# The hare and the tortoise — retold

The WTF moment

I am not an active stock market trader — most of my holdings operate on auto-pilot. Recently, however, as I was looking to do my annual rebalancing, I came across an interesting insight. This blog post is about what I learnt.

My understanding of the financial market can largely be summarized by the simple mental model — “Stocks have returned 13% over the past 10 years vs 7% for fixed-deposits (FDs), but beware! — stocks are riskier and can also go down more over the next 10 years”.

I suspect this is also how most people outside the financial industry interpret risk vs. reward — the reward was in the past, but the risk is in the future.

But, what if someone were to say — well yes, stocks returned 13% on average over the past 10 years, but sorry, they STILL made less than the 7%-FDs.

You would go WTF, right ? That just doesn’t mathematically compute.

### The Example Among Us

The Sensex, India’s popular stock market index, has gone up from 13827 on Jan 1, 2007 to 26,800 by Jan 1, 2017. Yay, O frabjous day! Callooh! Callay!”

As one might expect, its been a rocky 10 years, with annual returns (in percentages) being [47, -52, 80, 18, -25, 26, 9, 30, -5, 2]. The average of those 10 numbers is +13%. As we all know, despite the volatility, thats much better than a safe instrument (e.g. a bond) that returns 7% annually. Right ? Right ?

Wrong ! The boring 7%-fixed deposit trumped the Sensex, over the same period, taking the principal of 13,827 to 27,200 from 2007-2017. When a friend pointed this out, it seemed unbelievable.

Yet, here it is in Excel. When its a debate between Math and Excel, Excel wins, because you can touch each cell. Clicking is believing.

Even after working out the sheet, I thought it to be mathematically impossible for a hare running at 13% average speed to lose to a steady-7% tortoise. If one can’t even trust that average-13% > average-7%, then what was the point of all those years of math ?

### But Why ?

My first thought was that this must be because of the negative numbers — surely, it must be mathematically impossible to get this effect from positive numbers ? Not true — easy counter-examples with positive numbers exist [e.g. avg(121,1,..,1) = 13 vs. avg(9.3,9.3,…)=9.3 ]. Besides, the stock market frequently delivers negative returns. You can’t math-wish negative numbers away.

The realization quickly sunk in that it is because of the compounding effect (with simple interest, the average is indeed a true indicator of the final result). The compounding effect makes any two sequences-of-percentages non-comparable, because the annual returns are not additive.

Even though we desperately look for a convenient single number to compare two portfolios — averaged returns are not the answer — they are a very misleading (and dangerous) metric. As we saw with the Sensex example, a 13% average sequence can be worse than a 7% average sequence. Novice investors have been fooling themselves looking at average returns to compare funds/investment choices.

The second follow-on thought I had was that, surely, this must be a corner-case. Even though it has actually transpired over the last 10 years, it must be really hard to reproduce such an exceptional scenario, where an 7%-averaging instrument beats an 13%-averaging one. And again, I was wrong. As they say, nature doesn’t believe in corner-cases.

### On average, you win...

The key concept is dispersion (i.e. deviation from the mean). When the dispersion is 5% around a mean of 13%, it means that the annual yields (assuming a normal distribution), will range between 8%-18% (for one standard deviation). There will be some outliers beyond that range, but only a few.

I wrote up a little program that generates scenarios of various random annual yields, for each of the past 10 years — the mean being 13% with a certain dispersion D around that mean.
The dispersion was steadily varied from 0% (the steady case) to 50% (very volatile). For each of these dispersals (0%, 5%, …, 50%), the program creates a 1000 different random yield scenarios and checks how much the portfolio would have returned for that yield scenario.

On average, each of the 13%-portfolios returns 3.4x over 10 years, (e.g. [13,..,13]). As the dispersion increases, the annual yields become volatile, and some scenarios result in superb 10-year returns while other scenarios tank the portfolio (often, to zero). However, averaged across 1000 scenarios, we get the same 3.4x, independent of the dispersion factor (Note : this is much higher than what a 7% FD would return over 10 years — i.e. 2x). Ah, our faith in math is restored.

### But mostly, you lose

What is more interesting, however, are the number of scenarios in which the 7%-FD wins at each dispersion-level. When the dispersion is zero, the returns are steady [13,13..] and there is obviously no scenario in which the 7%-FD can win. Even if the dispersion is 5%, the 13%-equity fund is guaranteed to win every time.

However, when the dispersion increases, there are more and more scenarios where the steady-7% portfolio wins. When the dispersion is a modest 20%, the tortoise has a 24% likelihood of winning. And when the dispersion is around 40%, the tortoise is MORE than likely (56%) to win.

What is happening is that, as dispersion increases, the winning scenarios for the 13% portfolio get bigger and bigger, while the potential loss is capped. The hare’s good-looking average of 3.4x is mostly delivered by a few outlandishly good scenarios, while more and more of the “normal” scenarios start favoring the tortoise. One could call it the Rohit Sharma effect.

Thus a portfolio that returns 13% on average, but has a dispersion of ~37% is actually just as likely to beat a steady-7% portfolio, as it is to lose.

The financial industry, of course, knows this and uses a metric called the Sharpe-ratio to normalize all returns to the equivalent risk-free portfolio. They also use things like “what Rs 10K would grow to”. These are much better metrics. For more on this, refer to this link.

I didn’t know this, and perhaps you didn’t either. Now you do.

### Takeaways

1. Don’t look at average returns over time in isolation. Without the Sharpe-ratio, these are not representative.
2. The more volatile a market, the less meaningful the annual returns are after compounding. This means small-cap funds, sector funds etc. are even more prone to this effect. Similarly, the longer the time-frame, the more you should discount the average returns (this is counter-intuitive).
3. The risk with stock is not just about the future, it also has to be used to mentally discount the average returns of the past. When they say — “past returns are not an indicator of the future”, you should also read it as “past returns are not even an indicator of the past (after compounding)”.

### More Thoughts

First — The Indian financial markets have largely been copied from the west. Wisdom, investment strategies and even regulations about what fund prospectuses should disclose are largely copies. For the large part, this is good, because we should learn from other’s mistakes and adopt best practices.

However, these best practices don’t account for the fact that the Indian market is much more volatile than the West, and volatility brings strange effects which require different disclosure norms. What is a highly unlikely corner-case at a 20% standard-deviation, become a highly likely scenario at a 50% standard-deviation.

Second — India is a relatively small market in the overall global economy and hence it is easily influenced by larger global market forces. So even when you invest in the broad indian market you are already making a high-risk investment (relative to other global choices). So if you go niche in what is already a relatively small market, you are really doubling the factors that can cause volatility. Don’t go chasing waterfalls.

### Update

For the commenters.

1. The article is not arguing that one should invest in bonds rather than stocks. Just that any volatile instrument’s average annual return (arithmetic mean) is misleading. There are a lot of other factors to consider to make a stock vs bond choice (dividend, risk, liquidity, taxes etc). But returns tend to be a big (biggest?) factor in comparing investment choices and hence should be carefully reconsidered.
2. Volatility is not unique to the last 10 years. As per Murphy’s law and the Black swan theory, Shit keeps happening. See the last 38 years of the Sensex from inception.
3. Yes, CAGR (compounded annual growth rate) is a better metric but also has its own problems with volatility. Note that most Mutual funds recently quote their 1yr, 3yr, 5yr average returns (not their 4,6,7 or 10 year CAGRs). With volatile markets, cherry-picking the right periods can make a big difference.
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