The greatest shortcoming of the human race is our inability to understand the exponential function.

– Dr. Albert Bartlett, Professor of Physics, University of Colorado

Yesterday, Bill dealt with the subject of the exponential (non-linear) growth of populations, economies and money supplies… and the inevitable constraints on energy production.

What will win out in the end? The unstoppable force of growth and human innovation? Or the immovable object of resource limits? We can’t tell for certain. But we’re inclined to think that it will be the latter. Maybe we’ll get some form of stalemate.

Exponential growth is a key concept to understand. Not just in terms of the growth of populations and money supplies. Exponential growth is also exactly what we want with our investment portfolios. It is just another way of talking about the power of compounding, which I have written about at length in the past.

If something grows at a fixed percentage each year, the *amount* it grows in each successive year will get bigger over time (even though the *rate* of growth remains the same). And small differences in the yearly growth rate result in big differences in long-term results.

Let’s say an investment grows at 10% a year. Invest $100, and after one year you have $110. The investment has gone up $10. In year two it goes up another 10%, which is now $11. So the profit has increased from $10 to $11. *This is because there are profits on the earlier profits*.

This is what we call the power of compounding. Which as you should know by now is the most effective way to grow your wealth over time and, as Richard Russell famously put it, the “royal road to riches.”

Below is a chart of how exponential growth looks over 30 years (roughly a generation) using various growth rates from 1% a year (the lowest line) up to 10% a year (the highest line). Values are shown on the vertical axis and all start at 1. The years are shown along the horizontal axis.

The really important thing to grasp here is that the lines are curved, and not straight. This is how the power of compounding looks over the long haul. You could call it the “picture of compounding”.

A steady gain of 5% a year means that the initial value of 1 goes up to 4.33 after 30 years. A steady gain of 10% a year means that initial value rises to 17.45 over the same time. The yearly rate has doubled. *But the end result after 30 years is over four times as much. *The longer the time period, the bigger the effect.

Of course, there is price volatility along the way with any investment portfolio. You won’t get a perfectly smooth curve. But what matters for legacy investors is the end result. And the power of compounding still works its magic even when there is price volatility along the way.

Most people don’t understand compounding and exponential growth. This isn’t because they are stupid or uneducated. It’s just that it’s not intuitive to most people. We’re not wired to think in a non-linear way. But it’s increasingly essential to start to think this way if we are to truly understand the world we live in.

Today, I’m going to look at the subject of exponential growth… but with a twist.

If you know the rate at which something is growing, you can work out how long it will take to double in size (assuming the growth rate stays the same). The rate of growth is the percentage that it goes up each year. The number of years taken to double in size is called the “doubling time.”

The table below summarizes the doubling times using a range of different annual growth rates.

At lower rates of growth small differences in the rate make a big difference to the doubling times. If your investments beat price inflation and taxes by 1%, they will double in 70 years, measured in today’s money. If the outperformance is just two percentage points higher – at 3% – they will double in just over 23 years (again in today’s money).

There is an excellent rule of thumb for calculating the doubling time of anything. You take the number 70 and divide it by the yearly growth rate to give the number of years for something to double.

If you aren’t good with numbers, you can use a memory trick to help you. By creating mental images, we help our brains retrieve information more easily. So you could imagine 7 clocks all at 10 o’clock – combining the number of 70 with the idea of time. Or you could think of your 10 fingers and the seven days of the week. Just try to think of something that works for you.

Conclusion: To maximize long-term investment performance you need to keep investment profits up and costs and taxes down. This allows the power of compounding to work in your favor.

This chart shows the same data for the first 10 yearly growth rates in the above table – from 1% to 10% a year.

Looking at it this way, something jumps right out. The relationship between growth rates and doubling times is not a straight line, but a curve. More growth always leads to a shorter doubling time. But there are bigger absolute differences between the doubling times of lower rates of growth than between higher ones.

(You can also see this relationship in my earlier chart of exponential growth. Not every line gets up to a value of eight. But look at those that do. And compare the year at that point with the year when they crossed through the value of 4. The difference is the doubling time. You’ll see that it gets shorter and shorter as the growth rates increase.)

So in any investment portfolio there is a huge advantage to increasing your net return (after tax and inflation) from low single digits to high single digits. Of course, short-term price moves in a single week, month or year could be much better or worse. The crucial thing is that they average out at a higher level in the long run.

At the same time, at higher rates of growth each additional increment is still good news. But the effect on the doubling time is less. Moving from 20% a year to 25% a year – an increase of a whole 5% a year – only reduces the doubling time from 3.8 years to 3.1 years.

Conclusion: If you have a strategy to give already high returns, taking on the extra risk to squeeze even more out of your investments is probably not worth it.The priority is to get those first improvements. These are the low hanging fruit of enhancing your long-term investment returns, when viewed through the prism of doubling times.

