Question

year returns x y 1 15% 22% 2 29 30 3 10 10 4 -22 -27 5 10 21 using the returns shown above, calculate the arithmetic average returns, the variances, and the standard deviations for x and y. note: do not round intermediate calculations. enter your average return and standard deviation answers as a percent rounded to 2 decimal places, e.g., 32.16, and round the variance answers to 5 decimal places, e.g., .16161. average returns x % y % variances x y standard deviations x % y %

Explanation:

Step1: Calculate Arithmetic Average for X

The formula for arithmetic average (mean) is $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$. For X, the returns are 15, 29, 10, -22, 10. Sum: $15 + 29 + 10 + (-22) + 10 = 42$. $n = 5$. So $\bar{x}_X = \frac{42}{5} = 8.4\%$.

Step2: Calculate Arithmetic Average for Y

For Y, returns are 22, 30, 10, -27, 21. Sum: $22 + 30 + 10 + (-27) + 21 = 56$. $\bar{y}_Y = \frac{56}{5} = 11.2\%$.

Step3: Calculate Variance for X

Variance formula: $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}$ (sample variance, but since it's returns data, sometimes population variance is used, but let's check the problem. Wait, the problem says "variances" – maybe population? Wait, the formula for population variance is $\frac{\sum (x_i - \bar{x})^2}{n}$. Let's compute each $(x_i - \bar{x})^2$ for X:

- $(15 - 8.4)^2 = 6.6^2 = 43.56$
- $(29 - 8.4)^2 = 20.6^2 = 424.36$
- $(10 - 8.4)^2 = 1.6^2 = 2.56$
- $(-22 - 8.4)^2 = (-30.4)^2 = 924.16$
- $(10 - 8.4)^2 = 2.56$

Sum: $43.56 + 424.36 + 2.56 + 924.16 + 2.56 = 1397.2$. Population variance: $\frac{1397.2}{5} = 279.44$? Wait, no, wait, 15-8.4 is 6.6, squared 43.56; 29-8.4 is 20.6, squared 424.36; 10-8.4 is 1.6, squared 2.56; -22-8.4 is -30.4, squared 924.16; 10-8.4 is 1.6, squared 2.56. Sum: 43.56 + 424.36 = 467.92; +2.56 = 470.48; +924.16 = 1394.64; +2.56 = 1397.2. Then variance (population) is 1397.2 / 5 = 279.44? Wait, but that seems high. Wait, maybe sample variance? Wait, the problem says "variances" – let's check the problem statement again. The problem says "calculate the arithmetic average returns, the variances, and the standard deviations". Let's proceed with population variance (divide by n) first. Wait, but let's recalculate the sum for X's deviations:

Wait, X returns: 15, 29, 10, -22, 10. Mean is 8.4.

15 - 8.4 = 6.6 → 6.6² = 43.56

29 - 8.4 = 20.6 → 20.6² = 424.36

10 - 8.4 = 1.6 → 1.6² = 2.56

-22 - 8.4 = -30.4 → (-30.4)² = 924.16

10 - 8.4 = 1.6 → 1.6² = 2.56

Sum: 43.56 + 424.36 = 467.92; +2.56 = 470.48; +924.16 = 1394.64; +2.56 = 1397.2. Then variance (population) is 1397.2 / 5 = 279.44. Standard deviation is sqrt(279.44) ≈ 16.72%

Wait, but let's check Y.

Step4: Calculate Variance for Y

Y returns: 22, 30, 10, -27, 21. Mean is 11.2.

22 - 11.2 = 10.8 → 10.8² = 116.64

30 - 11.2 = 18.8 → 18.8² = 353.44

10 - 11.2 = -1.2 → (-1.2)² = 1.44

-27 - 11.2 = -38.2 → (-38.2)² = 1459.24

21 - 11.2 = 9.8 → 9.8² = 96.04

Sum: 116.64 + 353.44 = 470.08; +1.44 = 471.52; +1459.24 = 1930.76; +96.04 = 2026.8. Variance (population) is 2026.8 / 5 = 405.36. Standard deviation is sqrt(405.36) = 20.13%

Wait, but let's recheck the mean for X: 15 +29=44, +10=54, -22=32, +10=42. 42/5=8.4. Correct.

For Y: 22+30=52, +10=62, -27=35, +21=56. 56/5=11.2. Correct.

Now, let's check the variance calculations again.

For X:

(15-8.4)^2 = 6.6^2 = 43.56

(29-8.4)^2 = 20.6^2 = 424.36

(10-8.4)^2 = 1.6^2 = 2.56

(-22-8.4)^2 = (-30.4)^2 = 924.16

(10-8.4)^2 = 1.6^2 = 2.56

Sum: 43.56 + 424.36 = 467.92; +2.56 = 470.48; +924.16 = 1394.64; +2.56 = 1397.2. 1397.2 /5 = 279.44. So variance of X is 279.44 (as a decimal, but since returns are in percent, variance is in %², so 279.44%²? Wait, no, the returns are in percentage points, so when we calculate variance, it's (percentage points)^2. So the variance is 279.44 (in %²), and standard deviation is sqrt(279.44) ≈ 16.72%.

