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Consider the following table:
| Stock Fund | Bond Fund | ||
| Scenario | Probability | Rate of Return | Rate of Return |
| Severe recession | 0.10 | −43% | −12% |
| Mild recession | 0.20 | −17.0% | 12% |
| Normal growth | 0.30 | 17% | 6% |
| Boom | 0.40 | 31% | 4% |
a. Calculate the values of mean return and variance for the stock fund. (Do not round intermediate calculations. Round “Mean return” value to 1 decimal place and “Variance” to 4 decimal places.)
b.Calculate the value of the covariance between the stock and bond funds. (Negative value should be indicated by a minus sign. Do not round intermediate calculations. Round your answer to 4 decimal places.)
Covariance is a measurement of the relation of two different variables. It measures how much the variables alter together and how much one variable is the reason for the change in another variables. However, it doesn’t determine the relationship between the of the variables.
The variation could be positive or negative.
- Positive covariance is when variables be towards the exact direction.
- Negative covariance Variables tend to be in opposing directions.
Return on bond funds
Stock Fund
The formula below calculates the expected return.
Expected Return = E[R] = P1×R1 +/or p2×R2 +/or p3×R3+/or p4×R4
Expected Return of Stock Fund = E[RS] = 0.1% (-43%) + 0.0.2× (17%) + 3.3%17% + 4.4%31% = -4.3%+ (-3.4%) + 5.1% + 12.4% = 9.80%
Expected Return or Mean Return of Stock Fund = 9.8%
The formula below calculates variance
Variance = p1×(R1 – E[R]]2)2 + E2× (R2 – E[R]),2 + E3×(R3-E[R]),2 + P3×(R3-E[R]),2 + E4×(R4-E[R]),2
Variance of stock Fund = S2 = 0.0.1×(-43%-9.8%),2 +0.2×(-17%-9.8%),2 +0.3×(17%-9.8%)2+0.4×(31%-9.8%)2 = 0.02278784 +0.0143648 +0.0015552 +0.0179776 = 0.061776
The variance must be calculated in %-squared. To convert it to %-squared, we must multiply it by 10000.
Variance of stock fund in percent-squared = 0.061776×10000 = 617.76
Answer part a
Mean Return (%) = 9.8
Variance (%-Squared) = 617.76
Part b
Expected return = E[RB] = 0.1% (-12%) + 0.0.2×12% + 3.3%6% + 0.4×4% = 1.2% + 2.4% + 1.8% = 4.6%
The following data is available:
p1 = 0.1, p2 = 0.2, p3 = 0.3, p4 = 0.4
R1,S = -43%, R2,S = -17%, R3,S = 17%, R4,S = 31%
R1,B = -12%, R2,B = 12%, R3,B = 6%, R4,B = 4%
E[RS] = 9.8% Expected Return of the Stock Fund, E[RB] = 4.6% Expected Return of the Bond Fund
The formula below calculates the covariance between the bond and stock funds.
Covariance between stock funds and bond funds = Cov(S) = p1× (R1,S) – e[RS ])×(R1,B, – E[RB]), + p2× (R2,S) – e[RS ])×(R2,B, – E[RB]), + p3×[R3,S) – e[RS ])×(R3,B, E[RB]3_R4,B4_R4,B2_R4,B3_R4,B4_R4,B]]]]])
Cov(S,B) = 0.1×(-43% – 9.8%)×(-12% – 4.6%) + 0.2×(-17% – 9.8%)×(12% – 4.6%) + 0.3×(17% – 9.8%)×(6% – 4.6%) + 0.4×(31% – 9.8%)×(4% – 4.6%) = 0.0087648 + (-0.0039664) + 0.0003024 + (-0.0005088) = 0.004592
We need to multiply the covariance by 10000 to convert it to %-squared.
Cov(S),B in %-squared = 0.045292×10000 = 45.92
Answer b -> Covariance (%-Squared) = 45.92