Time Diversification: Fact or Fallacy? Historical, Theoretical and Empirical Analysis.

Zvi Bodie is a Professor of Finance at Boston University. In his paper appeared in 1995 in the Financial Analysts Journal, he analyzes the proposition that investing in common stocks is less risky the longer an investor plans to hold them. In his opinion, if the proposition were true then the cost of insuring against earning less than the risk- free rate of interest should decline as the length of the investment horizon increase. In his paper, he demonstrates that the opposite is true even if stock returns are “mean-reverting” in the long run. The case for young people investing more heavily in stocks than older people cannot therefore rest solely on the long run properties of stock returns. Moreover, for guarantors of money- fixed annuities, the proposition that stocks in their portfolio are a better hedge the longer the maturity of their obligations is unambiguously wrong.
His work starts with the measurement of the risk of stocks, analyzing then the cost of insuring against a shortfall using Black-Scholes Option Formula. After that, he tests the “meanreversion hypothesis” with the implications for individuals of his conclusions. The paper finishes with the implications of his conclusions for guarantor of money-fixed annuities, such Pension Benefit Guaranty Corporation (PBGC), a type of pension funds.
Zvi Bodie takes again a concept explained already in Chapter two: the risk of shortfall. A shortfall occurs when the value of a stock portfolio at the horizon date is less than some value determined by a specific “target” rate of return. A natural choice for this target rate of return is the rate of interest on default-free zero-coupon bonds maturing on the horizon date. Since such bonds are free of risk over the relevant time horizon, they represent a logical benchmark against which to measure the risk of stocks. So, for example, an investor with a time horizon of five years which wants to invest in risky assets, would have to compare the ex-post return on his portfolio with the return that a hypothetic 5-years Zero Coupon Government Bond. Unfortunately, Zero Coupon risk-free bonds are not available for long maturities: for example in Italy the Zero Coupon security with the highest maturity is the CTZ which expires after 24 months.
The basis for the proposition that stocks are less risky in the long run appears to be the observation that the longer the time horizon the smaller is the probability of a shortfall, like the data exposed in Chapter two seems to say.
If the ex-ante mean rate of return on stocks exceeds the risk-free rate of interest, it is indeed true that the probability of a shortfall declines with the length of the investment time horizon. For example, if the rate of return on stocks is lognormally distributed with a risk premium of 8 percent per year and an annualized standard deviation of 20 percent (the same numbers used in Chapter 2 based on Ibbotson & Associates post-war rate of return on US stock markets and short-term US Government Bonds), the probability of shortfall for an investor with an holding period of one year is 34 percent, while drops at only 4 percent if the time ho rizon increases to 20 years. As already mentioned, the probability of shortfall is a flawed measure of risk because it completely ignores how large the potential shortfall might be.