行政院國家科學委員會專題研究計畫 成果報告
台灣股票市場的權益溢酬(2/2)
計畫類別： 個別型計畫
計畫編號： NSC93-2416-H-004-005-
執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日
執行單位： 國立政治大學財務管理學系
計畫主持人： 周行一
計畫參與人員： 陳明憲
報告類型： 完整報告
報告附件： 出席國際會議研究心得報告及發表論文
處理方式： 本計畫涉及專利或其他智慧財產權，1 年後可公開查詢
中 華 民 國 94 年 11 月 1 日
行政院國家科學委員會補助專題研究計畫
結案報告
台灣股票市場的權益溢酬
計畫類別： 5 個別型計畫 □ 整合型計畫
計畫編號： NSC 92 － 2416 － H － 004 － 016 －
執行期間： 92 年 8 月 1 日 至 94 年 7 月 31 日
計畫主持人： 周 行 一
共同主持人：
計畫參與人員： 陳明憲 （政大財管所博士班 博士候選人）
成果報告類型(依經費核定清單規定繳交)：5精簡報告 □完整報告
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□赴國外出差或研習心得報告一份
□赴大陸地區出差或研習心得報告一份
5出席國際學術會議心得報告及發表之論文各一份
□國際合作研究計畫國外研究報告書一份
處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計
畫、列管計畫及下列情形者外，得立即公開查詢
□涉及專利或其他智慧財產權，□一年□二年後可公開查詢
執行單位：國立政治大學 財務管理學系
中華民國 94 年 10 月 31 日
Time-Varying Risk, Myopic Loss Aversion and Equity
Premium in Taiwan
Edward, H. Chow
*Department of Finance
National Chengchi University
Keywords: Equity Premium, Myopic Loss Aversion, Prospect theory
JEL codes: C22, G11.
Abstract
This research investigates equity risk premium in Taiwan from three points of views, expected return based on fundamentals, time-varying risk and average evaluation horizon according to the MLA hypothesis. Firstly, we find that the risk premiums estimated from the fundamentals suggested by Fama and French (2002) cannot explain the “Equity Premium Puzzle” in Taiwan. Secondly, The market price of risk has three characteristics, (1) the longer the holding period the more significant the estimates are; (2) the shorter the holding period, the smaller the estimated market prices of risk and (3) the more likely the estimated market price of risk could fall below zero. Finally, the evaluation period that investors are indifferent between risky portfolio and riskless one is ten months based on the MLA hypothesis.
Keywords: Equity Premium, Myopic Loss Aversion, Prospect theory JEL codes: C22, G11.
I. Introduction
Under the assumption of expected-utility maximizing investors with standard additively separable state preferences and constant relative risk aversion (CRRA), financial economic theory predicts the level of risk premium offered by a risky asset. Locus (1978) and Breeden (1979) examine that in equilibrium, each individual with homogeneous belief and CRRA will maximizes her/his intertemporal expected utility of consumption by acquiring a risky asset under the risk level she/he can afford. Mehra and Prescott (1985) study the U.S.. equity premium from 1889 to 1978 and state that the coefficient of relative risk aversion needed to justify the U.S. historical equity risk premium nears 30, implying it is too high to be reasonable1. Donaldson and
Mehra (2002) re-estimated the long-term real rate of return of the U.S. stock market, a relatively riskless asset, and that of the risk premium derived from the two estimates. He divided the sample into two periods including 1889–1978, a period does not include the recent bull period, and 1889–2000. The former’s annual real rate of return on equities was about 7 percent; the later’s real annual rate of return was about 7.9 percent. Otherwise, real return of riskless rate from 1889–1978 is only 0.8 percent and from 1889–2000 is about 1.0 percent. Thus, the equity risk premiums are about 6.3 and 6.9 percent, respectively. From the point of view of consumption-based asset pricing theory, Mehra and Prescott’s (1985) studies seemly induce a conflict between consumption-based theories and empirical evidence and could translate into these questions.
Why are returns on T-bills so much lower than those on equity? Is the phenomenon caused by the over-simplified assumptions of theoretical models or by the statistically calculated biases from different data sampling methodology? Most important and interesting, why and will anyone still desire to hold riskless assets (portfolios), even the long term evidence shows the
1 Not only Mehra and Prescoot, but other economists view the “unreasonably” implied coefficient of risk
aversion. Thus, it could take Mehra and Prescott almost six years to convince a skeptical profession for their paper (1985) to be published and attended by a lot of financial economists.
riskless asset pays worse less than that of equity? Many pieces of research have been conducted to provide possible explanations for the equity risk premium puzzle. Some deal with these issues with modifying consumption-based theory, Some handle these by addressing econometrics issues on empirical analysis, the others view the puzzle from behavioral aspects.
Firstly, several modifications dealing with the theoretical assumptions and modeling of Mehra and Prescott (1985) have been proposed.2 Epstein and Zin
(1989) introduce a new class of preference which allows a separate parametrization of risk aversion and intertemporal elasticity of substitution to improve the seemingly unrealistic assumption of constant relative risk aversion. Campbell and Cochrane (1999) and Barberis, Huang and Santos (2000) take habits formation, relaxing the assumption of time-separate, of individuals into account. Their findings point out that relative to the underlying consumption data, a model with stochastically time-varying risk aversion can increase the volatility of stock returns. Fama and French (1989) argue that stock returns can be interpreted as waves of irrational exuberance and pessimism as often as it is interpreted as business cycle risk or risk aversion. As investors want to be compensated for the increased volatility, it could raise the expected excess return on stocks, hence this comment partly resolves the equity premium puzzle. Constantinides and Duffie (1996) propose a model with heterogeneous and idiosyncratic income risk. They find that incomplete markets substantially enrich the implications of the representative-household model and conclude that the existence of household income processes which is consistent with given aggregate income and dividend processes matches the given equity and bond price processes. Their theory requires that the idiosyncratic income shocks must be uninsurable (such as job loss), persistent and heterogeneous, with counter-cyclical conditional variance. Applied to the explanation of equity risk premium puzzle, the existence of
2 Cochrane (1997) and Siegel and Thaler (1997) survey the literature on the equity risk premium puzzle
comprehensively. Cochrane (2001) analyzes equity risk premium under a unified framework of the stochastic discount factor methodology. Mehra (2002) summarizes main research directions in this field during the past two decades.
heterogeneous and idiosyncratic income risk confirms that the risk premium is highest in a recession because the stock is a poor hedge against the uninsurable income shocks, such as job loss, that are more likely to arrive during a recession.
