Risk, return, and investment horizon in emerging equity markets: evidence from India.

ABSTRACT

Time diversification suggests that equity investments over long horizons provide excessive premia after adjusting for risk. Encouraged by this idea, and the attractive equity premia delivered by the US equity market during the past century, pension fund and bank portfolios are making equity investments. However evidence from other markets shows that the US experience could be an exception. Before generalizations can be drawn from the US data, it is useful to collate and analyze the data from various other markets.

Fragile emerging markets, in particular, need to be careful while following the developed market trends. This paper attempts to present data from the Indian equity market and analyses the risk, return and investment horizon relationships embedded therein.

It is noticed that while long-term trends seemed to show attractive equity returns on risk-adjusted basis, the observed results deviate significantly from the results expected from the theory. The sub-period data defies the expected risk return relationships. Even in the long-term data there are trends that deviate from the theoretical expectations.

These findings indicate the need for careful examination of the equity market trends before safety-seeking-portfolios are encouraged to invest in equity.

1. INTRODUCTION

Indian equity market witnessed bullish trends during 1989-1994 drawing a large number of retail investors into equity investments. However, except for a brief period during the year 2000 the markets have remained subdued. During the period 1997-2002, there were just 302 public issues (Rs. 316.86 bn.) and 143 rights issues (Rs. 46.64 bn.). The secondary equity markets have also fared poorly with the equity index showing sideways movement in the range of 2800-3700 during the period 2001-2003. This is significantly lower than the trends observed during previous periods. This bearish phase has resulted in considerable decline in the value of the equity portfolio of the investors. While mutual funds and money managers attempted to convey the risk return profile of equity assets to the investors, it has been difficult for them to convince the investors about the attractiveness of equity investment as long-term proposition.

In relatively poorer economies, capital is scarce and it is of paramount importance that household savings are carefully channelised into investment avenues. Unless we are able to clearly delineate the risk, return and investment horizon relationships, it would be difficult for the investors to make informed investment decisions. This study is expected to be of use in analyzing policies that guide equity investment.

For example, policy decisions have been taken in India permitting banks and pension funds to invest in equity markets. It is not clear if these policies are in the right direction. Many eminent opinion leaders are not completely convinced about the safety of parking bank and pension money, even in small amounts, in equity.

It is useful to understand the historical trends in equity investments as it would help in making informed long-term asset allocation decisions. This paper provides a theoretical framework for examining the relationship between risk return and investment horizon. It further attempts to collect the available evidence in the Indian equity market to make inferences about the risk return relationships over varied investment horizons.

2. EQUITY RETURNS FROM VARIOUS MARKETS

The US equities furnished risk premium of about 6% over the 1889-1978 period (Mehra and Prescott, 1985). Siegel (1992) showed that over the period 1802-1990, the real compounded annual returns on equity in USA were about 6%, while the average nominal mean returns on stocks was 9% per year. In their best single year, stocks in the USA delivered a real return on 67% while in their worst single year they return -39% for a range of 160%. In the period 1802-1997, the standard deviation of the annual return from stocks was 18% (Campbell 1995). As the investment horizon increases, the dispersion in stock returns decreases. For example, the range for stock returns was 18% for holding period of 10 year; period, 12% for holding period of 20-year and 8% for 30-years. Standard deviations of real returns follow declining pattern when measured over long holding periods. In fact, it is seen that as holding period increases the stocks become less risky and the risk converges towards the risk of bonds. It appears that stocks appear to offer investors excessive returns over longer investment horizons. Siegel (1992) noted that equity appeared to be the best route to long-term wealth accumulation.

Jorian and Goetzmann (1999) studied thirty-nine markets over long time horizons. They found that real returns are the highest in the USA. They concluded that the high US equity premium seemed to an exception rather than the rule. For example, the German market showed only 1.91% equity return during most of last century. The story is similar for Japan where the post-war return on equity was 5.52% and the pre-war equity return was negative at -3.04%. Markets such as Portugal, Chile and Peru did not do well over long periods of analysis.

