Retirement Savings:  Stocks vs. Bonds

Professor Matthew Spiegel
June 29, 2005

A popular rule of thumb is to subtract your age from 100. The difference represents the percentage of stocks you should keep in your portfolio. For example, if you are age 40, 60 percent (100 minus 40) of your portfolio should consist of stock. However, you may want to modify the result after considering other factors, such as your age, risk tolerance, and financial goals.

The American Institute of Certified Public Accountants, http://www.360financialliteracy.org/Life+Stages/Retirement/FAQs/Investments/How+much+of+my+portfolio+should+I+keep+in+stocks.htm

Making Things Up as We Go Along

There is a tremendous amount of advice regarding how individuals should allocate their portfolios when saving for retirement.  One of the more popular rules of thumb is provided by the above quote from The American Institute of Certified Public Accountants (AICPA).  But where does it come from?  Why should you move more and more of your portfolio into bonds as you age? If you look at the web page from which the quote is drawn here is what they say:

Most professional planners advise that if you are saving for retirement, the younger you are, the more money you should put in stocks. Over the long term, stocks are more likely to provide favorable returns and capital appreciation than other commonly held securities. As you age, you have less time to recover from downturns in the stock market. Therefore, many planners suggest that as you approach and enter retirement, you should begin converting your more volatile growth-oriented investments to fixed-income securities such as bonds.

Well, that seems sensible enough.  If you are young and things go badly you can try to make it up later.  If you are older you may not have time.  But is this the right way to view the problem?  Does this line of analysis lead to the “right” answer?

A Bit of Analysis

To keep things simple assume your life takes place over just three stages: youth, middle age, and retirement.  The goal is to save during your youth and middle age for retirement.  Your savings can be placed into either bonds that pay 35% per period (think 15 year periods) or stocks.  Stocks are risky and may either gain or lose value over time.  In this example, they either return 614% half the time or −26% half the time.  On average this means stocks have a higher expected return than bonds, just like in real life.

For each $100 you place into stocks here is what those funds will do over time.

The arrows point in the direction of the market:  up means the market did well, and down poorly.  If you invest in bonds there is no risk and your $100 will grow to $135 when you are middle aged and $181 when you retire for sure.

To keep things simple, suppose you begin with $100 thousand and invest 100% of it into stocks when you are young.  Next suppose the market does poorly.  When you are middle aged you find that your retirement savings are now worth only $74 thousand.  What should you do?  The advice from the AICPA seems to lead in two different directions.  On the one hand you are supposed to invest about half of your money in bonds (because you are middle aged).  On the other hand the point of investing in stocks while you were young was that you could “make it up” later on.  But that would imply that you should put somewhat more than half of your funds into stocks so that you now have a shot at getting back to where you wanted to be.  Now, consider the other case in which the market rises during your youth.  At this point you have $714 thousand in savings. So now what is the proper interpretation of the AICPA’s advice?  On the one hand there is that absolute rule that says put half of your money in stocks and half in bonds since you are middle aged.  On the other hand, since there is nothing to make up perhaps you should put more than half you money into bonds?

We seemed to have reached a rather odd conclusion.  Imagine we follow the reasoning behind the rule.  Then, if the market goes down during our youth we should hold relatively more of our wealth in stocks.  On the other hand if the market does well we should hold relatively less of our wealth in stocks.  Is that right?  Poor people should hold stocks and wealthy people bonds?  One can certainly conceive of cases where that might be true.  But in general we believe wealthier people can afford to take on greater risks than poorer people.  On that basis then a rising market should lead most people to invest more stocks and a declining one less. 

Another aspect ignored by the 100 minus your age rule is that while young people have more time to make up losses, older people need not worry so much about making them up. Notice that if you are young there is some chance the market will fall over the next two “periods” causing each $100 you invest to decline to $54.  However, if you are middle aged there is only one investment period left.  At worst, each $100 you invest can only fall to $74. This is a very important point. While a longer time horizon gives you more time to make up losses it also gives you more time to lose even more! So it is not really clear that young people face less investment risk than older people.  In some ways they face more.

High Tech Mathematics to the Rescue

So what fraction of your retirement funds that should be invested in stocks and bonds as you age?  This is actually a somewhat formidable problem in the area of “dynamic optimal control.”  Roughly, you have a portfolio that you need to use in retirement.  You can “control it” by moving money in and out of stocks and bonds.  While the mathematics will leave most scratching their head, fortunately, you do not need to solve this problem yourself.  Others have already done so. 

