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Math  /  Data & Statistics

QuestionQ1: Risk and Reward (7 Marks) A- Ahmed wants to invest in one of these stocks A, B and C, which stock will he choose knowing that Ahmed is a risk-averse investor? \begin{tabular}{|l|l|l|l|} \hline Probability & \multicolumn{1}{|c|}{\begin{tabular}{c} Rate of return \\ A \end{tabular}} & \begin{tabular}{c} Rate of return \\ B \end{tabular} & \begin{tabular}{c} Rate of return \\ C \end{tabular} \\ \hline 0.15 & 0.02 & 0.20 & 0.20 \\ \hline 0.5 & 0.17 & 0.09 & 0.10 \\ \hline 0.35 & 0.08 & 0.02 & 0.08 \\ \hline \end{tabular}
B-Instead of investing in just one stock, Ahmed decided to invest \$10,000 in a portfolio, and was confused between two portfolios X and Y .
Portfolio XX consists of shares AA and BB, (where the investment in stock AA is $2,500\$ 2,500 ) Portfolio Y consists of shares A and C, (where the investment in stock A is $4,000\$ 4,000 ) Knowing that: The correlation coefficient between A and B=0.95\mathrm{B}=0.95 The correlation coefficient between A and C=−1.0\mathrm{C}=-1.0 The correlation coefficient between B and C=−0.5\mathrm{C}=-0.5 Which Portfolio should he choose? Explain why?

Studdy Solution

STEP 1

What is this asking? Which stock is best for a risk-averse investor, and which portfolio is better given some correlations? Watch out! Don't mix up risk aversion with risk-seeking!
A risk-averse investor wants the safest option, not the highest potential return.

STEP 2

1. Calculate expected returns
2. Calculate standard deviations
3. Choose the stock
4. Analyze the portfolios

STEP 3

The expected return is the sum of each possible return multiplied by its probability.
So for stock A: E[A]=(0.15⋅0.02)+(0.5⋅0.17)+(0.35⋅0.08)=0.003+0.085+0.028=0.116 E[A] = (0.15 \cdot 0.02) + (0.5 \cdot 0.17) + (0.35 \cdot 0.08) = 0.003 + 0.085 + 0.028 = \textbf{0.116}

STEP 4

Same idea for stock B: E[B]=(0.15⋅0.20)+(0.5⋅0.09)+(0.35⋅0.02)=0.03+0.045+0.007=0.082 E[B] = (0.15 \cdot 0.20) + (0.5 \cdot 0.09) + (0.35 \cdot 0.02) = 0.03 + 0.045 + 0.007 = \textbf{0.082}

STEP 5

And for stock C: E[C]=(0.15⋅0.20)+(0.5⋅0.10)+(0.35⋅0.08)=0.03+0.05+0.028=0.108 E[C] = (0.15 \cdot 0.20) + (0.5 \cdot 0.10) + (0.35 \cdot 0.08) = 0.03 + 0.05 + 0.028 = \textbf{0.108}

STEP 6

First, calculate the variance: Var[A]=0.15⋅(0.02−0.116)2+0.5⋅(0.17−0.116)2+0.35⋅(0.08−0.116)2 Var[A] = 0.15 \cdot (0.02 - 0.116)^2 + 0.5 \cdot (0.17 - 0.116)^2 + 0.35 \cdot (0.08 - 0.116)^2 Var[A]=0.0013824+0.0014596+0.0004536=0.0032956 Var[A] = 0.0013824 + 0.0014596 + 0.0004536 = 0.0032956 Now, the standard deviation is the square root of the variance: SD[A]=0.0032956≈0.0574 SD[A] = \sqrt{0.0032956} \approx \textbf{0.0574}

STEP 7

Variance of B: Var[B]=0.15⋅(0.20−0.082)2+0.5⋅(0.09−0.082)2+0.35⋅(0.02−0.082)2 Var[B] = 0.15 \cdot (0.20 - 0.082)^2 + 0.5 \cdot (0.09 - 0.082)^2 + 0.35 \cdot (0.02 - 0.082)^2 Var[B]=0.0020904+0.000032+0.0013464=0.0034688 Var[B] = 0.0020904 + 0.000032 + 0.0013464 = 0.0034688 Standard deviation: SD[B]=0.0034688≈0.0589 SD[B] = \sqrt{0.0034688} \approx \textbf{0.0589}

STEP 8

Variance of C: Var[C]=0.15⋅(0.20−0.108)2+0.5⋅(0.10−0.108)2+0.35⋅(0.08−0.108)2 Var[C] = 0.15 \cdot (0.20 - 0.108)^2 + 0.5 \cdot (0.10 - 0.108)^2 + 0.35 \cdot (0.08 - 0.108)^2 Var[C]=0.0012168+0.000032+0.0002688=0.0015176 Var[C] = 0.0012168 + 0.000032 + 0.0002688 = 0.0015176 Standard deviation: SD[C]=0.0015176≈0.0389 SD[C] = \sqrt{0.0015176} \approx \textbf{0.0389}

STEP 9

Stock A has the highest expected return (0.1160.116) and a relatively low standard deviation (0.05740.0574).
Stock C has a lower standard deviation (0.03890.0389), but also a lower expected return (0.1080.108).
Since Ahmed is risk-averse, he'll likely prefer **stock C** for its lower volatility, even though the return is slightly less.

STEP 10

Portfolio X has stocks A and B with a high positive correlation (0.950.95).
This means they tend to move together.
Diversification isn't as effective here.

STEP 11

Portfolio Y has stocks A and C with a perfect negative correlation (−1.0-1.0).
This means they move in opposite directions.
This offers excellent diversification and reduces risk significantly.

STEP 12

Since Ahmed is risk-averse, **Portfolio Y** is the better choice.
The negative correlation between stocks A and C provides a natural hedge, reducing the overall portfolio risk.

STEP 13

Ahmed should choose **stock C** and **Portfolio Y**.

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