I couldn’t resist this slightly geeky comment about a subject that comes up occasionally and to which I have not seen an answer elsewhere else, namely: if a financial model calculates interest expense (or income) by using the average of period start and end cash balances (including interest earned, thereby giving a circular reference), is this the same as using continuous compounding applied to the periodic cash flows (excluding interest) only?
The short answer is no, but the results are very close. For example, if the starting cash balance is $100, the reference interest rate is 10% per period., and there are no other cash flows in the period then: i) basing interest on starting balances, gives interest earned of $10, and a period end balance of $110, ii) if interest is interpreted as continuously compounded, the effective interest rate is EXP(10%)-1 or approx.. 10.5171%, giving an ending balance of approx. 110.5171… iii) using a circular reference in which interest at 10% is repeatedly applied to the average of the beginning and ending balances gives a final iterated balanced of 110.5263… [that is 10% is first applied to $100, to give $10 of interest and an ending balance of $110. Then, 10% is then applied to (100+110)/2 to update the interest, and this iterates until convergence.
In fact, the circular reference will always provide a higher figure than the continuously compounded approach for positive interest rates. The fact that the items are not identical and that the difference is positive is relatively easily to show by comparing the Taylor series for the exponential function with the Taylor series for exact solution of the algebraic equation that approach iii) implies [namely that the effective interest rate is implicitly i/(1-i/2)]. The difference between the two is (to first order) equal to one-twelfth of the cube of i. But we can leave the proof of that for another day or for “the interested reader”!! In any case, the circular reference approach is frequently used, but in my opinion, it should not be, … but that is for another blog (or read my book Principles of Financial Modelling 😊).