Secondly, many empirical researches have adopted new approaches trying to provide satisfactory explanations for the equity premium puzzle, one from improving econometrics issues (such as deals with the time-varying volatility with GARCH-family models, or takes survival bias problem into account), the other from hiring basic macroeconomic variables (such as Fama and French (2002)). Some treat the puzzle by varying volatility over time and model volatility by fitting the data to a popular used generalized autoregressive conditional herterhedasticity (GARCH) model (such as, Chou (1989) and Nijman (1993)). Brown, Goetzmann and Ross (1995) argue that the equity risk premium estimated from the U.S. data is conditioned on market survival, which induces unconditional premium including the possibility of a market failure might be lower. However, Li and Xu (2002) have recently shown that the survival bias is unlikely to be significant. Related studies by Campbell and Shiller (1988), Fama and French (1988) and Blanchard (1993) used prior dividends payout to predict the expected returns of security and showed that this proxy did explain the future expected returns much, although the explaining power is lower after 1990 (Goyal and Welch, 2000). Using the same data from CRSP, Siegel (1998), Schwert (1990) and Shiller (1989) estimated long-term performance of the U.S. stock market from 1802 to 1997. For example, Siegel（1998）showed that the annualized real return is 7.0 percent from 1802 to 1870, 6.6 percent from 1871 to 1925, and 7.2 percent from 1926 to 1997. Even during the period of the World War II with an unexpected inflation, the U.S. stock market still offered a 7.5 percent annual average real return. He also indicated that the real equity return not only behaved as a mean reversion pattern but compensated equity holders a stationary reward. Finally, researchers also attempted to resolve the equity risk premium puzzle with the help from behavior finance. Benartzi and Thaler (1995) firstly
combined loss aversion and mental accounting to propose “Myopic Lloss Aversion (MLA).” They suggested that investors care strongly about the market changes that occur over short periods, which explains both the observed portfolio holdings and the large equity premium. Loss-averse investors will find a risky portfolio even more risky if they receive or search information on its value more frequently. Evaluating the portfolio less "myopically" will reduce the risk viewed from these individuals. MLA (Benartzi and Thaler (1995)) combines two concepts together and forms a preference scheme trying to explain the premium puzzle. Using monthly data on stock returns and 5-year bond returns as a proxy of riskless asset in the United States over the period from 1926 to 1990, they derive prospective utilities of holding an all-stock and an all-bond portfolio for various evaluation periods, equally divided from one-month to twenty-four months, as the information feedback frequencies. They find investors in U.S. have an evaluation period of approximately twelve months and the large equity premium can be explained by MLA preferences, if investors evaluate their portfolios on the interval shorter than twelve-month. Their results “could” be explicated intuitively from the aspect that most investors tax their income on a yearly basis, and they also receive reports from their brokers, mutual funds and retirement accounts annually3. Barberis,
Huang, and Santos (1996) added the “house money effect” (that is, loss aversion is reduced following recent gains) to the MLA model and came to similar conclusions of Benartzi and Thaler (1995). Thaler, Tversky, Kahneman, and Schwartz (1997) conducted experiments to examine whether or not investors actually behave the way Benartzi and Thaler’s (1995) MLA model suggests. They found that the more often investors look at the market, the more risk averse they behave, which exactly matches what Benartzi and Thaler (1995) suggests.
Empirical evidences studied outside the U.S. market showed a similar puzzle. Jorion and Goetzmann (1999) studied inflation-adjusted stock market appreciation, excluding dividends, for 39 countries on the period of 1926-1999
3 The evaluation period of a portfolio manager should not be mistaken for her/his investment horizon.
Although the investment horizon may be five years, or any time interval, the time between portfolio evaluations could be twelve months.
and found that the median real appreciation rate was only 0.8 percent per year compared to 4.3 per year for the U.S.. There are reasons to suspect, however, these estimates of return on capital are subject to survivorship, as the United States could be the most successful capitalized system in the world; most other countries have been plagued by political upheaval, war, or related harmful events causing financial crises. Jorion and Goetzmann’s finding may indicate the importance of selection bias, (or survivorship bias) although it is possible that lower returns were compensated by higher dividend yields in that period. Similar empirical findings by Siegel (1992 a, b), Siegel and Montgomery (1995), Siegel and Thaler (1997), Welch (2000), and Fama and French (2002) for the U.S. security market, Baytas and Kakici (1999) for Japanese Nikko Securities market, and Jorion and Goetzmann (1999) for 39 countries indicate that an unreasonable large equity risk premium exists across countries and varies with time.
Perhaps due to the lack of data, Taiwan’s market is not included in Jorion and Goetzmann’s (1999) sample, although Taiwan has emerged as an important emerging market recently. The Taiwan Stock Exchange Corporation’s (TSEC) value weighted stock index (TAIEX) increases its weight on the Dow Jones World Stock Market Index and the MSCI Emerging Market Free Index series more4. In this paper, we will examine the equity risk
premium of Taiwan and will attempt to provide explanations for the well-publicized equity risk premium puzzle by comparing the nature of Taiwan’s equity risk premium with that of the U.S. market in depth. In addition, most of the emerging markets in Jorion and Goetzmann’s (1999) sample cover shortly interrupted periods of data. We will use the entire data series of the TSEC since its inception. In addition to available data from the Taiwan Economics Journal (TEJ), we have collected a broad dataset of Taiwan’s securities in variables such as market value, the distribution date and amount of cash dividend of each listed stock from 1966 to 1985, partially collected
4 For example, MSCI Emerging Market Free Index, consisted of the following 26 emerging market
country ,indices: Argentina, Brazil, Chile, China, Colombia, Czech Republic, Egypt, Hungary, India, Indonesia, Israel, Jordan, Korea, Malaysia, Mexico, Morocco, Pakistan, Peru, Philippines, Poland, Russia, South Africa, Taiwan, Thailand, Turkey and Venezuela. The rank of TAIEX’s weight in MSCI Emerging Market Free Index was raised to the first one in May, 2005.
manually from hard copied released by TSEC.
We think that there must be an academic research to fill up the gap on the Taiwan’s evidence into the literature on the well discussed field, the equity premium puzzle from about three proposed methodology. Since our dataset is not publicly available anywhere in Taiwan, it fills another contribution to the literatures in the equity premium research fields. By carefully and manually collecting the data from 1961 to 1966, we will do the expected equity premium and realized premium, and try to find the difference between them. Next, we model the time-varying volatility by GARCH process to prove, finally, we evaluate the MLA hypothesis and estimate the interval making Taiwan’s investors indifferently to hold the risky assets portfolio and riskless one.
