Calculate The Portfolio's Expected Return 016378

1a Given The Following Calculate The Portfolios Expected Return Va

Given the following, calculate the portfolio’s expected return, variance, and standard deviation. Investment E(R) Std Dev Weight Correlation Coefficient A 0....7 B 0...50 Assume the following changes occur. Investment E(R ) Std Dev Weight Correlation Coefficient A 0....8 B 0...b) As a Markowitz-efficient investor, which portfolio would you prefer, the original portfolio or this portfolio, and why? 1c) Once these changes occur, is this portfolio now riskless? Explain.

You have a portfolio of two assets, one with an expected return of 10% and a standard deviation of return of 9%, the other with an expected return of 10% and a standard deviation of return of 8%. Together, they have a covariance of -.0072. They are equally weighted in the portfolio. Is it possible to create a riskless portfolio under these conditions? Why?

Be as thorough as possible. 3a) In Capital Market Theory, what is the risk measure for an individual investment? 3b) Why is this so? 5) In Capital Market Theory, we assumed no transactions costs, which seemed unrealistic. However, there are instances where this assumption is not completely invalid. One such case is residential real estate, where only the seller typically pays a commission for a transaction. Assume the risk-free rate is 6%, sales commissions are 3%, and the SML is a positively-sloped line (i.e., normal). In words, graphs, or both, explain how this type of commission would alter the SML and the pricing of residential real estate assets.

Paper For Above instruction

Introduction

The process of constructing optimal investment portfolios requires an understanding of various financial theories and calculations. These include computing expected returns, variances, and standard deviations, assessing risk and return trade-offs, and understanding the implications of market assumptions and transaction costs. This paper critically examines these core concepts within the framework of modern portfolio theory (MPT), the Capital Asset Pricing Model (CAPM), and practical issues encountered in real estate investments.

Calculating Portfolio Expected Return, Variance, and Standard Deviation

The initial step involves calculating the expected return of a portfolio, which is a weighted average of individual asset returns. Assuming two investments, Asset A and Asset B, with expected returns and associated risk measures, the formula for expected return (E(Rp)) is:

E(Rp) = wA × E(RA) + wB × E(RB)

where wA and wB are the weights of assets A and B respectively. Variance (σ²p) incorporates the individual variances and the covariance between assets, expressed as:

σ²p = wA²σA² + wB²σB² + 2 × wA × wB × Cov(A,B)

The standard deviation (σp) is the square root of variance, representing the portfolio’s total risk.

In the given scenario, initial data, although incomplete, suggest analyzing how expected returns and covariance influence overall risk and return. Changes in the expected returns and correlation coefficients directly impact portfolio efficiency.

Preference Between Original and Modified Portfolios (Markowitz Efficiency)

Markowitz portfolio theory emphasizes selecting portfolios that optimize the trade-off between expected return and risk. An investor’s preference depends on the risk-return profile; a portfolio with higher expected return for the same or lower risk is preferred.

If the modified portfolio exhibits an enhanced expected return with unchanged or reduced variance, it becomes more attractive. Conversely, increased risk without corresponding return gains makes it less desirable. Moreover, the correlation coefficient's role means that a decrease in correlation can significantly reduce portfolio variance, enhancing efficiency.

Thus, a Markowitz-efficient investor would prefer the portfolio that offers the best risk-adjusted return, which could favor the modified portfolio if it improves upon the original.

Is the Portfolio Now Riskless?

A riskless portfolio is one with zero variance, implying no uncertainty in returns. Achieving this requires combining assets such that their variances and covariances cancel out entirely. Mathematically, this occurs when the covariance satisfies specific conditions relative to individual variances and weights.

Given the covariance data, if by adjusting weights or asset combinations, the variance approaches zero, the portfolio could become riskless. However, in practical and most theoretical contexts, perfect risklessness is rare unless assets are perfectly negatively correlated with weights chosen precisely. Therefore, unless the adjusted weights yield zero variance, the portfolio is not riskless.

Creating a Riskless Portfolio with Two Assets

In a scenario where two assets with identical expected returns but differing risks and a specific covariance are combined, the possibility of constructing a riskless portfolio depends on whether a linear combination can eliminate variance.

Given the assets' properties—a risk of 9% and 8%, expected returns, and negative covariance—the formula for riskless portfolios involves solving for weights that nullify the covariance effect. If the variance of the combined assets can be reduced to zero through appropriate weights, the portfolio becomes riskless.

Calculation shows that, with these parameters, it is theoretically possible to find weights that achieve a riskless position, provided the assets' covariance and variances align appropriately. Such a construction demonstrates the importance of diversification and negative covariance in risk management.

Risk Measures in Capital Market Theory

In CAPM, the primary risk measure for an individual investment is beta (β), which quantifies the sensitivity of the asset's returns to overall market movements. Beta assesses systematic risk, reflecting how fluctuations in the market influence individual security prices, unlike unsystematic risk, which can be diversified away.

Beta is derived from regressing an asset's returns against market returns and is fundamental because it links expected returns to market risk via the Security Market Line (SML). Thus, beta provides a standardized measure of risk that enables investors to evaluate whether an asset’s expected return justifies its market-related risks.

Why Beta is the Primary Risk Measure

Beta’s significance stems from the CAPM’s assumption that only systematic risk impacts investment valuation. Unsystematic risk can be diversified away in a well-constructed portfolio, leaving beta as the sole relevant measure of an asset's contribution to overall portfolio risk. Investors seek assets with appropriate betas to match their risk preferences, making beta an essential parameter in asset pricing and portfolio optimization.

Impact of Transaction Costs on the Security Market Line and Asset Pricing in Real Estate

The no transaction costs assumption facilitates the elegant graphical representation of the SML, where securities are priced efficiently according to their systematic risk levels. When transaction costs, such as commissions, are introduced—particularly asymmetric costs like in residential real estate where typically only the seller bears the commission—they distort the trade-off between expected return and risk.

Specifically, a commission effectively increases the overall transaction cost, shifting the effective expected return downward for the asset. For residential real estate, a 3% sales commission reduces the net proceeds for the seller, influencing both the perceived return and the risk-return relationship.

In the context of the SML, these costs would flatten the line, lowering expected returns for given levels of systematic risk because the net returns received are diminished. Graphically, the SML would shift downward or become less steep, reflecting higher costs of trade that inhibit market efficiency. Consequently, buyers and sellers may require higher gross returns to compensate for transaction costs, altering the asset's price and risk profile.

Moreover, in real estate, such commissions introduce a friction that must be considered when applying CAPM assumptions, impacting the theoretical pricing models and leading to potential deviations from idealized predictions.

Conclusion

Understanding the intricacies of portfolio construction and asset valuation involves grappling with the concepts of expected return, risk, covariance, and market assumptions. The theoretical foundation provided by Markowitz, CAPM, and other models offers valuable tools for investors, but real-world factors such as transaction costs and market imperfections require adjustments and nuanced interpretation. Recognizing these constraints and their implications for asset pricing, risk measurement, and portfolio optimization is essential for effective investment decision-making in both traditional securities markets and specialized assets like residential real estate.

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