A compound interest parable
A friend asked me to help her calculate the cost of her pension (over 2% per annum). I did a quick =FV(expect return %, number of years, annual payment amount) in Excel, showing the effect of a 7% CAGR falling to 5% CAGR over 35 years. I then told her the cumulative absolute amount of costs, rather than the 2% quoted. I was struck by the fact that I understand compounding, and have understood it for many years, but that I couldn’t make a rough “guestimate” in my head. I had to use Excel to help me. Try it yourself, assume, 7% CAGR falls to 5%, 35 years, and say £1000 annual payment each year, to keep the numbers simple. What’s the end figure? and what’s the absolute amount of cost if you deduct 2% per annum? Did you get close? The answer is at the bottom.
This is not an area where rules of thumb help, compounding is so unintuitive even to people who know how to do the calculation. The key insight is to make sure compounding works in your favour, not borrowing money on high APRs, paying off debt early, investing regularly and keeping your costs low.
This reminded me that just after the financial crisis in 2010, I wrote a parable of compound interest and published it as an appendix at the back of an equity research report on banks. I’d just finished reading Garrett Hardin’s book on ecology and population, who has a whole chapter on the subject and whose idea I have “borrowed”. Here’s the parable:
A historical perspective of money lending is useful to understanding banking. In 1930, J M Keynes expressed optimism for economic future based on restricting over-population, a fifteen hour work week and banning usury. Religious leaders have long been suspicious of money lending, but perhaps it is surprising that one of the 20th centuries greatest economists was too?
Up until 800 years ago, even a 5% interest rate, was viewed as pernicious by religious leaders, as low rates of growth compound over long periods of time to produce absurdly large numbers. In the 8th century Charlemagne declared usury a criminal offence and Pope Clement V totally banned the practice in 1311. However, gradually loopholes were exposed and in practice the bans were circumvented. Usury was first permitted on a tribalistic basis: it was permissible for Jew to charge Gentiles interest, and for Gentiles to charge Jews. Even today, devout Muslims who refuse to exact interests from fellow Muslims are quite willing to invest their oil revenues in the interest bearing financial instruments of the non-Muslims. This sheds a new light on the activities of former Barclays employees in the Middle East.
The word talent: comes from the Greek “talanton” a unit of money of weight; in Medieval Latin the sense was extended to ability through the influence of the parable of the talents (Matthew 25:14-30). In Chapter 25 of Matthew we are told that a master gave his three servants different amounts of money (talents). On returning from his travels, two of the servants had invested the money, and achieved a return – the third servant buried his talent in the ground and was punished by the master.
What happens next goes unrecorded. But imagine some wandering banker approaches the master and suggests depositing the talents in their bank saying “it is in an interest paying deposit account. For I shall grow your talent at 5% per year” No one quite knows the value of a talanton, but let’s say it was 2 grams of gold.
What weight of gold would the depositor have if the banker could “make it grow” at such a sustainable rate over 2 000 years?
A) Canary Wharf
B) Mont Blanc
C) Switzerland
D) the Atlantic Ocean
Anyone answering any of the above would be mistaken by many orders of magnitude. A 5% compound interest rate would grow to 4.78 x 10^42 grams. How great a mass is that?
The Earth has a mass of “only” 5.983 x 10^27 grams. Very little of that mass is gold. But let’s suppose it could be. To pay the beneficiary of a deposit account paying 5% 2000 years later 8 x 10^14 gold earths . (That’s 800 trillion Earths, made of solid gold). Assuming the global population is 6 billion, that is 133 000 golden earths for every man woman and child on the planet. Is it any wonder that bank failures, market crashes, repudiation of debts, confiscatory taxes, revolutions and inflation occur with such frequency ever since usury has been allowed?
No wonder religious leaders condemned the lending of money at interest, and until 800 years ago it was completely prohibited. In most human societies there are more losers than winners (more paupers than millionaires). In such a world, taking the part of the losers is a promising path to political power. Financial services attract talented people. They have a talent for using humble arithmetic to extract cash from their customers and shareholders, and they should be compensated for their talent in the same way that dentists are compensated for their talent in extracting teeth. If bankers could manage to get themselves thought of as humble, competent people, on a level with dentists, that would be splendid…
END
Photo of Mont Blanc massif by Marc Kargel on Unsplash
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Answer:
£1000 per year, 7% CAGR, 35 years = £138K
£1000 per year, 5% CAGR, 35 years = £90K – the difference is £48K or a 35% reduction of the final sum
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