Two different answers for same question in compound interest

Question

Here is the question A sum of 25000 is compounded annually at 6.5 percent per annum for 3 years.Calculate the amount received after 3 years?

When I solved it using this formula C.A :-Final Amount P:-Principle amount R:- Rate T:- Period

C.A=P(1+R/100)^T

C.A.=3019.87

While When I used the differential equation

So:-Initial Amount S:- Final Amount R:- Rate T:- Time Period

S=So * e^(RT)*

S=30382.77

Why are the two answers different shouldn't they should be equal?? Thank you.

Answer 1

The second answer represents the result of compounding continuously, while the first answer represents the result of compounding only once per year.

The more frequently compounding occurs at a fixed rate, the more interest is earned. Continuous compounding is more frequent than any periodic compounding (in the limit, the time between subsequent compoundings drops to zero).

The continuous compounding rate $\delta$ that gives the same result as the annual compounding rate $6.5\%$ can be determined by solving

$$e^{\delta} = 1.065$$ $$\delta = \ln 1.065 \approx 0.062975 = 6.2975\%$$

In other words, continuous compounding at $6.2975\%$ is equivalent to annual compounding at $6.5\%$.

Answer 2

The answers are different because the formulae are for different circumstances.

For the first formula, the interest is assumed to be paid yearly. So, the interest of the first day, starts earning interest only in the second year.

For the second formula, the interest of the first day starts earning further interest on the second day and so on.Thus, the final amount is larger.