Time diversification and changing volatility in an options pricing framework

ABSTRACT

We examine time diversification with changing asset volatility by evaluating the cost of insuring that a portfolio earns at least the risk-free rate of interest as the investment horizon lengthens. If stock returns are mean reverting, the cost of shortfall insurance will be less than if returns follow a random walk, since the annualized volatility of returns will be less under a mean-reverting process than for a random walk. Empirical evidence suggests that the degree of mean reversion is not sufficient to cause a decline in the cost of shortfall insurance. Furthermore, if bond returns are mean-averting, this implies that shortfall insurance is more costly than under a random walk. We derive theoretical conditions necessary for mean reversion to cause the cost of shortfall insurance to decrease with time.

1. INTRODUCTION

Time diversification, the idea that the risk of an investment will decrease with the length of the holding period, has recently been analyzed in an option-pricing framework. Bodie [1995] employed the Black-Scholes model to determine the cost of insuring against a stock portfolio earning less than the risk-free rate of interest. He demonstrated that the concept of time diversification does not hold, since the shortfall insurance cost may be viewed as a put option whose value increases with the length of the investment horizon. His analysis assumes that stock returns follow a random walk and are independent (zero autocorrelation) across time. One implication of this is that risk, as measured by the annualized standard deviation of stock returns, is constant over time.

Bodie further argued that mean reversion in stock returns does not affect his results. Although this statement is valid if stock price volatility is known, in reality, volatility is unknown and changing. In a world of stochastic volatility, under certain conditions, the implied volatility on a given option should equal the average volatility that is expected to prevail over the life of the option (Stein, 1989). This suggests that, if a measure of volatility is to be used to obtain an option's value, it should be estimated over a period whose length is consistent with the option's life. Since a number of previous studies have shown that stock return volatility decreases with the length of the investment horizon, it becomes an empirical question as to whether the volatility decrease is large enough to cause a decline in the cost of shortfall insurance (see Poterba and Summers [1988], Reichstein and Dorsett [1995], and Vanini and Vignola [2002]).

In this paper, we examine the issue of time diversification by specifically allowing for changes in the volatility of security returns as the length of the investment horizon changes. Our results indicate that stock return volatility decreases as the investment horizon increases, but the decrease is not large enough to cause the cost of shortfall insurance to decline. Bond return volatility actually increases with the length of the investment horizon making time diversification inapplicable.

2. THE INVESTMENT HORIZON AND THE COST OF PORTFOLIO INSURANCE

We analyze the cost of insuring against a stock or a bond portfolio not earning at least the risk-free rate as the investment horizon lengthens. This shortfall insurance can be valued as a put option in a Black-Scholes option-pricing framework. Since we wish to insure against the possibility of an investment in securities, S, earning less than the risk-free rate, the exercise price, E, is set to [Se.sup.rT]. Substituting into the put-call parity theorem yields P = C. In this special case, the put price expressed in terms of the Black-Scholes formula and stated as a fraction of the security price gives:

(1)

P/S : N(d1) - N(d2)

where: d1 - [sigma] [square root of T]/2

d2 = - [sigma] [square root of T]/2

N(*) = value of the cumulative normal density function

Equation 1 shows that the put price as a fraction of the security price depends only on the time until maturity of the option, T, and the underlying asset's volatility, [[PHI].sub.T].

Bodie shows that P/S increases with the investment horizon if the annualized standard deviation of returns, [[PHI].sub.T], is constant. Although assuming constant volatility for all time horizons is valid when the security volatility is known, in a world of stochastic volatility, the average volatility expected to prevail over the life of the option is a more appropriate measure. Thus, if stock returns are generated by a mean-reverting process and volatility decreases with the length of the investment horizon, it is possible that the cost of shortfall insurance could decline, since the put value decreases as volatility decreases.

