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Why is that we discount all the coupons and par amount of a bond at the same rate, our YTM?
If we think of each coupon/par as a strip/single bond on its own, wouldn't then each coupon payment be discounted at the rate for the maturity of that coupon payment? So a coupon payment due in 2 years would be discounted at the present rate for a 2 year bond?
Discounting all the bonds coupons at the same rate is to assume a flat yield curve, but that is not the case...
It is a mathematical convention that limits the yield to an average for a given price and assumed cash flows since bonds fluctuate in price thus affect the cash flows and rate.
If coupons were assumed to be invested in more of the same bond then those cash flows could be taken into account for a different YTM, but this could become computationally expensive with increasing complexity of the way bonds purchased with coupons are priced.
Bonds with longer duration yet identical maturities to shorter duration instruments trade at higher rates because of increased risk, implying a flat yield curve per security.
Bonds aren't the most senior securities in bankruptcy court. You can view the 1966 version of the law here. I'm not an expert so can't recite every caveat, but payment order in bankruptcy correlates to interest rate.
YTM is just trying to measure the return offered by a bond, taking into account the timings of cash flows. The discount rate that matches the quoted market price (net present value - NPV) of a bond is the YTM. All YTM of a bond (and the calculation of internal rate of return: IRR) does is to discount cashflows.
What you describe is to compute the NPV of a bond based on yield curves. This is also frequently done and for example what Interest Rate Risk In The Banking Book as well as Credit Spread Risk In The Banking Book (IRRBB & CSRBB) computations require. This is a lot harder though, and usually never produces the quoted price of the bond exactly.
The idea is to start with a risk free interest rate curve (nowadays ESTR and SOFR for EUR and USD), which has yields for all sorts of tenors, and would be interpolated to the exact cashflow dates of the bond. On top of that, you add a curve for credit spreads that are usually sector dependent (e.g. EUR Utility BBB rated - which means you look at bonds of EUR denominated utility providers, that are BBB rated and compute a spread curve for these bonds). On top of that, you would have a firm specific spread.
It is a lot more complex and adds little benefit for retail investors. One of the main benefits is that it allows you to get different risk metrics (various shifts in general yield; from flattening, to steepening yield curves, different credit risk shifts etc). Technically, it may also be used to discover mispricing of bonds but ultimately YTM itself will show that as well (relative to other comparable bonds). In case you are interested in such detail, and you do have access to Bloomberg (many universities and libraries like the New York Public Library have Windows PCs running Bloomberg Terminal software), you can look at ICVS for interest rate curves (SOFR, ESTR etc), CRVF - Credit for Credit spread curves.
A third, related concept is to compute flat spreads over certain curves (government, OIS swaps etc). YAS shows several of these by default.
Unlike claimed in another answer, YTM and yield curves are not directly related, as in a single YTM number does not imply that the yield curve is flat (or has any particular shape). A bond’s yield is a kind of average of the yield curve rates that correspond to its cash flows. The first equation in the screenshot from NYU.edu below shows a 1.5 year, 8.5% coupon bond which is discounted with zero rates (first part). The second one is with YTM. Using the rates 5.54%, 5.45% and 5.47% results in a bond value of 1.043066. This corresponds to a YTM of 5.4704%.
First, your assumption is wrong. Discounting all payments by the same rate and variable times does, in fact, produce a compounded (exponential) yield curve and not a linear one, because the yield rate isn't multiplied by the time factor, it's raised to the power of that time factor.
The present value formula is the future value formula (how much would you have if you invested P dollars at a periodic compound rate r over n compounding periods?), rearranged to solve for P. Because the future value formula is an exponential growth formula, the present value formula is similarly an exponential reduction formula.
As to why we don't discount longer-term investments based on rates for shorter-term investments, that's because this is a backwards way to think of it. The rates you see in the bond market are what the "market" as a whole is willing to accept as a yield over the life of a particular bond. That may or may not be what you are willing to accept as a yield for that particular bond. So, using market rates of short-term bonds to determine the desired rate of a longer-term bond will be unnaturally self-stabilizing, as long-term bonds eventually become short-term bonds as their maturity date nears.
Second, bonds that will mature in a short time are safer than bonds with longer maturation rates, because less can happen during the term of the bond, just as quantycuenta said. The longer the term, the more risk of default there is present in the investment, and therefore the higher the average yield demanded by the market.
Lastly, if you look at the price of long-term bonds being traded on the market (volume tends to be low compared to stocks, but bonds do get sold prior to maturity), and track the price of the bond as it progresses through its term, you'll see that, in fact, the price does follow an exponential curve. However, for coupon bonds, the slope of this curve is reduced, and for certain classes of bonds it can actually curve downward, if the coupon rate is greater than the yield demanded by the market. If the coupon rate matches the market yield rate exactly, the price of the bond will remain constant, because the coupon payments will avoid compounding of the interest payments. This doesn't mean the yield is flat; the payments, because they're made sooner, are worth more that later payments, because you can use money now to make more money later.
