Given the probability distribution for assets $X$ and $Y$, we need to compute the expected rate of return, variance, standard deviation, and coefficient of variation for each asset. Then, we need to determine which asset is a better investment.

1. Problem Description

Given the probability distribution for assets $X$ and $Y$, we need to compute the expected rate of return, variance, standard deviation, and coefficient of variation for each asset. Then, we need to determine which asset is a better investment.

2. Solution Steps

Asset X:

Returns: 8%, 9%, 11%, 12%

Probabilities: 0.10, 0.20, 0.30, 0.40

Expected Return:

$E(X) = (0.08)(0.10) + (0.09)(0.20) + (0.11)(0.30) + (0.12)(0.40)$

$E(X) = 0.008 + 0.018 + 0.033 + 0.048 = 0.107$

$E(X) = 10.7\%$

Variance:

$Var(X) = \sum [x_i - E(X)]^2 * P(x_i)$

$Var(X) = (0.08 - 0.107)^2(0.10) + (0.09 - 0.107)^2(0.20) + (0.11 - 0.107)^2(0.30) + (0.12 - 0.107)^2(0.40)$

$Var(X) = (-0.027)^2(0.10) + (-0.017)^2(0.20) + (0.003)^2(0.30) + (0.013)^2(0.40)$

$Var(X) = (0.000729)(0.10) + (0.000289)(0.20) + (0.000009)(0.30) + (0.000169)(0.40)$

$Var(X) = 0.0000729 + 0.0000578 + 0.0000027 + 0.0000676 = 0.000201$

Standard Deviation:

$SD(X) = \sqrt{Var(X)} = \sqrt{0.000201} \approx 0.014177$

$SD(X) \approx 1.42\%$

Coefficient of Variation:

$CV(X) = \frac{SD(X)}{E(X)} = \frac{0.014177}{0.107} \approx 0.1325$ or 13.25%

Asset Y:

Returns: 10%, 11%, 12%

Probabilities: 0.25, 0.35, 0.40

Expected Return:

$E(Y) = (0.10)(0.25) + (0.11)(0.35) + (0.12)(0.40)$

$E(Y) = 0.025 + 0.0385 + 0.048 = 0.1115$

$E(Y) = 11.15\%$

Variance:

$Var(Y) = \sum [y_i - E(Y)]^2 * P(y_i)$

$Var(Y) = (0.10 - 0.1115)^2(0.25) + (0.11 - 0.1115)^2(0.35) + (0.12 - 0.1115)^2(0.40)$

$Var(Y) = (-0.0115)^2(0.25) + (-0.0015)^2(0.35) + (0.0085)^2(0.40)$

$Var(Y) = (0.00013225)(0.25) + (0.00000225)(0.35) + (0.00007225)(0.40)$

$Var(Y) = 0.0000330625 + 0.0000007875 + 0.0000289 = 0.00006275$

Standard Deviation:

$SD(Y) = \sqrt{Var(Y)} = \sqrt{0.00006275} \approx 0.007921$

$SD(Y) \approx 0.79\%$

Coefficient of Variation:

$CV(Y) = \frac{SD(Y)}{E(Y)} = \frac{0.007921}{0.1115} \approx 0.0710$ or 7.10%

Comparing the Coefficient of Variation:

$CV(X) = 13.25\%$

$CV(Y) = 7.10\%$

Since the coefficient of variation of asset Y is lower than asset X, asset Y is a better investment because it has lower relative risk compared to its expected return.

3. Final Answer

Asset X:

Expected Return = 10.7%

Variance = 0.000201

Standard Deviation = 1.42%

Coefficient of Variation = 13.25%

Asset Y:

Expected Return = 11.15%

Variance = 0.00006275

Standard Deviation = 0.79%

Coefficient of Variation = 7.10%

Asset Y is a better investment.