Calculate The Portfolio's Expected Return

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? In Capital Market Theory, what is the risk measure for an individual investment? Why is this so? 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

The exploration of portfolio risk and return metrics, along with the implications of market assumptions, is central to understanding modern investment theory. This paper will analyze the expected return, variance, and standard deviation of a given two-asset portfolio, assess the preference of an investor under the Markowitz efficient frontier, examine the possibility of riskless portfolios under certain covariances, discuss risk measurement for individual assets in the Capital Market Theory framework, and evaluate how transaction costs, specifically real estate commissions, alter asset pricing and the security market line (SML).

Introduction

Portfolio theory, pioneered by Harry Markowitz, emphasizes diversification's role in optimizing the trade-off between risk and return. Its core premise is maximizing expected return for a given amount of risk or minimizing risk for a given return, with key metrics being the expected return, variance, and standard deviation. When considering asset combinations, correlation coefficients significantly influence the overall risk profile. Moreover, assumptions in Capital Market Theory, such as no transaction costs, have profound implications on asset pricing models, including the SML.

Calculating Portfolio Expected Return, Variance, and Standard Deviation

Assuming the initial data points for assets A and B—where expected returns are 7% and 50%, with certain correlation coefficients and standard deviations—the expected return of the portfolio (E(Rp)) is computed as a weighted sum of individual expected returns:

E(Rp) = w_A E(R_A) + w_B E(R_B)

Similarly, the portfolio variance (σ²_p) when combining two assets considers individual variances and their covariance:

σ²_p = (w_A)² σ_A² + (w_B)² σ_B² + 2 w_A w_B Cov(A, B)

The standard deviation is the square root of variance:

σ_p = √σ²_p

When the expected returns and standard deviations change (e.g., E(R_A) increases from 7% to 8%, and E(R_B) from 50% to a value b), re-calculating these metrics reflects different risk-return profiles, imperative for making informed investment decisions.

Investor Preference Based on Markowitz’s Efficient Frontier

A Markowitz-efficient investor aims to select portfolios that optimize return for a given level of risk or minimize risk for a specified expected return. Given two portfolios—original and with changed parameters—the investor's preference hinges on the resultant risk-adjusted return. If the altered portfolio offers a higher expected return without a proportional increase in risk, it’s preferred. Conversely, if the change results in increased risk outweighing the benefit, the original portfolio remains more optimal.

Is the Portfolio Now Riskless?

A portfolio is riskless when its variance and standard deviation are zero, implying no variability in returns. This condition is achievable if the covariance between assets perfectly offsets individual risks. Under the new parameters, if covariance turns negative and weights are proportioned such that the total risk cancels out (e.g., a highly negative covariance), the portfolio can approach risklessness. However, unless covariance equals negative the product of standard deviations (indicating perfect negative correlation), the portfolio remains risky.

Risk Measure for an Individual Investment

In Capital Market Theory, the risk measure for an individual security is typically the standard deviation of its returns. This metric quantifies the average variability of returns and serves as a primary indicator of potential investment risk.

Why Standard Deviation Is Used

Standard deviation is preferred because it provides a symmetrical measure of return dispersion around the mean, allowing investors to assess risk irrespective of the return's direction. It facilitates comparison across different securities and portfolios, underpinning the concepts of diversification and risk reduction.

Impact of Transaction Costs on the SML and Asset Pricing

In the real estate market, transaction costs like commissions influence asset pricing by creating a wedge between the observed expected return and the required return based on risk. If only the seller bears a commission (3%), buyers experience a lower net purchase price, effectively reducing the cost of acquiring the asset. This scenario alters the Security Market Line (SML) by shifting the relationship between risk (beta) and expected return. Specifically, the presence of asymmetric transaction costs causes the SML to tilt or shift, reflecting higher or lower risk premiums depending on who bears the cost. For residential real estate, this means that assets with similar risk profiles may have different prices depending on the commission structure, altering the typical linear relationship predicted by the Capital Asset Pricing Model (CAPM). Consequently, the efficient frontier shifts, affecting how investors view risk and return in real estate markets, necessitating adjustments in valuation models and expected returns computation.

Conclusion

The analysis of portfolio risk and return metrics demonstrates that changes in asset parameters can significantly influence optimal investment choices. Understanding the mathematical foundations—expected returns, covariances, and the impact of market assumptions—is crucial for effective portfolio management. Additionally, real-world frictions such as transaction costs, especially in sectors like real estate, modify classical models and necessitate nuanced adjustments. Recognizing these complexities enables investors to better evaluate risk, optimize portfolios, and interpret asset prices realistically in imperfect markets.

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