Suppose your expectations regarding the stock price are as follows:

State of the Market

Probability

Ending Price

Holding Period Return

Boom

0.28

$140

-18.5%

Normal Growth

0.40

$110

10.0%

Recession

0.32

$80

26.5%

Compute the standard deviation of the holding period return on the stock.

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Question

Suppose your expectations regarding the stock price are as follows:

State of the Market

Probability

Ending Price

Holding Period Return

Boom

0.28

$140

-18.5%

Normal Growth

0.40

$110

10.0%

Recession

0.32

$80

26.5%

Compute the standard deviation of the holding period return on the stock.

Click to Access Spreadsheet

Q04.xlsx
Download Q04.xlsx

BlackTom AI · Accepted Answer

Question restatement: You are given three market states with associated probabilities and ending prices, and you are asked to compute the standard deviation of the holding period return (HPR) on the stock. The states are Boom (probability 0.28, ending price $140, HPR -18.5%), Normal Growth (probability 0.40, ending price $110, HPR 10.0%), and Recession (probability 0.32, ending price $80, HPR 26.5%). The task is to find the standard deviation of the HPR values, not the price levels.

Option-by-option analysis:

Option A: 17.53%
- Step-by-step reasoning: First compute the expected HPR (the mean) using the probabilities and HPRs: mu = 0.28*(-0.185) + 0.40*(0.10) + 0.32*(0.265) = -0.0518 + 0.04 + 0.0848 = 0.0730, i.e., 7.30%. Next, compute each state’s squared deviation from the mean: for Boom, (-0.185 - 0.073)^2 = (-0.258)^2 = 0.066564; for Normal Growth, (0.10 - 0.073)^2 = 0.027^2 = 0.000729; for Recession, (0.265 - 0.073)^2 = (0.192)^2 = 0.036864. Weight these by their probabilities and sum: 0.28*0.066564 + 0.40*0.000729 + 0.32*0.036864 = 0.0186439 + 0.0002916 + 0.0117965 = 0.030732. The standard deviation is the square root of the variance: sqrt(0.030732) ≈ 0.1753, i.e., about 17.53%. This matches Option A’s numeric value.

Option B: 6.98%
- Why this is unlikely: This would imply a smaller dispersion around the mean than the actual deviations calculated above. The combined squared deviations, especially from the Boom and Recession states, contribute more than would be consistent with 6.98% after weighting, given the computed mean. The arithmetic shown for the variance components leads to a larger value than 0.0698.

Option C: 14.25%
- Why this is unlikely: This value is smaller than the calculated 17.53% based on the provided probabilities and HPRs, again because the variance components (especially the squared deviation for the Boom state and the Recession state) contribute enough to push the standard deviation above 14%. A quick check of the variance components confirms the mismatch.

Option D: 307.26%
- Why this is unlikely: A standard deviation exceeding 300% would require extremely large dispersion or incorrect units. The HPRs themselves are within about -18.5% to +26.5%, and with the given probabilities, the computed variance sits in the low single-digit to tens of percent range; 307.26% is inconsistent with these numbers and results from miscalculation or mix-up of units.

Option E: 10.05%
- Why this is unlikely: This value is closer to the mean and suggests a very small dispersion. However, the actual dispersion, driven by substantial deviations in the Boom and Recession states relative to the mean, yields a larger standard deviation (as shown in the calculations leading to ≈17.53%). The gap to 10.05% indicates an underestimation of the squared deviations or a misapplication of the variance formula.

Overall observation: The correct procedure is to compute the mean HPR from the state probabilities and HPRs, then compute the variance as the probability-weighted average of squared deviations from that mean, and finally take the square root. The numbers given in Options B–E do not align with that calculation, while Option A corresponds to the standard deviation derived from the proper formula and data. Note: If you recompute carefully, you should obtain a standard deviation of approximately 17.53%.

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