Our empirical results contributing the literature in explaining the equity premium puzzle are follows. First, the risk premium in Taiwan is contrary to the viewpoint of Fama and French (2002). In contrast of the evidences of Fama and French (2002), the average realized return, 18.52 percent, in Taiwan from 1961 to 2001 is about twice than that of U.S., 9.62 percent from 1951 to 2000. The similar results could also be found in the realized risk premium, 15.86 percent in Taiwan and 7.43 percent in U.S.. But, our results show more large values of the expected risk premiums from dividends and earnings growth models. In U.S., the average risk premium of dividends and earning models are just about 2.55 and 4.32 percent, respectively. Estimated value of risk premiums from the fundamentals suggested by Fama and French (2002) to explain the “Equity Premium Puzzle” seemly deeps the puzzle in Taiwan. The results could be contributed to (i) more violate dividend-payout policy and more earning dressing effects, (ii) the difference on time-varying risk premium induced by time-varying risk aversion between Taiwan and U.S. markets, (iii) different behavioral factors. Second, decomposing the risk premium to time-varying market prices of risk and time variation in the quantities of risk by well adapted GARCH model in time-varying volatility estimates, we find the longer the holding period the
more significant the estimates are, since unavoidable limitation that the power on estimating procedure could be reduced partially by the numbers of observations in a shorter holding period. By the way, the patterns on estimated market prices of risk trend to decrease as time passed. Finally, we find the portfolio evaluation period that beings indifference between the two portfolios is ten months, which is near but small the twelve month evaluation period of Benartzi and Thaler (1995) from 1926:01 to 1990:12, in Taiwan from 1967:01 to 2004:12. Furthermore, we find the indifferent evaluation period, twelve months, in Taiwan from 1967 to 1986 is equal to that in U.S. from 1926 to 1990, but the annual risk premium differs a lot, 2.98 percent, between them. The additional premium could be caused the different sample period limited to the evolving process in each security market, market risk, investing behavior or some factors the academic still can not figure out. It could also be a phenomenon that the structure of the composition on market participants has changed or that the investing behavior of participants was affected by some structural factors, such as an introduction of the electric ordering system, a faster diffusion mode of information by all communication tools and other social factors that could affect the mentality of the market.
The structure of this paper is as follows. Section 2 presents our sample and related statistics. In section 3, we estimate the expected risk premium proposed by Fama and French (2002) and realized premium calculated from TAIEX. We will show whether the foundation economic variables explain the puzzle. Section 4 introduces this papers contribution to the estimation concerning GARCH processes to test if the conditional volatility process fit the data more and if time-varying market volatility help to understand the puzzle and as a prepare to application of MLA framework in the next section. Section 5 hires the prospect theory to solve the puzzle. It is closely followed by the work of Benartzi and Thaler (1995) to estimate the return distributions on different periods by non-parameter bootstrapping method. Applied the results, we identify the portfolio evaluation period that brings indifference between the two portfolios. Conclusions are conducted in Section 6.
II. Data Source and Preliminary Results
In order to have an accurate picture of equity risk premium, we need to include cash dividends and its reinvestment effect in the Taiwan Stock Exchange Corporation Capitalization Weighted Stock Index (TAIEX). We manually collected cash dividends, daily closing prices and ex-dividend dates of individual stocks since the inception of the TSEC. Cash dividends of each stock ever listed on the TSEC are collected from the TEJ’s “Equity Data Base” for the period 1986 to 2002 and from the “Status of Securities Listed” and “TSEC Monthly Reviews” published by the TSEC for the period 1966 to 1985. We use TEJ’s “Equity Data Base” from 1986 to 2002 and “Status of Securities Listed” from 1962 to 1985 to calculate the daily market value on each stock. Ex-dividend dates are collected from the “TSEC Monthly Reviews” and “Daily Trading Reports” for the period 1971 to 2002, and from “United Daily News” and “Daily Trading Reports” of the TSEC for the period 1962 to 1970 manually. To remove inflation factor from the nominal return to a real term, we use monthly price index published by “Directorate General of Budget Accounting and Statistics Executive Yuan, R.O.C.”5 to adjust the nominal return. Based
on the price level of 1996, the price index is 16.88 in 1962 and 103.72 in 2001, which is about 6.05 times of that in 1962. In addition, we use the monthly deposit interest rate averaged from one-month Board Rate6 (rolled over each
month), calculated from the average on the five major commercial banks including, Bank of Taiwan, Taiwan Cooperative Bank, First Bank, Hua Nan Commercial Bank and Chang Hua Bank and disclosed on Web. site of Central Bank of R.O.C. from 1962 to 2001 as the proxy for risk free rate. The summary statistics are shown in Table 1.
< insert Table 1 about here >
The arithmetic mean and geometric mean of the ratio of cash dividend to market capitalization (i.e., the dividend yields) from 1962 to 2001 are about 3.94 percent and 2.54 percent, respectively. The dividend yield decreases with
5 Data source http://www.dgbas.gov.tw/lp.asp?CtNode=2848&CtUnit=331&BaseDSD=7 , 6 Interest rate published on the Central Bank website.
time with the highest value at 16.72 percent in 1962 and the lowest value at 0.51 percent in 1997. The same pattern can be found in the cash dividend payout ratio. The ratio decreases from 1962 to 1998, but increases in the recent three years from 1999 to 2002. Since dividend payout ratio is much lower in Taiwan than that in the U.S., our evidence suggests that dividend payout ratio may be partially contribute the difference on equity risk premium between Taiwan and the U.S..7 We incorporate the reinvestment effect of
cash dividend into the original TAIEX to obtain a new index with dividend reinvestment. The effect of cash dividend on the index is eye-catching. For example, the original TAIEX index was 92.48 points on Jan. 4, 1967 and 530.52 points on Jan. 4, 1979, while the dividend reinvested index grows from 530.52 points to 1057.51 points. The inflation rate in Taiwan shows slightly high than that, 4 percent with a standard deviation about 3.11 percent, in U.S. from 1951 to 2000.
A preliminary calculation of real rate of equity returns on TAIEX, riskless assets and equity premium from 1961 to 2002 on different holding periods including 20-, 10-, 5-years periods and two sub-periods divided by the highest historical price level are shown in Table 2.
< insert Table 2 about here >
The average risk premium on 20-years holding period equally divided from 1962 to 2002 is equal to 11.00 percent (the period from 1962 to 1982 is 6.84 percentage, and that from 1983 to 2002 is 15.15 percent); on 10-years holding period is also 11.00 percent (the highest is 27.89 percent and the worst is -4.35 percent), on 5-year is 4.15 percent (the highest is 34.35 percent and the worst is -27.57 percent), and on two period divided from the highest level of price index are 19.46 and -6.58 percent, respectively. Some interesting phenomenon shows in Table 2. The first, realized equity premium in Taiwan varies differently with the holding periods, longer the holding period, more the realized premium. Secondly, the average yearly equity premium (not reported)
7 Fama and French (2002), the real average dividend yields are 4.7%, 5.34% and 3.7% on the periods of 1872-2000,
is 11.00 percent, far more than that of the U.S., 6.90 percent presented by Mehra and Prescott (1985). Thirdly, the range of Sharpe ration, proxy for the relative risk aversion, is from -1.29 to 0.80 which implies a time-varying market risk (or index volatility). Finally, roughly consisting with Benartzi and Thaler (1995) MLA arguments, it shows that loss-averse investors will find a risky portfolio even more risky if they receive information on its value frequently based on the relation between different evaluation period and equity premium.