Many studies have shown that emerging markets are vastly different from those of developed markets in terms of risk, return, and liquidity patterns. Jiawei and Jianping (1999) found that emerging market indices have higher volatility. Lawrence (2002) highlighted the investors' experiences with falling liquidity and rising volatility. Campbell (1995) characterized emerging markets in Europe, Latin America, Asia, the Mid-East and Africa with high-expected returns as well as with high volatility. Seth (1998) noticed that emerging markets indexes are inefficient portfolios that lie beneath the efficient frontier. Jeffery and Ghose (1998) analyzed weekly holding period returns for 1990 to 1996 for fifty-one emerging market close-ended funds, which invest primarily in Asia and Latin America. During that period a broad index of U. S. stock returns had significantly higher returns and much lower variance than these funds. Kar, et al., (2000) showed that during 1985-1999, emerging markets displayed wider variation of the average mean returns.

Bekaert and Harvey (1997) discussed the sources of volatility differences between developed and emerging capital markets. Apart from the degree of diversification and concentration inherent in the index of each country, there were other sources such as stock market development/economic integration, microstructure effects, macro-economic influences and political risks. Aggarwal, et al., (1999) examined the type of events that cause large shifts in the volatility of emerging stock markets, and concluded that most of such events are local in nature. Grundy and Youngsoo (2002) found that price variability in heterogeneous information economy was 20% to 46 % higher than in an otherwise equivalent economy in which all signals are publicly announced.

It is clear that the financial market literature has recognized the existence of key differences between developed markets and the emerging markets. The large equity premium observed in the USA during 1871-1920 seems large not only in cross-country comparison but also by historical standards. It appears that the returns on US equities this century cannot be viewed as representative of global stock markets. In view of this, the evidence of time diversification and the excessive returns from equity over long investment horizons needs to be further examined by taking data from several other stock markets beside USA.

In this paper we attempt to study this phenomenon in the Indian stock markets and thereby add to the data base that might eventually be able to explain equity returns taking into account cross sectional data from across the global markets.

3. THEORETICAL CONCEPTS

The risk-return relationship between one-period and multi-period

Let

[x.sub.i] = random one-period-return during period i

[bar][x.sub.1] = mean of the distribution of [x.sub.i]

[bar][X.sub.N] = mean return for multi-period consisting of N one-periods

We assume that [x.sub.i] is distributed with finite mean [bar][x.sub.1] and finite non-zero standard deviation. Further, each draw is assumed independent of the other draws, all drawn from the same distribution.

[sigma]([x.sub.i]) [not equal to] 0 ; Covariance ([x.sub.i], [x.sub.j]) = 0

The random one-period return may be expressed as:

[x.sub.i] = [x.sub.1] + [[delta].sub.i]

Multi-period-return with N one-periods can be obtained by compounding the one-period-returns.

i.e., [X.sub.N] may be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For very small values of [x.sub.i] , this reduces to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When N is large, this is equal to N [bar][x.sub.1]

In other words, the mean of multi-period-return is N times the mean of one-period-return.

Variance of the multi-period-return is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For very small values of [x.sub.i] this is equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i.e. N[[sigma].sub.2]([x.sub.i])

Therefore, standard deviation of multi-period-return is [square root of (N)] times the standard deviation of one period return.

It is apparent that while return is N times the one-period return, the risk is only [square root of (N)] times the one period standard deviations. This shows that investments over longer periods offer higher returns and lower risks. This is termed as time diversification. These results are valid when the one-period return is very small compared to unity and the number of periods N is large. Further, the one-period return distribution should satisfy certain properties.

Academic research has generally tended to reject time diversification whereas practitioners have accepted its validity and economic significance. Krizman (1994) presented the arguments for and against time diversification. The rejection of time diversification by academicians is based on economic models of risk aversion. Under an expected utility function that exhibits constant relative risk aversion, an investor's optimal asset allocation is not a function of her time horizon (Thorley, 1995). If constant relative risk aversion were the right way to capture the risk behaviour, then time diversification would be a fallacy. However, alternate specifications such as decreasing relative risk aversion might lead to the feasibility of time diversification argument.

