Effect of Compounding on Principle Amount (and Ongoing Fund Injection)

It is well-known that the formula to calculate the effect of compounding on a principal amount is this:

(Principal Amount) x (1 + percentage of returns or interest per time unit)^(number of time unit)

So not withstanding fluctuations in returns or the commission fee charged, if an investor has put in a principal sum of $100,000 on a stock that gives 6% dividends annually and then reinvests into the same stock over a period of 5 years, his principal sum would have grown to $133,822.

However, this can only be applied to the case of starting out with a principal sum.

If you are actively periodically injecting funds into say, a Index Fund or an endowment plan, the calculation method is slightly different. The same applies to when you are paying off a long-term loan, such as a housing loan. Again this assumes no fluctuation in the percentage of returns or interest.

(Total Principal Amount) x (1 + percentage of returns or interest per time unit)^(number of time unit/2)

Notice the number of time unit is divided into 2. Remember how area of a triangle is calculated? You had to multiply its base by height and divide the result by 2.

 

(Source: Khan Academy)

The principle is the same - the number of time unit is divided by 2. This is to account for the reduction in effect of compounding on the funds that are injected later on (in the case of putting money into endowment plan or index periodically) or the effect on the amount over time as you pay off a loan.

If you have taken up a $300,000 HDB housing loan at a 2.6% interest rate (assuming no change) over 25 years, and you plan to stick to paying it off regularly (no early partial or full repayment), the total effective amount that needs to be paid off would be:

$300,000 x (1 + 0.026)^(25/2) = $569,908.68

This translates into a monthly repayment of $1899.70.

How about a case where both the periodic capital injection and having a principal sum "initially" applies?

(Total Principal Amount) x (1 + percentage of returns or interest per time unit)^(number of time unit before maturity/2 + number of time unit after maturity)

You inject $3000 yearly into a 20-year endowment plan that gives a return of 4% and it has reached its 20 years maturity, and you have the option of withdrawing all of it now, draw down a small amount periodically, or not to touch it at all until later on.

You decided to let it continue growing, intending to withdraw all of it in another 26 years.

Principal Amount = $3000 x 20 years = $60,000

$60,000 x (1 + 0.04)^(20/2 + 26) = $60,000 x 1.04^36
= $246,235.95

Your periodic capital injection applies prior to maturity. Upon maturity, this becomes your "initial" principal base sum to calculate the amount with 26 years of further compounding after maturity.

This also demonstrate the effect of compounding after maturity of an endowment plan. I imagine unless the person is very confident of beating the returns of an endowment plan, it may be a better idea to let it continue growing if there is no need to touch this sum yet.

In a later post, I'll re-visit the case of paying off a $300,000 HDB Housing Loan demonstrated earlier, and the effect of early repayment on his total effective amount to be paid.

Stay tuned, and thanks for reading!

Update: Check out the Part 2 of the Effect of Compounding series by clicking here!