Why might it be difficult to use past asset returns to predict future returns?
One possible reason is the efficiency of the markets that set asset prices, in the sense that those prices reflect all currently available information; hence, past return data does not include any additional useful information about future returns.
To see how efficient markets affect the properties of asset prices, we use the following simple model. Consider the price of one share of stock in a given company. Let Ωt denote all information available at timet; that is, Ωt consists of the values of all financial variables that have been observed up to and including timet. Because information accumulates over time,
Ω0⊂Ω1⊂. . .⊂Ωt⊂Ωt+1⊂. . .;
that is, the information available at time sis also available at times+hfor anyh= 0,1,2, . . . .
Let V denote a random variable representing the intrinsic value of one share of stock in this company; for instance,V may include properties of the stock such as future dividends and so on. LetPt denote the price of this share at timet,t= 1,2, . . . .Assume that
Pt= E(V|Ωt);
that is, we assume that the price of the stock at timet is the best predictor of the value of the stock based on the information available at that time.
Suppose we are interested in predicting the price of stock at time t+ 1, Pt+1, using the information available at time t. Using the interpretation of a conditional expectation value as the best predictor of a random variable, the best predictor ofPt+1is E(Pt+1|Ωt). BecausePt+1= E(V|Ωt+1), the best predictor ofPt+1based on the information available at timetmay be written E(Pt+1|Ωt) = E(E(V|Ωt+1)|Ωt). (3.1)
Iterated Conditional Expectations
The term on the right-hand side of 3.1 is known as an iterated conditional expectation, that is, the conditional expectation of a conditional expecta- tion. Note that E(V|Ωt+1) is the best predictor ofV using the information in Ωt+1 and E(E(V|Ωt+1)|Ωt) is the best predictor of that predictor using the information in Ωt.
Therefore, the end result, E(E(V|Ωt+1)|Ωt), is a predictor of V based on the information in Ωt calculated using a two-step process. Because the information in Ωtis also in Ωt+1, this final predictor must be at least as good as E(V|Ωt). But we know that E(V|Ωt) is the best predictor ofV using the information in Ωt. Therefore, we expect that
E(E(V|Ωt+1)|Ωt) = E(V|Ωt).
Note that the same argument holds if Ωt+1 is replaced by any set of information that includes Ωt.
The following proposition gives a formal statement of this result. A proof may be based on formalizing the argument described previously; the details are omitted.
Proposition 3.1. For any random variable V and for any information sets ΩtandΩt+h such thatΩt⊂Ωt+h,
E(E(V|Ωt+h)|Ωt) =E(V|Ωt) with probability1.
The Martingale Model
We now apply this result to asset prices. Recall that, according to our model, the price of a stock at timetisPt= E(V|Ωt), whereV represents the intrinsic value of the stock and Ωt is the information available at time t. The best predictor ofPt+1, the price of the stock at time t+ 1, using the information available at timetis E(Pt+1|Ωt); using Proposition 3.1, this may be written
E(Pt+1|Ωt) = E(E(V|Ωt+1)|Ωt) = E(V|Ωt) =Pt. (3.2) That is, the best predictor of tomorrow’s price of the stock is today’s price.
Clearly, the same argument works for any price in the future: for any h= 1,2, . . .,
E(Pt+h|Ωt) = E(E(V|Ωt+h)|Ωt)
= E(V|Ωt) =Pt.
A sequence of random variables P1, P2, . . . with this property is said to be amartingale with respect to Ω1,Ω2, . . . .Therefore, this is known as the martingale model for asset prices.
The martingale model for asset prices has important implications for the properties of the corresponding returns. Let Xt+1=Pt+1−Pt denote the change in the price from timet to timet+ 1 and consider E(Xt+1). Then
E(Xt+1) = E{E(Xt+1|Ωt)}
= E{E(Pt+1−Pt|Ωt)}
= E{E(Pt+1|Ωt)−E(Pt|Ω)}.
Note that Ωt, the information available at timet, includesPt; that is, the random variablePt is a function of the information in Ωt. It follows that
E(Pt|Ωt) =Pt;
furthermore, the result given in (3.2) shows that E(Pt+1|Ωt) =Pt. It follows that
E(Xt+1) = E{E(Pt+1|Ωt)−Pt}
= E(Pt−Pt) = 0.
Thus, price changes have an expected value of 0. Furthermore, the previous argument shows that
E(Xt+1|Ωt) = 0
so that, given the information available at timet, the predicted value of the change in price from timetto timet+ 1 is 0.
Let
Rt+1=Pt+1
Pt −1 = Pt+1−Pt
Pt , t= 0,1,2, . . .
denote the return at timet+ 1. Using the fact thatPtis a function of Ωt, E(Rt+1|Ωt) = E
Pt+1−Pt Pt |Ωt
= 1
PtE(Pt+1−Pt|Ωt) = 0; (3.3) that is, under the martingale model, the best predictor of the return in period t+ 1 using financial information in periods up to and including periodtis zero.
Note that this result also implies that the (unconditional) expected value of Rt+1 is also zero:
E(Rt+1) = E{E(Rt+1|Ωt)}= E(0) = 0.
Furthermore, the following result shows that the correlation of any two returnsRtandRs,t=s, is zero.
Proposition 3.2. Using the framework of this chapter, define the price of an asset by
Pt=E(V|Ωt), t= 0,1, . . . and letR1, R2, . . ., denote the corresponding returns.
Then, under the martingale model, for any t, s= 1,2, . . .,t=s, Cov(Rt, Rs) = 0.
Proof. Without loss of generality, we may assume that t < s. According to 3.3, E(Rt) = E(Rs) = 0 so that
Cov(Rt, Rs) = E(RtRs).
Note that
E(RtRs) = E{E(RtRs|Ωs−1)}
and thatt≤s−1,Rt, and Ps−1 are functions of Ωs−1. It follows that E(RtRs|Ωs−1) =RtE(Rs|Ωs−1) =RtE
Ps−Ps−1 Ps−1 |Ωs−1
=Rt 1
Ps−1E (Ps−Ps−1|Ωs−1)
=Rt 1
Ps−1(Ps−1−Ps−1) = 0,
using the fact that E(Ps|Ωs−1) =Ps−1, establishing the result.