Random walk myth

The random walk hypothesis (RWH) was introduced by Samuelson (1965), Fama (1970). The usual exposition is as follows: If all information is incorporated in the current market price, which equals the current equilibrium price, prices should not be expected to change, unless there is new information, i.e. news. Because new information is by definition unpredictable, price changes should be unpredictable, i.e. random walks. Although this may appear logical, the RW model is wrong. Proof that the EMH does NOT in general imply the random walk hypothesis for market prices was given a long time ago analytically by for example LeRoy (1973) and, Lucas (1978). The RWH really should be expunged from modern finance textbooks.

Below I describe my favorite stock market price example for showing why the RWH is wrong.

The efficient market hypothesis

The EMH implies that the correct current market price for a financial asset equals expected future cash flows (sale price and dividends) corrected for the current required return on investment:

i.e. P_t = (E_tP_t+1 + E_tD_t+1) / (1+r_t^e). Equally, the expected future price (incl. dividend) equals the current price corrected for required return (E_tP_t+1 + E_tD_t+1) = (1+r_t^e) P_t .

The random walk hypothesis

Theoretically, random walks are a special case based on special assumptions, including the rational expectations hypothesis. The classical form of the rational expectations hypothesis (REH) with respect to any variable x can be written as x_t+1 = E_tx_t+1 + e_t+1 . The actual value of variable x equals its expected value plus or minus the forecast error, where the case of no predictable forecast errors (i.e. "rational") implies expected value E(e_t+1) = 0 and correlation E(x_te_t+1) = 0. Note that this particular way of defining rational expectations is also referred to as the ‘perfect-foresight-with-error’ model. This is not the only way to define rational expectations and it makes very strong assumptions about the type of information available to economic agents, most likely not consistent with the real world. See my explanation elsewhere on the site.

Case with no dividends:

With a time-invariant constant required return r we have E_tP_t+1/P_t = (1+r). In logarithms and using as REH the formula ln P_t+1 = ln E_tP_t+1 + e_t+1 we find ln P_t+1 ≈ r + ln P_t + e_t+1 (i.e. log prices follow a “random walk with drift”).

Case with dividends:

In a stable equilibrium with constant required return and constant dividend growth rate we have E_tD_t+1/P_t = r – g (the Gordon growth model). We have P_t = E_tD_t+1/(r-g) and E_tP_t+1 = E_tD_t+2/(r-g) and E_tD_t+2 = (1+g) E_tD_t+1. This implies E_tP_t+1/P_t = (1+g). With rational expectations the realized (log) stock prices will again be a random walk with drift. Total return equals dividend yield (D_t+1/P_t) plus capital gain (DP_t+1/P_t ). Because D/P is constant the capital gain equals the dividend growth rate.

Some examples of efficient, rational and predictable stock price behavior

The model of a random walk with or without drift derived above is not generally true in reality. The RWH is conditional upon the assumptions of:

a) constant required return,

b) constant expected growth rate of dividends and only permanent shocks to dividend levels,

c) specific rational expectations hypothesis (i.e. perfect-foresight-with-error model).

The fundamental critique of the RWH: In general, required investment returns are not constant, dividend growth rates are not constant, dividend shocks are not always permanent, rational expectations need not follow the perfect-foresight-with-error model.

Dynamics of share prices and returns in different cases:

The dynamics of prices and returns can be complicated, with a crucial role for the concepts of 1) EX POST (after the fact, partly unexpected), 2) EX ANTE (before the fact, all expected), 3) TEMPORARY and 4) PERMANENT changes. Especially the difference between ex ante and ex post appears to create many misunderstandings! Financial markets are forward looking! Essentially, at any time t the observed ex post returns will largely be determined by changes that are needed for the creation of future required/expected returns. These necessary price movements can create so-called momentum or overshooting effects that are not consistent with random walk behavior but nevertheless follow from the efficient market constraint.

