Applying Compound Interest to Partial-Year Periods
Summary
The discussion explains how to use the standard compound-interest formula when an investment lasts a combination of whole years and months. The key is to express the total duration in years and ensure that the exponent counts the number of compounding periods. With monthly compounding, a duration of ten years and six months is represented as 10.5 years, giving 126 monthly periods. The same logic applies to other compounding frequencies: multiply the duration in years by the number of periods per year.
The answers confirm the proposed formula for monthly compounding and note that semiannual compounding uses two periods per year. The method assumes a fixed nominal annual rate divided by the compounding frequency and regular compounding intervals. It is a basic calculation explanation; it does not address changing rates, irregular deposit or payment schedules, fees, taxes, or continuous compounding.
Key ideas
- Convert the full investment duration, including months, into years before applying the formula.
- The exponent is the number of compounding periods, found by multiplying years by periods per year.
- Monthly compounding over ten and a half years corresponds to 126 periods.
- The same duration uses a different period count when the compounding frequency changes.
- The formula assumes a fixed rate and regular compounding intervals.
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Full text
# Compound Interest Calculation (Years + Months)
# Compound Interest Calculation (Years + Months)
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 by vega (score 0, accepted)
https://quant.stackexchange.com/a/18286
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 by Tom Sun (score 0)
https://quant.stackexchange.com/a/18279
If your compound period is monthly, then what you have is correct.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.