The first general solution to the retirement savings problem appears in a famous 1969 paper by Robert Merton.  There he showed that if investors have what is called “constant absolute risk aversion” then they should invest a constant fraction of their wealth in stocks regardless of their age!  So, what is “constant relative risk aversion” and do you have it? Strictly speaking, the answer is probably no.  The chance that this very particular mathematical characterization of a person’s attitude towards risk captures 100% of your views is very remote.  On the other hand, it is likely not that far away either.  Very roughly, imagine you have $100 and are willing to put $1 in a risky venture (1% of your wealth).  If your attitude towards risk can be described as constant relative risk aversion then if you started with $1,000 you would be willing to put $10 into the venture (again 1% of your wealth.)  That is the fraction of your wealth you are willing to invest in a risky asset is independent of your wealth.  In reality the wealthier you are the larger the fraction you are probably willing to invest in the risky venture.  So adapting all of this to our problem it seems that the fraction of your wealth that belongs in stocks does not vary much with your age, but might vary a bit with your wealth.  If you are rich more of your wealth should be in stocks and if you are poor more should be in bonds.

Before, I hear from somebody complaining that “Merton’s model is too simple.  It ignores taxes, short sale constraints and other costs faced by investors.  In real life you would not want to follow this advice.”  Well, it is true Merton’s model leaves out many important aspects of the real world.  But academic research marches on and Merton’s paper is not (big surprise) among the most recent on this issue.  For that we turn to a 2001 paper by Robert Dammon, Chester Spatt, and Harold Zhang.  They include within the investor’s problem both capital gains taxes, and short sale constraints.  What do they find?  Investors should increase the fraction of their portfolio in stocks over time!  No kidding. This, of course, is exactly the opposite of the advice given by the rule of thumb quoted at the top of this article. What is going on?  When an investor dies the tax basis on the estate’s capital holdings resets to their value at the time of death.  Translation: When an investor dies the heirs do not owe capital gains taxes on the estate’s holdings.  So if you can hold on to your stock investments until you die nobody will ever pay capital gains taxes on them.  As you get older this option to invest “tax free” (by dieing) becomes ever more valuable (since, alas, you are ever more likely to become its beneficiary). But again, there is one underlying force driving these results.  As you age while there is less time to make up for losses in the stock market there is also less time for it to go down significantly.  The same holds on the upside as well.  As you age it is less likely the market will run up significantly before you die.

What to Do?

The advice to invest a fraction of your portfolio equal to 100% minus your age in stocks appears to derive from casual intuition rather than careful reasoning.  So, I would not use that rule as a benchmark.  On the other hand the prescription to invest a constant fraction of your wealth in stocks, no matter your age, seems to scare many people.  Naturally, we think that given these conflicts your best bet would be to contact Alpha Investment Opportunities to discuss your options!  But, if you are looking for a “rule” you might consider the following:  Figure out today what fraction of your wealth you are currently comfortable with in the stock market.  Put that amount in stocks.  If the market goes down (thus leaving you with less wealth) you might want to reduce the fraction in stocks somewhat.  You can accomplish this by doing nothing.  An example will illustrate why.  Imagine you put $100 in stocks and $100 in bonds.  At that very moment you have 50% of your wealth in each asset.  Alas, the next day stocks drop 10% leaving you with only $90 in stocks.  You now have only 47% (90/190) of your wealth allocated to stocks which is 3% less than you started with.  Conversely, if the market goes up you might want to increase the fraction of your holdings in stocks (as you are now wealthier). Again, doing nothing will accomplish this quite nicely.  Then once a year look over your portfolio and see if the fraction in stocks is way up or down from where you started the year, and then move it back to somewhere near the original fraction.  The idea is to find a level of risk that you are happy with and then let your portfolio reflect your preferences.  What you should not do is let your portfolio simply reflect your age.

Bibliography:

Dammon, Robert, Chester Spatt, and Harold Zhang, 2001, “Optimal Consumption and Investment with Capital Gains Taxes,” Review of Financial Studies, 14(3), 583-616.

Merton, Robert, 1969, “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case,” Review of Economics and Statistics, 51(3), 247-57.