For the holding period of twenty years the equity risk premium is 6.84 percent in the first subperiod (from 1962 to 1982) and 15.15 percent in the second period (from 1983 to 2002). But in the second ten-year subperiod (from 1973 to 1982), contrary to the standard financial theory, the equity risk premium is negative. Our data also shows that during the subperiod prior to the highest point of TAIEX (12495.34) investors were compensated with a risk premium of 26.85 percent. In contrast, the equity premium was -0.49 percent afterwards for the period from 1990 to 2002. Since investors would not willingly accept negative equity risk premium, this disappointing result ought to be a surprise to investors. Our observation supports Sigel and Thaler’s (1997) idea that the issue of equity risk premium should be empirically studied under the long run. Our evidence also shows that Taiwan exhibits, in addition to a volatile market, volatile equity risk premium that is not matched by that of the U.S..
III. Expected and Realized Equity Risk Premium
Various techniques have been proposed to estimate ex ante expected returns on stocks. Welch (2000) found that the consensus equity premium estimate of academic financial economists lied between six and seven percent, depending on time horizons. Claus and Thomas (2001) computed the discount
rate which equated market prices to the present value of expected future cash flows. They found that, for the period 1985-1998 and for a panel of five countries, the equity premium was around 3 percent. Fama and French (2002) used dividends and earnings growth rates to measure the expected rate of capital gain and argued that the dividend and earnings growth estimates of the 1951-2000 equity premiums were close to the true expected value. They also show the equity risk premium forecasted by dividend growth model is more accurate than that forecasted by earnings growth model in the U.S. market. But there are important differences between market characteristics of Taiwan and the U.S., such as different degree of survival bias, dividend payout ratio and macroeconomic and political risk. We will take these differences into account when attempting to shed light on the cause of high equity risk premium.
Fama and French (2002) explain the equity premium puzzle by distinguishing the expected premium estimated from the fundamental economics, the dividend and earnings growth models, from the realized premium calculated from stock price index. They argue that the average stock return of the last half-century is a lot higher than expected and the results are caused from a decline in discount rates that produce a large unexpected capital gain and conclude the equity premium estimates from the dividends and earnings growth models, 2.55 percent and 4.32 percent, are far below the estimate from the average return, 7.43 percent. Their results seemly explain the puzzle in U.S. economic, but could it be applied in another country, such in Taiwan, an emerging market? We use the dividends and earnings growth approaches proposed by Fama and French (2002) to investigate the equity premium in Taiwan.
The core concept on Fama and French’s model is whether estimates, including realized and expected premiums, approach to the real equity premium in one economic. The average return on a board portfolio of stocks is typically used to estimate the expected market return. We estimate the realized average stock return by including average dividend yield and average
rate of capital gain as follows, 1 1 1 1
(
)
(
t)
(
)
(
t)
(
t t)
t t t t tD
D
P
P
A R
A
A GP
A
A
P
P
P
− − − −−
=
+
=
+
, (1) where,D
t represents the average dividend for the year t.P
t represents themarket index level at the end of year t.
GP
t=
(
P
t−
P
t−1)
P
t−1 is the rate ofcapital gain, and
A
( )
⋅
indicates an average value. Additionally, we define t tD P
as the dividend-price ratio,D P
t t−1 as dividend yield,E P
t t asearning-price ratio,
E P
t t−1 as ratio of earnings for year t to price at the end ofyear t−1 . The expected average stock return is estimated from two
fundamentals (dividends and earnings). The average stock return of dividend growth model is defined as follows,
1 1 1 1
(
div)
(
t)
(
)
(
t)
(
t t)
t t t t tD
D
D
D
A R
A
A GD
A
A
P
P
D
− − − −−
=
+
=
+
, (2) where,GD
t=
(
D
t−
D
t−1)
D
t−1 is average growth rate of dividends.8Another proxy of the expected premium model can be found through average stock return of earning growth model,
1 1 1 1
(
ern)
(
t)
(
)
(
t)
(
t t)
t t t t tD
D
E
E
A R
A
A GE
A
A
P
P
E
− − − −−
=
+
=
+
, (3) where,GE
t=
(
E
t−
E
t−1)
E
t−1 is average growth rate of earnings.9Using equation (1) through (3), we can estimate the realized and two expected yearly TAIEX returns and the results are shown in Table 3.10
8 Fama and French view the dividend-price ratio, Dt/Pt, follows a stationary (mean reverting) process.
Stationary property implies that if the sample period is long, the compound rate if dividend growth approaches the compound rate of capital gain.
9 As stated in footnote 4, if the earning-price ratio, Et/Pt, follows stationary, the average growth rate of
earnings is an alternative estimate of the expected rate of capital gain.
10 When estimating the expected return from equation (2) and (3), we should be care of dealing the issue that,
< insert Table 3 about here >
From Table 3, we can find estimates of the expected real equity premiums which were calculated from TAIEX as the market portfolio. The deflator is the Consumer Price Index from the same period. The real return for year t on one-month Board Rate (rolled over each month), rft , calculated from the average on the five major commercial banks including, Bank of Taiwan, Taiwan Cooperative Bank, First Bank, Hua Nan Commercial Bank and Chang Hua Bank11.
Beginning in 1961, we construct TAIEX book equity data, dividends, prices, and returns from Chow and Huang (2001). Real returns seem more relevant from the views of individual’s consumption, we estimate expected returns in real terms, and only results for real returns are shown. To use the dividends and earnings growth model as estimates of expected returns, it must be assumed that the market dividend-price and earnings-price ratios are stationary12. The first three annual autocorrelations of Dt/Pt are 0.764, 0.563,
and 0.388. The autocorrelations are large, but their decay is roughly like that of a stationary first-order autoregression process which matches the evidence provided by Fama and French (1988), Cochrane (1994) and Lamont (1998) that the market dividend-price ratio is highly autocorrelated but slowly mean reverting. The first three autocorrelations of Et/Pt for 1961-2000, 0.697, 0.364,
and 0.194, are again roughly like those of a stationary AR1 process.
We can find two estimates of expected risk premium from dividend and earning growth model and the realized risk premium from TAIEX in the last three columns in Table 3. The earning growth model provides more premium, say 19.64 percent averagely, than that of dividend model, 17.49 percent and
the conditional expected stock return and the conditional expected growth rates of dividends and earning (see, e.g., Campbell and Shiller, 1989). But if the stock return and the growth rates are stationary( they have constant unconditional means), the Dt/Pt and Et/Pt both are stationary. Thus, such as the average return in equation (1), the dividend and earning growth models in equation (2) and (3) provide suitable estimates of the unconditional expected stock return.
11 What one takes to be the riskfree rate has a bigger effect. But for our main task – comparing equity
premium estimates from (1), (2), and (3) –differences in the riskfree rate are an additive constant that does not affect inferences.
12 About discussion of the stationary property on dividends and earning growth models, please see Fama and
realized premium, 15.86 percent, and the similar phenomenon could be found in their standard deviations. The dividend and earning growth estimates of expected risk premium are more than the realized average risk premium about 1.63 and 3.78 percent, respectively.
