The under-appreciated phenomenon of compound interest

HSBC Private Bank (UK) Limited - April 2010

Everyone who has had a passing education knows of compound interest. And yet very few people are truly familiar with the concept. Many recognise the rule of thumb: “At ten per cent, you double your money in seven years” and could readily estimate a fairly accurate answer to the question: “How quickly do you double you money at 12 per cent?”. Which takes six years.

But the true glory of compound interest unfolds only in the longer term, and here, there are few people to whom its manifestations do not seem remarkable. Over 40 years, which is the typical extent of the active investing career of most people, ten per cent interest compounded annually will turn £1,000 into £45,000 whereas 12 per cent will turn your original £1,000 into…what? Write your answer in the margin now, please.

Compound interest is of course interest earned on interest, or we might say (so as not to confuse the interest which concerns us here with mere interest on loans) compound return is the subsequent return earned on earlier returns when they are reinvested alongside the initial investment. Thus, if you could invest £1,000 for 40 years at 12 per cent, you could draw off £120 a year to spend, and after 40 years, recover your original £1,000. Or you could leave each annual £120 within the investment pot and earn 12 per cent on that as well as on the initial £1,000, and treat all subsequent annual returns in the same way. And after 40 years you would recover not only your original £1,000 but a also a profit of £92,000 (which is more than twice as much as the same process would earn at 10 per cent and I suggest may be a long way from the figure you just wrote in the margin).

Albert Einstein almost certainly did not ever observe that compound interest is the greatest mathematical discovery of all time (the saying is routinely attributed to him). But if he had, it would certainly have struck a chord with the common man.

How does that work? Forty times £120 is £4,800. OK, so you get interest on it. But still, something tremendous must be going on, to turn that £4,800 turn into £92,000? Something tremendous is indeed going on, but that thing is simply interest on interest. That’s all it takes. There is in fact a pretty simple way of rationalising the huge difference in returns between X per cent and X+1 per cent over 40 years, which we will come to in due course.

Today, compound interest concerns us in two main manifestations. One is in the long term potential of investing in the stock market, where it is a simple option for the investor to roll up his returns and earn spectacular gains. The second is in the sphere of lending, wherein the terms of the loan will determine how often the interest is compounded. Is interest calculated and added to the interest-bearing principal every year, or every quarter? Or every day, or even more frequently? The interest on a £1,000 one year loan at 10 per cent will be £100 a year if the interest is compounded annually, £104 if compounded quarterly, £1,105.16 if compounded daily and, somewhat disappointingly, £1,105.17 if compounded at infinitely short intervals, (the discovery of the arithmetic behind this figure was landmark in the development of mathematics in the late 17th century). The standardisation of compounding is one important aspect in the derivation of consumer-friendly annual equivalent interest rates from the misleading headline rates historically quoted by lenders.

Compound interest has a colourful history. In ancient times, sages and legislators were much concerned with the application of compound interest - in fact, of any interest - to impecunious borrowers. The bible is generally very wobbly on the subject of interest on loans and is absolutely against usury, which could mean either an offensively high simple rate of interest or any level of compounded interest. The Hebrew word for “usury” was “nashak”, which also means snake bite. The Roman Republic often legislated to set a maximum rate of interest - at between zero (this particular rate, set in 342BC, did not last very long) and eight per cent. In 88BC Sulla - the dictator who set the model for the Roman Empire - raised the rate to 12 per cent as a legal maximum on mainstream loans (whereas only a few years later, Brutus charged some defeated Greek cities 48 per cent, rather to the chagrin of Cicero). This rate lasted until the fall of Rome 500 years later although it was doubtless often observed in the breach.

For most of the next 1,100 years, loan rates were governed by the Church’s varying interpretations of biblical injunctions against interest in general and usury in particular, and secular edicts defining maximum interest rates inspired by the Roman example. From time to time, monarchs, who tended to be major borrowers in order to finance wars, would combine interest rate maxima with any or all of default, repudiation or inflation. Such policies rendered interest rate maxima and the nicety of compounding relatively trivial.

