Rational Pricing - Pricing Shares

Pricing Shares

The arbitrage pricing theory (APT), a general theory of asset pricing, has become influential in the pricing of shares. APT holds that the expected return of a financial asset, can be modelled as a linear function of various macro-economic factors, where sensitivity to changes in each factor is represented by a factor specific beta coefficient:

where

• is the risky asset's expected return,
• is the risk free rate,
• is the macroeconomic factor,
• is the sensitivity of the asset to factor ,
• and is the risky asset's idiosyncratic random shock with mean zero.

The model derived rate of return will then be used to price the asset correctly – the asset price should equal the expected end of period price discounted at the rate implied by model. If the price diverges, arbitrage should bring it back into line. Here, to perform the arbitrage, the investor “creates” a correctly priced asset (a synthetic asset), a portfolio with the same net-exposure to each of the macroeconomic factors as the mispriced asset but a different expected return. See the arbitrage pricing theory article for detail on the construction of the portfolio. The arbitrageur is then in a position to make a risk free profit as follows:

• Where the asset price is too low, the portfolio should have appreciated at the rate implied by the APT, whereas the mispriced asset would have appreciated at more than this rate. The arbitrageur could therefore:

1. Today: short sell the portfolio and buy the mispriced-asset with the proceeds.
2. At the end of the period: sell the mispriced asset, use the proceeds to buy back the portfolio, and pocket the difference.

• Where the asset price is too high, the portfolio should have appreciated at the rate implied by the APT, whereas the mispriced asset would have appreciated at less than this rate. The arbitrageur could therefore:

1. Today: short sell the mispriced-asset and buy the portfolio with the proceeds.
2. At the end of the period: sell the portfolio, use the proceeds to buy back the mispriced-asset, and pocket the difference.

Note that under "true arbitrage", the investor locks-in a guaranteed payoff, whereas under APT arbitrage, the investor locks-in a positive expected payoff. The APT thus assumes "arbitrage in expectations" — i.e. that arbitrage by investors will bring asset prices back into line with the returns expected by the model.

The capital asset pricing model (CAPM) is an earlier, (more) influential theory on asset pricing. Although based on different assumptions, the CAPM can, in some ways, be considered a "special case" of the APT; specifically, the CAPM's securities market line represents a single-factor model of the asset price, where beta is exposure to changes in value of the market.