are the same as before. The 'e' is a constant and it's roughly e = 2.718... similar to how pi = 3.14... approximately.

I recommend using the calculator's stored value of 'e' to get as much accuracy as possible.

So,

A = P*e^(rt)

A = 10,000*e^(0.036*10)

A = 14,333.2941456052

A = 14,333.29

This exceeds 14,000 so we can use this account.

Comparing this to account 1's value (14,169.09), we can see that account 2 gets us the most money over the 10 year time period. Not only does account 2 have the higher interest rate, but it also has an infinitely higher compounding frequency as well. These two factors help contribute as to why account 2 beats out account 1.

This is why we should go for account 2.