Stangeland and Turtle (1999) suggested that there are important reasons why it is inappropriate to either always reject, or always recommend, the strategy of time diversification. With constant expected return, the annualized standard deviations over a long holding period of N years is the standard deviation over the period divided by the square root of N. Thus with constant expected returns, standard deviations of all assets would shrink as the investment horizon is increased; but the standard deviation of all the assets would shrink together. If, however, the standard deviation of stock returns shrink faster, it would be an indirect evidence of predictable variation in stock returns. The type of return variation that reduces long-term risk is known as mean reversion. If unusually good stock returns in one period lower the expectations in the next, then bull markets would be followed by bear market corrections and vice versa. The stock prices revert towards a long-run average or mean. Under these circumstances, stock market risk declines more rapidly with the investment horizon than the square root rule would imply.

Probability of the wealth from equity investing being lower than the wealth from risk-free investment

$1 invested in the beginning and held over N periods would result in average wealth of (1+ N [bar][x.sub.1])

The variance of multi-period wealth would follow the [square root of] N rule. i.e., the variance of the multi-period-wealth from equity investing would be [square root of] N[sigma].([x.sub.1])

Assuming normal distribution, we can compute the cumulative probability of obtaining wealth in excess of a specified value W as P(z>[z.sub.0]) where [z.sub.0] = (W - 1 - N [bar][x.sub.1]) / [square root of] N[sigma]([x.sub.1])

If $1 was invested in a risk free security and held over N periods it would offer a wealth of (1+[Nr.sub.f1]) where [r.sub.f1] is the one-day return from risk-free security.

Substituting (1+[Nr.sub.f1]) for W, we can calculate the probability of equity wealth being in excess of wealth from risk-free investment.

The [z.sub.0] value would be [square root of] N([r.sub.f1] - [bar][x.sub.1]) / [sigma]([x.sub.1])

It is clear that the probability of equity wealth exceeding wealth from risk free investment is determined by the following:

* N, the number of time periods

* [r.sub.f1], the risk-free one-period-return

* [bar][x.sub.1], the mean of one-period-returns from equity investment

* [sigma]([x.sub.1]). the standard deviation of one-period return from equity investment

The magnitude of the numerator in the above equation is the one-period equity risk premium. The algebraic sign is, however, negative. Therefore, we are interested in finding the probability of the normal variable exceeding [square root of] N(one-period-equity-risk-premium) /(standard deviation of the one-period equity return)

4. DATA USED

The Bombay Stock Exchange has been in existence for over 125 years. There are over 7000 equity shares listed in the Indian stock markets. The Bombay Stock Exchange Sensitive Index (Sensex) is a popular index that represents leading large cap stocks in the market. Sensex data is available for a much longer period compared to the other Indian equity indices.

We have taken the closing daily value of Sensex for the twenty-four year period beginning from 3rd April 1979 to 2nd April 2003. The dividend yield data is available only for the later part of this period. Therefore, we have used the index without making adjustments for the dividend yield. In all, there are 5242 data points indicating that there are 218.4 trading days (on average) in a calendar year.

5. METHODOLOGY USED

The holding period return for (0,t) is calculated as a simple return given by ([I.sub.t] - [I.sub.0])/[I.sub.0]. The holding period is varied from 1 day to 5024 days (i.e. 23 years). Based on this information, for each holding period, a number of statistics are generated as follow:

* The minimum holding period return

* The maximum holding period return

* The average holding period return

* The dispersion in holding period return

* The annualized values of the above parameters

6. INFERENCES

Chart 1 shows the movement of the BSE Sensitive Index during the period 03-04-1979 till 02-04-2003. It may be seen that the index has risen sharply after 1990. The average value of the index in the period 3rd April, 1979-31st March 1990 was 375.35 and for the period 1st April, 1990 to 28th March, 2002 it was 3302.00. The following table illustrates the differences in the returns generated during these two periods.

[GRAPHIC OMITTED]

It is seen that in Period II, the average returns have declined while the volatility has increased. The year 1990 is usually seen as the year in which liberalization was initiated in India. Significant amount of capital market reforms were attempted in the post 1990 period. This shows that long time series data about equity markets would reflect significant changes in the underlying structure of the economy over long periods. This underscores the need for careful interpretation of the time series data. This might mean that inferences made on the basis of specific time windows may or may not be representative of other periods.

