8-4 Alternative Theories

The CAPM pictures investors solely concerned with the level of uncertainty of their future wealth, but this is a too simplistic view.

Arbitrage Pricing Theory

Arbitrage Pricing Theory

APT does not ask which portfolios are efficient. It starts assuming that each stock's return depends partly on pervasive macroeconomic influences or factors and partly on noise. The return is assumed to obey the relationship:

$$\text{Return}=a+b_1(r_{factor1})+b_2(r_{f}actor2)+\cdots +noise$$

The theory does not say what the factors are.

a factor could be for example the oil price. the b term is how related the factor is to the stock price.

• coca cola price is less related to oil price than exxon mobile.
  • $b_{cocacola}<b_{exxonmobile}$

finding expected return for a stock.

1. Identify a reasonably short list of macroeconomic factors that could affect the stock's returns.
2. Estimate the risk premium on each of these factors.
3. Measure the sensitivity of each stock to these factors.

Risk in APT

For any individual stock there are two sources of risk:

1. the risk that stems from the factors. (something negative happens that affects the entire market)
   • cannot be eliminated by diversification.
2. risk that stems from company specific events.
   • diversification eliminates this risk.

the expected risk premium on a stock is affected by factor or macroeconomic risk it is not affected by specific risk.

Arbitrage pricing theory states that the expected risk premium on a stock should depend on the expected risk premium associated with each factor and the stock's sensitivity to each of the factors.

$\text{Expected risk premium}=r-r_f=b_1(r_{factor1}-r_f)+b_2(r_{factor2}-r_f)+\dots$

• If you plug in a value of zero for every b in the formula the expected risk premium is 0. A diversified portfolio that is constructed to have zero sensitivity to each macroeconomic factor is essentially risk free ("arbitrage") profit. (either by holding or shorting the portfolio based on it's return)

• A diversified portfolio that is constructed to have exposure to say factor 1, will offer a risk premium, which will vary in direct proportion to the portfolio's sensitivity to that factor.

The Three Factor Model

Three Factor Model

The three factor model is a shortcut on APT.

It already has three factors identified. return formula:

$$r-r_f=b_{market}(r_{marketfactor})+b_{size}(r_{sizefactor})+b_{book-to-market}(r_{book-to-marketfactor})$$

Book to Market

The book to market ratio compares a company's book value to its market value. The book value = value of it's assets - the value of the liabilities. Market value = the market price of one of its shares multiplied by the number of shares outstanding.

$$Book-to-Market=\frac{\text{Total Assets - Total Liabilities}}{\text{Market Cap}}$$

Factor	Measured By	estimated risk premium
Market factor	Return on market index minus risk free interest rate	7%
Size factor	Return on small-firm stocks minus return on large-firm stocks	3.2%
book to market factor	Return on high book to market ratio stocks minus return on low book-to-market stocks	4.9%

finding expected return for a stock.

1. ✅ Identify a reasonably short list of macroeconomic factors that could affect the stock's returns.
2. ✅ Estimate the risk premium on each of these factors.
   • listed in the table, they can be re-calculated with historical data.
   • done by finding the average of the yearly differences
3. Measure the sensitivity of each stock to these factors.
   • Some stocks are more sensitive than others to the three factors.
   • it's up to the investor to find an estimate of these sensitivities.

Examples:

• an increase of 1% in the return on the book-to-market factor reduces computer stocks by 0.21%
  • when value stocks (high book to market) outperform growth stocks (low book to market), computer stocks perform relatively badly.
  • gas and oil stocks do relatively well

this model is not widely used in practice.