Compound Interest Calculation (Years + Months)

Question

My question is with regards to the calculation of "Compound Interest". I have the formula below where I would get an answer to the total value of the investment over a period of "years".

• $A$ = Future value
• $P$ = Principal amount
• $R$ = Annual interest rate
• $N$ = Number of times compounded each year
• $T$ = The number of years the money is invested for

$$A = P\left(1 + \frac{R}{N} \right) ^{NT}$$

So for example if I have the following:

$P = 5,000\$$

$R = 5\%= 0.05$

$N = 12$ (Compounded monthly)

$T = 10$ years

The answer for A will be equal to $8,235.05

My question is how can I derive the equation above to account for the period of years and months? So, for example, how would I calculate $A$ if I had $T$ being equals to $T$ = $10$ years + $6$ months?

I think that the answer to the equation derivation is shown below but I'm not sure:

$$A = P\left(1 + \frac{R}{N}\right)^{N(10 + 1/2)}$$

Can anyone confirm if my calculations are correct?

Answer 1

Your answer is correct.

The point is you need to match the interest rate periods with the compounding periods. So if (R/N) is the rate for a 1-month period, then "NT" must be the number of compounding months. Since you are compounding for 10.5 years, this represents 126 months (10.5 * 12). If, on the other hand, your compounding is semiannual (as is usual with bonds), NT = 10.5 * 2.

Answer 2

If your compound period is monthly, then what you have is correct.