Historical Mean Return vs. Expected Return
To better understand this, think about supply and demand for the stock of company A in the stock market. Supply is met by investors who currently own the stock and could potentially sell into the market. Demand is created by investors who may want to purchase stock.
If the average investor views the stock for company A to be overvalued, what is most likely to happen to the price of that stock?
The stock market is risky. Although on average investors expect a return of 5% based on the price today and the forecast future price, the expected return rarely equals the __historical mean return__, or the actual return realized by investors.
Standard deviation of returns is one measure of risk. Consider two stocks: the standard deviation of returns is twice as high for stock A than it is for stock B, and the expected return for stock A is accordingly higher than for stock B.
Which stock do you think will likely have a larger difference between the expected return and actual return in a given year?
Not quite.
If the average investor views the stock as overvalued, there is an investment opportunity. If you own the stock, you will likely sell, and if you don't own the stock, you will seek to short sell.
Correct!
If the average investor views the stock as overvalued, that means the average investor views the current price as being too high based on total risk. More investors will be selling than buying, pushing the price of the stock lower.
Correct!
When the standard deviation of returns is higher for a stock, it reflects that future prices for that stock are more difficult to forecast. Consider that if you could perfectly forecast the price one year from today for a stock, the expected return would equal the actual return.
Differences arise between the actual return and expected return due to forecast error as unforeseeable events occur that surprise investors. By definition, surprises are random and unexpected. As they are random, positive and negative surprises are equally likely. It has also been shown that returns are roughly normally distributed, which means the magnitude of positive and negative surprises and their likelihoods are roughly equal.
These assumptions mean that over a long enough period (i.e., with an adequately large sample) positive surprises will offset negative surprises such that the mean historical return will equal the expected return demanded by investors over that period.
Think about a very small sample of one expected return and one realized return. What do you think the likelihood is that the two returns will be the same?
That response in incorrect.
When the standard deviation of returns is higher for a stock, it reflects that future prices for that stock are more difficult to forecast. Consider that if you could perfectly forecast the price one year from today for a stock, the expected return would equal the actual return.
That's not right.
If you view a stock as overvalued, you will react by making an investment which will affect the supply-demand equilibrium.
Correct.
The likelihood of those returns being equal is very small due to forecasting error. All that is needed is one negative or positive surprise to make the return different.
Now expand the sample to 10 return observations. The likelihood of positive and negative surprises canceling each other increases such that the mean historical return is more likely to equal the expected return.
If you expand the sample further to 1,000 returns, the likelihood that mean historical return equals the expected return increases further.
Incorrect.
The likelihood of those returns being equal is very small due to forecasting error.
Now expand the sample to 10 return observations. The likelihood of positive and negative surprises canceling each other increases such that the mean historical return is more likely to equal the expected return.
If you expand the sample further to 1,000 returns, the likelihood that mean historical return equals the expected return increases further.
No one knows how long of a period is necessary in order for mean historical returns to equal expected returns. Economists typically refer to this undefined period of time as the _long run_.
To summarize this lesson:
[[summary]]
The __expected return__ $$E(R)$$ is the nominal return an investor would demand to make an investment based on the real risk-free rate $$r_{rF}$$, expected inflation $$E(\pi)$$, and a risk premium $$E(RP)$$.
$$ 1 + E(R) = (1 + r_{rF}) \times [1 + E(\pi)] \times [1 + E(RP)] $$
Of course, the investor doesn't actually "demand" a given return in the literal sense, but if an adequate return is not offered, the typical investor will be unwilling to enter the investment.
Think about it this way. If the average investor perceives that fair compensation for bearing the risk of holding the stock of company A is 5%, she or he will forecast the price one year from now based on company fundamentals, and they will buy the stock only if the difference between the price today and the price in one year is greater than 5%.
If the stock is overvalued, the price today is too high and the difference between the price today and the expected price in one year is less than 5%. Investors that own the stock today will sell, pushing the price lower until an equilibrium price reflecting a 5% return is reached.
The price would increase
The price would stay the same
The price would decrease
Stock A
Stock B
The likelihood is low
The likelihood is high
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