Chart 2 shows how the value of $1 invested at the start of the holding period ([t.sub.0]) changes with the length of the holding period. It is seen that the average value of wealth at the end of the holding period shows an increasing trend.

[GRAPHIC OMITTED]

It may be seen that the dispersion of the wealth increases up to the twelfth year, remains more or less stable between 12th and 20th year and decreases thereafter. During the period 2001-2003 the index was moving in a narrow range. The convergence between the maximum wealth and the minimum wealth that we observe beyond the twentieth may have arisen on account of this range-bound movement. Kritzman (1994) suggested that the dispersion of terminal wealth diverges from the expected terminal wealth as the investment horizon expands. Our data shows that beyond the twentieth year, the mean value of wealth continues to increase where as the dispersion of the wealth begins to decrease. The reason for this unexpected behaviour lies in the basic assumptions made about the return generating process.

Chart 3 shows the pattern in the minimum wealth achieved over different holding periods. It is seen that up to the twelfth year there was a possibility of losing about half of the $1 invested. But, after the twelfth year, the minimum wealth is always more than $1. As explained in Chart 1, there is a significant difference in the index movement pattern after 1990. This accounts for the non-zero minimum returns for holding period in excess of twelve years resulting in preservation of initial amount invested.

[GRAPHIC OMITTED]

Chart 4 shows the annualized holding period returns (minimum, maximum, average) for different holding periods ranging from 2 years to 22 years. This picture is similar to the pattern suggested by Kritzman (1994). Kritzman noted that if returns are independent from one period to the next, the standard deviation of the annualized return diminishes with time. The distribution of annualized returns consequently converges as the investment horizon increases. The dispersion of the annualized return reduces as the holding period increases and converges towards zero for holding periods in excess of twenty years. In relation to the magnitude of dispersion, the annualized return is bounded in a narrow range. The implication of this pattern is that the mean value of annualized returns remains relatively stable whereas the standard deviation of the annualized return decreases as the investment horizon increases. The annualized minimum holding period return turns positive at about the twelfth year.

[GRAPHICS OMITTED]

Chart 4-1 shows the risk return relationship for shorter holding periods up to 5 yrs. It is evident that the annualized standard deviation of return decline steeply even as the annualized mean return increases. However, when we analyze the same relationship over other holding periods (see Charts 4-2; 4-3; 4-4 and 4-5), we notice that the pattern is not uniform across holding periods. From Chart 5 it is evident that the range between minimum and maximum is high compared to Chart 4. This is clear evidence of higher return volatility in short holding period.

Chart 6 shows the annualized average holding period return for holding period up to 23 years. It is seen that the annualized holding period return is in the range of 0.152 per year and 0.227 per year. There is slight upward trend till the twelfth year followed by a decline thereafter. This variation in trend can be attributed to the difference in the pattern of the index movement between the periods 1979-1990 and 1990-2003.

[GRAPHICS OMITTED]

Chart 7 shows the actual average end-wealth over varied holding periods and the corresponding expected period end-wealth derived from compounding the one-day holding period return. It may be noticed that the actual value is slightly above the expected value till about the 18th year and falls below the expected value thereafter.

[GRAPHIC OMITTED]

Chart 8 shows the actual standard deviation of end-of-period-wealth over varying holding periods and the theoretically expected standard deviations. The theoretically expected standard deviation has been calculated from the standard deviation of the one-day return distribution by using square root N formula. It is noticed that the actual standard deviation is significantly higher than the theoretically derived standard deviation. The gap between the actual and theoretical increases gradually and stabilizes around the fourteenth year. This steep difference between the actual and theoretical values might have arisen because we have defined the one period return as the simple return. The square root N formula makes certain assumptions about the return process (see section III). It is not clear if these assumptions hold true in reality.