His Poor Richard’s Almanac was a popular source of inspiration and optimism for eighteenth century Americans (including amongst countless other aphorisms, "A penny saved is a penny earned"). The almanac was also published in France, where Franklin later became a celebrity US ambassador. There, one of his fans, another polymath, Charles-Joseph Mathon de la Cour parodied Poor Richard. De la Cour imagined Poor Richard’s will, which pledged small sums to various public institutions which had to be left to collect interest for 500 years. Thereafter, the resulting billions were directed to be spent on lavish projects. Only an unreconstructed optimist could see the point in such a bequest.

Everyone who has had a passing education knows of compound interest. And yet very few people In 88BC Sulla - the dictator who set the model for the Roman Empire - raised the rate to 12 per cent as a legal maximum on mainstream loans (whereas only a few years later, Brutus charged some defeated Greek cities 48 per cent, rather to the chagrin of Cicero). This rate lasted until the fall of Rome 500 years later although it was doubtless often observed in the breach.

For most of the next 1,100 years, loan rates were governed by the Church’s varying interpretations of biblical injunctions against interest in general and usury in particular, and secular edicts defining maximum interest rates inspired by the Roman example. From time to time, monarchs, who tended to be major borrowers in order to finance wars, would combine interest rate maxima with any or all of default, repudiation or inflation. Such policies rendered interest rate maxima and the nicety of compounding relatively trivial. 

Step forward Ben Franklin! He was delighted by the idea and immediately amended his will to leave £1,000 to his two home cities, Boston and Philadelphia. Both funds were to be invested at five per cent for a hundred years, when a partial drawdown could be made. The balance was to be invested for a further hundred years by which time each City was projected to have £4,000,000 available for further public works. Both funds were indeed maintained for 200 years and although the outcomes fell short of Franklin’s projections, some $7m in all was realised when the funds matured in 1990.

A rather better-known tale of compound interest concerns the “Indians” who sold Manhattan Island to Dutch settlers for a reputed $24 worth of trinkets in 1626. This made the Indians look stupid or the Dutch treacherous depending on your point of view. But compounded at seven per cent since 1626, the original purchase price would now be worth $7 trillion, which would buy you a fair proportion of Manhattan’s land, so the Indians are reckoned to have had the last laugh. In fact, the story is better and sadder. The Indians had no concept of land ownership, and almost certainly considered they were selling not the freehold but a two year lease as it was their custom to occupy land for only two years before moving on to new pastures. On that basis, the Indians arguably negotiated a deal without parallel. More significantly, the local Indians had been dying in considerable numbers from imported European diseases for a hundred years and many more would be wiped out by a series of wars and massacres.

Both Ben Franklin’s will and the supposed good deal made by the Indians on the sale of Manhattan involve huge time frames, generating a make-believe aura which defuses the important message for savers and investors. The fact is, that applying compound interest to a realistic amount of money creates a giant payoff not for a beneficiary 200 or 400 years hence, but within 40years - the span of a typical working career - for the saver who puts down the original seed-money in time.

Most pension plans involve significant contributions in a saver’s middle and final working decades. But a relatively modest early sacrifice of spending generates a far more powerful result which is desperately important in the new age of defined contribution pension schemes. At typical long term rates of return on equity investments, £1 invested at age 25 will be worth six to eight times as much in the saver’s pension pot at age 65 (invested for 40 years) when compared with £1 invested at age 45 (invested for only 20 years - see second to last row of table). This tradeoff puts an interesting perspective on the supposed greater savings capacity of those of middle-age: they have to contribute £6 to £8 to get the same benefit as a 25 year old paying only £1.

For the investor with a 40 year perspective, the fact of the matter is that £1 invested now at say eight per cent compound is worth £22 on maturity and if he is lucky enough to earn nine per cent, the outcome is £31 or 40 per cent more. In fact, the investor with a 40 year perspective has a very simple and intuitive way to cut through the challenge of compound interest arithmetic: one per cent extra every year for 40 years is 40 per cent: for “1”, read “40”. Thus, for every extra point of return you earn, your final pot is worth 40 per cent more. In fact, the actual figure is a little higher (see bottom row of table)… and if you invest for longer, much higher.