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Given

principal initial amount
time number of years
rate interest rate as decimal n number of periods per time to compound principal by rate
a amount to add to principal at either beginning or end of each n period

where

principal is equal to 1
time is equal to 1
rate is equal to 0.03, or 3%
n is equal to 12, or monthly
a is equal to 10

The result that derived here is 124.68033078653431, which is equivalent to initial principal 121 compounded monthly, not initial principal 1 with addition a 10 made each n month then compounded.

What are the applicable mathematical formulas to determine the current value of the accrued interest and principal where the addition a is made a) at the beginning of each period, or b) at the end of each period?

When is the interest applied to the principal during the given period?

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Interest is computed at the end of each period. You can just consider each addition to be a new principal. The first $1$ becomes $1(1+r)^n$ at the end of $n$ periods. If you add $10$ at the end of the first period is becomes $10(1+r)^{n-1}$ because there is one less period to draw interest. If you add $10$ at the end of each period, you get a geometric series to sum: $10[(1+r)^{n-1}+(1+r)^{n-2}+\ldots (1+r)^1+1]=10\frac {(1+r)^n-1}{1+r}$

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  • $\begingroup$ What is the precise result? The online calculators which tried when attempting to verify the result that derived here provided different final amounts, which, in part led to the current question thecalculatorsite.com/finance/calculators/… : 123.00; depositaccounts.com/tools/compound-interest-calculator.aspx : 122.67. Also moneychimp.com/calculator/compound_interest_calculator.htm allows the addition to be made at either beginning or end of the period, which also provides different results for each option. $\endgroup$ – guest271314 Nov 20 '17 at 15:37
  • $\begingroup$ I am not going to debug online calculators. The above gives you a value to compare with. Alternately you can make your own spreadsheet. $\endgroup$ – Ross Millikan Nov 20 '17 at 15:42
  • $\begingroup$ Was able to get result 123.68577853630043 by adding a to initial principal at each iteration less than n periods per time principal = principal + a and multiplying principal by e^(rate * (rate * --n)), where n is decremented at each iteration less than inital n; and result 123.64550923392724 by multiplying principal by 1+rate^(rate*--n) $\endgroup$ – guest271314 Nov 20 '17 at 19:50
  • $\begingroup$ I just made a spreadsheet. With a deposit of $1$ at the start of the first month and a deposit of $10$ at the end of months $1$ through $12$ I get $122.6942436$ at the end of month $12$. I think your $123$ is clearly too high because you have $56$ on deposit on average, so the simple interest would be $1.68$. A little compounding gets to my number. $\endgroup$ – Ross Millikan Nov 20 '17 at 20:25
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    $\begingroup$ Yes, it is. I put the initial deposit in at the end of month 0 so I was consistently doing end of month deposits. Then yes, the 1 earns interest in month 1. New deposits do not earn interest because they come in at the end of the month, so the first 10 earns interest in month 2 and on. $\endgroup$ – Ross Millikan Nov 20 '17 at 21:57

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