What interest is needed for an investment to triple in four years if it is compounded (a) semiannually and (b) continuously?

Answer 1

Interest compounded semiannually fits the equation: A(t) = A₀(1 + r/n)^nt where A(t) is the amount of money you have at any time "t" (where t is in years so in this case t = 4), A₀ is the amount of money you start with (at time t = 0), "r" is the interest rate (decimal form), and "n" is the number of compounding periods per year (in this case n = 2). So, since you are tripling your money A(t) = 3A₀ making the equation:

3A₀ = A₀(1 + r/2)^2•4

3 = (1 + r/2)⁸

eight root of 3 = 1 + r/2

1.1472 = 1 + r/2

0.1472 = r/2

r = 0.2944 = 29.44%

Continuously compounded money fits the equation:

A(t) = A₀e^rt where, again where A(t) is the amount of money you have at any time "t" (where t is in years so in this case t = 4), A₀ is the amount of money you start with (at time t = 0), "r" is the interest rate (decimal form) so, our equation becomes:

3A₀ = A₀e^4r

3 = e^4r

ln(3) = ln(e^4r)

ln(3) = 4rln(e)

ln(3) = 4r(1)

r = ln(3)/4 = 0.274653 = 27.4653%