1* An immediate permanent change in dividends:

This type of shock creates volatility in ex post returns and resembles the RWH most closely. The price level will jump equal to the relative size of the jump in dividends. Ex ante, the required return for the new owner has not changed, but the previous owner gets large unexpected profits. Ex post historical returns alone tell us nothing about expected/required returns! A numerical example. Assume we start in a steady state. Dividends have been $1 for some time and are expected to remain the same, therefore the dividend growth rate g=0. The required return on the stocks is 10%, r=0.10. The steady state stock market price is P = D/(r-g) = 1 / 0.10 = $10. Now assume that at time t=0 news causes the expected dividends for periods t=1 and later to change to $1.50. At time t=0, the stock price increases to P = 1.5 / 0.10 = $15. An ex post return for the existing shareholders of 60%, although new investors will continue to only receive the expected future return of 10%.

2* Expected future temporary change in dividends:

This shock creates bubble type price movements. The price level jumps (but under-reacts to the dividend change) and gradually returns towards its old level over several future periods (“bubble”?). The ex ante returns are unchanged in all periods, but the ex post returns exhibit one-period spikes. A numerical example. Assume that at time t=0 news causes the expected dividends for time t=1 to t=4 to be (temporarily) increased to $1.50 (think of firms that temporarily pay special super dividends). For time t=5 and thereafter the expected dividends remain at the original $1. At time t=0 the share price will increase, creating ex post profits for existing shareholders. New investors still expect to earn a 10% return. We know that at time t=4 the share price will again be equal to the steady state price P = 1/0.10 = $10. But investors at time t=3 also know that they will still receive the higher dividend of t=4. Therefore, their price cannot be $10 because that would imply a return of 1.5/10 = 15% instead of the required 10%. In fact investors at time t=3 expect their market value to fall by 5% because 15% - 5% = 10%. In fact the stock price at time t=3 will satisfy the EMH condition (1+r₃) = (P₄+D₄)/P₃ or 1.10 = (10+1.5)/ P₃. Therefore P₃ = (10+1.5)/1.10 = 10.4545. Following this reasoning P₂ = 10.8636, P₁ = 11.24 and P₀ = 11.58. In other words, the stock price at t=0 increases strongly as a result of the news, and will decrease in the following 4 periods to return to its previous value of 10. Not a random walk but more like overshooting or bubble!.

3* Expected future permanent change in dividends:

This shock causes a price jump and prices that continue to drift to a new higher/lower level over several future periods (“momentum”). Assume that at time t=0 the expected dividends for t=4 and thereafter increase to $1.50 (permanently). You can calculate that the stock price jumps up at t=0 (P₀=13.7565) and will continue to increase in periods 1 (P₁=14.132), 2 (P₂=14.545), and 3 until P₃ = $15. Not a random walk but more like undershooting or momentum!

4* Change in expected/required returns (interest rate level, or risk premium):

Assume that the required return increases now and dividend payments are increased in the future. For example, we may assume that the company has a long-run target for its share price, and adjusts its dividends accordingly. With this shock the stock price falls and returns to its old level over several future periods (“bubble”). Assume that at time t=0 the required return on shares increases to 15% (permanently). Assume that dividends remain unchanged in periods 1, 2, and 3 but adjust to the new required return in period 4 and become $1.5 from then. Note that this firm is assumed to operate with a long-run stock price target of 10, i.e. it desires to keep the stock price close to some constant level. The stock price falls at t=0 ((P₀=8.86) and will subsequently increase in periods 1 (P₁=9.19), 2 (P₂=9.57), and 3 until P₃ = $10 again. Not a random walk but more like a negative bubble!

5* Expected future change in required return:

In this case the price level jumps and continues to drift to a new higher/lower level over several future periods (“momentum”).

Similar examples can be generated for other financial variables such as bond market prices, interest rates, exchange rates, etc. For example:

* Assume that monetary policy is expected to raise the overnight interest rate target of the central bank in 4 steps of 25 basis points in April, July, October and January from 3% to 4% (permanently). The expectations hypothesis of the term structure of interest rates will allow you to calculate the values of the 3-month, 6-month, 1-year, 5-year and 10-year interest rates. The longer maturity interest rates will jump now, and continue to increase over the next year. Not a random walk.

