RTF stock is expected to return 10.6 percent if the economy booms and only 4.2 percent if the economy goes into a recessionary period. The probability of a boom is 55 percent while the probability of a recession is 45 percent. What is the standard deviation of the returns on RTF stock?

A) 4.03 percent

B) 2.97 percent

C) 3.18 percent

D) 3.69 percent

E) 5.27 percent

Question

RTF stock is expected to return 10.6 percent if the economy booms and only 4.2 percent if the economy goes into a recessionary period. The probability of a boom is 55 percent while the probability of a recession is 45 percent. What is the standard deviation of the returns on RTF stock?

A) 4.03 percent

B) 2.97 percent

C) 3.18 percent

D) 3.69 percent

E) 5.27 percent

BlackTom AI · Accepted Answer

We start by restating the problem and the given data to keep the calculations clear.
- There are two possible returns for RTF stock: 10.6% (boom) with probability 0.55, and 4.2% (recession) with probability 0.45.
- We are asked to find the standard deviation of the returns.

Option A: 4.03 percent
- To evaluate, we’d need a relatively large dispersion. However, the mean return is not yet determined, and this value is noticeably higher than the spread suggested by the two outcomes. If we compute the actual standard deviation, we typically get a value around 3.2% given these two points, so 4.03% is likely too high.

Option B: 2.97 percent
- This is somewhat lower than the probable answer range. A standard deviation that small would imply less dispersion than what the two outcomes with probabilities imply. Since the outcomes are quite separated (10.6% vs 4.2%), the standard deviation should be above 2.9%, making this a less plausible option.

Option C: 3.18 percent
- Let’s compute precisely to check accuracy. First, compute the expected return (mean): E[R] = 0.55×10.6% + 0.45×4.2% = 5.83% + 1.89% = 7.72%.
- Next, compute E[R^2] using the squared decimal returns: (0.106)^2 = 0.011236 and (0.042)^2 = 0.001764. Then E[R^2] = 0.55×0.011236 + 0.45×0.001764 ≈ 0.0061798 + 0.0007938 = 0.0069736.
- Variance = E[R^2] − (E[R])^2 = 0.0069736 − (0.0772)^2 ≈ 0.0069736 − 0.0059578 = 0.0010158.
- Standard deviation = sqrt(0.0010158) ≈ 0.03187, i.e., about 3.19%. This aligns closely with 3.18%, making this option a good match.

Option D: 3.69 percent
- This would require a larger dispersion than computed. The calculated standard deviation is around 3.19%, so 3.69% overstates the volatility for these two outcomes with the given probabilities.

Option E: 5.27 percent
- This value is clearly too large given only two outcomes that are 6.4 percentage points apart (10.6% vs 4.2%). A standard deviation near 5.3% would imply far more variability than the data supports, so this is not plausible.

In summary, the two-point distribution with 10.6% and 4.2% and their respective probabilities yields a standard deviation about 3.19%, which corresponds to option C. The other options misestimate the spread either by overstating or understating the dispersion given the data, or by miscomputing the mean and squared terms necessary for the variance calculation.

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