What is the difference between present value and face value? When I search this question on Google is says they're the same in some cases and different in others. It says they are the same when the market interest rate is the same as the contractual interest rate.
I need help understanding what this means. I do not understand the difference between these types of interest rates, etc. And also does this have anything to do with discounting vs. coupon bonds, etc?
Suppose the face value of a bond is $M$ and its interest rate is $\tau$. This means it will pay $\tau \cdot M$ interest every year (other periods are also possible) and at the end of its run (its maturity) it will also repay the face value $M$. Government bonds are usually sold in auctions. Whatever ends up being the market price is considered to be the present value of the bond. Using the cash flow you can also calculate the yield of the bond. If and only if the face value and the present value are equal the yield will be equal to the interest rate.
For an example see the US treasury's website. Click on bonds to see face value, interest rate and maturity, click on the bond serial to see its price (present value).
Further reading here. (Link copied from a comment by @Jamzy .)
Think of it this way:
A dollar today is worth more than a dollar tomorrow
Why? Because today you can invest it, and have more money tomorrow!
How much more money? It depends on the interest rate. The interest rate on government bonds basically says: "Whoever you are, if you lend me this money I am going to give you back all of it plus a certain interest".
Since you can be sure that the bond will be repaid, and you're allowed to lend money to the government in any situation, what this means is that a dollar today is worth like a dollar tomorrow plus the interest the government gives!
There are a number of formulas for interest rates (but it's just a matter of convention).
Anyhow, if you lend today to the government $S$, and it will give you back tomorrow $S(1+i)$, then the face value is $S(1+i)$ (by definition) and the present value of $S(1+i)$ is $S$.
"And also does this have anything to do with discounting vs. coupon bonds, etc?"
It does.
Assume a bond without coupons, to be fully re-paid in a single payment. On it ("face value" $\equiv B$) the bond writes the amount to be paid, as well as the date of payment.
In such a situation, the "face value" includes both the principal amount and the interest. But the bond will be transformed to cash in say, one year's time. Then the "present value" of the bond is
$${\rm PV} = \frac {B}{1+r}$$
where $r$ does not equal the contractual yearly interest rate of the bond, but an interest rate that reflects current market conditions and expectations, as well as the current assessment of the solvency of the debtor at payment time.
Of course both "face value" and "present value" are very general terms, used in a variety of situations, in which their meaning is similar but not identical to the above described.
In short:
I don't usually think of these two in the same way, but I see why it can get confusing to do so. The difference between these two terms of a temporal (time) nature. With present value, you're thinking about the current value of the money that you're soon to receive; with face value, you're thinking of the money that you're currently receiving as a result of the maturity in its value.
Face value is the value of the item immediately, without regard for the future. For example, the "face value" of a $20 dollar bill is 20 dollars. I remember this because it is, literally, written on the face of the money.
The present value includes a valuation of the future of that money. If I can use that \$20 bill to obtain something else, such as risk-free bonds, then the present value of that $20 bill can be much higher, it might be worth many times that initial face value.
Here's a simple formula showing one of the possible relationships between the two terms:
$$Present \ Value = \Sigma_{t=0}^{\inf} \ FaceValue * i ^ t $$
Where t is the number of periods in the future you are looking, and i is the risk-free bond rate I mentioned already. If you have a sharp eye, you will notice we have assumed that people value all periods (today, tommorrow, etc.) equally. This is probably false, but simplifies things for now. More complex assumptions are available, (and more complex formula!), but the idea remains the same.
