Risk Calculation

In the previous chapter, we mastered the valuation of a risk-free project. We learned to discount future cash flows at the risk-free rate to calculate a Net Present Value. But the real world is not risk-free. The future is uncertain. A project's costs may be higher than expected, its revenues lower, or a competitor may emerge and disrupt the market.

This chapter introduces the tools we need to navigate this uncertainty. We move from the world of certainty to the world of probability. Our goal is to develop a framework for risk calculation—a way to measure, compare, and ultimately combine risky projects. This will lay the groundwork for determining their value in the chapters to come.

5.1 Probabilities in Economics

To calculate risk, we must first quantify uncertainty. This is done through the language of probability. In finance, we rarely know with certainty what the future holds, but we can often assign probabilities to different possible outcomes.

Consider a simple risky project: a startup seeking a €1 million investment. Based on market research, we estimate three possible scenarios:

• Success: A 20% probability of generating €5 million in profit.
• Moderate Success: A 50% probability of generating €1 million in profit.
• Failure: A 30% probability of losing the entire investment (€0 profit).

The sum of all probabilities must equal 100%. This set of possible outcomes, each with its own probability, is called a probability distribution.

From this distribution, we can calculate the most important single measure of a risky project: its expected value. The expected value is not the most likely outcome, but the probability-weighted average of all possible outcomes.

Expected Value = Σ (Probability of Outcome × Value of Outcome)

For our startup:

Expected Value = (0.20 × €5M) + (0.50 × €1M) + (0.30 × €0)

= €1M + €0.5M + €0

= €1.5 million

This tells us that, on average, if this project could be repeated many times, it would yield a profit of €1.5 million. However, it will never actually yield €1.5 million in a single trial. It will yield either €5M, €1M, or €0. This gap between the expected value and the actual possible outcomes is the essence of risk.

5.2 The Compensation of Risks

If a project has an expected value of €1.5 million, does that mean it is worth €1.5 million today? Not if it is risky.

Recall the principle of the time value of money: a euro today is worth more than a euro tomorrow. Similarly, in finance, we have a parallel principle: a certain euro is worth more than an uncertain euro. Investors are generally risk-averse. They dislike uncertainty and require compensation—a premium—for bearing it.

Therefore, the value of a risky project is not simply its expected future value discounted at the risk-free rate. It is its expected future value discounted at a higher rate, one that includes a risk premium.

Project Value = Expected Future Value / (1 + Risk-Free Rate + Risk Premium)

The size of the risk premium depends on the project's riskiness. A highly volatile project with a wide range of possible outcomes will require a larger risk premium than a stable, predictable one. Determining this risk premium is the central challenge of asset pricing.

5.3 Independence, Covariance, and Correlation

So far, we have considered a single project in isolation. But investors rarely hold just one asset. They hold portfolios. To understand the risk of a portfolio, we must understand how the individual projects within it interact.

This is where three statistical concepts become essential.

• Independence: Two projects are independent if the outcome of one has no influence on the outcome of the other. The success of a coffee shop in Paris and the success of a tech startup in Berlin are likely independent events. When risks are independent, they can cancel each other out in a portfolio.
• Covariance: This measures how two projects move together. A positive covariance means they tend to move in the same direction (both do well at the same time, or both do poorly). A negative covariance means they tend to move in opposite directions (when one does well, the other does poorly).
• Correlation: This is a standardized version of covariance. It ranges from +1 to -1.
•   Correlation = +1: Perfect positive correlation. The projects move in perfect lockstep.
•   Correlation = 0: No correlation. The projects are independent.
•   Correlation = -1: Perfect negative correlation. The projects move in exactly opposite directions.

The concept of correlation is the key to diversification. By combining projects that are not perfectly positively correlated, an investor can reduce the overall risk of their portfolio without necessarily reducing the expected return.

5.4 Optimal Projects

With these tools, we can now define what makes a project "optimal." A project is not judged in isolation, but by its contribution to a portfolio.

An optimal project is one that offers the highest possible expected return for a given level of risk, or the lowest possible risk for a given level of expected return. In the language of modern portfolio theory, it lies on the "efficient frontier."

