One of the fundamental principles in Finance is the relationship between risk and return. In general, it is reasonable to assume that investors are only willing to undertake additional risk if they are adequately compensated with extra return. This idea is rather fundamental considering that risky assets rarely produce their expected rates of return, which makes investors risk averse. This means that if they have to choose between two assets with equal rates of return, they are more likely to choose the asset with the lower level of risk. In other words, investors are willing to maximize their return on investment for a given level of risk.

Risk is an inherent factor of investment returns and is defined as the possibility of not meeting one’s investment objectives because of return uncertainty over time. Risk arises as a result of the volatility of the capital markets that has an impact on asset returns over time. Investment risk may be the result of (1) fluctuations in expected income caused by varying dividends, or missed interest payments, (2) fluctuations in the expected future price of the asset caused by changing economic conditions, and (3) fluctuations in the amount available for re-investment and in returns earned from re-investment caused by changes in tax rates, interest rates or asset returns.

The risk of an asset can be considered either on a stand-alone basis, where the asset’s cash flows are analyzed in isolation or in a portfolio context, where the asset’s cash flows are analyzed in comparison to other asset’s cash flows in the same portfolio.

(a) Stand-alone Risk

To illustrate stand-alone risk, we suppose that an investor buys $100,000 of short-term T-bills with an expected return of 5 percent. In this case, the risk is estimated accurately and therefore the investment is considered as being essentially risk free. If, instead, the $100,000 were invested in the stock market, the expected return would be uncertain and one might analyze the expected rate of return based on statistical evidence, but without being able to precisely predict it. The expected return could even be 20 percent, which would define the investment as risky.

Stand -alone risk is measured in correlation to the probability distribution of expected returns: the tighter the probability distribution of expected returns, the lower the risk of a given investment. To measure the tightness of probability distribution, finance professionals use the standard deviation (σ), a statistical measure of dispersion around a central tendency. The smaller the standard deviation, the less risky is the investment because the dispersion of expected returns is tighter and therefore the investment is less volatile.

Example of measuring risk on a stand-alone basis

The standard deviation (σ) is actually the square root of the variance of the probability distributions (σ^2). To calculate the variance of the probability distributions (σ^2), we need to know the expected rate of return (r), the probability of occurrence (P) for its return, and the deviation (r_{i}) – (r).

We assume that the demand for the products of company X is:

- strong, with a probability of occurrence (P) 0.30 andrate of return on stock (r_{i}) 100%, if this demand occurs,

- normal, with a probability of occurrence (P) 0.40 and rate of return on stock (r_{i}) 15%, if this demand occurs, and

- weak, with a probability of occurrence (P) 0.30 and rate of return on stock (r_{i}) -70%, if this demand occurs

To calculate the expected rate of return (r), we multiply each probability of occurrence (P) with the expected rate of return per demand. That gives:

- For strong demand: 0.30 x 100% = 30%

- For normal demand: 0.40 x 15% = 6%

- For weak demand: 0.30 x (-70%) = – 21%

Summing up the individual expected rates of return, we derive that the expected rate of return (r) = 30% + 6% + (-21%) = 15%.

Then, we need to calculate the deviation (r_{i}) – (r) of the expected rate of return (r) from each possible outcome (r_{i}).

- For strong demand: deviation = (r_{i}) – (r) = 100 – 15 = 85

- For normal demand: deviation = (r_{i}) – (r) = 15 – 15 = 0

- For weak demand: deviation = (r_{i}) – (r) = -70 – 15 = – 85

Then, we square each deviation and we multiply the result by the probability of occurrence (P) for its related outcome and sum up to obtain the variance of the probability distributions (σ^{2}).

σ^2 = (85)^2 x 0.30 + (0)^2 x 0.40 + (-85)^2 x 0.30 = 2,167.5 + 0 + 2,167.5 = 4,335

Consequently, the standard deviation (σ) is the square root of 4,335, which is σ = 65.84%

The standard deviation (σ) provides an indication of how far above or below the expected rate of return the actual return is. Between two investments, the project with the larger standard deviation is considered riskier because it has larger dispersion of expected returns indicating that the actual return may be significantly lower than the expected return.

However, there are cases that investors have to choose between two investments with one having the higher expected returns and the other having the lower standard deviation. In this case, another measure of stand-alone risk is the coefficient of variation (CV), which is the standard deviation (σ) divided by the expected return. The project with the lower coefficient of variation is the less risky.

(b) Portfolio Risk

Investors demand premium for undertaking risk. The higher the risk of an investment, the higher the expected return must be to induce investors to buy or hold a certain security. However, if investors are mostly preoccupied with the risk of their portfolio rather than the risk of their individual securities in the portfolio, then the model used to analyze the relationship between risk and return is the capital asset pricing model (CAPM).

Portfolio risk is associated to the systematic risk, which is measured by the beta coefficient (b) that indicates the volatility of a stock relative to the market. The beta coefficient measures the contribution of a stock to the risk to the portfolio, so basically beta is the theoretically correct measure of a stock’s risk. The market has by default beta equal to 1.0. In a diversified portfolio, stocks with beta = 1.0 are considered as risky as the market; stocks with beta < 1.0 are less volatile than the market; stocks with beta > 1.0 are more volatile than the market.

Measuring risk in a portfolio context

The standard deviation of a portfolio (σ_{p}) is measured by the standard deviations of its returns. Similarly like in the standard deviation of a single asset, to calculate the variance of the probability distributions we need to know the expected rate of return (r_{p}), the probability of occurrence (P) for the portfolio return, and the deviation (r_{pi}) – (r_{p}). Only here, instead of one asset, we calculate the asset as a portfolio of assets.

Two major concepts in portfolio analysis are (1) covariance (Cov) that combines the volatility of a stock’s returns with their tendency to rise or decline while other stocks rise or decline accordingly and (2) correlation coefficient (ρ) that standardizes the covariance.

Overall, the CAPM model is broadly used by financial professionals and investors to analyze the relationship between risk and return. However, CAPM can be expanded with the use of other parameters that are related to stock returns such as a firm’s size and a firm’s market/book ratio. For instance, the multi-beta model, unlike traditional CAPM model, suggests that market risk is measured relative to several risk factors such as inflation, bond default premium and bond term structure premium and not only to market returns. In this context, CAPM model is challenged and the multi-beta model may be the answer to CAPM’s limitations.

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