Understanding the Mathematics of Personal Finance

Investing: Risk versus Reward

If you kept all your savings in cash in a shoe box under your bed, you would be risking loss due to theft, fire, and so on. On the other hand, while your savings would never add up to anything other than exactly what you put into the shoe box, from a financial point of view, your savings would be absolutely safe. No stock market variations, bank failures, or whatever could impact your savings. Inflation, however, would slowly eat into the actual value of these savings, even though the dollar amount didn’t change.

Government-insured savings are, for all intents and purposes, perfectly safe. In addition, when your money is put into an insured saving product, you don’t have to worry about a burglar making off with it in the night. Interest rates on these products are not very high, but they’re usually a bit better than inflation. If total safety and security are your goals, then government-insured savings are the perfect choice. You won ’ t get rich, but you’ ll stay ahead of inflation and you ’ ll sleep well at night knowing that your money is secure.

At the other extreme, you could take all the money you want to put away for your old age and buy lottery tickets every week. In all likelihood, you’re throwing your money down the sewer, but you never know; somebody will win the lottery and get very rich quickly. Chapter 14 discusses this approach to “investing” in more detail.

The holy grail of investors is the perfectly safe investment that returns a signifi­cantly higher average annual percentage rate (APR) than a savings account. Remember that a holy grail is something that people spend their lives looking for, but never find. They know it’s out there somewhere, probably lying right next to the fountain of youth.

Before continuing on this topic, I should point out that I am writing this material early in March 2009. The stock markets are at their lowest levels in decades. Billions if not trillions of dollars have been lost, and nobody knows if the markets have reached bottom yet. Mainstays of American economic might such as General Motors are teetering on the brink of bankruptcy. Consequently, it’ s very hard for me to regurgitate the conventional wisdom about long-term growth of stock values being a historical truth.

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

Copyright © 2009 John Wiley & Sons, Inc.

Fortunately, I never intended to give stock market advice. For that matter, I’ve always wondered why people who know how to “pick good stocks” are spending their days being financial advisors for other people rather than picking these good stocks for their own accounts and quietly getting very rich. In an earlier book (Dworsky 2008), I discussed the concept of the “superior” fund manager and how it’s funda­mentally impossible to separate such a person from a crowd of randomly skilled fund managers. I’ll go ahead and present some of the approaches to measuring and coping with investment risk, hoping that by the time you’re reading these words, market stability has recovered. Keep in mind that when the stock market is rising, most portfolios will grow; when the market is falling, most portfolios will fall. If you’re fortunate enough to be investing during a period of rising markets, be grateful for your earnings but be very cautious about attributing your success to your own skills.

There are so many different ways to invest money that I can’t possibly present them all or give examples of more than a few of them in one short chapter. I’ll treat a few concepts, mention some other ideas, and refer you to the public library sections on investing for the rest.

The general theme of investing is that potential reward comes with risk. There is some mathematical discussions of risk below, but the basic idea is that the faster you want your money to grow, the bigger the chance that something will go wrong.


There are many different investment products available today. Stocks and bonds relate to a specific company. You are buying a small part of or loaning money to the company of your choice. A share of stock represents ownership of a part of the company that issued the stock. As long as the company is out there doing business or at least still has some assets, your investment will have some value.

The risk in a stock price, given fairly calm market conditions, is measured by the stock’s “volatility.” In statistical terms, considering the stock price to be a random variable, the return on an investment has an expected value and a risk that is quanti­fied by standard deviation. Expected value has been used several times already in this book. Standard deviation is a measure of how the stock’s price (in this case) has varied about the expected value. I’ll talk more about standard deviation below.

Figure 13.1 shows the performance of five hypothetical stocks over time for a period of 250 market days (about 1 year in actual time).1 I’ve “normalized” the value of these stocks to a starting price of $1. That is, I’ve divided the price of each stock by its price on day 1. This lets me compare stock price change by percent changes as compared with their starting values. If I wanted to look at, for example, a stock selling today for $100 and a stock selling for $5, then it doesn’t make sense to look at one share of each of these stocks. A $2 change in the $100 stock is noticeable but not exciting. A $2 change in the $5 stock is a major change in value. To compare these stock prices meaningfully, I have to look at equal dollar investments in each

When I talk about a stock’s price on a given day, I mean the closing price at the end of the day.


Figure 13.1 The price of five hypothetical stocks tracked for 1 year.

stock. I could compare the price of 50 shares of the $2 stock to the price of one share of the $100 stock, or I could look at what $1 would buy me of each stock, assuming that I could actually buy such pieces of shares of stocks. This latter choice is the mathematical equivalent of my normalization procedure.

At the end of the year, the average price of these five stocks is $1.105. If I had bought equal dollar amounts of all of these stocks on day 1 and waited a year, I would have earned slightly better than 10% APR on my investment. Had I put all of my money into stock #5, I would have earned about 37%. Had I put all of my money into stock #4, I would have lost about 4%. How do I know what to do?

I’ll review standard deviation and also introduce the idea of correlation. These are statistical indicators that are very useful in guiding your investing decisions. They are not guarantees of anything.

Figure 13.2 shows histograms of stock #1 and stock #5. These are histograms of the 250 data points making up the stock’s prices for the year. In the case of stock #1, all of the prices are bunched between $0.85 and $1.01, with the great majority bunched between $0.50 and $1.00. The average price for the year of this stock was $0.95. In other words, the stock price barely deviated from the average. The standard deviation of the stock price is a measure of how much the stock price deviated from the average. Without going through the math, the standard deviation for this stock was $0.030, approximately 3.2% of the average. The price from day to day probably reflected a bit of the company’s business success level, a bit of overall market sta­tistics, and a bit of random “noise.” This is not a volatile stock.

Now look at the histogram for stock #5. The average price for the year of this stock was $1.12. The histogram shows stock prices varying from about $0.85 to about $1.35. Although most of the time the price was close to the average, there is


considerable spread, or width, to this histogram. The standard deviation is $0.10, about 8.9% of the average. This is a much more volatile stock than stock #1. Looking back at Figure 13.1, this stock was performing far worse at about 30 days into the year than any other stock in the group. Its price was less than $0.90 while the other four stocks were about either $1.00 or greater. There would have been a strong temptation to dump this “loser” and put the money back into the other stocks. At the end of the year, however, stock #5 was clearly the best performer of the group. The point here is that “more volatile” is another way of saying “less predictable.”


The mix of stocks that you own make up what’s called your stock portfolio. In this simple example, the portfolio consists only of the five stocks shown, with the initial investment being an equal amount of money put into each of these stocks. Figure

13.3 shows this portfolio ’ s performance for the year, and Figure 13.4 shows the histogram of the portfolio’s daily prices. The standard deviation here is 0.026, about

2.5% of the average. Even with the volatile stock #5 in the mix, the portfolio standard deviation is smaller than even the very nonvolatile stock #1 alone. The price of the portfolio at the end of the year is $1.05. The trade-off for this increased stability is the reduction of the highest attainable profit. While increasing stability is desirable, the possibility of giving up profit is unfortunate. What we have to do now is look at how portfolios are selected.

If you were to buy $1’s worth of every stock in the stock market, your perfor­mance at the end of the year would be exactly that of the market itself. If you have faith in the future of the market and you want a very low-risk portfolio, this might be the way to go.

I f you want to add some oversight that hopefully makes things better, you will probably pick a small number of stocks that you believe will outperform the market. To add some stability to this portfolio, you should diversify you holdings. Diversification is the management of correlation of stock performances.


In Figure 13.5, I plotted the (250) daily prices of stock #1 versus the daily prices of stock #2. The points don’t seem to form a pattern, it’s just a jumble of dots. The line represents the “best-fit” line to these points.[32] This line is nearly horizontal. If you recall the definition of the slope of a line, this line has a slope close to zero. The jumble of points and the horizontal line tell us that these two sets of data (stocks #1 and #2 prices) have almost nothing to do with one another; that is, they are uncorrelated. Buying stock in these companies is indeed diversifying your portfolio. You are randomizing your choices and are not letting the fortunes of one company be related to the fortunes of another company in your portfolio.

Figure 13.6 shows the same type of plot for stock #1 and stock #3. In this case, there is a recognizable pattern, and the best-fit line has a slope of almost +1. These


Normalized stock #1 prices

Figure 13.7 Scatter plot of stock #4 prices versus stock #1 prices.

two data sets show a strong positive correlation. The correlation exists because I put it there for the sake of this discussion when I created the data sets. In the stock market, you could expect to see such a correlation between, say, two oil producers or perhaps two medical equipment suppliers. This correlation is telling us that strong forces that might affect all of the companies in a given business or in related businesses would exert a common “pull” on all of these companies’ stock prices. An example might be manufacturers of certain types of cholesterol-reducing medica­tions in a year when several studies report that taking these types of medications really increases your expected life span.

In terms of investing, you probably want to avoid “stacking your deck” with companies having a positive correlation unless you are very sure that these busi­nesses have a reason to expect good performance in a given year. If you get it right, you stand to make a lot of money. If a report comes out that these medications actu­ally might do you harm, all of these drug companies will probably fare poorly for a while.

Figure 13.7 shows the correlation between data sets #1 and #4. Again, I created these data sets and the correlations for my examples—there is no deeper meaning here. In this figure, a pattern is once again obvious, and the best-fit line shows a slope of almost -1. These data sets are anticorrelated. These two companies might be a lambskin winter coat manufacturer and a light sweater manufacturer. A harsh winter would be good for the winter coat manufacturer, while a mild winter would keep people out of the winter coat manufacturer’s stores and send them looking for light sweaters. In other words, there are factors that could drive one stock price up while simultaneously driving the other stock price down. Another example would be a manufacturer of a large SUV as compared with a manufacturer of a small hybrid car. Changing gasoline prices would significantly affect one of these businesses positively while affecting the other negatively. Buying stock in both of these com­panies might not make the most sense. If overall automobile sales are good, both companies could prosper—but a change in gasoline prices could cause one stock to go up while another stock goes down, and your portfolio just sees a profit cancelling a loss.

A simple way to diversify, rather than buying a few individual stocks, is to buy shares in a mutual fund. A mutual fund is a fund, managed by a professional manager, who buys selected stocks and then sells shares in the portfolio of these stocks. There are very many funds in the market. You can read about the “philosophy” of the fund manager for this particular fund. Many large fund companies offer several different funds, with choices in industries, risk level, social consciousness, “greenness,” and so on, for the buyer to choose among. Since these funds are professionally managed, they’re attractive for people who don’t have the time, the expertise, or the interest in closely following the stock market. Also, because these funds are investing the pooled resources of many people, they can diversify to a level that most individual investors cannot.

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