The standard deviation of Exxon Bonds that pay 12% for 50 percent of the time and 8% for the other half of the time would be 2.
The correct answer and explanation is:
Correct Answer:
Standard Deviation = 2%
Explanation (Approx. 300 Words):
To compute the standard deviation of returns, we need to measure how much the returns deviate from the average (mean) return. The standard deviation is a statistical measure of the dispersion or spread of a set of values.
In this case, we are told that the Exxon Bonds yield:
- 12% return 50% of the time
- 8% return 50% of the time
Step 1: Calculate the Mean (Expected Return)
The expected return μ\mu is the weighted average of all possible returns. μ=(0.5×12%)+(0.5×8%)=6%+4%=10%\mu = (0.5 \times 12\%) + (0.5 \times 8\%) = 6\% + 4\% = 10\%
Step 2: Calculate the Variance
Variance is the average of the squared differences from the mean: σ2=0.5×(12%−10%)2+0.5×(8%−10%)2\sigma^2 = 0.5 \times (12\% – 10\%)^2 + 0.5 \times (8\% – 10\%)^2 σ2=0.5×(2%)2+0.5×(−2%)2=0.5×0.0004+0.5×0.0004=0.0002+0.0002=0.0004\sigma^2 = 0.5 \times (2\%)^2 + 0.5 \times (-2\%)^2 = 0.5 \times 0.0004 + 0.5 \times 0.0004 = 0.0002 + 0.0002 = 0.0004
Step 3: Calculate the Standard Deviation
σ=0.0004=0.02 or 2%\sigma = \sqrt{0.0004} = 0.02 \text{ or } 2\%
Interpretation:
The standard deviation of 2% means the returns of Exxon Bonds vary by about 2 percentage points above or below the average return (10%). Since the return alternates between 8% and 12% with equal probability, it creates a symmetrical distribution around the mean. A 2% standard deviation reflects this regular fluctuation and is a relatively low level of risk, indicating the bond’s returns are predictable within a narrow range.
This simplified calculation assumes only two possible outcomes with equal probability, making the math straightforward and suitable for basic investment risk analysis.