For Y:

(22-11.2)^2 = 10.8^2 = 116.64

(30-11.2)^2 = 18.8^2 = 353.44

(10-11.2)^2 = (-1.2)^2 = 1.44

(-27-11.2)^2 = (-38.2)^2 = 1459.24

(21-11.2)^2 = 9.8^2 = 96.04

Sum: 116.64 + 353.44 = 470.08; +1.44 = 471.52; +1459.24 = 1930.76; +96.04 = 2026.8. 2026.8 /5 = 405.36. Standard deviation is sqrt(405.36) = 20.13%.

Wait, but let's check if we should use sample variance (divide by n-1=4). Let's try that. For X: 1397.2 /4 = 349.3. Standard deviation sqrt(349.3) ≈ 18.69%. For Y: 2026.8 /4 = 506.7. Standard deviation sqrt(506.7) ≈ 22.51%. But the problem says "variances" – maybe population variance? Wait, the problem is about returns, maybe it's population. Let's check the problem statement again: "calculate the arithmetic average returns, the variances, and the standard deviations for X and Y". It doesn't specify sample or population, but in finance, sometimes population variance is used for historical returns. Let's proceed with population (divide by n=5).

Now, let's summarize:

For X:

- Average return: $\frac{15 + 29 + 10 + (-22) + 10}{5} = \frac{42}{5} = 8.40\%$
- Variance: $\frac{(15-8.4)^2 + (29-8.4)^2 + (10-8.4)^2 + (-22-8.4)^2 + (10-8.4)^2}{5} = \frac{1397.2}{5} = 279.44000$ (rounded to 5 decimal places: 279.44000)
- Standard deviation: $\sqrt{279.44} = 16.72\%$ (rounded to 2 decimal places)

For Y:

- Average return: $\frac{22 + 30 + 10 + (-27) + 21}{5} = \frac{56}{5} = 11.20\%$
- Variance: $\frac{(22-11.2)^2 + (30-11.2)^2 + (10-11.2)^2 + (-27-11.2)^2 + (21-11.2)^2}{5} = \frac{2026.8}{5} = 405.36000$ (rounded to 5 decimal places: 405.36000)
- Standard deviation: $\sqrt{405.36} = 20.13\%$ (rounded to 2 decimal places)

Wait, but let's check the calculations again for variance:

For X:

(15-8.4) = 6.6 → 6.6² = 43.56

(29-8.4) = 20.6 → 20.6² = 424.36

(10-8.4) = 1.6 → 1.6² = 2.56

(-22-8.4) = -30.4 → (-30.4)² = 924.16

(10-8.4) = 1.6 → 1.6² = 2.56

Sum: 43.56 + 424.36 = 467.92; +2.56 = 470.48; +924.16 = 1394.64; +2.56 = 1397.2. Correct. 1397.2 /5 = 279.44. Correct.

For Y:

(22-11.2) = 10.8 → 10.8² = 116.64

(30-11.2) = 18.8 → 18.8² = 353.44

(10-11.2) = -1.2 → (-1.2)² = 1.44

(-27-11.2) = -38.2 → (-38.2)² = 1459.24

(21-11.2) = 9.8 → 9.8² = 96.04

Sum: 116.64 + 353.44 = 470.08; +1.44 = 471.52; +1459.24 = 1930.76; +96.04 = 2026.8. Correct. 2026.8 /5 = 405.36. Correct.

Standard deviation for X: sqrt(279.44) = 16.72 (since 16.72² = 279.4576, wait, wait, 16.72² is 16.72*16.72 = 279.5584. Wait, that's not 279.44. Wait, I made a mistake here. Wait, 16.7² = 278.89, 16.71² = 279.2241, 16.72²=279.5584. Wait, 279.44 is between 16.71 and 16.72. Wait, sqrt(279.44) = 16.71645... So approximately 16.72 when rounded to two decimal places.

For Y: sqrt(405.36) = 20.1335... Wait, 20.13² = 405.2169, 20.14²=405.6196. So 405.36 is between them. Wait, 20.1335² ≈ 405.36, so 20.13 when rounded to two decimal places.

Wait, but let's recalculate the square roots:

For X: variance is 279.44. sqrt(279.44) = 16.72 (because 16.72 * 16.72 = 279.5584, which is more than 279.44. Wait, no, 16.7 * 16.7 = 278.89, 16.71 * 16.71 = 279.2241, 16.72 * 16.72 = 279.5584. So 279.44 is 279.44 - 279.2241 = 0.2159 above 16.71². The difference between 16.72² and 16.71² is 279.5584 - 279.2241 = 0.3343. So 0.2159 / 0.3343 ≈ 0.646. So sqrt(279.44) ≈ 16.71 + 0.646*0.01 ≈ 16.7165, so 16.72 when rounded to two decimal places.

For Y: variance is 405.36. sqrt(405.36) = 20.1335, which is 20.13 when rounded to two decimal places.

Now, let's confirm the average returns:

X: (15 +29 +10 -22 +10) = 42; 42/5=8.4 → 8.40%

Y: (22 +30 +10 -27 +21)=56; 56/5=11.2 → 11.20%

Variances:

X: 279.44000 (5 decimal places)

Y: 405.36000 (5 decimal places)

Standard deviations:

X: 16.72%

Y: 20.13%

Answer:

For X:

• Average returns: $\boldsymbol{8.40\%}$
• Variance: $\boldsymbol{279.44000}$
• Standard deviation: $\boldsymbol{16.72\%}$

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