Before we start, please ensure you have read my previous post on visualizing risk.
As humans, we are inherently bad at visualizing risk. This can often lead us towards investment choices that reduce the probability of achieving our goals. For example, we are often influenced by amazing stock success stories and think that buying stock in potential unicorn companies will lead to good investment outcomes. This rationale is often expressed without consideration for the probability that a company will be able to achieve this estimated rate of growth. For example, Amazon stock (NASDAQ: AMZN) could be found for $15 in 2001 and, since then, has risen to over $3,500 in 2021 (Google finance, 2021). Each dollar invested in the company in 2001 would be worth over $230 in 2021 which is a staggering annual growth rate of over 30% per annum. Of course, Amazon is not alone in this growth story, numerous other companies have experienced similar success.
The issue with this mentality is that the probability of any small startup achieving similar growth rates to that of Amazon is very small. The world of tech startups is brutal, for example 70% of startups will be dissolved within the first 10 years (CB Insights 2018). Further to this less than 1% of tech startup companies will reach a valuation of over $1B (CB insights 2018) and considering the fact that Amazon is a $1.7T company simply reaching a market cap of $1B won’t be enough to give public investors astronomical returns.
Estimating the expected value
If we assume that 1% of stocks on the NASDAQ can achieve similar growth rates as Amazon over a 20-year period we can estimate the expected value of this investment.
| Cost of investment | $1 |
| Probability of success | 0.01 |
| Stock value if successful (value rises 200x) | $200 |
| Stock value if unsuccessful (value falls 50%) | $0.50 |
It turns out that this is a positive value which implies that if an investor has enough capital to invest enough times, they would likely make a profit in the long run. Bear in mind that with a 1% probability of success you’d need to invest A LOT of times i.e., invest in a huge number of companies for a decent chance of success.
What if we invest in more potential companies?
What about investing in multiple “potential Amazons” to increase the chance of success? Let’s see what the expected value becomes if we invest in a portfolio of 2 companies, where $0.50 is invested in each.
| Cost of investment in each stock | $0.5 |
| Probability of success of each company | 0.01 |
| Stock value if successful (value rises 200x) | $100 |
| Stock value if unsuccessful (value falls 50%) | $0.25 |
By diversifying our portfolio, we get the same expected value of $1.495 however the probability of making a profit almost doubles from 1% to 1.99%. This is a result of the fact that if 1 or 2 of the 2 companies in the portfolio are successful then we turn a profit of at least $99.25. The sacrifice made here is that to earn the maximum amount of $199 both companies need to succeed and the probability of that occurring is 1% x 1% = 0.01%. This example shows how diversification can be used to maintain expected return whilst increasing the probability of making a profit. Once again, the sacrifice that investors make is a decreased probability of making the highest possible return. A graphical representation of the probability distribution of a portfolio with 2 “potential Amazon” stocks looks like this:
Visualizing risk as a probability distribution
In the real-world successes are almost never binary – there is a huge range of returns that are possible and each outcome would be paired with a corresponding probability. Returns and probabilities existing on a continuum would generate a probability distribution that could look something like this:
You can see here that the probability distribution is skewed towards the left, this implies that over a 20-year period negative returns are much more likely than positive. While there is a chance to have 20%, 30% or even higher compounded returns because the probability of achieving them is so low, they aren’t visible on the graph.
To help with visualizing risk profiles I’ve done up a few other plots for some other investment ideas:
I really like the distribution shown in figure 3 as it somewhat articulates human irrationality. Despite a near 100% chance of a total loss of capital people are still willing to spend large amounts of money on lotto tickets for the chance of winning an 8-figure prize. Got to be in it to win it guess!
Comparing figure 4 and figure 5 shows the effect of diversifying an already broad-based index investment over time. We see that the highest return falls from approximately 30% to approximately 16%, we also see that the chance of producing a negative return is almost entirely eliminated. Both of these effects are the result of the returns becoming more concentrated around the median return of 7%.
Over extended periods of time I believe that it is quite likely that the expected value of a properly diversified portfolio tends to be higher than that of any individual stock. This is due to the idiosyncratic risk faced by an individual company causing a much wider spread of potential returns.
Conclusion
I wrote this post not to discourage anyone from looking for the next Amazon, Facebook or Fortescue Metals Group but to highlight the importance of understanding the probability of different investment outcomes. When assessing any investment opportunity, getting a rough estimate of the expected value and probability distribution of outcomes is an important component of making sound investment decisions. Doing so will help you manage risk to align with your risk tolerance and financial goals.
Diversification is a powerful tool that will concentrate returns more heavily around the mean and decrease the probability of a negative return. However, this comes at a cost of a decreased potential return. In the FFE house we believe that our diversified portfolio’s expected return will be sufficient to meet our goals. However, if your goals require a higher rate of return then you must understand that the higher this requirement is the further the probability of success decreases.
Engineer your freedom
References
Google finance, 2021, Amazon.com, Inc., Google finance 2021, available from: <https://www.google.com/finance/quote/AMZN:NASDAQ?window=MAX>
CB Insights, 2018, Venture Capital Funnel Shows Odds Of Becoming A Unicorn Are About 1%, CB Insights, available from: <https://www.cbinsights.com/research/venture-capital-funnel-2/>