The risk premium in Taiwan induces a very interesting result that is contrary to the viewpoint of Fama and French (2002). In contrast of the evidences of Fama and French (2002), the average realized return, 18.52 percent, in Taiwan from 1961 to 2001 is about twice than that of U.S., 9.62 percent from 1951 to 2000. The similar results could also be found in the realized risk premium, 15.86 percent in Taiwan and 7.43 percent in U.S.. But, our results show more large values of the expected risk premiums from dividends and earnings growth models. In U.S., the average risk premium of dividends and earning models are just about 2.55 and 4.32 percent, respectively. Estimated value of risk premiums from the fundamentals suggested by Fama and French (2002) to explain the “Equity Premium Puzzle” seemly deeps the puzzle in Taiwan.
Fama and French (2002) argue that the expected return estimates from the dividend and earnings growth models are more precise than the average realized return since the standard errors of the average stock return is higher than that of two expected growth models. Claus and Thomas (2001) also hold the same viewpoint. But, our empirical studies of Taiwan show the different results. The standard error of the dividend and earning growth estimates of the expected risk premium for 1961-2001 are 41.23 and 44.78 percent, versus 30.23 percent for the realized risk premium. Supporting the general sense that earnings growth is more volatile than dividend growth, our evidence of the standard error of the expected return from the earnings growth model, 44.78 percent, is slightly higher than that from the dividend growth model, 40.56 percent. But the standard errors are much larger that these values in U.S.. No matter from the standard errors or from the Sharpe ratio (not reported), we can find the standard errors of risk premium on dividend, earning growth model in Taiwan from 1961 to 2001 is about eight time and three time than
that of U.S. from 1951 to 2000; the standard error of realized risk premium is less double than that in U.S. Our empirical results that can not be judged from the fundamentals suggested by Fama and French (2002) could be contributed into, (i) more violate dividend-payout policy and more earning dressing effects, (ii) the difference on time-varying risk premium induced by time-varying risk aversion between Taiwan and U.S. markets, (iii) different behavioral factors.
IV. Decomposing and Modeling Time-Varying Risk Premium
The perceived riskiness of holding stocks is likely to vary substantially over different historical periods. If the perceived risk premium on stocks is adequately measured by the conditional variance, then it follows that the future risk premium is partly predicated. The combination of CAPM and an assumption that agents’ perceptions of future riskiness is persistent results in equilibrium returns being variable and in part predictable. With the aid of ARCH-type models proposed by Engle (1987), Bollersolve (1993), the market risk and related properties, such as changing pattern and error-correcting process of the market, could be easily described. Furthermore, the validity of the CAPM can be examined under the assumption that equilibrium returns for bonds and stocks depend on a time varying risk premium determined by conditional variances and covariance. Thus, we can understand the equity premium more by hiring time-varying volatility models. We test how persistence in the risk premium can, in principle, lead to the large swings in stock prices which are observed in Taiwan’s data. Modeling the time-varying risk premium, that could be caused by the time-varying variance, and estimating the degree of persistence in the risk premium by the concept CAPM and ARCH models are two major tasks in this section.
expected dividend, the discounted rate including risk-free rate and risk premium, could affect the valuation on a stock.
1 t t t j t j j P E
γ
D ∞ + + = ⎡ ⎤ = ⎢ ⎥ ⎣∑
⎦, (4) where(
)
1 11
j t j t i t i ir
RP
γ
− + + + =⎡
⎤
=
⎢
+
+
⎥
⎣
∏
⎦
, andD
t is the cash dividends at time t. tγ
, consisting of the risk-free rater
t plus a risk premiumRP
t , is thetime–varying discount rates. The stock values
P
t and equals to the excepteddiscounted future payments, namely the dividends. Schwert (1988, 1990) examines conditional volatility, the volatility in stock returns, on having obtained the best forecast possible for stock returns. He uses a fairly conventional approach, AR model plus monthly dummy variable, to measure conditional volatility for the U.S. stock market over the period 1859-1987, and finds the persistence in stock return volatility which could be caused from the fundaments and could influence the risk attitude in economic.
Merton’s intertemporal CAPM (1973 and 1980) states that the expected return of a portfolio on available and conditional information set
Ω
follows,[
]
21
|
1|
t t t t t t t
E R
+Ω = +
r
λ
E
⎡
⎣
σ
+Ω = +
⎤
⎦
r
RP
, (5)where
λ
is the market price of risk and 2 1 tσ
+ is the expected risk conditional on the information set at time t. The risk premium at time t can berepresented by 2
1
|
t t t
RP
=
λ
E
⎡
⎣
σ
+Ω
⎤
⎦
, and equals to the market price of riskmultiples the expected market volatility. Re-ranging above equation, we can find that the expected risk premium at time t,
[
]
21
|
1|
t t t t t
E R
+Ω − =
r
λ
E
⎡
⎣
σ
+Ω
⎤
⎦
, isfunctioned by the market price of risk, relating with the risk aversion of represented individual, and the expected volatility conditional on all available information set.
The realized risk premium at time t, is conditional on two terms, namely, the market prices of risk
λ
t and the expected value of quantities of risk2 1
|
t t
E
⎡
⎣
σ
+Ω
⎤
⎦
, and plus a residual,2
1 1
|
1t t t t t t
R
+− =
r
λ
E
⎡
⎣
σ
+Ω +
⎤
⎦
ε
+ , (6)where
ε
t+1 represents the noise term. We empirically operate equation (6) by,1
var (
)
1t t t t t t
R
+− =
r
λ
R
+
ε
+ (7)and assume the conditional variances to time-vary according to a Generalized AutoRegressive Conditional Heteroskedasticity Process, proposed by Bollerslev (1986).
2 2 2
0 1 1 1 1
t t t
h
=
α α ε
+
−+
β
h
− , (8)where
R
t=
(
P
t−
P
t−1)
P
t−113 is return from month t-1 to t, and tε
is assumed to be conditionally normal with zero mean and conditional variance 2t
h
. We usethe GARCH conditional volatility as the forecasting volatility on day t and could be modeled as 2
t
h
. From equation (6) to (8), we try to decompose the riskpremium to time-varying market prices of risk and time variation in the quantities of risk. It is popularly accepted that the GARCH model could estimate the volatility of financial variables well, thus we can investigate the relation between the market price of risk and the risk premium by the equation (8) and find a possible linkage between time-varying market price of risk induced by time-varying volatility and risk premium.
