Many academics agree that to beat the market, one needs to assume greater risk - as opposed to trying to predict the market.
Defining Risk: The Dispersion of Returns
| Business Conditions | Possibility of Occurrence | Expected Return (R) |
|---|---|---|
| “Normal” economic conditions | \(\frac{1}{3}\) | 10% |
| Rapid real growth without inflation | \(\frac{1}{3}\) | 30% |
| Recession with inflation (stagflation) | \(\frac{1}{3}\) | –10% |
$$ \mathbb{E}[R] = \frac{1}{3}(.10) + \frac{1}{3}(.30) + \frac{1}{3}(-0.10) = 0.10 $$ $$ Var(R) = \frac{1}{3}(0.30–0.10)^2 + \frac{1}{3}(0.10–0.10)^2 + \frac{1}{3}(–0.10–0.10)^2 = 0.0267 $$
Although only losses constitute risk, the distribution of upside vs downside is usually symmetrical (at least for well-diversified portfolios) such that variance to be a good measure of risk.
Documenting Risk: A Long-Run Study
On average (1926 - 2018), investors have received higher rates of return for bearing greater risk. Stocks have outperformed long-term treasury bills and inflation.
However, stocks have more variable returns, e.g. negative returns in 1930-32, Oct 1987, early 2000s.
Reducing Risk: Modern Portfolio Theory (MPT)
A toy example (suppose each season is equally likely):
| Return in Rainy Season | Return in Sunny Season | |
|---|---|---|
| Umbrella Manufacturer | 50% | -25% |
| Resort Owner | -25% | 50% |
$$ Corr = \frac{1}{2}(0.50 – 0.125) (–0.25 – 0.125) + \frac{1}{2}(–0.25 – 0.125) (0.50 – 0.125) = –0.141 $$
Investing in either business has an expected return of 12.5%. However, there could be many sunny/rainy seasons in a row. Investing in both businesses has an expected return of 12.5%, despite the seasons.
The caveat is that the businesses must be affected differently by the same condition, i.e. the correlation coefficient is as close to -1 as possible. Businesses that go perfectly in tandem have a correlation coefficient of 1.
The Role of Globalization in Portfolios
There’s is no perfect -1.0 correlation in the real world though. In 2007-09, all markets fell in tandem.
In practice, beyond 50 companies, the risk reduction from diversification tapers off. More risk is reduced if overseas companies are included. This is as close as we can get to a free lunch.
The principles of MPT look useful in other contexts, e.g. agriculture in Kenya which tends to succumb to erratic weather. I wonder if there are formal studies that treat crops and their yield as a portfolio problem. See ‘Agriculture in Kenya as a Portfolio Problem’ .