3. MEAN AVERSION, MEAN REVERSION, AND INVESTMENT HORIZON VOLATILITY

We compare the volatility for various investment horizons using the following ratio:

(2) RT = ([sigma].sub.T]/[square root of T]/[[sigma].sub.1]

where : RT = ratio of annualized standard deviation of returns for T--year holding period

relative to the standard deviation for a one--year holding period

[sigma]T = standard deviation of security returns for a T--year holding period

T = holding period in years

[sigma]1 = standard deviation of security returns for a one--year holding period

Note that the numerator of equation 2 represents the annualized standard deviation of returns for an n-year holding period. Thus, the ratio represents a comparison of the annualized standard deviation of returns for an n-year holding period to the standard deviation of returns for a one-year holding period.

Under a random walk, stock returns are independent and a large price movement in one period has no effect on price movements in following periods. This means that [R.sub.T] will be equal to one if returns follow a random walk and are independent.

However, under a mean-reverting process, a large upward or downward price movement is more likely to be followed by a price movement in the opposite direction. Thus, volatility is dampened across time by the negative autocorrelation of returns. This implies that if stock returns are mean reverting, [R.sub.T] will be less than one. [R.sub.T] will be greater than one if the volatility of returns increases with the investment horizon, which would result if returns follow a mean-averting process and are positively correlated.

To determine whether time diversification is efficacious for stockholders, we must determine the condition necessary for the cost of shortfall insurance to fall as the length of the planned holding period increases.

An examination of the put pricing model shows that P/S will decrease only if [d.sub.1] in equation 1 decreases, since [d.sub.1] and [d.sub.2] have the same magnitude and the normal density function is symmetric. Substituting [R.sub.T][[PHI].sub.l] into the equation for [d.sub.1], we see that [d.sub.1] decreases with increasing T if and only if:

(3) RT < 1/[square root of T]

That is, P/S will decline only if the ratio of the annualized n-year standard deviation to the one-year standard deviation decreases fast enough to offset the compounding of risk over time. Whether mean reversion or other effects result in stock returns being negatively correlated enough to have the cost of shortfall insurance decrease is an empirical question.

Although the discussion has focused on stock returns, the same empirical method can be applied to bond returns. However, previous research suggests that bond returns are mean averting. Under a mean-averting process, a large upward or downward price movement is more likely to be followed by a price movement in the same direction. Volatility is amplified across time by the positive autocorrelation of returns, so the standard deviation for an n-year holding period will be larger than if a random walk is followed.

4. SHORTFALL INSURANCE AND THE INVESTMENT HORIZON

We calculate [R.sub.T] for portfolios of small stocks, common stocks, long-term corporate bonds, and long-term government bonds using data from Ibbotson Associates 2002 Yearbook. Annual returns from 1926 to 2001 are used to construct three-, five-, and ten-year holding period returns. The standard deviations for the holding period returns are annualized and compared to the one-year standard deviation (see equation 2). For comparison purposes, 1/T is also computed for each holding period.

Our results, reported in Table 1, are consistent with those of other researchers. The stock portfolios exhibit mean-reversion, but the bond portfolios do not. For holding periods greater than three years, [R.sub.T] is less than one for the stock portfolios and greater than one for the bond portfolios. However, the degree of mean reversion for the stock portfolios is not sufficient to cause P/S to decrease. [R.sub.T] for a 10-year holding period is 0.70 for small stocks and 0.61 for large stocks--substantially larger than the value of 0.32 necessary for time diversification to cause the price of shortfall insurance to decrease.

Figure 1 illustrates the effects of a mean averting, mean reverting, and random walk return process on P/S directly. Equation 2 is used to compute P/S for various holding periods assuming that the annual standard deviation of returns is 0.20. The calculations are based on [R.sub.T] values of 1.10 and 0.90 for the mean-averting case (bonds) and mean-reverting case (stocks), respectively.

[FIGURE 1 OMITTED]

Figure 1 is consistent with our empirical results in that it shows that P/S increases with the holding period for all three return generating processes. It graphically shows that the relative cost of shortfall insurance for the mean-averting case compared to the random-walk case constantly increases. This suggests that a long-term investor in bonds should reduce his/her bond holdings as the planned holding period increases. Levy and Gunthorpe [1993] argue that if borrowing at the risk-free rate is allowed, the proportion of bonds in an investor's portfolio should increase with the length of the holding period. Although the cost of shortfall insurance increases with the holding period for the mean-reversion case, it is always less than it would be for the random-walk case. Related to this, Taylor and Brown [1996] use a different data set and find that the magnitude of mean reversion is great enough to cause the cost of shortfall insurance for a common stock portfolio to decrease with the investment horizon. This means that a long-term investor in equities would have larger holdings of stock for any given holding period under mean-reversion than for a random walk.

