How long will it take an investment to double in value if the interest rate is 11% compounded continuoulsy and what is the equivalent annual interest rate?
The best way to approach this is to use the traditional continuous compound interest formula:
`A(t) = A_0 e^(rt)`
Here, `A(t)` is the amount after a given amount of time, `A_0` is the initial amount you invest, `r` is the interest rate (11%), and `t` is the time in years.
Let's answer the first part, where we calculate the amount of time that is needed for the investment to double. In other words, we want to find when `A(t)` would be `2A_0`. So, let's substitute our given value of `r` and our value for `A(t)` into the continuous compound interest formula:
`2A_0 = A_0e^(0.11t)`
The problem asks us to solve for the amount of time, so let's go ahead and do that now. Start by dividing both sides by `A_0`:
`2 = e^(0.11t)`
Now, to bring down the exponent, we need to take the natural logarithm of both sides (see link below for more information on why we do this):
`ln2 = 0.11t`
Now, we simply divide both sides by 0.11 and solve in a calculator to get our result for t:
`6.30 = t`
So, in order for our initial investment to double, we need to invest it for around 6.30 years.
To find the equivalent annual interest rate, we use the same initial formula, with a slight exception. Here `t` will be 1 because we're looking at interest accrued each year. However, our `A(1)` will be slightly different:
`A(t) =A(1)= (1+r_e)A_0`
Here, notice that we are multiplying `A_0` by `(1+r_e)` . This coefficient, by definition, will give us the equivalent annual interest rate, `r_e`. Intuitively, we're simply seeing by what proportion of `A_0` we are increasing after the year is over. So, let's put these new values into our original interest equation and solve for `r_e`. Notice that `r` is still going to be 11%!
`(1+r_e)A_0 = A_0e^(0.11*1)`
Start by dividing both sides by `A_0`:
`1+r_e = e^0.11`
Now, let's subtract 1 from both sides:
`r_e = e^0.11 - 1`
Finally, let's just put that expression into a calculator to find our final result:
`r_e = 0.116`
So, our rough annual interest rate is 11.6%. It is not much of a change, but over time, that small difference can make a much bigger difference!
I hope this helps!