However, a more fundamental criterion is the project's contribution to the diversification of the investor's existing portfolio. A project that has a low, or even negative, correlation with the investor's other assets is extremely valuable. It acts as a hedge, smoothing out the overall portfolio returns and reducing total risk. Such a project might be accepted even if its individual expected return is modest, because its true value lies in its risk-reducing properties.

5.5 Leverage on an Optimal Project

In the previous chapter, we saw how leverage (borrowing) could magnify the returns of a risk-free project. The same principle applies to risky projects, but with a crucial difference: leverage magnifies both returns and risk.

Consider an optimal project with an expected return of 10% and a standard deviation (a measure of risk) of 15%. If an investor can borrow at 3% (the risk-free rate) and invest their own money alongside borrowed funds, the expected return on their equity will increase. However, the risk (standard deviation) of their equity position will also increase, and by a greater proportion.

The mathematics of leverage on a risky project is a powerful but dangerous tool. It can turn a moderately risky project into a highly speculative gamble. This is why leverage is often regulated, particularly in the banking sector, where excessive borrowing can lead to systemic collapse. The optimal use of leverage requires a careful balancing act: using just enough to enhance returns without creating an unacceptable probability of ruin.

5.6 The Composition of Optimal Projects

The ultimate goal of risk calculation is not just to evaluate individual projects, but to assemble a collection of them—a portfolio—that is itself optimal. This is the art and science of portfolio construction.

The key insight, for which Harry Markowitz won the Nobel Prize, is that the risk of a portfolio is not simply the weighted average of the risks of its components. It depends critically on the correlations between them.

The formula for the variance (risk squared) of a two-asset portfolio illustrates this:

Portfolio Variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂

Where:

• w₁ and w₂ are the weights of the two projects in the portfolio.
• σ₁ and σ₂ are the standard deviations (risks) of the two projects.
• ρ₁₂ is the correlation between them.

The last term is crucial. If ρ₁₂ is negative, it reduces the overall portfolio variance. This is the mathematical foundation of diversification. By combining projects that are not perfectly correlated, an investor can create a portfolio that has a lower risk than any of the individual projects within it. The whole is safer than its parts.

5.7 The Correlation Between All Optimal Projects

If every rational investor seeks to hold optimal portfolios, a profound consequence emerges. They will all be buying the same kinds of assets: those that offer the best trade-off between risk and return. This process of collective buying and selling will drive asset prices to a point where, in equilibrium, the expected return of every project is determined not by its own risk in isolation, but by its correlation with the overall market portfolio.

This is the central insight of the Capital Asset Pricing Model (CAPM). A project that has a high correlation with the overall market (high "beta") will be risky for most investors, because it tends to do poorly precisely when the rest of their portfolio is also doing poorly. Investors will demand a high risk premium to hold it.

Conversely, a project that has a low, or negative, correlation with the market is incredibly valuable as a diversifier. Investors will accept a lower expected return for it, bidding up its price. In this way, the market price of risk is established. The risk of an individual project is not its own volatility, but its systematic risk—the risk it contributes to a well-diversified portfolio.

5.8 Risk and Time

Finally, we must integrate our understanding of risk with our earlier understanding of time. In Chapter 4, we discounted future cash flows at the risk-free rate. For risky projects, we must now discount them at a risk-adjusted rate.

This rate, often called the cost of capital, is the sum of the risk-free rate and a risk premium that reflects the project's systematic risk. The further into the future a cash flow is, the more it is discounted, and the more sensitive its present value becomes to the chosen discount rate.

This creates a profound link between risk and time. A small increase in the perceived risk of a long-term project can wipe out a huge amount of its present value. This is why long-term investments, such as infrastructure projects or research and development, are so sensitive to changes in interest rates and market sentiment. They are exposed to both the devaluation of the future (time) and the magnification of uncertainty (risk).

In conclusion, risk calculation is not just about measuring volatility. It is about understanding the relationships between assets, the benefits of diversification, and the market's collective judgment of which risks deserve compensation. Armed with these tools, we are now ready to tackle the ultimate question: how to place a precise value on a risky project.

Go to 👉 Next Page | 👉 Main Page

E-cyclopedia Resources by Kateule Sydney is licensed under CC BY-SA 4.0