How to Value Bonds

In this article, we’re going to explore how to value bonds (aka fixed income valuation). To keep things relatively simple, we’ll focus on one specific fixed income security – straight bonds (aka vanilla bonds).

We do also have another article that explains how to value a zero coupon bond. And a separate article on perpetual bonds (aka Consols).

If this is the first time you’re looking at bonds, you want to start here where we explain What is a Bond? in detail.

Do also check out this article which explains how bonds work.

Before we get into bond valuation / bond pricing though, let’s start with the basics.

What is Fixed Income Valuation?

Firstly, what is fixed income valuation?

In a nutshell, it’s the valuation of fixed income securities.

It’s about putting a price on a fixed income asset (hence the term bond pricing).

It’s ultimately a case of identifying how much the bond is worth to us today, or at some other point in time in future.

As with all assets, the value of a fixed income asset is equal to the present value of its future cash flows.

More on this point later on, when we learn how to value bonds.

What is a straight / vanilla bond?

Straight or vanilla bonds are your standard / typical bonds, that pay a series of Coupons (interest payment), followed by a Par value (aka Face value, lump sum) at maturity.

The payoff timeline for a straight / vanilla bond, for instance, is akin to a “traditional” bond’s payoff timeline.

It consists of a coupon payment every year, or every six months, or every quarter; depending on the frequencies that the bond promised at issuance.

And then you get the bond’s Par value / Face value on the bond’s maturity date.

So how do we go about pricing a straight / vanilla bond?

In other words, how to value bonds?

We achieve this by discounting the bond’s future cash flows back to the present.

Put differently, we value the bond by estimating the present value of future cash flow.

It looks like this…

$P = \sum_{t=1}^T \frac{C_t}{(1+YTM)^t} + \frac{Par_T}{(1+YTM)^T}$

Where:

$C_t$ refers to the coupon payment (interest payment) at time $t$
$YTM$ refers to the Yield to Maturity (aka bond’s interest rate, or bond yield), and
$T$ refers to the year in which the bond matures or expires.

If the equation’s freaking you out, please don’t let it freak you out.

It’s actually easier and more straightforward than it looks.

The first part of the equation is just discounting all the coupons back to the present (to today).

And the second part is discounting the bond’s par value to today.

That’s it.

Alternative Expressions for the Bond Valuation formula

If we were to expand this equation out, here’s what we’d have…

$P = \frac{C_1}{(1+YTM)^1} + \frac{C_2}{(1+YTM)^2} + \cdots + \frac{C_T + Par_T}{(1+YTM)^T}$

Note that we can discount the final coupon ( $C_T$ ) and the Par value / Face value together, because they both occur at the same time, $T$ .

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In other words, they’re both discounted over the same time period.

Now, if 3 conditions hold, we can simplify this equation even further.

The conditions are as follows:

the coupon payments remain constant
the bond yield or YTM remains unchanged, and
the maturity of the bond is finite (i.e., it’s not a Consol / Perpetual bond)

If these three conditions hold, then the bond price can be estimated like this…

$P = \frac{C_1}{YTM} \left(1 - \frac{1}{(1+YTM)^T} \right) + \frac{Par_T}{(1+YTM)^T}$

All we’re doing, is exploiting the fact that if these three conditions hold, the cash flow stream of the coupon payments is an annuity.

We can therefore apply the formula for the present value of an annuity, which is…

$PV_{Annuity} = \frac{CF_1}{r} \left(1 - \frac{1}{(1+r)^T} \right)$

The only difference is that rather than a generic cash flow $CF$ , we have a specific cash flow, the coupon $C$ .

And rather than a generic discount rate $r$ , we have a specific discount rate, $YTM$ ; the Yield to Maturity of the bond.

Note that the YTM is nothing but the interest rate of the bond.

We provide a formal proof of why this holds in our Bond Valuation course but that’s a tad bit outside the scope of this particular article.

Okay, so that’s how fixed income valuation works; specifically, the valuation of a straight / vanilla bond.

Let’s go ahead and apply the equation so it makes even more sense, with an example.

Fixed Income Valuation Example (Hills Inc.)

Consider Hills Inc., which is evaluating a bond that’s offering 5% coupons with a $1,000 par value and a five-year maturity.

What is the fair price of this bond if the appropriate yield (interest rate) is 4%?

How do we go about solving this and estimating the bond value?

Start by getting the relevant information. We know we’re going to need a Coupon Payment, Par value, a Yield to Maturity, and a maturity date / timeframe.

The bond’s Face value / Par value is $1,000

The coupon rate ( $CR$ )is 5% per the question.

Given a coupon rate of 5%, the coupon is $50.

How did we get that?

Because the coupon payment is nothing but the coupon rate multiplied by the Par value.

In our case then…

$C = CR \times Par$

$C = 0.05 \times \$1,000$

$C = \$50$

We know that $YTM = 4% \equiv 0.04$

And we know that the bond matures in 5 years, so $T = 5$

So we’ve got everything we need. And we know that the price of a straight / vanilla bond is estimated as…

$P = \frac{C_1}{(1+YTM)^1} + \frac{C_2}{(1+YTM)^2} + \cdots + \frac{C_T + Par_T}{(1+YTM)^T}$

Now it’s just a simple case of plugging in the numbers into the equation as…

$P = \frac{\$50}{(1+0.04)^1} + \frac{\$50}{(1+0.04)^2} + \cdots + \frac{\$50 + \$1,000}{(1+0.04)^5}$

Solve for that, and you’ll find that the price of the bond is approximately equal to $1,044.52

Now, this is perhaps not the most efficient way of solving this.

There is an alternative approach we can take, so let’s consider that.

