Finance Course – Lesson 8: Uncertainty, Mean-Variance (MV) & Coefficient of Variation (CV) Explained

Added: 2026-06-01

[00:00:00]Lesson eight. Investment decisions under uncertainty, risk, expected value, variance, and risk measures. This lesson marks the transition from investment analysis under certainty to analysis under uncertainty. Until now, all cash flows, discount rates, and project durations were known with precision. In reality, future outcomes are uncertain, and this lesson introduces the statistical and financial tools needed to evaluate risky investments. Expected value, variance, standard deviation, the coefficient of variation, the sharp ratio, and the certainty equivalent.

[00:00:34]One.

[00:00:35]From certainty to uncertainty. Under certainty, we knew exactly how much we would invest, how much we would receive, and what the cost of capital was. Using these known inputs, we applied three criteria.

[00:00:47]NPV greater than zero, IRR greater than R, and PI greater than one. Under uncertainty, cash flows are no longer single known values. Instead, each future cash flow can take multiple possible values, each with an associated probability. The question becomes, how do we make investment decisions when outcomes are random? The answer requires two things, a way to summarize the distribution of outcomes, and a way to price risk. Two.

[00:01:18]The risk-return relationship. A fundamental principle of finance is that higher risk demands higher expected return. Consider a spectrum of investments.

[00:01:28]A bank deposit carries virtually no risk and offers a low return. Government bonds carry slightly more risk and offer a slightly higher return. Corporate bonds are riskier and offer higher yields. Stocks are the riskiest and historically provide the highest average returns. This relationship is captured in the cost of capital formula.

[00:01:49]K equals RF plus RP, where K is the cost of capital, RF is the risk-free rate, and RP is the risk premium.

[00:01:58]The additional return investors demand for bearing the project's risk. Higher risk means a higher risk premium, which increases K, which in turn reduces NPV.

[00:02:08]Three, risky versus risk-free assets, a two-state world. To build intuition, consider a simplified world with only two possible states.

[00:02:18]The market goes up or the market goes down. A risk-free asset pays 110 regardless of which state occurs. If it costs 100 today, the risk-free rate is 10%. A risky asset pays 130 if the market goes up and 90 if the market goes down. It's expected payoff is 110, the same as the risk-free asset. However, because its outcome is uncertain, rational investors will pay less than 100 for it, demanding compensation for bearing the risk. Four, expected value. The expected value of a random variable X is the probability weighted average of all possible outcomes.

[00:02:57]E of X equals the sum of each outcome times its probability. For the risky asset with outcomes 130 and 90 at equal probability, E of X equals 130 * 0.5 + 90 * 0.5, which equals 110. Five, variance and standard deviation.

[00:03:18]Variance measures how spread out the outcomes are around the expected value.

[00:03:23]Standard deviation is the square root of variance. Standard deviation is expressed in the same units as the original variable, making it more interpretable. In this course, standard deviation is the primary measure of risk. The rationale, if risk is defined as uncertainty about outcomes, then the wider the distribution of possible results, the riskier the investment. Six, mean-variance preferences. Investors prefer higher expected returns and lower variance. In a mean-variance diagram, the ideal investment sits in the upper left corner, high return, low risk. When one investment has both a higher expected return and a lower variance than another, it dominates. The choice is clear. However, when one investment has a higher return but also higher variance, there is no dominance and additional tools are needed. Seven, coefficient of variation. The coefficient of variation or CV is defined as standard deviation divided by expected value. A lower CV means less risk per unit of expected return, which is preferred. CV is useful when investments have different scales or different expected returns. It normalizes risk relative to return, providing a ratio that enables comparison across very different investments. Eight, Sharpe ratio. The Sharpe ratio is defined as the excess return, the expected return minus the risk-free rate, divided by the standard deviation. A higher Sharpe ratio means more excess return per unit of risk, which is preferred. The Sharpe ratio differs from CV in two key ways. First, it uses excess return rather than total return. Second, the ratio is inverted, return over risk, so a higher Sharpe ratio is better, whereas a lower CV is better. Nine, certainty equivalent. The certainty equivalent or CE is the amount of money an investor would accept with certainty in exchange for a risky gamble. It captures the investor's personal attitude toward risk. The certainty equivalent is always less than or equal to the expected value for a risk-averse investor. The more risk-averse the investor, the lower the CE relative to the expected value. 10, risk premium from the certainty equivalent. The risk premium embedded in an asset can be derived from the certainty equivalent.

[00:05:46]Risk premium equals expected value minus certainty equivalent. If the expected payoff is 110 and the certainty equivalent is 90, the risk premium is 20. This means the investor requires 20 units of extra expected payoff as compensation for bearing the uncertainty. Summary: Risk in finance is measured by variance or standard deviation of returns. The cost of capital equals the risk-free rate plus a risk premium that compensates investors for bearing uncertainty. Expected value and variance are the two summary statistics that characterize a risky investment. When one investment dominates another in both dimensions, the choice is clear. When there is no dominance, the coefficient of variation and the Sharpe ratio provide additional ranking criteria. The certainty equivalent is the certain payment that makes an investor indifferent to a risky gamble, and the difference between the expected value and the certainty equivalent is the risk premium. These concepts form the foundation for all further analysis of investment under uncertainty.