* Assume that monetary policy causes the interest rate in country A to go up by 1% relative to foreign country B for a temporary period of 2 years. After 2 years the interest rate differential is expected to be zero again. The uncovered interest rate parity condition tells us that the exchange rate of country A will appreciate by 2% today and is expected to depreciate by 1% in each of the two following years to return to its previous level. After two years, interest rate parity requires that the expected depreciation is zero. No random walk, but the classical overshooting model of exchange rates.

Various examples of predictions that may or may not be compatible with EMH

1. Short-term interest rates. Proposition: By end of this year the short (3-month) rate will be higher/lower than today.

Answer: Short rates are not durable assets. There is no opportunity for arbitrage through time. But we will see effects in other markets (e.g. term structure of interest rates, stock market).

2a. Share prices. Proposition: The AEX index will be higher next year.

Answer: Return on shares consists of dividend and capital gain. When dividend yield is below required return (as is the case in the real world) there must be expected capital gains.

2b. Share prices. Proposition: The AEX index will be higher tomorrow by 1 percent.

Answer: A 1 percent capital gain for one day equals 365 percent capital gain annually. This is not consistent with normal average required returns (except in countries with hyperinflation). Note that for short periods transaction costs can complicate matters. If transaction costs are 0.5 percent: buy + sell = 2x 0.5 percent = 1 percent. Nobody has incentives to arbitrage away an expected capital gain of 1%. In this case the predictability of 1% capital gain is not economically significant.

3. Long term interest rate. Proposition: The 10-year government bond rate will be higher/lower next month!

Answer: The expectation theory of the term structure of interest rates tells us that long rates are (approx) a weighted average of current and future expected short rates. R(k,t)= Σ(i=0,k-1) w(i) R(1,t+i) + term premium, where k remaining maturity of the long rate, and w(i) weights for future short rates. Consequently Et R(k,t+1) - R(k,t) = α(k) [R(k,t) - R(1,t)] + constant. Predictable long rates are possible. However, predictability is limited by the size of capital gains caused by interest rate changes. Duration formula: dP/P = - D. dR/(1+R). Too large interest rate changes will generate capital gains that are not consistent with normal expected returns on short investments (the short rate).

4. Exchange rates. Proposition: The dollar/euro exchange rate will be higher next year.

Answer: Nominal exchange rates change consistent with purchasing power parity and interest rate parity. IRP tells us that nominal exchange rate appreciates if foreign interest rates are higher than domestic rates. However, too large changes for short time periods will generate returns that are inconsistent with normal annualized returns.

Conclusion

Predictability in prices/returns is allowed and does not directly violate the EMH, but restrictions apply.

1: Predictability is possible but arbitrage restrictions across markets must be considered. Markets do not exist in a vacuum, markets are generally not segmented.

2: Predictability is limited by the size of (annualized) capital gains. Predictable changes over time cannot be too large.

3: There can be no abnormal returns using ex ante, publicly available, widely known, and transparent information. But different types of “inside information” may exist. Also, abnormal return is a relative concept: relative to other assets (accounting for common changes in time-varying returns), relative to risk (time-varying risk premiums).

Random walks in practice: the dominance of news

There are good reasons to state that “asset prices should approximately follow a random walk in the short run, that is, tomorrow’s change in prices should, for all practical purposes be unpredictable”. (Note the emphasis!)

Remember that expected returns over short time periods are normally extremely small. For example, a 10% annual return equals a 10/365 = 0.03% daily return. These small expected and predictable returns are easily dominated by the much larger effects of news on current prices. However, it should be emphasized that the RW is not a necessary requirement for an efficient market. Neither necessary nor sufficient. Even if prices were shown to be random walks, it does not prove that prices are actually rationally correct. Bubbles can also be random walks. The RW is therefore also not a sufficient condition for an efficient market. Thus, understand this quote: “Of course, as Leroy (1973) and Lucas (1978) have shown, the unforecastability of asset returns is neither a necessary nor a sufficient condition of economic equilibrium. And, in view of the empirical evidence in Lo and MacKinlay (1988b), it is also apparent that historical stock market prices do not follow random walks .” (Lo/MacKinlay, 2001 p.115)