Random Walk Hypothesis: Assumptions and Tests | Financial Economics

In this article we will discuss about:- 1. Introduction to Random Walk Hypothesis 2. Random Walk Assumptions 3. Schematic Presentation 4. Test 5. Essence 6. Limitations.

Introduction to Random Walk Hypothesis:

There are theoretically three approaches to market valuation, namely, efficient market hypothesis, fundamental analysis and technical analysis. Under fundamental analysis, the share value depends on the intrinsic worth of the shares, namely, its earnings potential. If the efficient market hypothesis is used, then the market becomes perfect and the entry into the market by buyers at any time gives equal benefit to all.

The prices are determined in a random manner by competitive forces and perfect information flow and are independent of the past prices. This informa­tion is not only free and perfect, but it is absorbed fully and immediately by the market. But in actual practice, the information flow is not free and perfect and markets are not, therefore, efficient. Where the efficient market hypothesis does not hold, technical analysis is applicable.

Under technical analysis, the prices move in a predictable manner and in waves and trends. The present prices are the result of past trends and can accordingly be predicted. Thus, by the use of analytical tools of charts and curves, the price trends can be studied and future trends can be predicted to decide on when to buy and sell.

Investment refers to purchase of claims on money or financial assets used in the productive process in the economy. Investment, if it is in productive assets, should lead to an increase in output and income in the economy. In the macro sense, the investment income multiplier of Keynes operates whereby additional investment leads to an addition to output and income. Investment and income are related by a constant, namely, investment income multiplier or incremental capital-output ratio, which is the ratio of additional output due to the additional capital used in the productive process.

Besides, the aggregate capital-output ratio should be distinguished from the incremental capital-output ratio. While the former relates to the output created through a given capital input, the latter refers to the incremental output generated through a given increase in capital input. Corresponding to these two concepts, there are two multipliers, namely, aggregate investment multiplier and incremental investment multiplier. These refer to the increase in incomes due to a given increase in investment.

Mathematically, multiplier M = ΔY/ΔI where Y is income and I is invest­ment. An increase in I (ΔI) will lead to a rise in Y (ΔY), as a multiple of I. This multiple is called investment multiplier by Keynes. A clear understanding of this concept will also explain the capital-output ratio. An increase in capital is invest­ment itself while an increase in output as a result of an increase in capital input will lead to an increase in output and thus income in the macro sense.

If the incremental capital-output ratio is given by K, then the increase in output ΔO can be set out as- ΔO = K x ΔI.

Thus, K = ΔO/ΔI, which is similar to the equation set out above for the investment multiplier.

Thus, the first theoretical tool is the Savings Investment Theory, which postulates that savings flow into investments which in turn lead to a multiple increase in output and income through what is called the capital-output ratio or investment multiplier. Thus, savings promote capital formation and economic growth through increase in output and incomes of the country. The mobilisation of savings for capital formation is through the capital market comprising the new issues market and the stock market.

The role of capital market is thus primarily to promote economic growth. The instrument through which this process is carried out is through the sale of corporate securities, which are claims on financial assets. These instruments are issued by the corporate sector to raise capital through such securities, as equities, preference shares, debentures, etc. The purchase and sale of these securities is carried on in the stock and capital markets, which impart liquidity to these investment instruments and thus promote the flow of public savings into these financial markets.

Secondly, the market behaviour also depends on the players and their role in trading. An analysis of the market price behaviour is thus possible through the number of buyers and sellers available and the free flow of correct and unbiased information into the market. The Market Efficiency Theory or Random Walk Theory and many other theories explain how prices behave in the market in the macro sense.

Competitive market conditions with a large number of buyers and sellers and with free and perfect flow of information will result in correct price formation in which prices tend to move near to their true intrinsic values of shares. In the absence of the above conditions, the share price movements may be erratic and biased; they may be overvalued or undervalued at any point of rime. The insider information, rumours, cornering of shares and semi-monopoly conditions would all lead to imperfections in the market and price formation would be unrelated to the prevailing fundamentals of the company and its shares.

