you invest $20,000 in an account where the interest is compounded continuously and in 5 years you have $32,000

Question

find nominal interest rate and effective rate

Answer 1

 

Hi Kay,

 

According to the formula for continuous compounding,

 

     32,000 = 20,000 · e^5r

 

where r is the nominal interest rate.

 

     e^5r = 32,000/20,000

 

     5r = ln(32/20)

 

     r = (1/5)·ln(32/20)

 

       = 0.094

 

So, the nominal interest rate is 9.4%.

 

As e^r = (1 + i), where i is the effective interest rate,

 

     i = e^r - 1

 

       = e^0.094 - 1

 

       = 0.0986

 

So, the effective interest rate is 9.86%.

 

 

 

Answer 2

Use the formula V = Pe^rt (V = P times "e" raised to the "rt" power), in which V = final value = 32000, P = principal or initial value = 20000, e = constant 2.71828..., r = annual percentage rate, and t = number of years = 5.

 

32000 = 20000 * (e ^ rt)

 

First, divide out the 20000.

 

1.6 = e ^ rt

 

Next, take the natural log of both sides of the equation.  The rt is no longer an exponent.

 

ln (1.6) = rt * ln (e)

 

ln (e) = 1, so that term drops out.

 

ln (1.6) = rt

 

0.47 = rt

 

We know t = 5.

 

0.47 = 5r

 

0.094 = r

 

Annual percentage rate = 9.4%.

 

Check:  e ^ 0.47 = 1.6

1.6 * 20000 = 32000