whole world, in fact – at both short- and long-term time scales. That the efficient market hypothesis is able to say anything universal about valuation is in fact quite remarkable. And it does it by ignoring as much of the particulars as possible.
The Risk of Stocks
The most important feature of a stock is the uncertainty of its returns. One of the simplest models of uncertainty is the risk involved in flipping a coin. Figure 2.1 illustrates a similarly simple model, a binomial tree, for the evolution of the return on a stock with return volatility σ and expected return μ over a small instant of time Δt.4 The mean return during this time is μΔt, with a 50 % probability that the return will be higher, μΔt + σ
Figure 2.1 A Binomial Tree for the Future Returns of a Stock
The volatility σ is a measure of the stock's risk. If σ is large, then the difference between an up-move and a down-move will be significant.
This simple model turns out to be extremely powerful. By adding more steps, as in Figure 2.2, and shrinking the size of Δt, we can mimic the more or less continuous motion of prices, much as movies produce the illusion of real motion by changing images at the rate of 24 frames per second. Assuming successive returns are uncorrelated with each other, in the limit as Δt → 0, the distribution of returns at time t becomes normally distributed with mean total return μt and standard deviation of returns σ
Figure 2.2 Binomial Tree of Returns with Four Steps
Figure 2.3 Examples of Normal Distributions
The key feature of this model of risky securities is that the entire behavior of the security is captured in just two numbers, the expected return μ and the volatility σ. This assumption, a very strong one, will be used later, in combination with the law of one price, to derive some famous results of neoclassical finance, in particular the capital asset pricing model (CAPM), and later, the famous Black-Scholes-Merton option pricing formula.
The symmetric distribution of our simple model is at odds with the observed return distributions of almost all securities, which are characterized by negatively skewed distributions and fat tails. Nevertheless, the binomial model is a reasonable starting point for modeling risk. Though the actual behavior of securities is more complex and unpredictable, the binomial model provides an easily accessible intuitive and mathematical treatment of risk. Actual risk is wilder than the model and the normal distribution can accommodate. This should never be forgotten. We will investigate some more ambitious models, which go beyond these assumptions, later in this book.
Riskless Bonds
In the binomial model in the limit when σ is zero, the up-move and the down-move are identical, and risk vanishes. We refer to the rate earned by a riskless security as the riskless rate, often denoted by r. The riskless rate is ubiquitous throughout economics and finance and is central to the replication and valuation of options.
Figure 2.4 shows the binomial tree for a riskless security. The two branches of our tree, though we've kept them separate in the drawing, are identical. No matter which branch we take, the end value is the same.
Figure 2.4 Binomial Tree for a Riskless Security
For any risky security, the riskless rate must lie in the zone between the up-return and the down-return. If this were not the case – if, for example, both the up- and down-returns were greater than the riskless return – you could create a portfolio that is long $100 of stock and short $100 of a riskless bond with zero net cost and a paradoxically positive payoff under all future scenarios in the binomial model. Any model with such possibilities is in trouble before it leaves the ground, because it immediately provides an opportunity for a riskless profit, an arbitrage opportunity that violates the principle of no riskless arbitrage.
How do we determine the riskless rate in practice? One possibility is to use the yield of a bond with no risk of default, such as a U.S. Treasury bill, commonly considered to be entirely safe. Rather than talking about borrowing or lending at the riskless rate, in fact, we often talk about buying or selling a riskless bond. The problem of determining the riskless rate is then a problem of defining and then finding a riskless bond. While this may sound simple, in practice agreeing on what number to use for the riskless rate can become complicated, especially in crisis-ridden markets. Here we will simply assume the riskless rate is known.
The Key Question of Investing
We never know what the future holds. An extremely important question in life as well as in finance is how to act in the face of risk or uncertainty. In finance, as outlined in the previous section, we think about securities in terms of their anticipated risk and return. The key question of investing can therefore be stated as follows:
What anticipated possible future reward justifies a particular anticipated risk?
The law of one price states that securities with identical payoffs under all possible circumstances should have identical prices. For the binomial model described earlier, the payoffs for a security are entirely characterized by its volatility σ and its expected return μ. Within the binomial framework, on which we will focus for now, the key question of finance then becomes:
What is the relation between μ and σ?
To answer this question, we must think more deeply about risk and return.
Some Investment Risks Can Be Avoided
The law of one price states that securities with identical payoffs under all possible circumstances should have identical prices, and therefore identical expected returns. It is tempting to reformulate the law of one price to say that securities with identical risks should have identical expected returns. It's not quite that simple, though. Not all risks are the same. The risk of a security depends on its relation to other securities. Two securities with the same numerical volatility σ might, for example, have different correlations with the Standard & Poor's (S&P) 500 index, and, therefore, when one hedges their exposure to the S&P 500, they would have different risks. In other words, when more than one stock exists, σ alone is not an adequate characterization of risk.
In
