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What are Perpetual Bonds? How to Value Them?

What are Perpetual Bonds? How to Value Them?

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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.

What are Perpetual Bonds

Firstly, what are perpetual bonds exactly?

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

There is a theoretical possibility of a Perpetual Bond having a Par Value (aka Face Value) like regular bonds / plain vanilla bonds , but this is never paid.

So we don’t tend to pay any attention to the “par value” for a Perpetual Bond / Consol.

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 this debt instrument, which can be thought of as the “King” of all long-term bonds, it looks something like this…

Slide showcasing the payoff timeline to explain what are perpetual bonds

You’ve got the bond price P that you pay today, in exchange for getting a series of coupons every year for an indefinite period of time (forever).

Realistically, no corporation in their right mind would issue this kind of security to an investor (neither to institutional investors nor retail investors).

Governments these days don’t tend to do these types of bond issues, either.

And why would they? Would you like debt obligations that last for an infinite period of time?

The closest equivalent but frequently-issued bond in the real world would probably be long-term bonds like a 30-year Government Bond for example. Importantly, these aren’t consols.

But in the past, governments certainly have issued Consols.

And there are calls by some investors like George Soros for the EU to issue Perpetual Bonds.

So they are still relevant to some extent.

Okay, but now that you’re no longer wondering what are perpetual bonds, we can explore the valuation of perpetual bond (aka Perpetual Bond Pricing).

Related Course: Bond Valuation Mastery

This Article features a concept that is covered extensively in our Bond Valuation course.

If you’re interested in learning and mastering Bond Valuation, then you should definitely check out the course.

How To Value Perpetual Bonds

Valuing perpetual bonds is simple. It’s actually the easiest bond to value.

The perpetual bond formula is as follows:

    \[P = \frac{C}{YTM}\]

Where P refers to the price of the bond. C denotes the Coupon Payment, and YTM reflects the Yield to Maturity.

But why is this the formula for perpetual bonds?

To answer this, we want to head back to the fundamentals of bond pricing.

When it comes to valuing a Perpetual Bond, we can start with the general equation for the price of a bond (aka plain vanilla bonds).

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

Here:

  • P denotes the intrinsic or fair price of the bond (which should be equal to the market price under efficient markets)
  • 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.

Just as a side note, remember that the YTM is not the same as the Current Yield. Not for bonds in general, anyway.

Simplifying the Formula for Bond Price

Now, if 3 conditions hold, we can simplify the Formula for Bond Price even further.

The conditions are as follows:

  1. the coupon payment (and coupon rate) remains constant
  2. the yield or YTM (aka discount rate, interest rate) remains unchanged, and
  3. 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 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 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 formula 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 into 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 Perpetual Bond 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.

Incidentally, the formula for valuing perpetual bonds above is essentially the Present Value of a Perpetuity.

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 any more 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.

Related Course: Bond Valuation Mastery

Do you want to learn how to value bonds from scratch? And become a PRO at it while you’re at it?

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