What is the formula for compound discount with simple discount over the fractional period. I could not find the formula in my book and when I looked online the ones that had the formula where answers to a question in my book which I did not want to see the answer in order to get the formula. The formulas that I know are: Note that $d$ stands for the discount rate.
$a(t)^{-1}=(1-d)^t$ for compound discount
$a(t)^{-1}=(1-dt)$
$d_{m}=\frac{i_{m}}{1+i_{m}}$
I am wondering why no one has answered am I asking a question that does not make sense?
Edit: my question is in response to the following:
Method A assumes simple interest over final fractional periods, while Method B assumes simple discount over final fractional periods. The annual effective rate of interest is 20%. Find the ratio of the present value of a payment made in $1.5$ years computed under Method A to that computed under Method B
Let d be the discount rate, t be the total time of investment, and k be the largest integer such that t > k . First we calculate the compound interest over time k, then multiply that by the simple interest for time t-k. Thus your accumulation function is:
$ a(t) = (1 -d)^{-k} (1-d(t-k)) $
Similarly, the accumulation function for compound interest with interest rate i :
$ a(t) = (1 + i)^k (1 + it) $
Bonus points if you don't forget to take $ (a(t))^{-1} $ for present value in your question like I did.