The theoretical analysis of the market for its price behaviour would enable the investor to understand the market and make the right investment decisions regarding corporate securities.

The third theoretical tool in investment analysis is the fundamental analysis which explains why prices are what they are. This is an analysis of fundamental factors affecting the market in the macro sense, namely, economic, industry and company analysis. In the micro sense, the price of a share can be analysed through security valuation to find out the intrinsic value of a share and to examine whether a share is overvalued or undervalued.

In the security valuation the most important tool is the ratio analysis or examination of the balance sheet and profit and loss accounts of the company, whose share is being examined. The examination of these fundamentals will enable us to locate the undervalued shares and overvalued shares and to decide what shares to buy and what to sell.

The next question is when to buy and when to sell. This leads to the fourth theoretical tool, namely, technical analysis, which is an analysis of the price behaviour of the aggregate market and of individual shares with the help of charts on price, trading volume and moving averages.

An analysis of the price behaviour of the individual scrip historically in the background of the market price index behaviour will help to locate the turning signals indicating the likely changes in trends and suggesting the buy and sell points in the charts. The Dow Theory and Elliot Wave theory are some of the theories in this analysis, which explain the price behaviour in the past and help us forecast the likely behaviour in the immediate future.

Yet another type of analysis is the analysis of risk and return which are the two major characteristics of any investment. This is sought to be achieved by the use of portfolio theory and portfolio management. In the analysis, the choice of scrips is decided by an analysis of risks involved in relation to the return in the background of the market risk and market return. A diversified portfolio of scrips is decided by an analysis of risk involved in relation to the return in the background of the market risk and market return.

A diversified portfolio of scrips with varying degrees of risks in the upward and downward directions would lead to a minimisation of risk for the selected basket of scrips in the market. The degree of risk of the whole basket would, of course, depend upon the asset preferences, income requirements and other investment characteristics which an investor would choose, given his likes, prefe­rences, needs etc.

Broadly the modern portfolio theory depicts the choice of scrips in terms of its risk-return characteristics and maximisation of returns and minimisation of risks involved. The greater the risk taken the larger is the reward. Capital Assets Pricing Model is a hypothesis explaining the valuation of assets in a portfolio.

Random Walk Assumptions:

The price movements under Random Walk Theory are randomly distributed, in such a way that the present steps are independent of past steps and in view of such random movements entry into the market any time gives same returns for the same risk to the investors.

This theory is based on the following assumptions:

(1) Market is perfect and free without trade restrictions.

(2) Market absorbs all the information quickly and efficiently.

(3) Information is free and costless and is quickly available to all at the same time.

(4) Information is unbiased and correct.

(5) Market players can analyse the information quickly and the information is absorbed in the market through buy and sell signals.

(6) Demand and supply pressures are absorbed in the market through price changes. Such absorption leads to quick and prompt movements in prices which are random in fashion.

Schematic Presentation of Random Walk Hypothesis:

All the theories are integrated to help decision-making by investors. Firstly, the basis of markets is the money flow, which is represented by the funds flow theory, resulting in the emergence of stock and capital markets. Looked at from a different angle, the savings of the public are channelled into investment, which for an individual at micro level leads to claims on money and future cash flows. For the country as a whole, the savings flow into investment, which helps the growth process in the economy. This is explained by the Savings — Investment Theory.

Secondly, an investor to make a right decision to purchase or sell shares has to know the correct and fair value of a share. To explain why share prices fluctuate and what the fair prices are, the theories used are Market Efficiency Theory or, Random Walk Theory and Capital Assets Pricing Theory. All these present markets as efficient due to free and perfect information flows and their absorption by the markets.

In actual practice, information is not perfect and markets are not efficient. Prices depend inter alia on a host of psychological and emotional factors. The theory of Trend Walkers explains the market trends as set by a few trend setters or leaders followed by a mass of trend walkers.

