Value: 10.00 Points Consider The Following Multifactor (APT)
value: 10.00 points Consider the following multifactor (APT) model of security returns for a particular stock
Consider the following multifactor (APT) model of security returns for a particular stock. The model includes three macroeconomic factors with specified factor loadings (betas) and risk premiums. The first part asks for the expected return of the stock assuming it is fairly priced, given the current risk-free rate and the factor risk premiums. The second part asks for the revised expected return when actual factor changes differ from expected changes, introducing surprises in the macroeconomic environment. The goal is to apply the Arbitrage Pricing Theory (APT) framework to calculate these returns, considering the impact of macroeconomic factor changes on the stock's expected return.
Paper For Above instruction
The Arbitrage Pricing Theory (APT) offers a flexible asset-pricing framework that explains security returns based on multiple macroeconomic factors, rather than relying solely on the market portfolio as in the Capital Asset Pricing Model (CAPM). The model posits that the expected return of a security depends on its sensitivities (betas) to various economic factors, each with its associated risk premium. This approach enables investors to understand and quantify how different economic conditions influence asset prices, and it allows for a more nuanced analysis of expected returns and surprises in macroeconomic variables.
Expected Return Calculation under Fair Pricing
In the first part of this problem, the key is to compute the expected return of the stock based on the known factor loadings and risk premiums, complemented by the risk-free rate. The general form of the APT model is:
\[
R_i = R_f + \sum_{j=1}^n \beta_{ij} \times \text{Risk Premium}_j + \epsilon_i
\]
Where:
- \( R_i \) = expected return of the stock
- \( R_f \) = risk-free rate (T-bill rate)
- \( \beta_{ij} \) = sensitivity of stock to factor \( j \)
- \( \text{Risk Premium}_j \) = risk premium for factor \( j \)
- \( \epsilon_i \) = idiosyncratic error term (assumed to be zero for the fair valuation estimate)
Given data:
- Risk-free rate, \( R_f = 4\% \) or 0.04
- Factors:
- Inflation: \( \beta_{inflation} = 0.9 \), Risk premium: 8%
- Industrial production: \( \beta_{prod} = 0.5 \), Risk premium: 11%
- Oil prices: \( \beta_{oil} = 0.2 \), Risk premium: 7%
Calculating the expected return:
\[
R_{expected} = R_f + (\beta_{inflation} \times 8\%) + (\beta_{prod} \times 11\%) + (\beta_{oil} \times 7\%)
\]
\[
R_{expected} = 4\% + (0.9 \times 8\%) + (0.5 \times 11\%) + (0.2 \times 7\%)
\]
\[
R_{expected} = 4\% + 7.2\% + 5.5\% + 1.4\%
\]
\[
R_{expected} = 4\% + 14.1\% = 18.1\%
\]
Thus, the expected rate of return, assuming the stock is fairly priced, is 18.1%.
Revised Expectations Considering Actual Factor Changes
In the second part, the actual macroeconomic environment deviates from expectations. The actual changes in factors are:
- Inflation: expected 7%, actual 5%
- Industrial production: expected 5%, actual 7%
- Oil prices: expected 3%, actual 0%
The surprises are calculated as:
\[
\text{Surprise}_j = \text{Actual}_j - \text{Expected}_j
\]
- Inflation: \( 5\% - 7\% = -2\% \)
- Industrial production: \( 7\% - 5\% = +2\% \)
- Oil prices: \( 0\% - 3\% = -3\% \)
The impact of these surprises on the expected return is:
\[
\Delta R = \beta_{inflation} \times (\text{Actual}_\text{inflation} - \text{Expected}_\text{inflation}) + \beta_{prod} \times (\text{Actual}_\text{industrial} - \text{Expected}_\text{industrial}) + \beta_{oil} \times (\text{Actual}_\text{oil} - \text{Expected}_\text{oil})
\]
\[
\Delta R = 0.9 \times (-2\%) + 0.5 \times (+2\%) + 0.2 \times (-3\%)
\]
\[
\Delta R = -1.8\% + 1\% - 0.6\%
\]
\[
\Delta R = -1.4\%
\]
Therefore, the revised expected return is:
\[
R_{revised} = R_{expected} + \Delta R = 18.1\% - 1.4\% = 16.7\%
\]
Conclusion
The analysis demonstrates the sensitivity of asset returns to macroeconomic surprises, and its importance in portfolio management and risk assessment. The initial expected return of 18.1% reflects the fair valuation based on known risk premiums, while the revised expected return of 16.7% accounts for macroeconomic deviations, emphasizing that unexpected changes in economic factors can significantly alter expected security returns.
References
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