Find the accumulated value of $2,480 at the end of twelve years if the nominal interest rate was 2% convertible monthly for the first three years, the nominal rate of discount was 3% convertible semiannually for the next two years, the nominal rate of interest was 4.2% convertible once every two years for the next four years, and the annual effective rate of discount was .058 for the last three years.

In order to properly tackle this problem, we must understand the relationship between the nominal annual rate and real (effective) annual rate. 

To do this:
  -First you take the nominal rate, divide by the number of times it's compounded (converted) per year.
   -Then, add one to that number, and raise that number to the power of how many times you compound per year.

Here is the method in practice:
First 3 Years: 
Nominal rate= 2% ÷ 12 times/yr = 0.001667
Effective rate = 1.001667 ^12 = 1.020184

Next 2 Years (Discounting)
3% ÷ 2/yr = .015
1.015 ^ 2 = 1.061364

Next 4 years (Interest)
.042 ÷ .5 (once every 2 years) = .084
1.084 ^ (1/2) = 1.041153

The last 3 years are already expressed as an effective rate, so we don't need to convert them. The annual rate is:
1.058

I kept the 1 in the numbers (1.058 instead of 5.8% for example) so that it's easier to find the final number

Take every relevant number and raise it to the power of the number of years it's compounded for. For discounting, raise it to a negative power.
First 3 years: 1.020184 ^ 3 = 1.061784
Next 2 years: 1.030225 ^ -2 = .942184
Next 4 years: 1.041163 ^ 4 = 1.175056
Last 3 years: 1.058 ^ -3 = .84439

Multiply these numbers (include all decimals when you do this calculation)
1.062 * .942 * 1.175 * .844 = .992598

This is our final multiplier to find the effect on our principal:
.992598 * 2,480 = 2461.64

Answer is 2461.64

Nucleus