Thirdly, the market prices and individual share prices are explained by the fundamental factors, namely, economy, industry and company analysis. This leads to factors determining the market prices and their movements around the intrinsic worth of the shares. This process is helped by the balance sheet analysis of companies and application of ratio analysis and other tools of financial management.

Fourthly, an investor should know the timing of investments, when to buy and sell. This decision is helped by the technical analysis of markets, incorporated in the Dow Theory and moving averages, Elliot’s Wave Theory, etc.

Lastly, the portfolio theory provides the linkage of markets to investment. An efficient portfolio is to be developed by the investor by making proper investments to minimise the risks and maximise the returns. The portfolio management helps the investment process by applying the principles of Portfolio Theory to build up an efficient portfolio through a diversified basket of scrips and by using the concept of Beta. Security evaluation and risk return assessment are linked to the investment process. What securities to buy and when to buy so as to build up an efficient portfolio are all interlinked to result in the buying and selling of shares in the market and trading.

Thus, the whole theoretical framework results in a practical application to the buying and selling and investment process in the market.

Investment and Time Value of Money:

A bird in hand is worth two in the bush. Money today is more valuable than the same tomorrow or a few days hence. This is because money has alternative uses and opportunity costs. In fact time is money. If it is invested, it would bring a return and better the investment, the better is the return. To part with money is a risk which should be rewarded by a return.

The present value of a delayed pay off may be found by multiplying the pay off by a discounting factor, which is expressed as the reciprocal of 1 plus a rate of return. Thus, the discounting factor = 1/1+ r, where r is the rate of return that investor thinks adequate for his parting with money. Money has also a psychological satisfaction and value and to part with money is to part with a value which can only be compensated by a return.

Present Value Method:

Thus, the present value (PV) = 1/1+r XC, where C is the expected cash flow or pay off in the period. If there are more than one period, then,

where, C₁, C₂ etc., are returns in periods 1 and 2 etc., and r remains as the same rate of return during all these periods. If r is the required rate of return or reward for the risk of parting with money and it remains the same throughout all the periods considered, and C₁, C₂, C₃ etc., are the cash receipts in all the periods, then PV = C/r or PV x r = C. Thus, given the required rate of return, the needed cash flow can be derived.

In the above equation, r refers to the nominal return. The real return is the nominal return divided by the rate of inflation or rise in prices. Money is losing its value by the degree of inflation, year after year. In the eighties, the average rate of inflation per annum was around 8%; the real return is negative if it is less than 8% and if r is 8%, then the real return is zero. Only if it is more than 8%, the real return becomes positive.

Thus, the return r, required to compensate the lender or investor of money, should give a minimum of inflation rate (or riskless return) plus a reward for risk for parting with liquidity. This reward for risk element varies from instrument, maturity period, the creditworthiness of the issuer of the instrument and a host of other factors.

In India, the Bank rate (8% in October 1999) or the long-term rate of government securities (around 12%) could be considered the risk-free return and anything above this is the return for risk. Thus, the rate of preference shares and fixed deposits is kept by the Government fixed at 14% and the rate of PSU bonds (taxable) at 13%-14%. The yield on equities should be definitely more than these rates as they are subject to higher risks.

D.C.F. (Discounted Cash Flows):

The present value of future flows should thus yield a return which includes a riskless return (for the degree of inflation) plus a return for the risk shouldered in the investment. The same can be understood as discounted cash flows of the future periods to the present period. Thus cash flows (C₁, C₂, C₃, etc.) in periods 1, 2, 3 etc., should be discounted to the present period by the required rate of return (r) as-

 

Assuming the cash flow of C₁, C₂, C₃, etc., as constant (C) and the rate of returns is the same throughout the periods when these returns come, the present value can be set out as having a pay-off period of [PV/C], say, 6 to 10 years, depending upon the return of 15% to 20%. If the rate of return is 20%, then the pay-off period is 5 years. This concept of pay-off period shows how many years one has to wait to get back the present value of investment at the required rate of return. The shorter the period of waiting, the better is the investment. It is obvious that alternative investments have to be weighed before deciding on any investment.