Taiwan monthly sock index, returns and conditional standard deviation on period from 1967:1 to 2003:6 are shown in Figure 1. It is interesting to check the first graph in Figure 1. Two major stages, one is from 1961 to 1987 and
13 Another way to measure the dependent variable, vart(Rt), is to use the variance of the risk premium rather
than that of return. But according to fact that, contrasting to the volatility of market return, that of riskless rate nearly remains a constant, we use market return as the dependent variable in mean equation of GARCH process.
the other is from 1988 to 2005, could be roughly broken during the period from 1987 to 1988 accompanying with the largest volatility of return shown in the second graph. The average of TAIEX on a monthly base is about 3034.87 with a standard deviation 903.24; the average monthly return is about 1.43 percent with a standard deviation 10.20 percent implying an inconceivable picture. The violate characteristic in Taiwan’s stock market can also be found in the maximum and minimum values, say 50 percent and -39 percent, in middle graph. From the third and forth moments, monthly return in Taiwan shows right-skewing and leptokurtosis distribution. The conditional volatility pattern estimated from GARCH (1, 1) process is shown in the third graph of Figure 1. It consolidates the possibility of a time-varying characteristic on volatility which could induces a time-varying risk premium based on a rudimentary grasp of the conditional volatility pattern in Figure 1.
< insert Figure 1 about here >
The empirical framework from equation (7) and (8) is to investigate whether the market price of risk varies with different holding periods and different entry-timing into the market. If it does, we could understand the role of the time-varying risk on the risk premium, the changing process, and the cognitive and learning process of the market to a time-varying risk and could try to explain the equity premium more beyond the consumption-based models. We first estimate the parameter
λ
, the market price of risk, both from equation (7) and (8) on different holding periods, namely 30-, 20-, 10-, 9-, 8-, 7-, 6-, 5- years. In each holding period, the market price of risk is estimated rolling-over on a sequence of entering-time into the market. For example, on a 30-years base we estimate parameters of the price of risk from 1967 to 1996, 1968 to 1997, …, 1975 to 2004, and observe totally nine estimates of the market price of risk. The similar procedure is applied to other holding periods. Empirical results of different holding periods and different entry-timings are shown in Table 4 and plotted in Figure 3. There are two estimates, the market price of risk and related t-value, are shown in each holding period in Table 4 including nine different holding periods from 1967 to 2000. The empirical results show three major characteristics. Firstly, the longer the holding periodthe more significant the estimates are, since unavoidable limitation that the power on estimating procedure could be reduced partially by the numbers of observations in a shorter holding period. Secondly, the patterns on estimated market prices of risk trend to decrease as time passed. For example, for 10-years holding period, the maximum monthly market price of risk, 1.97, happens in the first 10-year holding period from 1967 to 1976; the minimum value, -0.29, locates in the period from 1994 to 2003; and most estimates around from 0.8 to 1.2. Theoretically, the value of the market price is always great than zero when the market is in equilibrium. Finally, the shorter holding period, the more possibility the estimated market price of risk could falls below zero.
< insert Table 4 about here > < insert Figure 3 about here >
What happens to a negative market price of risk? Someone bears the unwilling of reduced consumption and the uncertainty of future reward today and could be punished in future (in this example, an investor invested Taiwan stock market from 1994 to 2003). But nobody could promise that market can fully and automatically adjust toward the equilibrium state in the condition of filling lots of kinds of disturbances from political and economic factors in a macro-view and behavioral factors in a micro-perspective. We infer a negative market price of risk to the possibility that the expectation of market’s participations at timestamp t, is limited to form a full and accurate forecast about a investing horizon, t+n. Another explanation could be the disturbance term,
ε
t+1, in equation (7), when most empirical evidences do not support what theoretical models say. In the estimates of the market price of risk from 1994 to 2003, first-order autocorrection (AR(1)) nears 0.145 (not reported) of the residual process, comparing with a normal price of risk, say from 1975 to 1984, the AR(1) of the residual process nears 0.009 (not reported), it is clear to infer that the different error correction process, possibly induced from different degree of market factors related to risk cognition, between two equal investing horizons.V. How well the Myopic Loss Aversion explains the TAIEX
Equity Premium
Mypotic loss aversion (MLA) was presented and applied firstly to the equity premium puzzle by Benartzi and Thaler (1995) who use preference scheme combining mainly two experimentally observed behavioral concepts, namely loss aversion and mental accounting which are well illustrated by the famous example of Samuelson (1963)14.
In their proposed MLA solution to the equity premium puzzle, Benartzi and Thaler (1995) hire cumulative prospect theory and emphasize the role of loss aversion for the individual decision maker and contribute to the literature on important implication of preference scheme brings to portfolio evaluation periods, i.e., the mental accounting loss averse investors perform. Empirical evidences from consumption-based CAPM think that the empirical risk premium is too large to be unexplained, but according to the MLA, the size of risk premium depends on how frequently an investor checks her/his portfolio. The more an investor checks the market or changes his/her portfolios, the more risk averse she/he behaves, thus acquires lots of risk premium more than the
14 Samuelson (1963) gives an example of the effects of loss averse and mental account on the portfolio
evaluation by an experiment which offers his colleague of a fifty-fifty chance at winning $200 or losing $100. The colleague refuses this bet, since he would feel a $100 loss more than a $200 gain (loss aversion), but at the same time he express the willingness (mental accounting) to take on one hundred such bets. The loss aversion is easily to understand, but the mental accounting effect remains unclear. Samuelson assume that his colleague has the following value function.
0 ( ) 2.5 0 x if x u x x if x ≥ ⎧ = ⎨ < ⎩
With this utility function the bet is rejected since the prospective utility is negative, since the utility of a single bet is 0.5*200 + 0.5*25*(-100) =-25, but what about a game of two bets? The attractiveness of this gamble will depend heavily on the mental accounting of the problem. If the two bets are treated separately the game has double the unattractiveness. However, if the two bets are compounded into a single bet, having outcome/probability set {$400, 0.25; $100, 0.50; -$200, 0.25} and having a value of perspective utility 25, it will have positive expected utility and be accepted. As it turns out, compounding any number of this bet greater than one will be favorable for the colleague as long as he does not have to monitor the separate bets being played. Thus, Samuelson (1963) proves in a theorem that if an individual turns down a bet at every level of wealth, then accepting a multiple gamble is inconsistent with expected utility maximization. A parallel to the above example is a loss averse investor choosing between stocks and bonds. The evaluation period will be crucial for the investor’s attitude towards the risk of the investment under the utility function setting. If the decision maker evaluates the portfolio on a daily basis, a portfolio consisting of stocks will be unattractive, since stock returns go down almost as often as they go up, from day to day, and losses are mentally doubled. On the other hand, consider a long evaluation period of say ten years. The investor can rest assured that stocks most surely will increase in value every ten years. Hence, a stock portfolio can be an unattractive investment if evaluated very often, but an attractive one over longer evaluation periods. Relatively risk-free bond portfolios are not affected by this phenomenon to the same extent since they do not display losses as frequently.
consumption-based pricing theory can explain. The above argument brings two questions to mind. First, how loss averse are financial investors? Second, assuming a reasonable level of loss aversion, how short an evaluation period will be in accordance with the observed equity risk premium?