A great example of long-term performance… irrespective of short-term price volatility… is the Russia Prosperity Fund, which I wrote about last week.

Short-term price volatility in this fund has been huge at times, which goes with the territory in Russian stocks. But long-term results have been spectacular. The average growth rate over 15 years was 23% a year. At that rate the doubling time – when $100,000 would turn into $200,000 – is just 3.4 years.

Of course, we have to contend with taxes and price inflation as well. Assuming we’re left with 60% of this return after investment taxes of 40% (this figure could be lower)… and that yearly price inflation was 5% (this figure could be higher)… this would leave an 8.8% a year net return over inflation (23% times 60% minus 5%).

This is well out to the right hand side on my chart above. In fact, it’s about an eight-year doubling time. This would mean that, measured in today’s money and after taxes, a $100,000 investment would be worth $200,000 in just eight years. That would be a great result.

Of course, many investors pay lower tax rates on bond coupons, dividends and capital gains on their stocks. But in a world of distressed government finances, I don’t expect this to last long. As a result, it’s a bad idea to plan long-term investments on the basis of temporary tax breaks.

Conclusion: Assuming these tax and inflation rates are about right, you would need 8.33% gross return (before taxes and inflation, but after fees) just to break even. A standard balanced portfolio of developed country stocks and bonds is unlikely to deliver this over the long run.

Let’s say you have 50% of your investments in 10-year Treasury bonds and the other 50% in the S&P 500. Your T-bond has a yield to maturity of 2%, at time of writing. Over 10 years you will make exactly 2% gross a year with this investment – 50% of the portfolio in this example.

But you need a gross return of 8.33% just to break even. So your S&P 500 stocks will have to make up the difference. This means the S&P 500 would have to return 14.66% over 10 years for the net result to be 8.33% a year gross.

Think about it…

The US economy is weak. And there is a huge “tail risk” to growth from a default in the euro zone. Top-line revenue growth at US corporations is weak. And recent profit growth has largely come from cost cutting and margin expansion.

Meanwhile, net profit margins are now at the highest level for decades – a level from which they will almost certainly fall. Interest rates are pinned to the floor. So there is no room to further stimulate the economy or goose up P/E ratios by cutting rates further. Plus, dividend yields are low by historical standards – at about 2.4% for the S&P 500.

Does this look like an environment where the S&P 500 can return nearly 15% a year over the next decade? Are you feeling lucky?

Our own stock market investments – dominated by emerging markets – have fallen in price this year. And the emerging markets in general have had a torrid year. But it is important to remember that the value of our investments, broadly speaking, remain unchanged. (By value I mean “fair value” – an estimate of the real worth of the stock of these companies, based on their net assets and earning power.)

Conclusion: We can’t know the future. But any short-term price action is not something, as legacy investors, we should be too concerned about. We need to keep focused on the far mountain peaks and not on the ditch that we are currently crossing. Investors who lack patience, guts and determination will be forced to take their chances as short-term traders in one of the most volatile periods in market history. They have about as much chance of building long-term wealth as serial coin flippers.

Using our simple formula, let’s think about price inflation. If the inflation rate is 3%, the doubling time is 23.3 years. This is close to the average official (CPI-U) inflation rate in the US since the end of 1987, which is also 23 years ago. So “official” prices have doubled since then.

Because they think in a linear way, most people wrongly assume that this doubling would take 50 years. For something to double it has to go up 100%. So the thinking goes that if you add 2% a year then it takes 50 years to double. But because inflation is exponential – *because it compounds* – the doubling time is much shorter.

Put another way, a bundle of dollar bills in a safety deposit box would buy half as much in 23.3 years if this inflation rate continues. The bills earn no interest. But consumer prices would double.

Actual (non-official) price inflation is substantially above official rates. Government statisticians use various tricks to fiddle the numbers. The result? Inflation-linked pension payments and the like grow slower than actual price increases. Bond investors are conned into accepting lower coupon payments. And “real” (inflation adjusted) GDP growth is systematically overstated.

Now, let’s say inflation is 5% instead of 3%. Divide 70 by 5 to get the doubling time… and we’re down to 14 years. Just 14 years for a stack of cash to lose half of its purchasing power! This is why many people view inflation as a “stealth tax” – it’s just as dangerous to your wealth..

Some say that inflation is already running way above this level, most notably John Williams at shadowstats.com. He calculates a current rate of around 7% a year. This would mean prices double in just 10 years if the rate stays the same.

So holding cash is a losing proposition in the long run. This is why we recommend you hold cash only as “short-term parking”. Cash may lose some of its value relative to consumer prices. But we continue to believe that market price volatility will bring down equity prices much faster in the short run.

