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Suppose you invest a dollars to earn an annual interest rate of r percent (as a decimal). After t years, the value of the investment with interest compounded yearly is A(t)=a(1+r) t . The value with interest compounded continuously is A(t)=a . e rt
a. Explain why you can call e r-1 the effective annual interest rate for the continuous compounding.

Respuesta :

For continuous compounding, the effective annual interest rate is e^r - 1.

What do we mean by continuous compounding formula?

  • When an issue expressly states that the amount is "constantly compounded," the continuous compounding formula should be used.
  • This formula employs the mathematical constant "e," whose value is approximately 2.7182818.

The following is the continuous compounding formula:

  • A = Pe^rt
  • Where P equals the starting sum, A is the total sum, r equals the interest rate, t = time and e is a mathematical constant.

So,

Given: A(t) = a(1+r)^t and A(t) = a.e^rt.

Let the effective rate be R.

Then,  

  • a(1+R)^t = a.e^rt
  • (1+R)^t = e^rt
  • (1+R) = e^r
  • R = e^r - 1

Therefore, for continuous compounding, the effective annual interest rate is e^r - 1.

Know more about continuous compounding formula here:

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