Benartzi and Thaler (1995) use a procedure on estimating loss aversion provided by Tversky and Kahneman (1992), i.e. 2.25,
0 ( ) 2.25 0 x if x u x x if x ≥ ⎧ = ⎨ < ⎩ , (9)
and find an investor with prospect theory preferences will be indifferent between an all-stock portfolio and an all-bond portfolio if the evaluation period is twelve months. At this horizon of evaluation, there is a kind of market equilibrium where investors are content with the risk-return relationship of stocks and bonds. The large equity premium can be understood as compensating the investor for her/him fearing of stock portfolio losses, as well as her/his myopic way of evaluating the portfolio based on a shorter period than twelve months. But what evaluating frequency can fill the gap between the realized risk premium and that consumption-based theory predicts and how to bring it into a empirical? To determine the evaluation period that makes the loss averse investor indifferent between the historical returns on stocks and bonds, Benartzi and Thaler (1995) derive prospective utilities of holding these assets with lengths between evaluations for the investor. If the agent evaluates her/his portfolio every six months, her/his utility of holding a stock portfolio is derived using six-month data on assets returns. In order to apply equation (9), we must have the possible payoffs on stocks and riskless assets with corresponding probabilities, at each data frequencies. There are many ways to determine these distributions and Benartzi and Thaler use a non-parametric bootstrap approach.
The basic idea of the approach is that returns are assumed independent over time. Using the high frequency monthly data, we estimate an n-months return by drawing n returns at random (with replacement), and derive the low
frequency return assuming these n returns are consecutive. For example, drawing the four monthly returns, r1, r2, r3, and r4, and using these four high frequency returns, the low frequency 4-months return can be calculated as,
1 2 3 4
(1+r)(1+r)(1+r)(1+r) 1− . This procedure is performed 100,000 times to obtain a
smooth n-month return distribution, since n increases, the number of possible combinations increase, and a more continuous distribution of low frequent returns can be obtained. A histogram over these returns is then derived using an interval size of choice, so that we can associate the possible returns (midpoint of every histogram interval) with a specific probability (the frequency of returns in each interval divided by the total number of returns). Choosing for example twenty intervals, just as in Benartzi and Thaler, will construct an empirical distribution over twenty possible outcomes for the portfolio. Investing in a risky stock portfolio can, in this way, be seen as a gamble over twenty specific outcomes with predetermined probabilities, which allows for equation (9) to be directly applicable. Thus, using historical data on any portfolio we can derive the perspective utility of holding this portfolio at an evaluation period of choice. All we need to do is decide on how loss averse the investor behaves and how often she/he evaluates her/his portfolio.
What do the estimated distribution look like? Figure 3 illustrates estimated distribution for 1-, 2-, 3-, 6-, 9-, 12-, 18-, and 24-months returns when using the non-parametric bootstrap. Monthly stock returns from 1967:1 to 2003:6 are used. Since we expect the portfolio mean to be larger over longer holding periods and also, show a greater uncertainty, compared with shorter portfolio horizons, the distributions display larger means accompanying with larger standard deviations and become more outspread with a greater aggregation supported by the skewness on each distributions. The paper does not figure any clear picture in kurtosis, which dose not show a consistent pattern, although there might be such. Overall, the non-bootstrap method of estimating stock return distributions shows changes in the first three unconditional moments, as the aggregation level increases, though the forth moment shows an unclear changes.
< insert Figure 4 about here >
From Figure 4, the returns and related distributions are available and ready to calculate the horizon of evaluation, which represents a kind of market equilibrium where investors are content with the risk-return relationship of stocks and bonds. We use equation (9) as the prospective utility scheme, just as Tversky and Kahneman (1992) and Benartzi and Thaler (1995), to identify an indifferent holding horizon between stock and riskless assets. Results are shown in Figure 5. The figure displays two graphs, one for the riskfree rate and another is stock return estimated from non-parametric bootstrapping method. The portfolio evaluation period that beings indifference between the two portfolios is ten months in Taiwan from 1967:01 to 2004:12. This is near but small the twelve month evaluation period of Benartzi and Thaler (1995) from 1926:01 to 1990:12. The fact that indifferent holding horizon in Taiwan (ten months) is large than that in U.S. (twelve months) reflects a shorter horizon of market equilibrium where investors are content with the risk-return relationship of stocks and bonds under prospective utilities scheme, though the sample period is different, Taiwan is from 1967 to 2004 but U.S. is from 1926 to 1990. To check the relation between the equity premium and the evaluation periods in advanced, we divide the sample in Taiwan into two subsets, one is from 1967 to 1986, another is from 1987 to 2004 based on the roughly identification that there could exist a structural change at 1986. (We can check this argument in Figure 1)
< insert Figure 5 about here >
For these data, Table 5 shows annual risk premium and indifferent holding horizon in Taiwan and U.S.. Whole sample period provides an annual risk premium about 12.69 percent with an evaluation period, ten months, while the first sub-period from 1967 to 1986 suggests 9.38 percent and twelve months, respectively and the second one from 1987 to 2004 offers 15.49 percent and seven months. Borrowing the statistics from Benartzi and Thaler (1995), the annual risk premium and evaluation period are about 6.40 percent and twelve
months. Two things can be noticed. First, there exists a negative relation between the equity premium and the equilibrium evaluation period, ie., evaluation period decreases as the risk premium increases, even if we could not identify the causality between equilibrium evaluation periods and risk premium. We think that the relationship between theses two variables is interesting and could be a dynamic process, thus dealing the issue further is our next study. Second, the indifferent evaluation period, twelve months, in Taiwan from 1967 to 1986 is equal to that in U.S. from 1926 to 1990, but the annual risk premium differs a lot, 2.98 percent, between them. The additional premium could be caused the different sample period limited to the evolving process in each security market, market risk, investing behavior or some factors the academic still can not figure out. The economic meaning of two evaluation periods from twelve months to seven months tells us the investors ask more risk premium at the second stage than that on the first stage and implies loss-averse investors check their portfolios more frequently than they did before. It could also be a phenomenon that the composition of market participants has changed or that the investing behavior of participants was affected by some structural factors, such as an introduction of the electric ordering system, a faster diffusion mode of information by all communication tools and other social factors that could affect the mentality of the market. To investigate these factors is quite interesting, though it could beyond the content of this paper. Dealing these issues into the equity premium is another further research from us.
< insert Table 5 about here >
VI. Conclusion
Equity premium plays an important role not only in academics but also in practice. When demand meets the supply in equilibrium, how large is the
equity premium should be required? The multidimensional equity premium had greatly perplexed the academics since Mehra and Prescott’s research using U.S. data first appeared in 1985. Many pieces of research have been conducted to provide possible explanations for the equity risk premium puzzle. But the issue is definitely not settled yet!