And if your home currency is the US dollar, you’ve actually seen the value of your cash rise relative to other major currencies since we recommended you increase your cash allocation in the Family Wealth Portfolio back in February. You can see this clearly from the chart below of the US Dollar Index, which measures the exchange rate of the dollar versus six other major trading partner currencies.

Remember: cash = bullets. This is something Rick Rule emphasized at our Global Partners’ Reunion in Courtomer two years ago. And this simple insight is one of the keys to Rick’s success as a resource investor. When prices fall to “bargain counter” levels you need cash “ammo” to buy them with. The profits come as those prices rise back up towards fair values.

But large amounts of cash held for the *long run* are a sure loser..

Price inflation is a hot topic in Argentina, where I live. The government claims a yearly inflation rate of about 10%, which is supposed to be acceptable. But anyone with a pulse knows that prices have been rising at something like a 25% clip in recent years. At 10%, prices double every 7 years. At 25% they double every 3.1 years!

(Note here that dividing 70 by 25 gives a result of 2.8 years. As the annual growth rate gets bigger the formula becomes less accurate. This is why I described it as a “rule of thumb”. It’s not perfect. But it’s 96% accurate up to 10% annual growth and still 90% accurate at 25% a year. That’s “good enough for government work,” as an old work colleague once said to me.)

An inflation rate of 25% is enough to notice changes in the cost of living with relative ease. I moved to Argentina in October 2008. Since then the service charge on my apartment… which has always been my biggest expense… has gone up over 100% in the local currency. It has more than doubled in just over three years. So has the price of a cup of coffee – up from about 6 pesos to about 12 pesos.

But the Argentine peso has lost value against the dollar as well. My costs have gone up about 50% over three years when measured in dollars. A lot less. But still significant. That means my dollar inflation rate is about 14.5% a year. And my doubling time for dollar prices is just over five years. Another peso devaluation would come in handy…

Of course, exponential growth can work against you too. Here are some more doubling times at *current *rates of growth:

**1) World Population **– The amount of people drawing air on planet Earth grows 1.1% a year. The doubling time is under 64 years. At this rate, this means the planet will add 7 billion people in two generations

**2) World Energy Use **– Energy use across all sources grew 2.2% a year since 1985, measured using millions of tons of oil equivalent. At that rate it will double again in less than 32 years – or about one generation.

As Bill pointed out yesterday, this increase in demand is happening at the same time as reserves are depleting and new energy is becoming harder to extract. It’s not the cost that matters most, but how much energy is required to produce new energy. If it takes a barrel of oil to make a barrel of oil, it’s game over for oil production.

**3) Social Security Claims – **The number of retired Americans on Social Security rose 4.4% a year between 1999 and 2009. At that rate the doubling time is just under 16 years. This refers to number of claimants, not dollar amounts. (The dollar amount grows faster due to inflation adjustments. It also doubles faster.)

Since 2007, the rate of claimant growth has accelerated. By comparison, the US population is growing at just under 1% a year. At current benefit and tax levels, even the US Social Security Administration expects payments to be greater than new contributions from 2014 onwards. This means the money will run out by 2037, according to their calculations. I’d be willing to bet that the reality is worse.

**4) US Federal Debt –** This grew about 9.1% year over year. At that rate it will double in less than eight years. This illustrates the urgency of controlling the US budget deficit. And the fatal math applies to most other countries in the developed world.

Of course, these growth rates could change. If food supplies don’t grow fast enough, many people will starve. And population growth will fall. New technologies or higher energy prices could reduce the growth rate of energy usage… and even send it into reverse. And it’s not impossible (although from our perspective, it is highly unlikely) that the US, Europe and Japan could all have growth spurts, balance their budgets and start reducing their debts to manageable levels.

I’ve explained in the past how severe price deflation would have to occur for a buy-and-hold investor in US Treasurys of any maturity to preserve the buying power of his investment (after taxes and at current yields to maturity).

But the energy crunch – where rising demand crashes into depleting reserves – makes this unlikely. So we continue to steer clear of bonds..

If this wasn’t bad enough, there’s another feature of government bond markets that provides yet more cause for concern. Bond maturities are piled up at the short end. Something like a third to a half of all bonds issued by the US, Western Europe and Japan are due to mature in the next three years.

Governments have not only borrowed too much, but they have also allowed their debts to get concentrated at short maturities. A government solvency crisis may be about to turn into a government liquidity crisis. This is more reason to expect money printing during a stretch of years where there could be an energy price shock. Price inflation could be surprisingly strong in coming years.

Conclusion: More reasons to like gold. We’re sticking with the 20% recommended allocation. Gold’s doubling time has been around 3.3 years during the bull market of the last decade, and the yearly rate of price rises has been steady. I expect this to continue… and even accelerate… in coming years.

Until next week,

Rob