I am learning some financial terms and am having trouble understanding what a discount $d$ is.
Numerically, I understand that it is defined as $\frac{i}{1+i}$ but I do not intuitively understand what that is.
I tried to look online, but all explanations look very similar and it does not help. Can someone help me out?
What I understand so far is the fact that interest works as $$FV = PV(1+i)^t$$ for compound interest.
On
At it core, the idea of discount is simply relating the value of money at one time to its value at another. There's a lot of potential factors to quantify exactly what the discount rate should be (inflation, risks, opportunity costs, etc.), but the core idea is basically the same.
A simple example - let's say I live in a world where I can get a risk-free 1-year bond at 10%, so that if I buy the bond for \$100 today I will definitely get \$110 next year. Then, when I look at any potential cash flow today vs. next year, I should be totally indifferent as to whether I receive \$100 now, or \$110 next year; they are worth exactly the same to me, because I can convert one directly into the other. This convertibility tells me that any financial transaction I enter where I pay or receive money in the future, I need to apply some discount factor to the later money to understand what that's worth to me today. Hope that helps.
Discounting is as you have correctly surmised the opposite of compounding interest.
In terms of a financial instrument, I'll use government bonds.
If I buy a new issue with a maturity of 10 years, typically I will pay the face value of the bond, lets say it is €1000.
A bond has two incomes, the coupon, and the principal. The coupon is the interest paid at regular intervals during the life of the bond until maturity. And the principal is my €1000 paid back when the bond matures in 10 years time. In effect I get regular interest payments on a loan until its maturity, and then lent amount back at the end.
To work out the present value of the future value of the €1000 principal in 10 years time I need to discount for inflation. Each year that inflation goes up, the value of my €1000 goes down.
Does that make intuitive sense?
With certain bonds, for example US 1 year Treasury Bills, there are no coupon payments. In that case, the face value of the bond may be \$1000, but when you buy the bond you will pay a discounted value. In effect you loan \$950*, and you get back $1000 for each bond you buy.
*No you will never get a rate of return as good as this, it's just an example. :)