GB550 Unit 5 Risk and Return Analysis Example

Volatile but Mostly Diversifiable: A Unit 5 Risk and Return Analysis of a Barge Operator's Shares

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Purdue University Global

GB550: Financial Management

Unit 5 Risk and Return Analysis

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[Date]

The barge operator, its return history and every statistic are composite and illustrative; the models and sources cited are real, and nothing here is investment advice.

Summary Table

Window: sixty monthly returns over five years, the shares against a broad market index.

Mean monthly return: shares 0.92 percent; market 0.81 percent.

Annualized standard deviation: shares 29.8 percent; market 15.6 percent.

Correlation with the market: 0.62.

Beta: 0.62 × 29.8 / 15.6 = 1.18 (1.184 unrounded).

R-squared: 0.62 squared = 0.384, so 38.4 percent of the shares' variance moves with the market.

Residual standard deviation: 29.8 × √(1 − 0.384) = about 23.4 percent.

Standard error of beta: about 0.20, giving a 95 percent band from 0.80 to 1.57.

Adjusted beta: two-thirds of 1.184 plus one-third of 1.0 = 1.12.

What the Numbers Say About Risk

The shares are nearly twice as volatile as the market, but most of that volatility is not related to the market at all. Only 38.4 percent of the variance moves with the index; the rest, a residual deviation near 23.4 percent, comes from events specific to this company: river levels that strand barges, swings in grain exports that change demand for hopper capacity, and the loss or win of single large contracts.

Most of what makes these shares volatile is the river, not the market, and a diversified investor is not paid to bear the river.

An investor holding many stocks sees such company-specific swings offset one another. A drought that hurts this operator barely moves a broad portfolio. The capital asset pricing model therefore pays only for the part of risk that moves with the market, which beta measures (Fama & French, 2004).

A Short Portfolio Illustration

Consider an investor holding 90 percent in the market index and 10 percent in the barge shares. Portfolio variance is 0.81 × 15.6² + 0.01 × 29.8² + 2 × 0.9 × 0.1 × 0.62 × 15.6 × 29.8, which equals about 257.9, so the portfolio's standard deviation is about 16.1 percent. Adding a stock with a 29.8 percent deviation raises the portfolio's risk by only half a point, because its company-specific swings are mostly diluted away and only its market-related part remains.

How Precise Is Beta?

A beta from sixty months is an estimate with error. Its standard error of about 0.20 puts the true figure, with 95 percent confidence, somewhere between 0.80 and 1.57, a range that spans a stock less risky than the market and one considerably more risky. That width is typical for a single company and is one reason analysts often use industry betas or adjusted figures.

Blume (1975) found that betas estimated in one period tend to move toward 1.0 in the next, which is the basis for the common adjustment that weights the raw figure two-thirds and 1.0 one-third. For this company the adjustment turns 1.18 into 1.12. Many data providers report adjusted betas by default, so a paper should say which kind it uses.

Choices Behind the Estimate

Three choices shape the beta, and all three are laid out here for testing. The window is five years of monthly returns, a common standard that balances having enough observations against using data from a company that has since changed; a two-year window of weekly returns would reflect the current fleet more closely but would be noisier. The market index is a broad one rather than a transportation index, because the investor the model describes holds the whole market. And the beta is computed from the company's own returns rather than taken from a data provider; a published figure, cited with its provider and date, would be an equally acceptable choice and a useful check.

The estimate could also change as the business changes. The tank barge contract in Unit 8 adds steady petrochemical revenue less tied to grain exports, which may lower the company-specific part of risk without much effect on beta. More debt, examined in Unit 9, would raise beta, because a larger share of operating swings would fall on shareholders.

Required Return

Under the capital asset pricing model, investors demand a Treasury-like base return plus a reward for market risk scaled by beta. Using a ten-year Treasury yield of 4.4 percent and a market premium of 5.0 percent:

On the unrounded beta of 1.184: 4.4 + 1.184 × 5.0 = 10.32 percent; on the rounded 1.18 carried to Unit 6, 10.30 percent.

On the adjusted beta of 1.12: 4.4 + 1.123 × 5.0 = 10.01 percent.

Both are higher than the 9.8 percent supplied in Unit 4, so the dividend model values there would fall slightly if rerun at these rates, widening the gap to the $22.80 market price.

Limits

The model rests on assumptions Fama and French (2004) show are only partly supported: beta explains less of the variation in returns across stocks than the theory predicts. The premium of 5.0 percent is itself an estimate that varies by source. Brealey et al. (2023) nonetheless note that the model remains the most common way firms estimate the cost of equity, and Unit 6 uses the unadjusted 1.18 for that reason while reporting the adjusted figure beside it.

References

Blume, M. E. (1975). Betas and their regression tendencies. Journal of Finance, 30(3), 785-795. https://doi.org/10.1111/j.1540-6261.1975.tb01850.x

Brealey, R. A., Myers, S. C., & Allen, F. (2023). Principles of corporate finance (14th ed.). McGraw Hill.

Fama, E. F., & French, K. R. (2004). The capital asset pricing model: Theory and evidence. Journal of Economic Perspectives, 18(3), 25-46. https://doi.org/10.1257/0895330042162430