INTRODUCTION For a given set of securities, any number of portfolios can be constructed. A rational investor attempts to find the most efficient of these portfolios. The efficiency of each portfolio can be evaluated only in terms of the expected return and risk of the portfolio as such. It is known as “Portfolio Analysis”. It is based on “Mean-Variance” Analysis.

MEAN-VARIANCE ANALYSIS Mean-variance analysis refers to the use of expected returns, variances, and co variances of individual investments to analyze the risk-return tradeoff of combinations (i.e., portfolios) of these assets. Assumptions of mean-variance analysis All investors are risk averse. Investors minimize risk for any given level of expected return, or, stated differently, investors demand additional compensation in exchange for additional risk. Investors may differ in their degree of risk aversion, but the key is that all investors are assumed to be risk averse to some degree.

MEAN-VARIANCE ANALYSIS Expected returns, variances, and co variances are known for all assets. Investors know the future values of these parameters. Investors create optimal portfolios by relying solely on expected returns, variances, and co variances. No other distributional parameter is used. For example, often returns are assumed to follow a normal distribution in which skew ness and kurtosis can be ignored. Investors face no taxes or transaction costs. Therefore, there is no difference between before-tax gross returns and after-tax net returns, placing all investors on equal footing. Mean-variance analysis is used to identify optimal or efficient portfolios. Before discussing the implications of efficient portfolios, we must first be able to understand and calculate portfolio expected returns and standard deviations.

Expected Return for an Individual Security The expected rate of return from exceptional data (probability model) for a single risky asset can be calculated by using the following formula: Where, E(R) = Expected return from the stock R i =Return from stock under state “i” P i =Probability that the state “i” occurs n= Number of possible states of the world

State of the worldProbability (P i )Return (R i ) Expansion % Normal % Recession % Example: Expected Return for an Individual Security

State of the world Probability (P i )Return (R i )Expected Return (P i R i ) Expansion %(0.25)(5.00)=1.25% Normal %(0.50)(15.0)=7.50% Recession %(0.25)(25.00)=6.25% Expected Return =( )%=15.00% Solution: Expected Return for an Individual Security

Expected Portfolio Return The expected return on a portfolio of assets is simply the weighted average of the returns on the individual assets, using their portfolio weights. The general formula for the expected return on a portfolio of N assets with returns R i and weights w i are Where, E(R p )=Expected return on the portfolio W i =Weight of security “I” in the portfolio E(R i )=Expected return on security “i” N=Number of securities in the portfolio

Portfolio Risk The total risk associated with owning portfolio, the sum of systematic and unsystematic risk. Unsystematic risk may be significantly reduced through diversification. Diversification: The process of accumulating different securities from different industries The variance or standard deviation of an individual security measures the riskiness of a security in absolute sense. For calculating the risk of a portfolio of securities, the riskiness of each security within the context of the overall portfolio has to be considered. This depends on their interactive risk, i.e. how the returns of a security move with the returns of other securities in the portfolio and contribute to the overall risk of the portfolio. Covariance measures the same.

Expected Return and Standard Deviation for a Two-asset Portfolio The weights (w 1 and w 2 ) must sum to 100% for a two-asset portfolio. Expected Return

Expected Return and Standard Deviation for a Two-asset Portfolio Variance The covariance, Cov 1, 2 measures the strength of the relationship between the returns earned on assets 1 and 2. The covariance is unbounded (ranges from negative infinity to positive infinity), and, therefore, is not a very useful measure of the strength of the relationship between two asset's returns. Instead, we often scale the covariance by the standard deviations of the two assets to derive the correlation, ρ 1, 2.

Expected Return and Standard Deviation for a Two-asset Portfolio The covariance equals ρ 1, 2 σ 1 σ 2 where ρ 1, 2 is the correlation of returns between the two assets. Therefore, the variance of the two-asset portfolio can be written:

Example: Expected Return and Standard Deviation for a Two-asset Portfolio Characteristics for a Two-Stock Portfolio Tata Steel ACC Amount Invested Expected Return11%25% Standard Deviation15%20% Correlation 0.30

Example: Expected Return and Standard Deviation for a Two-asset Portfolio Portfolio Returns for Various Weights of Two Assets wTwT %80.00%60.00%40.00%20.00%0.00% wAwA 20.00%40.00%60.00%80.00%100.00% RPRP 11.00%25.00% σPσP 15.00%20.00%

Example: Expected Return and Standard Deviation for a Two-asset Portfolio Portfolio Returns for Various Weights of Two Assets wTwT %80.00%60.00%40.00%20.00%0.00% wAwA 20.00%40.00%60.00%80.00%100.00% RPRP 11.00%13.80%16.60%19.40%22.20%25.00% σPσP 15.00%13.74%13.72%14.94%17.10%20.00%

