How to Quantify Risk
21 Feb 2025
When an investor invests in an asset, they expect to receive a return on that investment. But the set of possible outcomes is not deterministic: the asset may perform better than expected, or it may perform worse.
We say that an asset is high-risk if the spread of returns is wide. In statistics, this spread is measured by the variance or standard deviation. Thus, one way to estimate the risk of an investment is to collect historical data on past returns, and calculate the variance or standard deviation of these returns.
However, it is important to note that investors are more concerned with the future risk. Thus, although historical data can provide some guidance, we need to make sure that we adjust these estimates to account for the future. For instance, if the economy has just entered a recession, we may expect the future risk to be higher than what the short-term historical data suggests.
When we invest in an individual stock, that stock generates some return. The exact return is uncertain. In other words, there's risk involved.
The same is true when we invest in multiple stocks. Of course, we would get a different return: the amount is simply the weighted average of the returns of the individual stocks. But the story is different when it comes to risk. When we combine multiple stocks in a portfolio, the risk of the portfolio is not the weighted average of the risks of the individual stocks. The reason for this is that the risk of a stock has two components: systematic risk and specific risk. When we combine multiple stocks in a portfolio, the specific risks are diversified away, and we are left with only the systematic risks. Note that this is true only if the stocks are not perfectly positively correlated with each other, but this assumption usually holds in practice when we our portfolio consists of a large number of stocks.
Thus, when we are dealing with a portfolio of stocks, we only need to concern ourselves with systematic risk; all specific risks have been diversified away. This portfolio, which is free of specific risks, we call the market portfolio, because it is representative of the entire market. Now, investors don't usually hold the actual market portfolio. Instead, they use proxies that approximate the market portfolio, such as the S&P 500 index.
The market portfolio is a theoretical construct, and there's more that we can say about it. First, let's assume that investors are rational: they want to maximize their returns while minimizing their risks. Thus, given an investment opportunity set, i.e. a set of all possible portfolios that can be constructed from all available stocks, investors would choose the set of portfolios that generate the highest return for a given level of risk. This set of portfolios has a name: we call it the efficient portfolios. On a graph of expected return against risk, the efficient portfolios lie on a curve called the efficient frontier.
Now, given that investors can choose from the efficient portfolios, their next decision is to decide how much risk they are willing to take. Risk-tolerant investors, or lions, who demand higher returns will choose portfolios that lie at the highest end of the efficient frontier. Risk-averse investors, or chickens, who are willing to accept lower returns will choose portfolios that lie at the lower end of the efficient frontier.
But we can do better. Suppose that, in addition to stocks, investors can also invest in risk-free assets. In this case, the investment opportunity set for the rational investor becomes a straight line that cuts through various points on the efficient frontier. This line is called the capital market line.
Now, investors have two knobs which they can turn and play with. They can either move along the efficient frontiers, by mixing and matching different stocks. Or they can move along the capital market line, by mixing and matching stocks and risk-free assets. In both cases, the goal is the same: maximize returns while minimizing risks. In fact, if we assume that investors are rational, then there is one portfolio that is better than all others, in that it generates the optimal return for a given level of risk. This portfolio is represented by the point where the capital market line is tangent to the efficient frontier. Remember that the graph is a plot of expected return against risk. The slope of any line on this graph represents the return gained per unit of risk. As investors, we want portfolios that lie on the line with the steepest slope, because then we are getting the most return that we can for a given level of risk. The most bang for our buck. And the capital market line is steepest at the point where it is tangent to the efficient frontier.
We have a special name for the portfolio that is on the tangency point between the capital market line and the efficient frontier: the tangent portfolio (unsurprisingly). If we assume that investors are rational, and that they have the same information, and that they can borrow and lend at the risk-free rate, then all investors will hold the same portfolio: the tangent portfolio. Once they hold the tangent portfolio, they can reduce their risk exposure by allocating more to the risk-free asset, or increase their risk exposure by borrowing money to invest in the tangent portfolio.
We have seen that the only risk that matters in a well-diversified portfolio is systematic risk. Whenever we add a stock to our portfolio, we are not adding any specific risk, they're all diversified away. The only risk that the stock contributes is its market risk.
The market risk of a stock is measured by its beta. Beta tells us how much the stock moves in relation to the market. If the market moves up, a high-beta stock moves up, and it moves up more than one-for-one with the market. A low-beta stock also moves up with the market, but less than one-for-one. A stock with a beta of 1 moves up exactly in line with the market, whereas a stock with a beta of 0 moves independently of the market. In theory, betas can be negative. Negative beta stocks move in the opposite direction of the market.
The crucial point is that the beta of a portfolio is the weighted average of the betas of the individual stocks in the portfolio. Note that beta and risk are not the same thing. Beta measures the sensitivity of a stock to the market, whereas risk is a more general concept that describes the spread of possible outcomes. The risk of a portfolio is not the weighted average of the risks of the individual stocks in the portfolio.
CAPM has a simple formula that relates the expected return of a stock to its beta:
ri - rf = βi(rm - rf)
In words, the risk premium that investors expect to earn from holding a stock is proportional to the product of the market risk premium and that stock's beta. CAPM is useful. Investors use it to determine the expected return of a stock, given its beta. Financial managers use it to determine the cost of equity, which is a key factor in evaluating project NPVs.
There are several assumptions behind CAPM: