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Milevsky: How Much in Risky Stocks vs. Safe Cash?

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From a purely mathematical point of view, the “stocks are safer over long periods of time” argument ­— before Paul Samuelson took an axe to it — went as follows. Historically, there was a 35% chance that a diversified portfolio of stocks would underperform safe cash during any one-year period. Moreover, the stock market’s behavior from year-to-year exhibited the same independence as consecutive coin tosses. The coin has no memory of its past behavior, and neither, so it was believed, did the stock market.

Put these two assumptions together and you get the following: Over a two-year period there is a (0.35)(0.35) = 12.25% Probability of Shortfall. Over a three-year period there is a (0.35)(0.35)(0.35) = 4.3% probability of shortfall, etc. Continuing this logic, by year 10, there was a 1 in 40,000 chance you would regret investing in a portfolio of stocks versus safe cash. This mathematically contrived justification was labeled time diversification, and became another rallying cry of the “buy, hold and prosper” crowd. A longer time period diversified away risk. The stock market was a casino in which the odds were heavily in the gamblers’ favor.

Enter Paul Samuelson who argued that if an investment was risky over one year, time didn’t make it any safer. The only safe asset for the long run was a risk-free inflation-adjusted government bond. Anything else carried risk, and that risk didn’t disappear in the long run, or even the very long run. It therefore didn’t matter how old or young you were. The fact that the probability of shortfall declined quite rapidly with time didn’t imply that you should skew your portfolio more heavily towards stocks when you were young vs. when you were old.

In many scholarly articles he argued that that very small shortfall probability was offset by the enormous pain and disutility of a loss. His great flash of genius was to demonstrate that these two effects exactly balanced themselves out. Ergo, the optimal amount of stocks vs. safe cash was time-invariant.

So — you might wonder — why is the financial industry encouraging people who are saving for retirement to invest more in the stock market and take on more risk? Is there any justification for this advice?

Human Capital

Despite Paul Samuelson’s warning about the riskiness of stocks, which I have taken to heart, I personally remain heavily invested in them, for better or for worse. The reason I have made this seemingly risky decision is because I am many years away from retirement. In fact, I have a very long time horizon before I plan to stop working at (planned) age 70, and start collecting my pension income from the university.

Now, if you have been paying any attention at all, your first reaction might be: “Didn’t he read anything he just wrote above? Professor Samuelson proved that time shouldn’t matter.”

But the fact is that I actually agree 100% with Professor Samuelson, and my behavior is perfectly consistent with his recommendations. The key to reconciling the above-stated ‘Samuelsonian’ time invariance and the investment advice doled out by industry — and my own behavior — is the concept of human capital.

There are two types of assets that a person might possess on their personal balance sheet. The first is financial capital, which is quite visible and market-based. Think of money in the bank, bonds in a retirement account or stocks in a portfolio. You can sell them anytime and immediately consume the proceeds. This is your financial capital.

But you also own another asset, and that is your income-earning ability. This asset is called your human capital value. Researchers such as Professor Gary Becker have computed how much an extra year of education can impact your earnings profile over your life. Evidently, the rate of return from investing in education is between 8% and 15% depending on your field and major. For now the important takeaway is as follows: If you define your wealth broadly enough to include both human and financial capital — and your human capital is relatively safe ­— then even Paul Samuelson will agree that it makes sense to have some money in the stock market even if you are very risk-averse. The question is how much.

The Equation Explained

Here are the six factors or variables that should determine how much money you should risk and invest in the stock market:

1. The amount of financial capital (i.e. money in the bank) that you have accumulated already, which is denoted by the letters FC. This is measured in dollars.

2. The value of your human capital, which is measured in dollars and denoted by the letters HC. Make a conservative estimate of what you think you will earn between now and retirement ­— net of any income taxes and required expenditures — and then compute its present economic value. For young college graduates, this number could be in the millions.

3. Your expectation of the rate at which you (or the experts) think stocks will grow over time, denoted by the letter mu (μ). Historically this number has been around 7% for stocks, once inflation is accounted for, although it might be as low as 5% in today’s economic environment.

4. Your expectation for the volatility of stocks, denoted by sigma (σ), which has been around 20% for a well-diversified portfolio of stocks. This is a purely objective number, best determined by a statistician. Some argue that it might be as high as 25% or even 30%.

5. The risk-free rate (r), which has been between 1% to 4%, and probably closer to the low-end of the range today. It represents the alternative to stocks.

6. Your risk aversion, denoted by (γ), is a purely personal value that only you can determine.

Notice that this equation says nothing (explicitly) about age, time horizon, mortality rates, or even if stocks have been doing well recently. Whether you are 20, 50 or 80, Samuelson’s asset allocation equation tells you to consider your attitude towards risk, your total wealth and the long-term view of the market — but not the time horizon. It is nowhere to be found explicitly in the equation.

When you think about it though, time is really embedded inside of Samuelson’s equation, because the human capital value gets smaller, and eventually hits zero, as you age. Once you reach retirement — defined as the date when you stop working, permanently — and you have no more human capital, and all your wealth is tied-up in financial capital, then perhaps stocks really are too risky for you. This is your call.

Samuelson’s Life and Impact

Paul Samuelson was born on May 15, 1915 in Gary, Indiana, and died on Dec. 13, 2009. He eventually enrolled at the University of Chicago, home of Milton Friedman and other conservative economic theorists. Although he was turned off by the Chicago school’s blind defense of free and efficient markets, his friendship and intellectual rivalry with Milton Friedman continued for the rest of his life.

In 1935 he earned his B.A. degree and headed east to Harvard University, where he earned his Master’s degree in 1936 and his Ph.D. in 1941. During his time at Harvard he fell under the spell of the Keynesian approach to economics, which argues that governments should get involved in financial markets and not leave economic affairs to Adam Smith’s invisible hand. Samuelson often called himself and used the term “cafeteria Keynesian,” claiming to pick and choose only what he liked from the theory. In fact, in 1960 he advised newly elected John F. Kennedy — as the U.S. was heading into a recession — to reduce taxes. Kennedy the Democrat was shocked at the advice, but was convinced of the merits by Samuelson.

Paul Samuelson was recognized as an economics prodigy from very early on in his career. In 1947, the American Economics Association awarded him the (inaugural) John Bates Clark medal, which is bestowed to economists under the age of 40 and considered to be the second most prestigious award in the field. The first most prestigious award is the Nobel  Memorial Prize in Economic Sciences, which he was awarded in 1970.

Remarkably, it was only in his 50s that Paul Samuelson turned his research attention to matters of finance, the stock market and asset allocation. In articles  published in the late 1960s he laid the foundations upon which Robert Merton and Myron Scholes built their famous option pricing equation, which garnered their own Nobel prizes. Indeed, if anybody deserved a second Nobel award, it would have been Paul Samuelson.


Reprinted by permission of the publisher, John Wiley & Sons Canada, Ltd., from The 7 Most Important Equations for Your Retirement, by Moshe A. Milevsky. Copyright © 2012 by Moshe A. Milevsky.


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