[GRAPHIC OMITTED]

The implication of this divergence between the theoretical expectations and the market data is that the dispersion of the period-end-wealth is magnified several times. A comparison between Chart 4 and Chart 8 highlights the point that although dispersion of annualized return converges with the passage of time, the dispersion of period-end-wealth diverges from the expected value. As investors are interested in wealth accumulation, it would be useful to focus on the notion of time diversification from the perspective of losing money. The Indian data shows that while the average period end wealth is about $25, the standard deviation associated with this number is about $8. Though the return distribution is normal over long horizons, the thumb rules of normal distribution could be used to arrive at order-of-magnitude-probabilities of losing money. It is evident that there is about 18% probability of receiving $32 or less (assuming a mean of 40 and standard deviation of 8).

Chart 9 depicts the actual and theoretical coefficient of variation in end-of-the-period-wealth ([sigma]/[mu]). The actual value is far in excess of theoretical value for most holding periods. Chart 10 presents the ratio of the actual and theoretical coefficient of variation. The ratio is consistently above unity. It increases gradually till about the 20th year and then falls steeply.

[GRAPHIC OMITTED]

Chart 11 shows the relationship between the holding period wealth and the dispersion in the holding period wealth for various holding periods. Single period asset pricing theory suggests a linear relationship between risk and return. Chart 11 seems to indicate that the coefficient of linearity is not constant across holding periods. The risk seems to increase with wealth up to about 11 years and then falls steeply even as the wealth continues to increase.

[GRAPHIC OMITTED]

Assuming that an investor has the options of investing either in equity index or in a risk-free security, it is possible to estimate the probability of equity investment generating wealth lower than that from risk-free securities (see the derivation in section III). Chart 12 shows the results graphically for Indian data. Indian equity index offered an average daily equity premium ranging from 3.25 basis points (assuming [R.sub.f] = 10% p.a.) to 4.95 basis points (assuming [R.sub.f] = 6% p.a.) per day.

The graph shows that for shorter investment horizons of, say 5 years, there is a near 20 to 30% probability that the risk free investment might outperform equity investments. When investment horizon is about 20 years, there is 10% probability of inferior equity performance. The attractive investment horizon for equity investments seems to depend upon two factors:

* The daily equity risk premium

* The probability of superior equity performance required

In the USA the risk premium over the century was computed as 6% (Mehra & Prescott, 1995; Siegel, 1992). This would approximately be equal to 3 basis points per day. Jorian and Goetzmann (1999) have stated that this is an exception rather than the rule. Given this evidence, it is quite feasible that emerging market equity investments could result in higher probabilities of inferior equity performance even over long horizons. Suggestions to permit banks and pension funds to invest in equity market need to be examined in the context of this data. It is likely that more than once out of ten instances, such strategy could underperform risk-free investment.

7. CONCLUSION

This paper attempted to understand the historical trends in the India equity market. Specifically, it studied the risk-return relationship over varied investment horizons. The following are the main findings:

* The risk-return relationship in equity investments exhibits different patterns across different time segments.

* These differences reflect the changes in the underlying structure of the economy when studies span extended periods.

* Inferences based upon specific time windows may or may be representative of other periods. Therefore, extrapolation of such results should be done cautiously.

* The annualized holding period returns (maximum, minimum, average) tend to converge as the holding period increases.

* The annualized minimum holding period return is in excess of zero after the twelfth year of investment.

* The annualized return is highly volatile for short holding periods, say up to 5 Years.

* The dispersion of holding period wealth seems to increase up to a certain holding period and then decreases. The pattern of decrease is not in alignment with the square-root-N-rule.

* The average wealth at the end of the holding period is fairly close to the theoretical value expected. However, the dispersion of the wealth is significantly higher than the value suggested by the square-root-N-rule.

* The single-period asset pricing theory does not seem to hold true in multi-period situation.

* There could be a substantial probability of equity investments under performing risk-free investments even over long investment horizon.

* These findings indicate the difficulties involved in generalizing the finding about risk-return-investment horizon across markets and across time segments in the same market.

* Policy decisions that encourage equity in pension funds and banks need to be reviewed carefully.

TABLE 1

                            Period I                  Period II
                     03-04-1979 - 31-03-1990   01-04-1990 - 28-03-2002

Mean of daily               0.000924                  0.000732
  returns
Standard deviation          0.014075                  0.019562
  of daily returns

END NOTES

(1) 1US$ = approximately Rs. 48

(1) Equity premium is measured as the difference between the return offered by an equity index and that offered by the treasury bill.

