Exactly how risk is measured is a complicated issue. Before you can begin managing a portfolio you have to look at individual investments. Originally, this task was done using a calculation called the capital asset pricing model (CAPM). The CAPM is now seen more as an unrealistic view of investing, but it’s still valid as a starting point.

The CAPM leads to the use of the arbitrage pricing theory (APT), which is more flexible and has gained more favor as a functional approach to quantifying portfolio management.

## Capital asset pricing model (CAPM)

Take a quick look at the CAPM equation and hopefully that will clear up CAPM and APT a bit.

r_{s}=r_{f}+ β(_{a}r_{m}–r_{f})

where:

r_{s}= the rate of return demanded to invest in a specific asset

r_{f}= the rate of return on risk-free assets

β_{a}= the level of risk on a specific asset

r_{m}= the rate of return on a market portfolio (an investment portfolio that perfectly matches the investment market)

By subtracting the risk-free rate from the market rate of return (which is best accomplished by investing in an index portfolio that matches something like the S&P 500), you’re determining what market premium is being offered for investing in risky assets.

In other words, the market premium is the amount of financial return you can generate by managing a market portfolio of risky assets instead of only risk-free assets. By multiplying the risk of just a specific asset, you’re calculating the risk premium being offered by a specific investment.

Because investors want a return premium that’s higher than the risk-free rate, you add the risk-free rate back into the equation, and you get the rate of returns demanded by investors to entice them to purchase an investment under CAPM.

## What beta measures

The risk symbol beta (β) is not actually a measure of risk. It’s actually a statistical measure of volatility. What that means is that a particular investment can fluctuate far above normal market returns and still be considered high-risk, even though it’s making huge gains compared to the market.

Meanwhile, another investment can be consistently only 0.01 percent under market returns and still be considered low-risk, even though it’s losing value compared to the market. Beta is measured as follows:

β = [Cov(r_{s},r_{m})] / σ^{2}_{m}

Beta measures the amount that the value of an individual asset changes in response to a change in market value. *Cov* is short for covariance and refers to any movement in one variable that is inherently linked to movement in another variable.

Consider stocks for a moment. If the stock market increases by 10 percent and an individual stock increases in value 20 percent in response to the change in value of the stock market, then the beta of that particular stock is 2, or twice as volatile as the stock market itself.

So, beta doesn’t measure actual risk of loss at all, it just measures volatility. Beta is still useful, but only when used properly, which doesn’t include using it to measure risk. What this equation does allow you to do is better understand the movements of an investment.

Figuring out the covariance alone tells you how much volatility in an investment is simply in response to market volatility, and how much of it is unique to that investment (the proportion of variance not accounted for by the covariance is unique to the investment). This knowledge allows you to more accurately predict volatility.

Measures of risk used by *value investors* (investors who focus on finding undervalued and overvalued assets) utilize primarily measures that compare accounting values to market prices. For example, in stocks, measures of risk include the balance sheet and a comparison of the book value to the current market price as well as the book value per share to the market price per share.

When an investment’s price is higher than its value, it’s considered overpriced, making it more likely to lose value. When the investment’s price is lower than its value, it’s considered underpriced. This analysis is done in conjunction with assessments of the quality of the underlying asset of the investment, in order to ensure that the market truly is overpricing or underpricing rather than anticipating qualitative traits, such as amazing management.

## Arbitrage pricing theory

Arbitrage pricing theory (APT) is far more flexible and effective than CAPM. Instead of worrying about returns on a market portfolio, APT looks for differentials in the market price of a single investment and what the market price of the same investment actually should be.

You can actually think of it in terms of volatility measures, similar to the way beta should be used. The expected returns on an investment change in response to other factors and the sensitivity that the investment has to that factor.

When the price of an investment is lower than the price predicted by the model, you should purchase it because the prediction is that it’s undervalued and will generate more value or increase in value in the future. If a price is higher than the price predicted by the model, the investment is considered overvalued and you should sell it.

Use the proceeds of the sale either to purchase an investment whose market to expected price differential is negative (meaning that the market price is below the expected price) or to purchase risk-free investments until you find such an opportunity.

The model itself is very easy to understand and just as easy to customize, which is nice:

r_{s}=r_{f}+ β_{1}r_{1}+ β_{2}r_{2 }+ … β_{n}r_{n}+ ε

where

r_{s}= the return on a specific investment

r_{f}= the return on risk-free investments

β = the change in returns in response to a change in a variable

r= the variable that influences returns on an investment

ε = an error variable that accounts for temporary market deviations and shocks

In CAPM, beta measures the amount of change in value that an investment experiences in response to a change in the market. In APT, beta actually measures something similar: It measures the amount of change in returns caused in response to a change of a particular variable.

That variable can be interest rates, the cost of oil, GDP, annual sales, changes in market value, or anything else that influences the returns on an investment. It’s a little bit like guess-and-check. You try out different factors to determine whether changes in that factor correlate with changes in the value of the returns in an investment. If an investment’s price is low compared to the price estimated by the model, it’s a good investment to make.

Note that this model doesn’t really take risk into consideration at all. That’s because it’s not using measures of probability to determine the value of the investment. Rather, it’s looking for differentials in value of the current market price and the price that the investment should have.

Because that’s the case, the only risk to be concerned with is market risk; the state of being over- or undervalued has already been established rather than relying on probabilities of risk.