Annual = Years = 6.64; Actually 7 years
Monthly = Years = 6.33; 6 Years, 4 months
Daily = Years = 6.30; 6 Years, 111 days
Continuously = 6.30; 6 Years, 110 days
The formula for compound interest is
FV = P*(1 + R/n)^(nt)
where
FV = Future Value
P = Principle
R = Annual interest rate
n = number of periods per year
t = number of years
For this problem, we can ignore p and concentrate on the (1+R/n)^(nt) term, looking for where it becomes 2. So let's use this simplified formula:
2 = (1 + R/n)^(nt)
With R, n, and t having the same meaning as in the original formula.
For for the case of compounding annually
2 = (1 + R/n)^(nt)
2 = (1 + 0.11/1)^(1t)
2 = (1.11)^t
The above equation is effectively asking for the logarithm of 2 using a base of 1.11. To do this take the log of 2 and divide by the log of 1.11. So
log(2) / log(1.11) = 0.301029996 / 0.045322979 = 6.641884618
This explanation of creating logarithms to arbitrary bases will not be repeated for the other problems.
The value of 6.641884618 indicates that many periods is needed. 6 is too low giving an increase of
1.11^6 =1.870414552
and 7 is too high, giving an increase of 1.11^7 = 2.076160153
But for the purpose of this problem, I'll say you double your money after 7 years.
For compounding monthly:
2 = (1 + R/n)^(nt)
2 = (1 + 0.11/12)^(12t)
2 = (1 + 0.009166667)^(12t)
2 = 1.009166667^(12t)
log(2)/log(1.009166667) = 0.301029996 / 0.003962897 = 75.96210258
And since the logarithm is actually 12*t, divide by 12
75.96210258 / 12 = 6.330175215
Which is 6 years and 4 months.
For compounding daily:
2 = (1 + 0.11/365)^(365t)
2 = (1 + 0.00030137)^(365t)
2 = 1.00030137^(365t)
log(2)/log(1.00030137) = 0.301029996 / 0.000130864 = 2300.334928
2300.334928 / 365 = 6.302287474
Continuously:
For continuous compounding, there's a bit of calculus required and the final formula is
FV = Pe^(rt)
where
FV = Future value
P = Principle
e = mathematical constant e. Approximately 2.718281828
r = Interest rate
t = time in years
Just as before, we'll simplify the formula and use
2 = e^(rt)
Since we have the function ln(x) which is the natural log of x, I won't bother doing log conversions.
rt = ln(2)
0.11 * t = 0.693147181
t = 0.693147181 / 0.11
t = 6.301338005
