# Thorough Description of Yield to Maturity?

After some studies for the last month, I have encountered a concept that has just completely eluded me, and that is Yield to Maturity (YTM).

Can someone explain the purpose and reasoning behind this concept?

My background is that I am going into a Finance Masters program after coming from a Math background.

I understand coupon rates, present value, maturity dates, and the general working of bonds and all that, but how does YTM work?

The general description my book uses is this, which barely helps:

The equation makes sense all right, but why would anyone need to figure out a discount factor (y in the equation) that makes the future cash flows of the bond equal to its price?

In addition, what is even meant by the "bond price"? As to this point in my studies only a bond's principal, face value, and par have ever been mentioned (and they're all the same thing).

I grasp that YTM is somehow supposed to account for other nominal yield calculations' deficiencies by incorporating Present Value, but it still confuses me.

• "Why would anyone need to figure out a discount factor (y in the equation) that makes the future cash flows of the bond equal to its price?" Because that's the factor used to compare investments of all types. If the YTM of a bond is 1%, but the average annual return on the stock market is 7%, why would anyone buy a bond? [The answer is because the bond is lower risk] Jul 18, 2017 at 16:46

What is a bond price? A bond is an asset, and like any tradeable asset it has a price. If I hold \$10K face value of a certain GM bond, then I would be willing to sell it at some price, which may be more or less than \$10K. Whoever is willing to sell it for the lowest amount determines the price. The price is determined by the market, just as all prices are. It's what you can sell a bond for. Bond prices may be quoted in various funny ways, like as a discount or premium relative to the face value or as a premium over a treasury, but at the end it all should be converted to how much you have to pay today. In this case, it's how much you would pay today to get a set of future coupon and principal payments.

What is Yield to Maturity? A bond is a contract entitling you to a certain set of predefined cash flows. If you take that set of cash flows and discount them using a single rate at all maturities such that the discounted value is equal to the price, the single rate you have identified is the YTM. Mathematically, this is the same as finding the IRR (internal rate of return) of some set of cash flows. In this case the cash flows are the coupons and principal repayment.

Other bond concepts. Note that the other aspects of a bond, like maturity, coupon rate, and face value, are immutably written into the bond contract. All they do is define what payments the bond entitles the owner to. They don't say how much someone would pay today in order to be entitled to those payments. One can't know how much a future payment is worth without discounting. If you know the appropriate discount rate at every relevant maturity, you could calculate the fair price of a bond. That's the other direction. YTM looks at the market price and associated cash flows and imputes what single discount rate would make that price fair.

What is YTM good for? Recall what I said about IRR above. Why would anyone want to know what discount rate equates the cash flows of a project to its cost? Because it's an easy way to summarize how profitable the project is expected to be. YTM is a quick way to summarize the yield one would get on a bond if they were to buy it today and hold to maturity. If one bond has a higher YTM than another, than heuristically we believe it pays out more and should be associated with greater risk if the market is working properly. It can be used to compare bonds or to look at how changes in bond prices are affecting expected yields. Ask yourself, how would you compare two different bonds with different maturities and coupon rates? Which one is riskier or more profitable? The simplest way to summarize this information is with the yield to maturity. YTM is used frequently enough that when you just say a bond's "yield," people will assume you are talking about its yield to maturity.

What is YTM not good for? One thing to be wary of is using YTM as a discount rate. It looks like a discount rate but it works for that bond and that bond only. In reality each individual coupon payment has a true discount rate, and the discount rate at each horizon is different from each other horizon. Those are true discount rates that can be applied to any cash flow of similar risk to get the right price. We can think of YTM as some kind of average of those discount rates that produces the correct price for that bond only. You should never use it for discounting something else.

• Ok, I think I almost got it. You asked me how I would compare two bonds with different maturities and coupon rates? I would calculate the present values of all the respective cash flows, both coupons and par at maturity, and sum them up, and choose whichever had the higher amount. Why isn't this used rather than YTM? Jul 18, 2017 at 18:23
• That's a valid approach but it's hard to know what the right discount factors are if the bonds have any risk at all. This is analogous on the corporate side to using IRR instead of PV analysis. The latter is more theoretically correct but the former is easier and very common. Jul 18, 2017 at 18:28
• Also note that the PV analysis you suggest doesn't inform you about which bond is riskier, which you might want to know. All it really tells you is the price. If you averaged the discount rates you used, you would get an indicator of risk, but that's very similar to YTM. Jul 18, 2017 at 18:32
• So, I have two more questions that I think will pull together all the issues I have left with YTM. One, since YTM is an estimate, does that mean that the actual discount factor for each cash flow will be slightly above or below the Yield, depending on market conditions, but still overall balances out? Second, so the method I described could work, but it does not incorporate risk like YTM does; is there, however, a way to statistically calculate the "likeliest" discount factors for each cash flow using my method with a normal curve or whatnot, thus incorporating some idea of "risk"? Jul 19, 2017 at 17:44
• There are many important ways of computing/estimating discount factors, some of which you will learn in your program. However, that's quite a different problem than YTM solves. YTM is really just there to quickly summarize how much a bond will earn per unit time based on its current price (which in turn reflects its risk). I wouldn't try and stretch YTM to do other things nor would I try and find an alternative measure. YTM is the standard for this purpose. Not perfect, but good enough and ubiquitous. Jul 19, 2017 at 17:52