(1) Return is sometimes taken as In(It/[I.sub.0]). As the investment horizon increases, the simple return would furnish higher values compared to the logarithmic returns.

(1) [W.sub.t] = [(1+[x.sub.1]).sup.N]

Where

[x.sub.1] is the average of the one-period returns

N is the number of one-periods in the holding period t

[W.sub.t] is the value of initial investment of $1 at the end of the holding period t.

(1) This has been calculated as follows:

[x.sub.a] = [(1+[x.sub.t]).sup.M/t] - 1

Where

[x.sub.a] is the annualized average holding period return

[x.sub.t] is the average holding period return

t is the holding period

M is the number of holding periods in a year

(1) The computations for expected period end wealth have been made as follows:

[W.sub.t] = [(1 +[x.sub.1]).sup.t]

Where

[W.sub.t] is the expected period-end-wealth

[x.sub.1] is the average daily return

t is the number days in the holding period

(1) Madhusoodanan (1998) studied the Indian data for the period April 1979 to June 1997. He used logarithmic returns. He notices that the standard deviation of actual data is below the expected value for investment horizons in excess of 1000 days.

REFERENCES

Aggarwal, Reena, Carla Inclan, and Ricardo Leal, "Volatility in Emerging Stock Markets", Journal of Financial and Quantitative Analysis, Vol. 34, No. 1, March 1999.

Bekaert, G., and Harvey C. R. "Emerging Equity Market Volatility", Journal of Financial Economics, 43, 1997

Born, Jeffery A, Subrata Ghose "Asian and Latin American Emerging Market Closed--End Funds: Return Diversification, Emerging Markets Quarterly, Vo. 2, No. 3, Fall 1998,

Campbell Harvey R., "Predictable Risk and Returns in Emerging Markets", The Review of Financial Studies, Vol. 8, No. 3, 1995

Grundy D. Bruce and Kim Youngsoo, "Stock Market Volatility in Heterogeneous Information Economy", Journal of Financial and Quantitative Analysis, Vol. 37, March 2002

Hu, Jiawei, Jianping Mei, "The Return and Risk of Property Stock Index" Emerging Markets Quarterly, Vol. 3, No. 1 Spring 1999

Jorian Philippe, Goetzmann William N. "Global Stock Markets in the Twentieth Century" Journal of Finance, 54(3), 953-980, 1999

Kar Pratip, Raju M. T., Patil R. Prabhakar & Kiran Karande, "Stock Market Volatility--A Comparative Study of Select Markets" Working paper series, SEBI, Jan 2000

Kritzman Mark "... About Time Diversification", Financial Analyst Journal, Jan-Feb 1994

Masters J. Seth, "The Problem with Emerging Market Indexes", The Journal of Portfolio Management, Winter 1998.

Mehra, Rajnish, and Edward Prescott, The Equity Premium: A Puzzle? Journal of Monetary Economies 22, 133-136, 1988

Madhusoodanan, T. P. "Investment Horizon and Volatility: An analysis of the Indian Stock Market" The ICFAI Journal of Applied Finance, Vol. 4, No. 1, Jan 1998

Siegel, Jeremy "The Equity Premium: Stock and Bond Returns since 1802" Financial Analyst Journal, 48, 1992

Speidell, Lawrence S, "What is Emerging in Emerging markets?", The Journal of Investing, Vol. 11, No. 2, Summer 2002

Stangeland A. David and Harry J. Turtle "Time Diversification: Fact or Fallacy", Journal of Financial Education, Fall 1999

Thorley R. Steven "Time Diversification Controversy", Financial Analyst journal, May-June 1995

Vaihekoski, Mika "Portfolio Construction in Emerging Markets", Emerging Markets Quarterly, Vo. 4, No. 2, Fall 2000.

Prof. G. Sethu earned his doctorate at Indian Institute of Management, Ahmedabad, India in 1994. Presently he is dean of the UTI Institute of Capital Markets, India.

Dr. Rachana Baid earned her doctorate at Osmania University, India in 2002. Presently she is an assistant professor at the UTI Institute of Capital Markets, India.