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# 金融代写|FINC6009 Portfolio Theory

## FINC6009课程简介

This unit covers several aspects of modern/post-modern portfolio theory. An introduction to mathematical optimisation techniques in the presence of uncertainty is covered and results from modern portfolio theory to the Capital Asset Pricing Model derived. The unit also examines other popular models such as the Arbitrage Pricing Theory and Black-Litterman Model and concludes with some topical examples from industry. There is a degree of mathematical sophistication associated with this unit and consequently, students should be comfortable with a mathematical approach. However, the required mathematical tools are covered in the unit.

## Prerequisites

At the completion of this unit, you should be able to:

• LO1. describe portfolio and investment problems in a mathematical context to enable an advanced approach to portfolio construction.
• LO2. provide a detailed explanation of the mathematical and economic principles of risk and return and demonstrate how that relates to portfolio theory.
• LO3. apply the principles of portfolio theory to solve practical investment problems utilising EXCEL and/or other software packages
• LO4. contrast different approaches to portfolio theory and explain the strengths and weaknesses of the various models covered.

## FINC6009 Portfolio Theory HELP（EXAM HELP， ONLINE TUTOR）

Once portfolio analysis extends beyond the two-asset case, we observed in Chapter Two of the theory text that the data requirements of portfolio analysis become increasingly formidable. If the covariance is used as a measure of the variability of returns, not only do we require estimates for the expected return and the variance for each asset in a portfolio but also estimates for the correlation matrix between the returns on all assets.
In the absence of today’s computer technology and software, the gravity of the problems that confronted 1950s academics and analysts should not be underestimated.

Required:
To prove the point, suppose an insurance company’s equity fund manager wishes to assemble an efficient portfolio from all the companies listed on the Financial Times 30 Share Index.
How many distinct covariance terms would enter the Markowitz variance calculation?
How does this figure compare with a portfolio drawn from the entire FT-SE 100 ?
What about the data requirements for more comprehensive stock exchange indices with which you are familiar, including those outside the UK?

Based upon the pioneering work of Markowitz, we have explained how rational (risk-averse) investors in efficient capital markets, characterised by normal (symmetrical) distribution of returns, who require an optimal portfolio of investments, can maximise their utility with reference to the relationship between expected returns and their dispersion (risk) associated with the covariability of returns within a portfolio.
Any combination of investments produces a trade-off between the two statistical parameters that define a normal distribution: the expected return and standard deviation (risk) associated with the covariability of individual returns. According to Markowitz, an efficient diversified portfolio is one that minimises its standard deviation without compromising the investor’s desired rate of return, or vice versa.

The Markowitz model of portfolio selection is theoretically sound. Unfortunately, even if we substitute the correlation coefficient into the covariance term of the portfolio variance, the mathematical complexity of the variance-covariance matrix calculations associated with multi-asset portfolios still limits its application. So, let us move on to the theoretical resolution of very real practical problems in the following chapters.

So far throughout the Theory and Exercise texts we have referred to wealth maximisation and shareholder wealth maximisation as the normative objectives of investors generally and strategic financial management specifically. If you have downloaded the author’s previous SFM material (2009) from Bookboon or read any other finance texts with which you are familiar, you will appreciate that this translates into security price maximisation based on models of price behaviour (debt as well as equity).

Moreover, you will have noted that all these models are based on the Separation Theorem of Irving Fisher (1930) which underpins the development of Capital Market Theory (CMT). It is a two-period consumption-investment decision model, whereby corporate management and fund managers acting on behalf their clientele can make an investment decision without prior knowledge of their individual risk-return profiles.

The Separation Theorem of John Tobin (1958) is equally significant in the development of CMT and its most recent component, Modern Portfolio Theory (MPT). Based on the pioneering work of Fisher nearly thirty years earlier, he was the first academic to define a portfolio of risky investments that any risk-averse investor would wish to hold without prior knowledge of their risk attitudes (hence the term Separation). However, Tobin’s theorem and the validity of its conclusions (like that of Fisher’s) still depend upon the assumptions of a perfect capital market.
Required:
The perfect capital market assumptions that validate the conclusions of Tobin’s Seperation Theorem (like much else in finance) should be familiar to you. If not: seek them out, jot them down and consider their significance before we continue.