Thus, any two investments can be compared in terms of:

1. Actual rate of return above the risk-free return as compared to return on other similar assets;

2. Pay-off period;

3. Net present value which is equal to the present value of cash flows in future minus the actual investment in the present period (NPV).

Test of Random Walk Theory:

i. Mutual Fund Performance Test:

One of the tests of the validity of Random Walk Hypothesis is that of the Mutual Funds, because they are expected to have better access to insider information or at-least have better information due to their research expertise. But evidence shows that mutual funds do not out perform the market normally. But there are excep­tions. The normal return of Mutual Funds in India is around 15% to 20% and most of them are quoted at a discount on Net Asset Value (NAV). Thus, the mutual fund performance in India partially supports the efficient market hypothesis.

ii. Serial Correlation Tests:

A number of researchers like FAMA, FISCHER, JENSEN, etc., in the USA and many experts in India have conducted tests for serial correlation of the price changes. These tests are run to find correlation between the present price changes and the price changes in any past period, with a lag of one day to a few days. The average daily price changes for stocks, in B.S.E. Sensex are not found to be significantly correlated with any lagged series.

iii. Run Tests:

Run Tests are also conducted in which the absolute numbers are replaced by signs. These merely count the number of runs, namely, consecutive price changes or signs in the same direction and their repetition at a later date. These tests revealed their distribution in random manner.

iv. Filter Tests:

In this analysis, filters are fixed at some percentage change and price move­ments are observed. Some mechanical trading strategies run on these filter levels on the premise that once a movement in prices has exceeded a fixed level of price movement called resistance or support levels, the security price will move in the same direction.

Thus, if the closing price of a security on a daily basis has moved up atleast by 5%, then strategy calls for buying that security until the price starts moving down then, the investment strategy is to sell and go short, until the price fell by the same percentage (5%) and then start covering up the short position. The selection of filter levels from 0.5% to 20% was adopted by Researchers and it was found that the results, of this filter strategy are no better than those with buy and Hold strategy or simple formula plan of buying at regular intervals.

This is particularly true, if the total transactions costs are also included in the prices. All the above tests mildly confirm the validity of the Random Walk Hypothesis. Under Indian market conditions, the strong form of Market Efficiency Theory does not seem to have validity but a weak form of this hypothesis has been found to be applicable to our conditions.

Even in the developed markets like the U.S., research results do not give any unambivalent conclusion regarding the validity of the semi-strong and strong forms of Market Efficiency Theory.

Essence of Random Walk Theory:

There is a lot of misunderstanding of this Theory. This is identified firstly with Market Efficiency Theory. Perfect market efficiency is taken as the basis for random walk of prices. Random Walk Hypothesis says nothing of the reasons for price movements or the valuation of stocks. It does not depend on perfect market conditions or perfect market absorption of all information. What only this Model postulated on the basis of empirical tests is that successive price changes are indepen­dent of the past changes. That means by implication that prices will average out and reflect the intrinsic value of a security.

The implication of the validity of this Theory is that Technical analysis is relegated to a secondary position and analysis of fundamentals is brought to greater focus in the valuation of securities. Security Analysis should therefore concentrate only on the intrinsic worth of a share, its fundamentals, like E.P.S. P/E, etc. The developments in the field of fundamentals like dividend announcements, Earnings Reports, Bonus or Rights announcements and financial results will have influence on stock prices and they need to be studied to the extent that this hypothesis supports the market efficiency theory and asserts that the markets are efficient to some extent in the absorption of all information, or superior expertise in analysis and better understanding of the affairs of the company in question. Better information and expertise in analysis and interpretation can give better results or enable one to outperform the market in India, as some analysts have shown. Hence, the Funda­mentalist school has more relevance in India.

