2.3. Bond cash flows and other parameters. Yield to maturity

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来自 Moscow Institute of Physics and Technology 的课程

Principles of Corporate Finance – A Tale of Value

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Moscow Institute of Physics and Technology

Principles of Corporate Finance – A Tale of Value

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课程 2（共 5 门，Specialization Understanding Modern Finance）

The study of Corporate Finance seems to be a very generic part of business education. Still, it either falls in the trap of intimidating formulas or is superficially journalistic. Both extremes preclude the understanding of the core finance ideas, concepts, and models. This Course is an attempt to avoid the above extremes. We discuss the core basis and mechanisms of modern corporate finance in a learner-friendly way. We will analyze the market’s most fundamental problems, realize the intrinsic interests and preferences of investors, reveal the true meaning of specific financial terms, and uncover important issues that are so often ignored in choosing and valuing investment projects. The learners will gain insight into the essence of corporate finance. They will be able to use the obtained knowledge and skills to successfully advance in their career at a financial institution, as well as in the area of financial management at non-financial businesses.

从本节课中

Applications of NPV. Valuing Bonds and Stocks

Week 2 of the Course is devoted to the applications of NPV. In the first part of the week we use NPV to study riskless debt. Of special attention will be the challenges in valuing even riskless bonds. We discuss bond parameters and the special role of yield to maturity. Then we demonstrate how the NPV approach helps determine spot and forward interest rates. The second part of Week 2 deals with the core concepts in valuing equity. We introduce the idea of the common stock value as a function of its cash disbursements. Then we present some formulas that are used to value common stock on the basis of NPV. We focus on growth as a major contributor to the stock value. We analyze growth drivers and the mechanism of growth. On an example we reveal the influence of investments on the stock value. Finally, we pose some questions with respect to NPV approach.

2.1. Introduction – a simple way to value bonds and stocks3:07

2.2. Valuing riskless debt6:37

2.3. Bond cash flows and other parameters. Yield to maturity10:12

2.4. Calculating spot rates8:01

2.5. Forward rates – the idea and calculation8:53

2.6. Valuing common stocks – the expected rate of return5:13

2.7. The general stock valuation formula. The “tail”7:09

2.8. The model of growth. PVGO5:55

2.9. A constant growing company – an example10:05

2.10. – What’s ahead?2:41

与讲师见面

Konstantin Kontor
Director and Professor of Finance and Strategy
American Institute of Business and Economics (AIBEc)

Now let's proceed with bond cash flows and present value.

Again, I will briefly recall.

So this is a pattern, so 0, 1, t, c,

c, c, and then C + F here.

So the PV formula looks like this,

PV of the bond is equal to C over 1 + r1.

If here we have the rate of r1 + C2 over 1

+ r2 squared + the general firm is Ck over 1 + rk for

the kth hour, and then the final payment is C.

I'm sorry, here, these Cs are equal,

like this, C + F over 1 + r t to the tth power.

That would be too much of a good thing, if we.

So the formula looks great,

if we knew all these r1, r2,

r3, and so on to rt.

And if we did know all of them, we could say, great.

This should be equal to the market price of the bond and 0.1.

The problem is that on the market we observe ps,

we do not observe rs as anything seen.

So our problem is to somehow extract rs from market prices of bonds.

Now, you can immediately see that the use of this formula is,

well, you cannot get all these Rs from not only one equation.

But you can have many equations and

these equations are not linear, so it's kind of difficult to find these Rs.

And on top of that the key story is that these rs are expected and

the long of the horizon the wider the range of these expectations.

So we can see that, unfortunately, even for riskless bonds, we see

that we cannot use our standard shortcuts, because these are all different.

And we have to somehow find these rs, that are by the way called spot rates,

and that really creates a problem.

Like I said, we do not expect anything we do not expect anything else.

But the question is, let's say I'm an investor and I look at this formula.

And I say, well, that's really cool but

I wish I understood what these rs are and normally investors have

more specific and more straightforward question to answer.

And the question goes like, if I held this bond for a certain period of time,

what would be my realized rate of return over this period of time?

And there's this specific case when I hold the bond until maturity,

because investors normally do not have to hold bonds until maturity.

We talked about that in greater detail in our capital market scores.

What the idea is that bonds are public liquid instrument.

So basically people buy and sell them freely, and

therefore even if I bought a bond at the moment of the issue.

Then I might change my mind, and then sell it in market at any point in time.

Then my return would be calculated from the time that I bought it

to the time when I sold it.

But oftentimes, people buy bonds to hold them until maturity and

that specifically is the case with bonds with no options.

Because for example, if I bought a callable bond, and I did have an idea

to hold it until maturity, but it might be called by the issuer.

So this question of how much will I get if I held these

bonds to maturity, actually can be treated

in a way by introducing special parameter.

I will specifically start from the next flip chart,

and that would be called yield to maturity.

Let's see, what if, in the general formula, I forcefully

replace all these rk with one and the same parameter.

You can say, well wait a minute, you just told us that you can not,

assume that rs are equal.

Well, they are not, this is a special parameter.

This parameter is called Y, or yield to maturity.

And this is not any kind of an average or,

this is a very complex severance if you will.

So I feed the formula and say PV is equal to C over

1 + Y plus C over 1 + the same Y squared

+ the general term will be C over 1 + Y to the K power.

And the final payment will be C + F 1 + Y to the tth power.

So this is sort of an artificially designed formula.

But the good news about this formula is that, first of all, from this equation,

if we set this PD to the price of the bond, we can say that we sold it.

Because these Cs are, for bonds known.

P we observe on the market.

We can solve this equation, well, this is a bad equation, it's nonlinear.

It has a.

But it can be solved using numerical methods, and

we can receive this Y as a single thing.

Well, people who studied Calculus and

advanced algebra can say, well, this equation has two solutions,

but it can be shown that for us only one would be a good one.

Without going deep with what this good means.

So the important thing is the meaning of this yield to maturity.

And again, the meaning is, like I said, this is what will be my

expected annual

rate of return if I held the bond from the moment I bought it, until maturity.

This is very nice, and again this is a great advancement for

us in the ability to use the approach.

However, the yield to maturity is widely used, but

unfortunately as some problems.

Well, and all these problems, they stem from the fact that when I replace

the more general formula with a known RK, with this formula.

I sort of compress the information.

It can be said in a metaphorical way that Y is

a very specific average of all these expected rs.

And therefore the information has been lost.

And therefore if I see yields to maturity,

that is not enough to oftentimes find these market prices.

So the story goes like this, we observe expected rs from them we must calculate

the right PV based on the formula that was on the previous list on the flip chart.

Only then from PV, we can go to yield to maturity,

not the other way around, because on the first way, we have a lot of information.

And then we get it from crest.

We cannot go backwards, because from only one parameter,

you can by no means find all these rs.

So often times people say, well, that yield to maturity is sort of equivalent

to the price of the bond, because you can find one from the other.

Well, this is not perfectly right,

it's sort of approximately right, in most cases.

But you have to really keep in mind that

on this way you might find some aggravations.

And therefore like I said, strictly speaking P

over 1 is not the function of yield maturity.

But as I said before, the good news is that most often

we can use the yield to maturity as a very nice proxy.

Or better to say approximation to

the realized return that investors would actually get.

I'm wrapping up here, and in the next episode, we will go a little bit further.

Remember, I told you that we observe prices, and we need these RK.

How can we extract these expected rates of return, RK from observed prices?

This is called calculating spot rates,

where we'll study the example of how this is done in the very next episode.

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