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We have come a long way. We began by defining wealth and income, then traced the origins of money. We learned to value a risk-free project and built a toolkit for measuring risk. We even quantified the cost of risk . Now, we arrive at the culmination of our journey: determining the value of a risky project. This chapter synthesizes everything we have learned. It introduces the crucial distinction between diversifiable and irreducible risk , presents the final formula for Net Present Value under uncertainty, and explores the profound implications of this framework for understanding everything from corporate finance to the value of cryptocurrencies . 7.1 Only Irreducible Risk Has a Cost In Chapter 5, we learned about diversification . By combining projects that are not perfectly correlated, an investor can reduce overall portfolio risk. This process has a profound implication for the valuation of individual projects. Consider two types of risk associated with any project: Diversifia...

In the previous chapter, we developed the tools to calculate and measure risk. We learned about probability distributions , expected values , covariance , and the power of diversification . But we left a critical question unanswered: what is the cost of risk ? We stated that investors are risk-averse and require a risk premium , but we did not quantify it. This chapter tackles that challenge head-on. We will explore how to calculate the cost of risk, examine common mistakes in doing so, and finally, consider the other side of the coin: the potential benefit that risk can bring. 6.1 The Cost of Risk The cost of risk is not a physical cost like a raw material or a wage. It is an opportunity cost. It represents what investors give up, or the extra return they demand, to bear uncertainty. To understand this, imagine two projects: Project A: A risk-free bond that guarantees a return of €100 in one year. Project B: A risky venture with an expected return of €100 in one year, but with a...

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 p...

In the preceding chapters, we built the foundational concepts of wealth, income, and money. We now arrive at the central question of finance: how do we determine what something is worth? Specifically, how do we calculate the value of a project that promises a future return? This chapter tackles the simplest version of this question: valuing a risk-free project . While true risk-free projects are rare in practice, understanding their valuation is essential. It establishes the fundamental principles of the time value of money and provides the benchmark—the risk-free rate —against which all risky investments are measured. 4.1 The Language of Value To begin, we must adopt the precise language of finance. When we speak of the value of a project, we are not guessing its price. We are performing a calculation based on its anticipated financial outcomes. This calculation revolves around two core concepts: costs and benefits. Every project, whether it's building a factory, launching a...