Gordan’s Hypothesis:

M. J. Gordan has postulated a hypothesis called Bird — in the Hand Model, in which dividend policy becomes relevant, in the real world, as dividends today are more valuable than dividends in future. That means that internal rate of return = r and appropriate discount rate = k may or may not be equal. If r = k, the introduc­tion of risk in the model, makes K >K_t-1, and K cannot be held constant. If this factor is also taken into account, namely, today’s dividends are taken as more valuable than tomorrow’s, then two possibilities occur- (i) r > k or r < k; rates of return r, on each successive investment goes upto a point and then drops, due to operation of Laws of Returns. (ii) K is itself changing due to the principle that dividends today are more valuable than those in future due to the introduction of risk.

Walter Model:

The Model of Walter is based on the hypothesis that dividends effect the share price. The reasoning is that the firm’s internal rate of return (r) and its cost of capital (k) may be different. If it is cheaper to raise funds from external sources, then it is better to pay out dividends and raise resources from outside. If the internal rate of return (r) exceeds cost of capital (k) then the firm should payout the earnings as dividend. Since these two sources of funds have different returns, dividend payouts do matter and the value of the firm is affected by dividend payout policy. Investor prefers to invest funds outside the company, if r > k.

M. M. Hypothesis:

Miller & Modiglian Hypothesis postulates that dividend payout has no effect on share prices. What is relevant for share valuation is its earnings capacity. Whether earnings, cash flows and/or dividends are taken, the result is the same for the above purpose. If dividends are not paid they are used in the firm for investment and total return on capital employed is more relevant for valuation of the firm than what is paid as dividend for the shareholder.

But this hypothesis is based on some assumptions:

(a) Perfect markets, and no uncertainty.

(b) No taxes and no transactions costs.

(c) Fixed investment policy of the firm.

(d) Return on capital employed is the same as the cost of capital acquired from outside say r = k, which means that internal rate of return is the same as the cost of capital.

Under the above assumptions arbitrage process operates in a way that it is the same for the company (or for investor) if they use the retained earnings or external funds for investment. This approach is cost of capital approach.

Equation for M & M model is-

Where D stands for dividends or earnings and k for required rate of return and g is the rate of growth of dividends.

That Dividend Payout has financial signaling effect on share value in the minds of investors is ignored here. The effect of taxes, costs of transactions and a number variables and their impact is ignored in this analysis.

Walter’s formula:

Walter’s equation can be set as-

E = earnings, D = dividends per share

K = market discount rate or required rate of return

g = growth rate of dividends or earnings.

Walter took into account both dividends and earnings, as they both affect the share prices.

Earnings Model:

The market value of a share is determined by the present value of all anticipated future earnings flows.

Et is earnings per share during various periods. I_t is investment per share during the period and the other variables are the same as above.

Graham and Dodd Model:

According to Graham & Dodd, the dividends of a firm determine its share value and the equation can be set out as-

P = M (D + E/3) + A

M = Total earnings of firm

E = Earnings per share

A = Adjustment for Asset Values (say Book value per share)

D = Dividend per share

Stock Valuation:

P/E — Ratio and its Determinants

P/E — Ratio is the market price divided by the earnings per share.

P/E — Ratio is one of the models used for comparing prices of shares, like the

Dividend discount model or earnings.

Discount Model:

Consider the Dividend Discount Model which is given as-

Po = D₁/r-g

Dividend D can be written as Eb, where E is earnings and b is payout ratio, then

Po = Eb/r-g, and rearranging, we have

P/E = b/r-g

Then P/E ratio can be taken as a function of three variables — payout ratio, Discount rate and growth rate of earnings.

