The following table shows the nominal returns on Brazilian stocks and the rate of inflation. The table contains all available data. [table]
Year | Nominal Return (%) | Inflation (%) 2012 | 0.1 | 7.8 2013 | -15.0 | 7.9 2014 | -13.0 | 8.4 2015 | -43.4 | 12.7 2016 | 68.2 | 8.3 2017 | 28.9 | 4.9
[/table] a. What was the standard deviation of the market returns? (Use decimals, not percents, in your calculations. Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) b. Calculate the average real return. (A negative answer should be indicated by a minus sign. Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places)

Question

BlackTom AI · Accepted Answer

Standard deviation: compute the mean nominal return (use decimals). Mean = (0.001 + (−0.15) + (−0.13) + (−0.434) + 0.682 + 0.289)/6 = 0.258/6 = 0.0430 (4.30%).

Step-by-step for the standard deviation (population formula dividing by n, as used in this problem):
1) Compute deviations and squared deviations:
- (0.001 − 0.043)² = (−0.042)² = 0.001764
- (−0.15 − 0.043)² = (−0.193)² = 0.037249
- (−0.13 − 0.043)² = (−0.173)² = 0.029929
- (−0.434 − 0.043)² = (−0.477)² = 0.227529
- (0.682 − 0.043)² = (0.639)² = 0.408321
- (0.289 − 0.043)² = (0.246)² = 0.060516

2) Sum of squared deviations = 0.765308.

3) Variance (population) = sum / n = 0.765308 / 6 = 0.1275513333.

4) Standard deviation = sqrt(0.1275513333) = 0.357141... = 35.7141% → rounded to 2 decimals: 35.71%.

Average real return: use the exact formula r_real = (1 + r_nominal)/(1 + inflation) − 1 = (r_nominal − inflation)/(1 + inflation).

Compute real returns year by year (using decimals):
- 2012: (0.001 − 0.078)/(1.078) = −0.071428571
- 2013: (−0.150 − 0.079)/(1.079) = −229/1079 ≈ −0.212233958
- 2014: (−0.130 − 0.084)/(1.084) = −107/542 ≈ −0.197415781
- 2015: (−0.434 − 0.127)/(1.127) = −561/1127 ≈ −0.497787000
- 2016: (0.682 − 0.083)/(1.083) = 599/1083 ≈ 0.553093000
- 2017: (0.289 − 0.049)/(1.049) = 240/1049 ≈ 0.228789323

Sum of these real returns ≈ −0.196982987. Average real return = sum/6 ≈ −0.032830498 = −3.2830498% → rounded to 2 decimals: −3.28%.

Why the correct option is B and the others are wrong:
- Option A (36.71, −3.38) uses a larger SD and a more negative average real return — likely from different rounding or using sample variance (n−1), which gives a different SD and incorrect rounding for the real average.
- Option C (34.71, −3.18) is too low for SD and too high (less negative) for the real return.
Conclusion: The calculations give standard deviation = 35.71% and average real return = −3.28%, so the correct choice is B (35.71, −3.28).

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