You Own A Portfolio With 1,300 Invested In Stock A And 21

you Own A Portfolio That Has 1300 Invested In Stock A And 2100 Inv

Analyze the financial scenarios and calculations involved in portfolio management, expected returns, and Capital Asset Pricing Model (CAPM) applications. This includes determining the expected returns on a portfolio with given investments, balancing investments to achieve targeted returns, assessing portfolio risk, calculating beta values of stocks within a portfolio, and deriving market return or risk-free rate based on given data.

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Investment decision-making hinges critically on the understanding of expected returns and risk, central to portfolio management. The initial scenario involves a portfolio comprising investments in Stock A and Stock B, with fixed amounts of $1300 and $2100 respectively. The expected returns on these stocks are 10% and 16%. To determine the expected return on this portfolio, the weighted average method is used, proportional to individual investments relative to the total investment.

Calculating the total investment, we have: $1300 + $2100 = $3400. The weightings for each stock are therefore:

  • Stock A: $1300 / $3400 ≈ 0.3824 (38.24%)
  • Stock B: $2100 / $3400 ≈ 0.6176 (61.76%)

The expected return on the portfolio is then computed as:

Expected Return = (Weight of Stock A × Return of Stock A) + (Weight of Stock B × Return of Stock B)

= 0.3824 × 10% + 0.6176 × 16% = 0.03824 + 0.09882 = 0.13706 or approximately 13.71%.

This expected return indicates the average return anticipated from the investment portfolio, combining the performance expectations of its constituent stocks. It reflects a moderate risk-return tradeoff based on current expected returns.

In the second scenario, an investor has $10,000 to allocate between Stock X with an expected return of 14% and Stock Y with 11%. The goal is to construct a portfolio with an expected return of 12.4%. The question is how much money should be invested in each stock to achieve this target. This involves solving for the weight of Stock X (w) and Stock Y (1 - w).

The expected return formula is:

Expected Return = w × Return of X + (1 - w) × Return of Y = 12.4%

Substituting known values:

12.4% = w × 14% + (1 - w) × 11%

Rewrite as:

0.124 = 0.14w + 0.11 - 0.11w

Combine like terms:

0.124 - 0.11 = (0.14 - 0.11)w

0.014 = 0.03w

Solution:

w = 0.014 / 0.03 ≈ 0.4667 or 46.67%

Therefore, approximately 46.67% of the total $10,000, which is about $4667, should be invested in Stock X, and the remaining $5333 in Stock Y to achieve an expected return of 12.4%.

Moving to the third scenario, which involves calculating the expected return for a portfolio based on the probabilistic outcomes of different economic states. Given the probabilities of recession (30%) and boom (70%), along with their respective returns, we compute the expected return by summing the products of each state’s probability with its return.

If the return in recession is -8%, and in boom is unspecified, suppose the return in a boom is R. Then, the expected return (ER) is:

ER = (Probability of Recession × Return in Recession) + (Probability of Boom × Return in Boom)

= 0.30 × (-0.08) + 0.70 × R = -0.024 + 0.70R

Without the specific return for the boom state, a numerical result cannot be exact. However, if provided, the calculation simply involves substituting R and computing the weighted average.

The subsequent scenario discusses a portfolio diversified across three stocks—G, J, and K—with specific weights and expected returns as 9.2%, 12%, and 15.7%. The expected return of the portfolio is calculated similarly to the previous weighted-average approach:

Expected Return = (WeightG × ReturnG) + (WeightJ × ReturnJ) + (WeightK × ReturnK)


= 0.20 × 9.2% + 0.35 × 12% + 0.45 × 15.7%


= 0.20 × 0.092 + 0.35 × 0.12 + 0.45 × 0.157


= 0.0184 + 0.042 + 0.07065 = 0.13105 or approximately 13.11%

This expected return gives insight into the overall profitability of diversified investment across these stocks, indicating a balanced yet growth-oriented portfolio. It underscores the importance of weightings in shaping the overall expected return based on individual stock prospects.

The fifth scenario involves risk measurement through beta, with a portfolio equally invested in a risk-free asset and two stocks, where one stock has a beta of 1.42, and the total portfolio's risk equals that of the market. Since the portfolio is equally risky as the market, the weighted beta of the overall portfolio is 1.0, and symmetry in weights applies due to equal risk exposure.

Let beta of the other stock be β. The overall beta is:

0.5 × 1.42 + 0.5 × β = 1.0

0.71 + 0.5β = 1.0

0.5β = 0.29

β = 0.58

This indicates the second stock must have a beta of approximately 0.58 to keep the portfolio's risk equivalent to the market, illustrating how individual asset risks influence overall portfolio volatility.

Finally, using the Capital Asset Pricing Model (CAPM) to determine unknown market or risk-free rates involves rearranging the formula:

Expected Return = Risk-Free Rate + Beta × (Market Return - Risk-Free Rate)

For the sixth case, with a stock expected return of 10.9%, a beta of 0.85, and a risk-free rate of 4.6%, solving for the market return (RM) gives:

10.9% = 4.6% + 0.85 × (RM - 4.6%)

0.109 - 0.046 = 0.85(RM - 0.046)

0.063 = 0.85(RM - 0.046)

RM - 0.046 = 0.063 / 0.85 ≈ 0.0741

RM ≈ 0.0741 + 0.046 = 0.1201 or 12.01%

Thus, the expected market return should be approximately 12.01% to justify the stock's expected return given its beta and the risk-free rate.

In the seventh scenario, with a stock expected return of 12.5%, beta of 1.15, and an expected market return of 11.5%, the calculation solves for the risk-free rate:

12.5% = Rf + 1.15 × (11.5% - Rf)

0.125 = Rf + 0.115 × 1.15 - 1.15 Rf

0.125 = Rf + 0.13225 - 1.15 Rf

0.125 - 0.13225 = Rf - 1.15 Rf

-0.00725 = -0.15 Rf

Rf = 0.00725 / 0.15 ≈ 0.0483 or 4.83%

Therefore, the risk-free rate should be approximately 4.83% to reconcile these values within the CAPM framework.

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