Empirical evidences studied outside the U.S. market also show a similar puzzle. Jorion and Goetzmann (1999) study 39 countries for the period of 1926-1999 and find that the median real appreciation rate was only 0.8 percent per year compared to 4.3 per year for the U.S.. Perhaps due to the lack of data, Taiwan’s market is not included in Jorion and Goetzmann’s (1999) sample, although Taiwan has emerged as an important emerging market recently. This research investigates the risk premium in three respects. First, we estimate the expected return based on fundamentals. Second, we explain the relation between the time-varying risk and the premium. Finally, we use MLA hypothesis to estimate the average evaluation interval in Taiwan.
We study the equity premium in Taiwan from 1961, when the TSEC commenced its operation, to 2004. The risk premium in Taiwan has a different picture in Taiwan than reported in Fama and French (2002). For the U.S. data given the evidences of Fama and French (2002), the average realized rate of return of 18.52 percent in Taiwan from 1961 to 2001 is about twice as much as that of U.S. 9.62 percent from 1951 to 2000. Similar results could also be found in the realized risk premium, 15.86 percent in Taiwan and 7.43 percent in U.S.. But, our results show greater values of the expected risk premiums from dividends and earnings growth models. In the U.S., the average risk premium of dividends and earning models are just about 2.55 and 4.32 percent, respectively. Thus, the estimated risk premiums based on the fundamentals suggested by Fama and French (2002) seems to make the puzzle more perplexed in Taiwan. The results could be contributed to (i) more violate dividend-payout policy and more earnings window dressing effects, (ii) the difference in time-varying risk premium induced by time-varying risk aversion between Taiwan and U.S. markets and (iii) different behavioral factors. In
addition, decomposing the risk premium to time-varying market prices of risk and time variation in the quantities of risk through well adapted GARCH model, we find that the longer the holding period the more significant the estimates are. However, due to the unavoidable limitation that the power of estimating procedure could be reduced partially by the numbers of observations in a shorter holding period, we believe that one can still make useful inference of our results by looking at the sign of the estimated coefficient. Also, the estimated market prices of risk decrease with time. Finally, we find that the portfolio evaluation period in which under investors are indifferent between the risky portfolio and riskless asset is ten months, which is smaller than the twelve month evaluation period found by Benartzi and Thaler (1995) for the period from 1926:01 to 1990:12. Furthermore, we find that the indifferent evaluation period, twelve months, in Taiwan from 1967 to 1986 is equal to that in U.S. from 1926 to 1990, but the annual risk premium differs a lot, 2.98 percent, between them. The additional premium could be caused by the different sample period, market risk, investing behavior or some factors that academics still has figured out.
Table 1
Summary Statistics of Dividend Yield and Inflation Rates
Real returns of risky and riskless portfolios including cash dividends are calculated from the year normal year return by removing inflation factor. We use monthly price index published by “Directorate General of Budget Accounting and Statistics Executive Yuan, R.O.C.” to adjust the nominal return. Real return
t Rof TAIEX 1962 to 2002 1962 to 1982 1982 to 2002 Arg. average 3.94% 6.38% 1.50% Geo average. 2.54% 4.98% 1.30% Dividend yields Standard deviation 4.22% 4.84% 0.80% Arg. average 4.74% 7.44% 1.89% Inflation rates Standard deviation 7.94% 10.37% 1.71%
Table 2
Real Return of TAIEX, Riskless and Equity Risk Premium from 1962 to 2002
Real returns of risky and riskless portfolios including cash dividends are calculated from the year normal year return by removing inflation factor. We use monthly price index published by “Directorate General of Budget Accounting and Statistics Executive Yuan, R.O.C.” to adjust the nominal return. Real return
t
Rof TAIEX is
calculated by
1 1 1
(D Pt t−) (+ Pt−Pt−) Pt− , and considering inflation rate for the year t is πt=CPI CPIt/ t−1−1, where t
CPI is the price level at the end of the year t. The Sharpe Ratio is defined as, (Rt−rft) / (σ Rt).
Period Return of Real
Stock Index Real Return of Riskless Rate Equity Risk
Premium Sharpe Ratio
1962~1982 14.44% 7.6% 6.84% 0.1731 Holding Period of 20 Years 1983~2002 21.49% 6.33% 15.15% 0.2992 1962~1972 22.74% 4.72% 18.03% 0.4742 1973~1982 6.14% 10.48% -4.35% -0.1054 1983~1992 35.20% 7.31% 27.89% 0.4577 Holding Period of 10 Years 1993~2002 7.78% 5.36% 2.42% 0.0674 1962~1967 18.04% 4.32% 13.72% 0.3493 1968~1972 18.24% 7.01% 11.23% 0.7538 1973~1977 8.86% 9.62% -0.76% 0.1549 1978~1982 -17.42% 10.13% -27.57% -1.2927 1983~1987 40.98% 6.63% 34.35% 0.8062 1988~1992 14.81% 8.12% 6.69% 0.2017 1993~1997 15.04% 7.01% 8.03% 0.3996 Holding Period of 5 Years 1998~2002 -10.21% 2.31% -12.52% -0.3265 1962~1989 (28 years) 26.85% 7.39% 19.46% 0.4178 Two Sub-periods Divided by the Highest Historical Price Level 1990~2002 (13 years) -0.49% 6.09% -6.58% -0.1805 Note:
1. Taiwan’s stock market reached its highest level at 12495.34 points on Feb. 10, 1990. 2. We use Sharpe ratio to proxy for a measure of the relative risk aversion of the market.
Table 3
Annual Equity Premium and Related Statistics for the TAIEX from 1961 to 2004
The inflation rate for the year t is
1
/ 1
t CPIt CPIt
π = − − , where CPIt is the price level at the end of the year t. The real return for year t, on one-month Board
Rate,
t
rf , (rolled over each month) calculated from the average on the five major commercial banks including, Bank of Taiwan, Taiwan Cooperative Bank,
First Bank, Hua Nan Commercial Bank and Chang Hua Bank. The normal values of book equity and price for TAIEX index at the end of year t are Bt and
t
P. Normal TAIEX dividends and earnings for year t are Dt and Et. Real rates of growth on dividends, earning and stock price are represented by the
following estimators,
1 1
( / )( / ) 1,
t t t t t
GD = D D− CPI− CPI − GEt =(Et/Et−1)(CPIt−1/CPIt) 1− andGPt =(P Pt/ t−1)(CPIt−1/CPIt) 1− . The real dividend yields is
1 1 1
/ ( / )( / )
t t t t t t
d p− = D P− CPI− CPI . RDt =dt/pt−1+GDt is the dividend growth estimate of the real TAIEX return for year t. REt =dt/pt−1+GYt is the earnings
growth estimate. Rt is realized real TAIEX return. RP D( t)=RDt−rft and RP E( t)=REt−rft are the dividend and earnings growth estimates of the real
equity premium for year t. RP R( t)=Rt−rft is the real equity premium from the realized real return. The first and second moments of all variables are
expressed as percents. The variables, mean and standard deviation, include Taiwan and U.S. markets which are shown in the parentheses, are placed in the second and sixth rows..
