How long (in years) would it take for an investment of $1000 to double in value if it earns 5% compounded weekly?

Answer 1

The compound interest formula is : A = P (1 + r/n)^(nt)

A = final amount

P = principle or starting amount

r = interest rate (remember 5% = 0.05

t = time period (days, weeks, months, years, etc.)

n = number of times applied per time period

While it's pretty easy to plug in the numbers and get an answer, I think it might be more educational to try and derive the formula using a little bit of basic math and logical thinking.

What does it mean for money to be compounded at 5% weekly? It means that it will grow 5% annually (per year) and that 5% will be divided equally among the 52 weeks of a year.

So 5% weekly actually means 5%/52 or 0.0961% every week. That's one of the things that makes compound interest confusing.

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Advice: Students usually have a hard time with percentages and percentage growth. Students should always review these topics before trying compound interest problems.

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$1 compounded annually at 5% = $1.00 * 1.00+0.05 = $1.05

Pretty easy, right? Let's try weekly compounding on $1

$1 compounded weekly at 5% = $1.00 * 1.00+0.000961 = $1.000961 (1st week)

$1 compounded weekly at 5% = $1.00 * 1.00961 * 1.00961 = $1.00192 (2nd week)

$1 compounded weekly at 5% = $1.00 * 1.00961 * 1.00961 * 1.00961 = $1.00288 (3rd week)

This could take a while if we do all 52 weeks. Thankfully there is a pattern. Every week we just multiply by 1.000961 which adds a little interest. Repeated multiplication is where exponents come in. Let's look at week 4:

$1 compounded weekly at 5% = $1.00 * (1.00+0.000961)^4 = $1.00384 (4th week)

We can finally calculate week 52 which is the end of the first time period, T, one year.

$1 compounded weekly at 5% = $1.00 * (1.00+0.000961)^52 = $1.05121 (52nd week)

So to compare

$1 compounded annually at 5% = $1.05 at the end of the year

$1 compounded weekly at 5% = $1.0512 at the end of the year

As we can see 5% compounded weekly results in just a teeny bit more growth. With rounding there is no difference at all! That's becuase we started with such a small number. If our original investment was $1000, that would be an extra $1. If our original investment was $10,000 that would be an extra $12, $100,000 would yield an extra $121 during the year compared to a regular ole' 5% compounded once per year account.

Alright, let's solve our problem.

We start with $1000 and it grows by 0.0961% each time (every week). Let's see how many weeks it will take to get to $2000. Let's write an algebra equation to make it easier to solve.

1000*(1.00961)^? = 2000

Essentially we just need to figure out how many times to compound to get to our final destination. A proper equation with a variable looks like this:

1000*(1.00961)^w = 2000 (where w is weeks)

Simplifying:

1.00961^w = 2000/1000

w = log_1.009612

w = 72.473

This means it will take more than 72 weeks and less than 73 weeks for our investment to double.

How many years is that? 72 weeks / 52 weeks = 1.39 years

That seems awfully fast. Remember up above we found that $1000 compounded at 5% once a year would give us $1050 and compounded weekly at 5% would give us $1051.

Can you spot the error?

It's always good to be able to do a sanity check at the end for all your answers. It turns out we accidentally moved our decimal place. We used 0.00961 instead of 0.000961. Trying again:

w = log_1.0009612

w = 721.62

That's 13.87 years. Much better!

Let's plug our numbers into the formula now that we understand how it works:

2000 = 1000 (1 + 0.05/52)^52t (where t is in years and n is weeks)

2000 = 1000 (1 + 0.000961)^52t

If you try to solve this equation using algebra, you'll find that the last step is to divide by 52 which means your answer will be in years.

Hope that helps!

Answer 2

1000 ( 1 + 5/100 * 1/ 52)^t = 2000

(1 + 0.01923076923)^t = 2

t Log (1.0923076923) = Log(2)

t = Log 2 / Log(1.0923....)

t = 7.85....