Financial maths- compound interest rate and equivalent discount rate

Question

I need to find a formula for the compound interest rate i equivalent to a discount rate of d, if the money is discounted over n years. I know that i=d/(1-d), but not how the no. of years comes into it, or how i=d/(1-d) is derived. I'm sure the information is on the internet somewhere- so sorry for asking here- but I'm finding it quite confusing and if someone could give me any help that would be great. Thanks.

Answer 1

If you substract $d$ percentage from $x$ you have to calculate $(1-d)x$. Now you can ask yourself for what value of $i$ it is the same value if you discount $x$ one time ?

You get the equation:

$(1-d)x=\frac{1}{1+i}x$

x is cancelling out

$(1-d)=\frac{1}{1+i}$

Taking the reciprocal on both sides

$\frac1{(1-d)}=1+i$

$\frac1{(1-d)}-1=i$

$\frac1{(1-d)}-\frac{1-d}{(1-d)}=i$

$\frac{1-(1-d)}{(1-d)}=i$

$i=\frac{d}{1-d}$

If $d=0.2$ then $i=0.25$