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Alternative solution

Of course, it goes without saying that this is an inefficient way of fixed income valuation.

Because the…

coupon payment remain unchanged,
discount rate or the yield is constant, and
the bond’s maturity date is finite

Given that these three conditions are true, we can say that the coupons cash flow stream is an annuity.

So we can apply the formula for the Present Value of an annuity and make our lives easier.

What does that look like?

$P = \frac{C_1}{YTM} \left(1 - \frac{1}{(1+YTM)^T} \right) + \frac{Par_T}{(1+YTM)^T}$

As before then, just plug in the numbers into the equation to get…

$P = \frac{\$50}{0.04} \left(1 - \frac{1}{(1+0.04)^5} \right) + \frac{\$1,000}{(1+0.04)^5}$

Solve for that, and you’ll get the same result as above. The bond price is approximately equal to $1,044.52

Does that make sense?

If it doesn’t make sense, please read the example again, and only then proceed on to the next example.

We’re going to assume that you’re okay.

Fixed Income Valuation (Alternative Example: Watson Plc)

Let’s move on now, and look at Watson Plc, which wants to issue 10,000 10 year 6% £100 bonds.

What is the price of each bond if the yield is 8%?

How much money will Watson Plc raise, if all the bonds are sold?

Now, we want to take a minute to discuss the way this question is written out.

Oftentimes we see people looking at this writing and going, what on earth is this?!

And why is it written this way?!

So, why is it written this way?

Mainly because it makes us folks in Finance look clever.

But also because once you get used to it, it is a little easier to write it out this way, as opposed to specifying every single thing.

Writing Schema for Bonds

This way of writing out the bond’s characteristics is pretty standard.

In this case, Watson Plc wants to issue 10,000 bonds.

The 10,000 then, refers to the number of bonds that they’re issuing.

Then we’ve got a “10 year” right next to it.

The “10 year” there, refers to the fact that this bond has a 10 year maturity; or the lifetime of the bond.

6% that’s right next to it refers to the coupon rate.

And the £100 reflects the Par value.

Importantly, this is not us trying to complicate things.

This is just the way it’s written; pretty standard across the world of finance.

You will get used to it as you go about solving more problems.

But the key thing is to always be able to extract the relevant information.

A common mistake people make is to end up using the coupon rate as the yield.

You’ve got to make sure that that doesn’t happen to you.

Okay, so now that you know what the characteristics of the bond are, you should be able to solve the question on your own.

We’d recommend that you pause reading the article now and try solving it on your own.

The process is exactly the same as in Hills Inc., above.

Okay, we’re going to assume you did that.

Let’s go ahead and solve this question together now.

Solving for the price

As always, we start with the general equation for the price of a bond.

$P = \sum_{t=1}^T \frac{C_t}{(1+YTM)^t} + \frac{Par_T}{(1+YTM)^T}$

In this particular example, we know that the…

coupon remains constant,
yield remains unchanged, and
the lifetime of the bond is finite (at 10 years)

So we can go ahead and just apply the annuity formula to discount the coupons. And discount the Par value as a single cash flow.

$P = \frac{C_1}{YTM} \left(1 - \frac{1}{(1+YTM)^T} \right) + \frac{Par_T}{(1+YTM)^T}$

As before, just plug in the numbers into the equation to get…

$P = \frac{£6}{0.08} \left(1 - \frac{1}{(1+0.08)^{10}} \right) + \frac{£100}{(1+0.08)^{10}}$

How do we get £6 as the Coupon?

Well, the question said the coupon rate is 6%.

The coupon can be obtained by multiplying the coupon rate with the par value.

So in our case, that’s going to be…

$C = 0.06 \times £100 = £6$

When you solve for the pricing equation above, you’ll end up with a price equal to approximately £86.58

Now, the second part of the question asked us how much money Watson Plc would be able to raise, given that they were issuing 10,000 bonds.

With 10,000 bonds, they’d be able to raise £865,800

We got this by simply multiplying the Price (£86.58) by the number of bonds (10,000).

A note on assumptions

Importantly though, we made a rather simplistic assumption; that there’s no transaction costs involved.

The assumption is of course totally flawed because that’s not how the real world works.

Given a price of £86.58, they’ll actually end up raising less than that because they’ll have to pay the bankers, the exchange, lawyers, etc.

Strictly speaking then, £865,800 is not entirely correct.

But in a theoretical sense, that would be the amount they can raise.

We’ve now learned how to calculate the price of the bonds using the formula-based approach.

Let’s see how we can do the same thing on Excel, too.

How to value bonds on Excel®

You can use Excel’s “PRICE” function to estimate the bond value.

To do this, you want to start by setting up your spreadsheet to include all the information.

Notice from the image below that we’ve set the bond’s Face Value (or “Redemption” as it’s called on Excel®) to £100

The “Settlement” is set randomly to the 1st of January 2020. You can set it to any date you want; it really doesn’t matter.

The maturity date is set to 31st December 2029 because that’s 10 years later.

Remember that the question for Watson Plc said the bond has a 10-year maturity.

That’s why we want to set the maturity to 10 years after the “settlement” date.

The coupon rate was 6%, and the yield is set to 8%

Finally, the frequency is set to 1 because we’re dealing with annual coupon payments.

Once you’ve set up the spreadsheet, it’s just a simple case of calling the “PRICE” function and putting in the different parameters.

Note that the full documentation for the PRICE function is accessible here.

You’ll find that the result is identical to what we got when we solved for it manually.

The price of the bond according to Excel is also approximately equal to £86.58

And that’s it!

That’s how fixed income valuation works. And now you know how to value bonds!