FinToolBox - Special Case of Multiple Cash Flows: Perpetuity

Special Case of Multiple Cash Flows: Perpetuity

The word, "perpetuity", derives from the Latin adjective perpetuus ("continual" or "uninterrupted"). So, it natural to guess it is related to something that goes on and on. As a matter of fact, it means a sequence of equal cash flows that goes forever in our financial context.

In another words, perpetuity is an annuity which goes forever.

Annecode: Perpetuity and Consol

Believe or not, perpetuity may play a much longer role than you can probably imagine — it has been active since the early age of capitalism until today.

US is the the world's first country to officially issue government bond (through Bank of England) in 1694. In 1815, at the end of the Napoleonic Wars, British government debt reached a peak of £1 billion, which was more than 200% of its GDP (By the way, it is £1,876.8 billion at the end of the financial year ending (FYE) 2020). While, remember it was back to the time when the Pound is backed by gold — it roughly costs £6.5 a year to raise a kid in 1740 (Ralph Turvey, the cost of living in London, 1740-1834).

To solve this huge debt overhang issue, in 1752, Prime Minister Henry Pelham converted the entire outstanding stock of British debt into consolidated annuities that would become known as consols. Consol is a perpetuity debt, which means the government of UK doesn't need to redeem the principle of the debt, but only need to pay the interest of the debt on annual basis, until forever.

In 1870, US also issued consol. Below is a US government 4% consol bond.

Learning by Example: Perpetuity

At first, you would probably feel very daunted trying to figure out the PV of the perpetuity — yes, we can calculate each cash flow's PV and add them up. However, since there are literally, not figuratively, infinite CFs in a perpetuity. How can we add infinite PVs together?

Well, it turns out, yes, we can! It is actually very simple (through geometric series). The formula is listed below, where c is the equal cash flow per period and r is the interest rate per period. Remember, PV is one period ahead of your first payment, just like in the annuity.

Now let's go through an example. Suppose the interest payment for the consol bond of UK government is £50 billion in by the end of the year 2021 (which is very close to the real £48 billion in 2018) and the interest rate in UK 2% per year. What is the total debt worth of the UK consol bonds at the beginning of year 2021?

So, it turns out, in our hypothetical world, the total value of the UK consol debt is about £1.25 trillion. In reality, this is not far from the £1.9 trillion general government debt of UK. In US, this number is about $27 trillion by the end of Oct. 2020.

Page updated

Google Sites

Report abuse