Does passive investment strategy always yield index return?

Passive investment is a popular strategy. Many people invest in index funds. It is generally understood that passive investment strategy yield index return minus management fee. Is it true? We will look at a simple example.

Suppose there are two stocks, A and B, in the stock market. Each stock has one hundred shares. The share prices of A and B at each year are listed in the following table

Suppose we invest two dollars in the index fund at the beginning of each year. At year four, we have invested six dollars (two dollars times three years) in the market. Since the market level at the beginning of year four is identical to the beginning of year one, we might assume the total wealth at the beginning of year four is six dollars as well. In the following, we will do a simple calculation.

At year one, share prices of A and B are the same. We invest one dollar each in stock A and B. After three years, the share prices are the same. The final wealth from this investment is two dollars, one dollar from each stock.

At year two, share price of A doubles the share price of B. We invest 4/3 dollar in stock A and 2/3 dollar in stock B. After two years, the share price of stock A is halved. The wealth invested in stock A becomes 2/3 dollar. The final wealth from this investment is 4/3 dollars, 2/3 dollar from each stock.

At year three, share price of B doubles the share price of A. We invest 2/3 dollar in stock A and 4/3 dollar in stock B. After one year, the share price of stock B is halved. The wealth invested in stock B becomes 2/3 dollar. The final wealth from this investment is 4/3 dollars, 2/3 dollar from each stock.

Total wealth from three years’ investment is 4 and 2/3 dollars, (2+4/3+4/3 = 4 and 2/3). This is less than 6 dollars we invested, although the index level at the end of investment period is the same as the beginning.

This phenomenon is easy to understand. Stock A and B each experiences bubble during the investment period. When bubble burst, the investors will lose money.

If the shares are underpriced sometime during the investment period and recover in the end, the investors will make money.

Which scenario is more likely, overvaluation or undervaluation of a stock? Company executives’ pay and stock option values are tied to stock prices. When companies engage in mergers and acquisitions, higher stock prices help them to get better deals. Fees for investment bankers and fund managers are often proportional to asset values. Politicians and the general public prefer high stock valuations. Stock analysts wish to maintain good relations with the companies they cover.

Overall, most parties prefer a high valuation of stocks. This makes the overvaluations of stocks more likely than undervaluation. For retail investors, this means that their long term performance in passive investment are more likely to underperform the market index.