The Whitbeck and Kisor Model has calculated the undervalued and overvalued portfolios for 1960-61, quarter wise. As per their study (V.S. Whitbeck and M. Kisor — “New Tool in Investment Decision-Making,” Financial Analyst Journal, May-June 1963) one method of “beating the market” is to estimate the undervalued portfolios compared with the market portfolio as given by Standard & Poor 500 Index and choose such companies in the portfolio. The undervalued portfolio is calculated on the basis of P/E ratios, estimated and the actuals. The performance of each portfolio is compared with that of the Standard and Poor market index. Undervalued portfolio gives the above market return, called “Beating the Market.”

Malkiel and Cragg Regression Model:

Malkiel and Cragg used the regression model to suggest that the cross sectional regression models explain a good deal of the cross sectional variance in P/E ratios at any particular point of time. The dependent variable in the regression equation is P/E, the three independent variables were expected earnings growth rate (g), expected payout ratio (b) and the risk measure (β).

The regression equations for 5 years, studied by them are shown below:

As for each year, the R² ranged from 0.70 to 0.85. The authors have concluded that there is a good relation between P/E and the independent variables g, b and p. Similarly, one can use ROE and other variables like ROA (ROE is return on equity and ROA is return on assets, etc.).

Cootner’s Price-Value Interaction Model:

Stock Valuation Models take into account future cash flows discounted to the present time.

The Standard Model can be represented in the following equation:

where t = 1,2,3…etc.

i = Discount rate expected

If these incomes or cash flows vary in every period, then Cf₁, Cf₂, Cf₃… etc., will be discounted to the present time.

Paul Cootner had suggested that security prices can be viewed as series of constrained random fluctuations around their intrinsic value. There are two classes of investors. Of these, the more important professional investors who have better avenues to get information and analyse them, make estimates of intrinsic value and if its market price goes very much off the intrinsic value, they start buying or selling. The second class of investors, whom he calls Naive Investors follow the ‘hot tips’ or the mass media and buy and sell following the path set by the professionals.

Actual prices in the market move around the intrinsic value, for which the trend is set by the professionals and naive investors follow them. Buying and selling pressures emanate from the mass of the naive investors but the professionals snatch the cream of the gains, as they have the better information to foresee the changes in intrinsic value and book profits when necessary. This market is then called the intrinsic value random walk market.

Martingale Models:

A Martingale Process is a mathematical process in which the conditional expectation of the (n+1)st value equals the nth value in same set of data. A Martingale does not require that successive members of a series should be either independent or identically distributed. The Random Walk Hypothesis is a special case of Martingale Models. It is a Mathematical Model in which a series is both independent and identically distributed.

In a Martingale Model, the rates of returns follow the equation given below:

Where _Jr_t is the return for security J at period t. This equation represents symbolically the weakly efficient market hypothesis. The market absorbs all the price information fully and suggests that next period’s returns is expected to equal or reflect the last period’s return. In the semi-strong form of efficient market hypothesis, future returns are independent of the past data. This stronger form of Martingale process is represented by the equation.

E (_jr_t+1 / Nt) = _J^rt

Where Nt represents all public news and public information at the time period t.

Limitations of Random Walk Theory:

According to Random Walk Theory, security’s intrinsic values change and market prices move randomly around these intrinsic values. The new information affecting the market arrives at random intervals. This new information will force the analysts to re-estimate the intrinsic value and again the stock prices move randomly around the new intrinsic value.

This Theory thus states that security prices move randomly in a continuous fashion to set new equilibriums. There may be upward or downward movements and changes take place in a random manner. If the trade barriers are imposed, by the Stock Exchange authorities then these changes may be lower, reflecting barriers (support lines) or upper reflecting barriers (resistance lines) etc. The movements in share prices thus move generally within a narrow band, in a random fashion and these trends are changed from time-to-time with the flow of new information.

These random steps are based on the market absorption of the information. With perfect absorption, there will be continuous moves to equilibrium and this is what Samuelson had in mind when he referred to continuous equilibrium model, under perfectly efficient market conditions leading to perfectly efficient prices. This is based on the assumption of continuous flow of market information, which will change the stock prices, following the changes in the estimates of intrinsic value of the company’s shares.