What are Perpetual Bonds

What are Perpetual Bonds, aka Consols? In this article, we’ll explore this; and also learn how to value perpetual bonds.

Let’s get into it.

A Perpetual Bonds is a fixed income security that pays a series of coupon payments (interest), forever.

It might have a theoretical Par Value (aka Face Value) like regular bonds / straight bonds, but this is never paid.

So we don’t tend to pay any attention to the “par value” for Perpetual Bonds / Consols.

The theoretical par value can be used to identify the coupon payment, if for instance, you only know the coupon rate, but not the actual coupon payment.

But other than that, we’re only really interested in the coupon payments.

If we look at the payoff timeline for a this debt instrument, it looks something like this…

You’ve got the bond price $P$ that you pay today, in exchange for getting a series of coupons every year, forever.

Realistically, no corporation in their right mind would issue this kind of a security to an investor.

But in the past, governments certainly have issued Consols.

So they are still relevant to some extent.

Okay, but now that you’re no longer wondering what are perpetual bonds, we can explore how to value these perpetual securities.

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How to value Perpetual Bonds

When it comes to pricing Consols, we can start with the general equation for the price of a bond.

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

Here:

$C_t$ refers to the coupon payment (interest payment) at time $t$
$YTM$ refers to the Yield to Maturity (aka interest rate), and
$T$ refers to the bond’s maturity date.

So you’ve got the present value of the coupon payments, plus the par value discounted to the present.

Simplifying the equation

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

The conditions are as follows:

the coupon payment remain constant
the yield or YTM (aka discount rate, interest rate) remains unchanged, and
the bond’s maturity date is finite (i.e., it’s not a Consol / Perpetual bond, it does NOT have infinite maturity)

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

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

Yes, we’re ignoring the fact that we’re talking about a Consol for now.

All we’re doing, is exploiting the fact that if these three conditions hold, then the cashflow 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 payment $C$ .

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

We provide a formal proof of why this holds in our course on Bond Valuation Mastery and in our Financial Math Primer for Absolute Beginners course, but that’s a tad bit outside the scope of this particular Article.

Now again, we realise that the condition was that $T$ needs to be finite.

Whereas in this article, we’re looking at a Perpetual Bond or Consol, which by definition has an infinite $T$ or an infinite maturity.

But just for simplicity, we’re going to work with the annuity equation.

Because the intuition makes really good sense, and you can see how this transforms into the equation for the price of a Consol.

So just stay with us for a second.

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Proving the equation for Consols

Let’s think about what happens when $T$ tends to infinity in the equation here:

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

Humour us for a second. Let’s substitute $T$ with $\infty$

The equation then transforms to this…

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

Now, note that the $(1+YTM)$ in the annuity part of the equation, as well as the single cashflow part, are being raised to the power of infinity.

When you raise $(1+YTM)$ to the power of $\infty$ , that’s going to make it become a really massive number.

In fact, it’ll take it to something close to infinity. Or put simply, it will become infinity.

In other words…

$(1+YTM)^\infty = \infty$

So the pricing equation above will transform into this…

$P = \frac{C_1}{YTM} \left(1 - \frac{1}{\infty} \right) + \frac{Par_\infty}{\infty}$

And what’s anything divided by infinity? Well it’s 0 of course.

So the equation simplifies to this…

$P = \frac{C_1}{YTM} \left(1 - 0) + 0$

And that then simplifies to this…

$P = \frac{C_1}{YTM}$

So all you’re left with then, is the fact that the price of a Consol is simply equal to the coupon at time 1, divided by the YTM.

And that’s literally it.

So we’ve gone from the general equation for the bond price, to the equation for the price of a Consol, which is literally just nothing but the coupon divided by the YTM.

Nice and simple, isn’t it?

Let’s see what this looks like when we apply it with an example.

Valuing Perpetual Bonds (Example)

Consider the Government of Utopia, which is issuing Consols with a $60 annual coupon payment (interest payment).

What is the fair price of this bond if the appropriate yield (interest rate) is 5%?

How do we go about solving this?

We can start with the equation for the price of the Consol, which we now know is this…

$P = \frac{C_1}{YTM}$

In our case, the Coupon is $60 and the Yield (or Yield to Maturity) is 5%.

$P = \frac{\$60}{0.05}$

So you’ve got $60 divided by 0.05, which is equal to $1,200.

The price of these bonds is equal to $1,200. Nice and simple.

Let’s just look at one more example so you’re really clear with this.

Valuing Perpetual Bonds (Alternative Example)

Consider the Republic of Bliss, which is considering buying 50,000 Consols / perpetual debt for €16 million.

The Consols promise perpetual annual coupon payments of €30.

Advise the Republic of Bliss on whether it should buy these Consols, given a YTM of 8.5%

Notice that this is pretty much similar to the previous example with Utopia.

The only difference here is that they’re buying a certain quantity at a certain price.

What you need to do is see whether the price that they’re paying (€16 million for 50,000 Consols) is fair, given the intrinsic price / fair price of the bonds.

In other words, you need to see whether the bond is undervalued or overvalued.

If it’s overvalued, then you’d advise against buying the bond.

And if it’s undervalued or fairly priced, then you’d be happy to go ahead.

Pause reading the article now and try solving it on your own!

Okay, we’re going to assume you did that. Let’s go ahead and solve it together now.

We start with the equation for the price of a Consol as…

$P = \frac{C_1}{YTM}$

Just a quick note by the way. We’re calling it $C_1$ , but you can call it $C$ , or indeed $C_{10}$ . It doesn’t matter, because the coupons remain constant.

If it doesn’t remain constant, then this equation doesn’t hold anymore anyway!

So the subscript 1 is just there for reference, but you don’t really need it.

Anyway, the coupon payment is €30 and the yield (interest rate) is 8.5%, which is 0.085. Plugging these into our equation looks like this…

$P = \frac{€30}{0.085}$

Solve for that, and you’ll end up with a price approximately equal to €352.94

Now, in terms of advising them…

50,000 bonds should cost €17.647 million given the fair price of €352.94 euros.

The Republic of Bliss is able to buy it for €16 million euros, which clearly is a great deal, right?!

They’re essentially paying €16 million for something that’s worth €17.647 million.

In other words, they essentially saving €1.7 million.

So this is a great deal for them. Good luck finding this kind of deal in the real world!

But anyway, hopefully this all makes sense. And you now know what are perpetual bonds, and you know how to value this fixed income security.

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