Do Stocks Provide a Positive Expected Return?

The classic answer is that when we buy a share of the company, we are providing financial capital to help grow and run the business — we are performing a service and should expect to be paid for it.

And like making a personal loan to a friend, there is a risk that you may get only a portion of your money back or none at all.

So in essence the higher expected return of stocks, much like the interest rate on a loan, is there to compensate you for that risk.

But does it?Mean Annual Return of S&P 500 Including Dividends (1962–2017): 10.

9%Mean Risk Free Rate (1962–2017): 5.

2%The average returns look decent but just looking at averages can trick us.

The following chart compares annual stock returns to the risk free rate.

Three things jump out:S&P 500 returns exhibit much higher volatility than cash returns.

Cash has outperformed the S&P 500 many times.

Pre-1990, cash seemed to outperform stocks more often than post-1990.

Post 2008, stocks have easily beat cash because risk free rates have been locked to zero by the Federal Reserve.

Stocks returns vary much more than cash returnsTesting our Hypothesis with some SimulationsInstead of calculating test statistics and running a formal hypothesis test, let’s visualize the process by running some simulations (a very similar analysis in spirit).

I ran 5,000 one year simulations with the following assumptions:Stock returns are normally distributed with an expected return of 10.

9% and standard deviation of 15.

2%.

The risk free rate (cash return) has an expected value of 5.

2% and standard deviation of 3.

4% but cannot go below 0%.

The above values were estimated using historical data (1962–2017).

One could argue that the high inflation (and high interest rate) 1970s were an anomaly but we are already data constrained so I have decided to use as much data as possible in my analysis.

Stock and cash returns are independent.

I did not inflation adjust any of these values (everything is in nominal terms).

Cash beats stocks in 35% of the simulations that I ran!.That’s not great.

So in any given year, the chances that the S&P 500 will outperform cash is just 2 out of 3.

For a visualization, check out the histogram of my simulation results below.

Note the vertical orange line at x=0; that happens because I don’t allow the nominal return of cash to go below 0% (and neither should you, if your bank tries to charge you for holding onto your money, it is time to change banks).

Thus, there is a large amount of instances where cash return equals 0%.

Stocks look pretty risky…That’s not all.

Recall that we estimated our expected returns and standard deviations using only 56 years of history — that is not a lot of data.

So there is a second level of uncertainty — uncertainty around our estimates of the population parameters (for simplicity, we will focus on addressing the uncertainty around expected returns and ignore the uncertainty around the standard deviations).

We can attempt to capture this uncertainty by applying a standard error to our estimates of expected returns.

The standard error is basically the standard deviation of our estimate of the true mean.

That is, the standard error tries to answer the question — if we ran this experiment many times (pretend we can) and calculated the expected return each time, how much variation would we observe in this expected return estimate?We can calculate the standard error (SE) as:SE = Sample_Standard_Deviation / sqrt(Sample_Size)SE_S&P500 = 15.

22% / sqrt(56) = 2.

03%SE_Cash = 3.

36% / sqrt(56) = 0.

45%Modifying our Simulation for Uncertainty Around Our ParametersSince there is no practical limit to the number of simulations we can run, let’s just express our uncertainty around the true value of the expected returns as an additional source of randomness.

Here is our setup:Run 5,000 experiments.

Where each experiment consists itself of 5,000 one year simulations.

Except now the expected returns of the S&P 500 and cash are no longer fixed; rather they are themselves random variables (that are distributed according to the Student’s t distribution) whose standard deviation is equal to their respective standard errors.

So we are just repeating what we did earlier (the earlier histogram) 5,000 times (like a nested loop), each time allowing the variables we are uncertain about (expected returns) to change.

Let’s take a look to see how that looks (remember the vertical orange line is there because I don’t allow negative cash returns, thus there are many instances where cash return = 0%).

The shape looks about the same as before, just smoother because we ran so many simulations.

Looks about the same as beforeLooking at the stats, nothing has really changed (cash still wins 35% of the time on an annual basis).

So was this all for nothing (it was actually useful for my own intuition to check whether uncertainty around estimates would drive even more variance — seems like it did not)?Or maybe we didn’t ask the right question…Stocks Look Better when you Extend your Investment HorizonBuy and hold is a popular investment strategy for a reason.

We shouldn’t judge our investments in terms of days, weeks, or even months.

We are investing to build long term wealth and to fund our retirements.

Recall that previously we found that cash wins 35% of the time (out of our 5,000 simulations).

Let’s take a look at how the amount of time that we buy and hold for (the holding period) impacts the cash win percentage:The longer your holding period, the more likely that stocks will beat cashIf we buy and hold stocks for 5 years, the probability that cash wins drops from 35% to 25%.

If we buy and hold for 10 years, it drops to 18%.

And if we buy and hold for 20 years, the probability of stocks underperforming cash goes all the way down to 12%.

Still not ideal, but that is significantly better.

So at a long enough horizon, it seems like a reasonable bet to say that stocks will return better than cash — with a 20 year investment horizon, stocks beat cash 7 out of 8 times.

Let’s build some intuition into why a longer holding period makes stocks more attractive relative to cash.

The main reason is that as you increase your holding period, expected returns start to matter more than volatility (standard deviation).

Let’s look at the equation for expected return over a five year holding period:5_yr_return = (1 + annual_return)⁵ – 1And here is the standard deviation (stdev) of your cumulative return if you held an investment asset for five years:5_yr_stdev = annual_stdev * sqrt(5)Notice anything cool?.If you take away just one thing from this post, I want it to be the following point.

This is probably the most important concept in all of finance:Expected returns compound exponentially with time but volatility (standard deviation) scales with just the square root of time.

This means that as more years pass, the volatility of your investments get swamped by the returns, provided that the expected returns are high enough (returns outweigh volatility over long periods of time).

You will hear this referred to as the Power of Compounding.

Let’s check this out visually:Return starts to take off relative to volatilityThe previous chart makes stocks seem like a no brainer but remember, we care less about the absolute return of stocks and more about the excess return of stocks over cash.

And the same compounding benefits that benefit stocks help cash as well.

Stress TestIt would be too easy to just say, “Hold stocks for 20 years and you are good to go.

” So let us stress test our assumptions a little bit.

Earlier, we calculated the standard error for stocks (the standard deviation around our estimate of the expected return) to be 2.

03%.

So it would not be surprising at all if the actual realized return for stocks turned out to be 1.

0% lower (9.

9% we estimated).

What happens to our probabilities of cash beating stocks across various investment horizons if stock returns turned out lower than we expected?.Unsurprisingly, things get worse across the board and now even at a 20 year horizon, cash beats the S&P 500 nearly 1 out of 5 times.

Gets worse but longer holding periods still helpConclusionSo does the S&P 500 tend to outperform One Year Treasury Bills?.Generally yes.

But in any given year, we should expect wildly diverging results (stocks win only approximately 2 times out of 3).

And even held over 20 years, we cannot be extremely confident that we are better off investing in stocks (stocks win roughly 7 times out of 8).

So why bother with stocks then?.As I mentioned above, while cash is risk free in the nominal sense, it is extremely exposed to the ravages of inflation.

Stocks, on the other hand, because of their higher expected return and the ability of companies to raise prices on their goods and services are relatively less exposed to inflation.

So stocks remain one our better bets to build real (inflation adjusted) wealth over long time horizons — but as we have seen, they are in no way a sure bet.

Sources:Data for S&P 500 returns obtained from Robert Shiller’s website.

Data for Treasury Bill rates obtained from FRED.

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