Determine an interest rate less than 15%, a period of investment greater than two years, and a regular payment that will result in the total amount of interest you earn being equal to the total amount of money you put in? (for example, under what conditions will you have a future value of $1 000 000, having earned $500 000 interest?)
In effect what you are asking is, what is the relation between time and interest rate for money to double.
Before the ease of calculations, we used a procedure called the "rule of 72" which gave an approximation for that effect.
e.g. for a rate of 6% it would take appr 12 years for money to double, 6*12 = 72
check: 1.06^12 = 2.012 , almost doubled
or , 8% and 9 years, 1.08^9 = 1.999
so let's take your max of 15%, it would take 72/15 or 4.8 years
check : 1.15^4.8 = 1.96
and at the low end of 2 year, the rate would be 72/2 or 36% according to that 72 run, but that is way too high.
oobleck kicks in here with (1+r)^n = 2
pick any n>2 and you can find r
pick any r < .15 and you can find n
e.g. If r = .12 or 12%
1.12^n = 2
n = log2/log1.12 = 6.11 or about 6 years
Notice that still validates the rule of 72, 6*12 = 72
you want (1+r)^n = 2
so, for example, if n=5, you want (1+r)^5 = 2, so r = 14.87%
To determine the interest rate, period of investment, and regular payment that will result in the total amount of interest being equal to the total amount of money you put in, we need to use the future value formula and solve for the interest rate.
The future value formula is given by:
FV = P * (1 + r)^n
Where:
FV = Future Value
P = Regular Payment
r = Interest Rate per period
n = Number of periods
In this case, we want the future value (FV) to be equal to the total amount of money put in. So we have:
FV = Principal Amount + Interest Earned
Let's say the principal amount is denoted by P0.
Therefore, FV = P0 + P0 = 2 * P0
We also know that the future value is $1,000,000, and the interest earned is $500,000. Thus, we have:
$1,000,000 = 2 * P0
To solve for P0, we divide both sides by 2:
P0 = $1,000,000 / 2
P0 = $500,000
Now that we know the principal amount, we can proceed to find the interest rate.
Using the future value formula, we have:
$1,000,000 = $500,000 * (1 + r)^n
Let's assume the period of investment is denoted by n.
To isolate the interest rate (r), we need to rearrange the formula:
(1 + r)^n = $1,000,000 / $500,000
(1 + r)^n = 2
Now, we can solve for the interest rate by taking the logarithm of both sides. Let's assume we use the natural logarithm (ln):
ln[(1 + r)^n] = ln(2)
Apply logarithm properties:
n * ln(1 + r) = ln(2)
Now, we can solve for the interest rate (r) by dividing both sides by n and taking the exponential:
ln(1 + r) = ln(2) / n
1 + r = e^(ln(2) / n)
Subtract 1 from both sides to isolate r:
r = e^(ln(2) / n) - 1
This expression will give you the interest rate (r) required to have a future value of $1,000,000, with an interest earned equal to the principal amount of $500,000, provided you have a period of investment greater than two years.