I am currently studying a chapter called "An Economic Interpretation of e" in my Economics class and we are finding amounts of compounded interest. I am not actually looking for help on the problems but rather I have a mathematical question. I have a few different equations in the book to use, but I am getting two different answers, one from one equation and one from all the others.
(Also sorry if I'm explaining any obvious things past this, I just want to be as clear as possible)
My first equation is V1 = Ae^rt where:
A = principle;
e = e (2.71828);
r = rate;
t = years
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My other equations are V2 = A( 1 + r/m ) ^mt where:
A = principle;
e = e (2.71828);
r = rate;
t = years;
m = compounding periods per year
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Lastly: V3 = A[(1 + 1/w) ^w] ^rt where:
all the variables are the same as the previous equation with the addition of w = m/r.
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Using the values:
A = $80; r = 10%; m = semi annually (2); and t = 4
I use the same values in all three equations and get the values:
V1 = 119.34597
V2= 118.1964
V3= 118.1964
My question there for is, which is better? Why is it different? Is it because e is causing the numbers to round differently or is e actually more accurate? I understand that V2 and V3 are just alternatives forms of each other, but the book explains V1 as:
V1 =(Triple Bar)= lim [m-> infinity] V(m) = Ae^rt
So why is it different from the other equations, and why a greater value? This is purely curiosity so let me know if I need to explain anything else about my question as this is not a formal question and might be missing some information... thank you!
V2 is the exact formula,
V1 is just an approximation which should be used only if like you are compounding the interest after each day, so you can't use the formula of V2, (it will be difficult to calculate something to the power of $365*t$)
As m tends to infinity, the closed form tends to e
its the general formula of calculus,
$$\lim _{n\rightarrow \infty } \left( 1+{\frac {x}{n}} \right) ^{n} = {{\rm e}^{x}}$$
so in this case m is very large, (each day compound counts to m=365)
so you can use formula of V1, but note that, lesser the value of m, more is the inaccuracy produced, more you are compounding it.