The arbitrage pricing theory (APT) was developed primarily by **Ross**. It is a one-period model in which every investor believes that the stochastic properties of return of capital assets ate consistently with a factor structure.

If equilibrium prices offer no arbitrage opportunities over the static portfolio of the assets, then the expected returns on the assets covariances with the factors.

The **arbitrage pricing theory** (APT) is a substitute for the capital asset pricing model (CAPM) in that both assert a linear relation between assets expected returns and their covariance with other random variables. (In the CAPM, the covariance is with the market portfolio return).

The covariance is interpreted as a measure of risk that investors cannot avoid by diversification.

The slope coefficient in the linear relation between the expected returns and the covariance is interpreted as a risk premium.

## Risk Factors in Arbitrage Pricing Theory (APT)

Following are the risk factors involved in APT:

### 1. Confidence Risk

Confidence risk in the unanticipated changes in investors’ willingness to undertake relatively risky investments.

It is measured as the difference between the rate of return on relatively risky corporate bonds and the rate of return on government bonds both with 20-year maturities, adjusted so that the mean of the difference is zero over a long historical sample period.

In any month when the return on corporate bonds exceeds the return on government bonds by more than the long-run average, this measure of confidence risk is positive.

*The intuition is that a positive return difference reflects increased investor confidence because the required yield on risky corporate bonds has fallen relative to safe government bonds.*

Stocks that are positively exposed to this then will rise in price. (Most equities to have positive exposure to confidence risk and stocks generally have greater exposure than large stocks).

### 2. Time Horizon Risk

Time horizon risk is the unanticipated change sin investors desired time to payouts.

It is measured as the difference between the return on 20-year government bonds and 30-year treasury bills again adjusted to be mean zero over a long historical sample period.

*A positive realization of the time horizon risk means that the price of long term bonds has risen relative to the 30-day treasury bill price.*

This is signals that investors require lower compensation for holding investments with relatively longer times to payouts.

The **price of stocks** that are positively exposed to the time horizon risk will rise to an appropriate decrease in their yields. (Growth stocks benefits more than income stocks hen this occurs).

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### 3. Inflation Risk

Inflation risk is a combination of the unexpected components of shot and long-run inflation rates.

Exacted future inflation rates are computed at the beginning of each period from available information, historical inflation rates, and other economic variables that influence inflation.

**For any month**, inflation risk is the unexpected surprise that is computed at the end of the month like, it is the difference between the actual inflation for that month and what has been expected at the beginning of the month.

Since most stocks have negative exposures to inflation risk, a positive inflation surprise causes a negative contribution to return whereas a negative surprise (a deflation shock) contributes positively towards return.

### 4. Business Cycle Risk

Business cycle risk represents unanticipated changes in the level of real business activity.

The expected values of a business activity index are computed both at the beginning and end of the month using only information available at those times.

**Then, business cycle risk is calculated as the difference between the end end of month value and the beginning of month value.**

A positive realization of business cycle risk indicates that the expected growth rate of the economy measured in constant price has increased.

**Under such circumstance**s, firms that are more positively exposed to business cycle risk like, firms such as retail stores that do well when business activity increases as the economy recover from a recession, will outperform those such as utility companies that do not respond much to increased levels in business activity.

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### 5. Markets Timing Risk

Market timing risk is computed as that part of the S&P500 total return that is not explained by the first four macroeconomic risks and an intercept term.

Many people find it useful to think of the **APT** as a generalization of the CAPM and by including this market timing factor the CAPM becomes a special case.

If the risk exposure to all of the first four macroeconomic factors was exactly zero, then market timing risk would be proportional to the S&P500 total return.

**Under these extremely unlikely conditions**, a stock’s exposure to market timing risk would is equal to its CAPM beta.

Almost all stocks have positive exposure to market timings risk and hence positive market timing surprises increase returns and vice versa.

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## Assumptions of Arbitrage Pricing Theory (APT)

Following are the important assumptions of arbitrage pricing theory:

- The investors have homogenous expectations.
- The investors are risk-averse and utility maximizes.
- Perfect competition prevails in the market and there is no transaction cost.

**The APT theory does not assume:**

- Single period investment horizon,
- No taxes,
- Investors can borrow and lend at the risk-free rate of interest, and
- The selection of the portfolio is based on the mean and variance analysis.

These assumptions are present in the CAPM.

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