5. CONCLUDING COMMENTS

We examine time diversification in an option-pricing framework when security returns do not follow a random walk. If stock return volatility decreases with the length of the investment horizon, time diversification could theoretically reduce the absolute cost of shortfall insurance for a stock portfolio. The conditions necessary for this are derived.

Our empirical results show that the degree of mean reversion in stock returns is not sufficient to reduce the cost of shortfall insurance. However, the cost is still less than it would be for the random-walk case for any given investment horizon. For bonds, mean aversion causes the price of shortfall insurance to be greater than it would be if the returns followed a random walk.

TABLE 1: VOLATILITY AND THE INVESTMENT HORIZON *
RATIO OF ANNUALIZED N-YEAR VOLATILITY TO ONE-YEAR VOLATILITY

                                  Holding Period

                                    5 years   10 years

Small Stocks                 1.00     .94        .70
Common Stocks                 .92     .61        .61
Long-term Corporate Bonds     .98    1.27       1.15
Long-term Government Bonds    .94    1.23       1.02
1/T                           .58     .45        .32

* This table gives the ratio, [R.sub.T], of the annualized volatility
for three-, five-, and ten-year holding periods to the one-year
volatility for various types of securities (calculated using equation
2). Annual returns from 1926 through 1991 are obtained from Ibbotson
Associates data.

REFERENCES

Bodie, Zvi. "On the Risk of Stocks in the Long Run," Financial Analysts Journal 51, 1995, pp. 1822.

Levy, Haim and Deborah Gunthorpe. "Optimal Investment Proportions in Senior Securities and Equities Under Alternative Holding Periods," Journal of Portfolio Management 19, 1993 pp. 30-36.

Poterba, James and Lawrence Summers. "Mean Reversion in Stock Prices Evidence and Implications," Journal of Financial Economics 22, 1988, pp.27-60.

Reichstein, William, and Dovalee Dorsett. "Time Diversification Revisited," The Research Foundation of Charted Financial Analysts, Charlottesville, Va., 1995.

Stein, Jeremy. "Overreactions in the Option Market," Journal of Finance 44, 1989, pp. 1011-1024.

Stocks Bonds, Bills and Inflation: 2002 Yearbook Chicago: Ibbotson Associates, Inc., 2001.

Taylor, Richard and Donald Brown, "The Risk of Stocks in the Long Run: A Note," Financial Analysts Journal 52, March/April 1996, pp. 69-70.

Vanini, Paolo and Luigi Vignola. "Optimal Decision-Making with Time Diversification," European Financial Review 6, January 2002, pp. 1-30.

Author Profiles:

Dr. Ronald Best is a Professor of Finance and Associate Dean at the University of West Georgia in Carrollton, Georgia. He has published numerous articles in finance and economics, including publications in the Journal of Finance, Financial Review, and Journal of Financial Research. He completed his Ph.D. at Georgia State University in 1992.

Dr. Charles W. Hodges is an Associate Professor of Finance at the University of West Georgia. He has published in numerous journals, including the Journal of Financial Economics and Financial Analysts Journal. He consults extensively and is Managing Partner of Fairlie Poplar Associates. He completed his Ph.D. at Florida State University in 1993.

Dr. Robert C. Yoder is an Assistant Professor of Computer Science at Siena College in Loudonville, New York. He completed his Ph.D. at the University of Albany in 1999.

Dr. James A. Yoder is a Professor of Finance at the University of West Georgia. He is among the more prolific authors in financial economics, including such journals as Financial Analysts Journal, Journal of Portfolio Management, and Financial Review. He completed his Ph.D. at the University of Florida in 1988.

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