Example: Expected Return and Standard Deviation for a Two-asset Portfolio Expected Return and Standard Deviation Combinations

Expected Return and Standard Deviation for a Three-asset Portfolio

Example: Expected Return and Standard Deviation for a Three-asset Portfolio Tata SteelACCBHEL Amount Invested Expected Return11%25%30% Standard Deviation15%20%25% Correlation Tata Steel and ACC0.3 Tata Steel and BHEL0.1 ACC and BHEL0.5

Factors affecting Return and Risk of a Portfolio The return and risk of a portfolio depends on two sets of factors  The returns and risks of individual securities and the covariance between securities in the portfolio  The proportion of investment in each security The first set of factors is parametric to the investor in the sense that he has no control over the returns, risks and co variances of individual securities The second set of factors are choice variables in the sense that the investor can choose the proportions of each security in the portfolio.

Expected Return / Variance Combinations PortfolioExpected ReturnVariance A B C D E F G

CorrelationDS % AllocationDB% AllocationE(R P )σPσP

22 Effects of Correlation on Portfolio Risk

23 Effect of Number of Assets on Diversification

How does diversification reduce risk? Mr. Desouza instead of put his money in a single company choose to invest equally in shares of two companies as per below. Moonlight Ltd. a manufacturer of sunglasses and Varsha Limited a manufacturer of rain coats. If the monsoons are above average in a particular year, the earnings of Varsha Ltd. would be up leading to an increase in its share price and returns to shareholders. On the other hand, the earnings of Moonlight Ltd. would be on the decline, leading to a corresponding decline in the share prices and investor’s returns. If there is a prolonged summer the situation would be just the opposite. The table below gives the returns on the two stocks on three weather conditions:

The formula for the variance of an n-asset portfolio is very complex, but the formula is simplified dramatically for equally-weighted portfolios (e.g., each w = 1/n): Calculating the variance for an equally-weighted portfolio Consider two equally-weighted portfolios, A and B, in which the average asset variance equals 0.40 and the average covariance equals Portfolio A comprises three assets, and Portfolio B comprises 100 assets. Calculate the variance of each portfolio. 25

Example: Diversification Weather Conditions Return on Moonlight Stock-% Return on Varsha Stock % Return on Portfolio (50% Moonlight + 50% Varsha) % Rainy02010 Normal10 Sunny20010 The portfolio earns 10% no matter what the weather is because of diversification.

Strategies of Diversification Portfolio diversification refers to the strategy of reducing risk by combining many different types of assets into a portfolio. Portfolio variance falls as more assets are added to the portfolio because not all asset prices move in the same direction at the same time. Therefore, portfolio diversification is affected by the: Correlations between assets: Lower correlation means greater diversification benefits. Number of assets included in the portfolio: More assets mean greater diversification benefits.

In case of perfectly positive correlated (+1) securities, the risk of the portfolio will not be reduced below the risk of the least risky investment in the portfolio. Here diversification provides only risk averaging. In case of perfectly negative correlated (-1) securities, portfolio risk can be considerably reduced and sometimes even eliminated. But, in reality, it is rare to find securities that are perfectly negatively correlated. In case of uncorrelated (0) securities, diversification reduces risk and is a productive activity. Lower the correlation of the securities in the portfolio, less risky the portfolio will be. As correlation coefficient declines from +1 to -1, the portfolio risk also declines. Correlation and Diversification Advantage

An investor can make the portfolio risk arbitrarily small by including a large number of assets with negative or zero correlation in the portfolio. But, in reality, no assets show negative or even zero correlation. As a result, adding assets to a portfolio results in some reduction in total portfolio risk but not in complete elimination of risk. Most of the reduction in portfolio risk occurs by the time the portfolio size increases to 25 or 30 assets. Portfolio with More Than Two Assets

Number of Assets in a Portfolio and the Standard Deviation of Portfolio Return Standard Deviation of Return Number of Assets in the Portfolio Standard Deviation of the Market Portfolio (systematic risk) Systematic Risk Total Risk Unsystematic (diversifiable) Risk

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Minimum variance portfolios offer the minimum variance for each level of expected return 3232 An efficient portfolio offers the highest expected return for a given level of risk Efficient portfolios are a sub-set of minimum variance portfolios

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3 pair-wise correlations… 1 0

n stocks; equal weight in each sock Use average variance and average covariance Assuming: Equal weight Socks have the same variance Stocks have same pair-wise correlation we get…

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Assumptions:

Portfolio A is better than B if… 4343 The Sharpe ratio again…

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