Figure 1. Low Risk Portfolios
The dataset contains 490 out of S&P 500 stocks from Oct. 1st, 1998 to Dec. 31st, 2007.
To avoid dealing with split and inverse-split, only the adjusted prices is used.
Initially, we have $1,000,000. Portfolio is built at the first trading day of each month and is sold up at the last trading day of each month. Each of selected stocks is bought with equavilent amount of money. Share of stocks is rounded down, if the calculated value is not integer, so that we may have a small amount, typically less than $10,000, of marginal cash each month.
The stocks are chosen according to their volatilities. The volatilities are calculated right at the first trading day of each month and take only the last month's prices into account. Then the volatilies are sorted ASCENDINGLY.
It is worth noting that if we talk about the stocks whose volatilites rank from 90% to 100%, we're talking about the ones with top 10% highest volatilities. On the other hand, if we choose the stocks whose volatilities rank from 0% to 20%, we're choosing one in five that have the lowest volatilities.
First we investigate the monthly payoffs of different portfolios in terms of volatilities. At early Oct. 1998, we have no idea what the volatility of each stock is, so we just hold cash and earn interest for a month. At the beginning of Nov. 1998, we can calculate the volatilities of stocks of last month, build ten mutually disjoint portfolios based on different levels of volatilities and hold the each of the portfolioes until the end of the month. Then we start over at the beginning of next month and so on. Figure 1 and 2 illustrates our monthly return of these portfolios.
There are two reference lines: the yellow one is monthly return of keeping all money in bank while the red one is buying all available stocks which can be regarded as an stock index.
It is showed that low risk portfolios can do a little bit better than interest rate but far from satisfied comparing to the movement of the whole market.
According to Figure 2, it seems that high risk portfolios always outperform the interest rate. Moreover, if the volatility is great enough, say greater than 70% number of all stocks, the portfolio would be better than the index.
It is worth noting that the most risky portfolio yields nearly 9 times of initial money and its overall return at any time never falls lower than any other portfolios, even it suffers from several significant monthly losses such as late 2001.
Why we earn so much from the most risky portfolio? The answer is our fortune (payoff) comes from fortune (luck). Oct. 1998 is a right time for us to go into the stock market. If we have waited two years before we bought stocks, leaving our money in the bank, things would be totally different. This case is showed by Figure 3 & 4.
If we stick to low risk, we are better off in terms of final return by waiting two years. Though from 2002 to 2004, the value of our capital has fluctuated dramatically around interest rate line, we never have our money less than $1000,000 which we started from.
As for high risk portfolios, it's another story. They performs awfully worse comparing to Figure 1. Not only can't we earn so much final returns, but also we nearly lost half of initial money at the end of 2002. And that is the truth of Risk.
Actually, the shape of curves after Nov. 2001 are exactly the same between Figure 1 and Figure 3 as well as between Figure 2 and Figure 4. If we move upward the curves in Figure 4, they would match perfectly with curves in Figure 2. Things are good when we buy stocks at the very beginning, because we have earned a lot (due to luck) before suffering a great loss but no greater than our previous gain. Comparatively, things are not that good if we start to buy stocks two years later, because we coincidently came across a depression at that time.
To expose the real features of risk, we should get help from Monthly Return Rate, as described in Figure 5 & 6.
We can see clearly that higher volatility implies more dramatical fluctuation of monthly return rate.
Figure 7, 8 & 9 depict the differences of final return rates with regard to different entering time. In each figure, there is a baseline whose y coordinate is about 1.55 which is the final return rate of holding cash all along.
Again, we see that higher risk portfolios fluctuate more significanly than lower risk ones. Is there an optimal entering time or can we work out a comparatively good time to go into the market? Would it be good for us to buy stocks during financial crisis like today? Is it a gamble or something else? Hopefully I could have some ideas about it later.