The assumptions of a perfect capital market (like the assumptions of perfect competition in economics) provide a sturdy theoretical framework based on logical reasoning for the derivation of increasingly sophisticated investment and financial models. Perfect markets benchmark our understanding of individual and corporate wealth maximisation strategies, irrespective of risk attitudes and the return trade-off.

• Large numbers of individuals and companies, none of whom is large enough to distort market prices or interest rates by their own action, (think perfect competition).
• All market participants are free to borrow or lend (invest) at the risk-free rate of interest, or to buy and sell financial securities.
• There are no material transaction costs, other than the prevailing market rate of interest, to prevent these actions.
• All investors have free access to financial information relating to all existing and future investment opportunities, including a firm’s projects.
• All investors can invest in other companies of equivalent relative risk, in order to earn their required rare of return.
• The tax system is neutral.

In our theory text we observed how the movement of individual prices, or returns, relative to a market portfolio defined by beta factors can be calculated in several ways, depending on the information available.
Required:

Define the formulae for $\beta$ and explain their inter-relationships.
Summarise the theoretical and practical informational requirements for a beta calculation.
Given the following data set, relating to the risk-return characteristics of Coldplay plc and its market listing, use the appropriate formula to derive the company’s beta factor.
Correlation between the returns for Coldplay plc and the market portfolio
0.80
Standard deviation for Coldplay plc:
0.20
Standard deviation for the market portfolio:
0.15

(1) The beta factor:
We began Part Three of our theory text by defining the relationship between an individual investment’s systematic risk and market risk measured by $\beta_{\mathrm{j}}$, its beta factor (or coefficient).
$$\text { (33) } \beta_{\mathrm{j}}=\frac{\operatorname{COV}(\mathrm{j}, \mathrm{m})}{\operatorname{VAR}(\mathrm{m})}$$
This factor equals the covariance of an investment’s returns, relative to the market portfolio, divided by the variance of that portfolio.

In Chapter Five we also explained that given the relationship between the covariance and the linear correlation coefficient, the covariance term in Equation (33) can be rewritten as:
$$\operatorname{COV}(j, \mathrm{~m})=\operatorname{COR}(j, \mathrm{~m}) \cdot \sigma j \sigma \mathrm{m}$$
So, we can also define a theoretical value for beta as follows:
$$\beta_{\mathrm{j}}=\frac{\operatorname{COR}(\mathrm{j}, \mathrm{m}) \cdot \sigma \mathrm{j} \sigma \mathrm{m}}{\sigma^2(\mathrm{~m})}$$
And simplifying:
$$\beta_j=\frac{\operatorname{COR}(\mathrm{jm}) \mathrm{s} \mathrm{j}}{\sigma(\mathrm{m})}$$
(2) Informational $\beta$ requirements:
If statistical information on the appropriate variance (or standard deviation) and covariance (or correlation coefficient) is readily available, the calculation of beta is extremely straightforward using any of the previous equations. Ideally, $\beta$ should be determined using forecast data (in order to appraise future investments). In its absence, however, an estimator using least-squares linear regression can be used as an approximation. This plots a security’s historical periodic return against the corresponding return for the appropriate market index. Moreover, the media also publishes betas provided by the financial services industry for stock exchange listings
(3) The derivation of $\beta$ for Coldplay plc:
Given the statistical information for Coldplay and the market portfolio (the correlation coefficient, company and market standard deviations, respectively) the beta factor can be defined using Equation (36) as follows:
\begin{aligned} & \text { (36) } \beta_j=\frac{\operatorname{COR}(\mathrm{jm}) \mathrm{s} \mathrm{j}}{\sigma(\mathrm{m})} \ & =0.80 \times 0.20 \ & 0.15 \ & =1.07 \ & \end{aligned}

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