CHAPTER 3: PORTFOLIO RISK AND RETURN FIN 552 INVESTMENT AND PORTFOLIO ANALYSIS

Maryam Q Othman

3.1 WHAT IS PORTFOLIO? It is a grouping of financial assets such as stocks, bonds, commodities, currencies and cash equivalents, as well as their fund counterparts, including mutual fund, exchange-traded fund and closed funds. A portfolio can also consist of non-publicly tradable securities like real estate, art and private investments.

MORKOWITZ PORTFOLIO THEORY • Developed by Harry Markowitz who derived the expected rate of return for a portfolio assets and an expected risk measure. • Showed that the variance of the rate of return was a meaningful measure of portfolio risk under a reasonable set of assumptions: i.

Investors consider each investment alternative as being represented by a probability distribution of expected returns over some holding period.

ii.

Investors maximize one-period utility and their utility curves demonstrate diminishing marginal utility of wealth.

iii.

Investors estimates the risk of the portfolio on the basis of the variability of expected returns.

iv.

Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and expected variance (std dvtn) of return only.

v.

For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected return, investors prefer less risk to more risk.

MORKOWITZ PORTFOLIO THEORY • Markowitz formulae cover: • The measurement of risk on single asset. • The measurement of risk on a portfolio. • Covariance (COV) • Correlation Coefficient (rij)

RISK OF A PORTFOLIO • The risk of any single proposed asset investment should not be viewed independent of other assets (Markowitz). • New investments must be considered in light of their impact on the risk and return of the portfolio of assets. • The financial manager’s goal is to create an efficient portfolio, therefore we need a way to measure a return and risk of a portfolio of assets. • The return on a portfolio is a weighted average of the returns on the individual assets from which it is formed.

EXPECTED RETURN OF A PORTFOLIO Portfolio’s expected return: 𝐸 𝑅𝑝 = ෍ 𝑊𝑗 𝑥 𝑅𝑗 where; 𝑅𝑗 = expected return on each asset (asset j) 𝑊𝑗 = weight/proportion of the portfolio’s total dollar value represented by asset j.

COVARIANCE (COV) Without Probabilities 𝐶𝑂𝑉𝑖,𝑗 σ 𝑅𝑖 − 𝐸 𝑅𝑖 [𝑅𝑗 − 𝐸 𝑅𝑗 ] = 𝑛 Where; Ri = annual return on asset i Rj = annual return on asset j E(Ri) = expected return on asset i E(Rj) = expected return on asset j n = number of observations P = probabilities

With Probabilities 𝐶𝑂𝑉𝑖𝑗

= ෍ 𝑅𝑖 − 𝐸 𝑅𝑖

𝑅𝑗 − 𝐸 𝑅𝑗 (𝑃)

CORRELATION / COEFFICIENT OF CORRELATION (ri,j) • We use the correlation coefficient (ri,j) to obtain a relative measure of a given relationship. • Formally, ri,j is a statistical measure of the relationship if any, between series of numbers representing data of any kind from returns to test score. • It is used to indicate the similarity and dissimilarity in the behaviour of two variables/assets: • Correlation, ri,j; 𝑟𝑖,𝑗

𝐶𝑂𝑉𝑖,𝑗 = 𝜎𝑖 𝜎𝑗

TYPES OF CORRELATION Positively correlated • If two assets move in the same direction

Perfectly positively correlated • Two positively correlated assets that have ri,j of +1.00

Negatively correlated • If two assets move in the opposite direction

Perfectly negatively correlated • Two negatively correlated assets that have ri,j of -1.00

Uncorrelated • Two assets that lack any interaction and therefore have ri,j close to zero or zero.

STANDARD DEVIATION OF A PORTFOLIO Portfolio’s standard deviation: 𝜎𝑝 =

𝑤𝑖 2 𝜎𝑖 2 + 𝑤𝑗 2 𝜎𝑗 2 + 2𝑤𝑖 𝑤𝑗 𝑟𝑖,𝑗 𝜎𝑖 𝜎𝑗

OR 𝜎𝑝 =

𝑤𝑖 2 𝜎𝑖 2 + 𝑤𝑗 2 𝜎𝑗 2 + 2𝑤𝑖 𝑤𝑗 𝐶𝑂𝑉𝑖,𝑗

Where; w = weight/proportion of the portfolio’s total dollar value represented by each asset.

DIVERSIFICATION • The concept of correlation is essential to develop an efficient portfolio. • To reduce overall risk, it is best to combine or add the portfolio, assets that have a negative or low positive correlation. • Combining negatively correlated assets can reduce the overall variability of returns (risk).

• The variability can be eliminated and if not it will at least be reduced. • Combining uncorrelated assets can reduce risk but not as effectively as combining negatively correlated assets but more efficient than combining positively correlated assets.

DIVERSIFICATION Ranking of the combination of assets in a portfolio:

Perfect negative correlation

Negative correlation

Positive correlation

Perfect positive correlation

Uncorrelated

CORRELATION, DIVERSIFICATION, RISK AND RETURN • In general, the lower the correlation between assets returns, the greater the potential diversification of risk (minimize the risk). • For any pair of assets, there is a combination that will result in lowest risk (std dvtn) possible. • This will depend on the ‘degree of correlation’ between those assets.

• Diversification: combining two or more securities in a portfolio (not to increase the return) but to reduce or eliminate the risk. It is all about managing risk.

DETERMINATION OF RIGHT WEIGHT FOR MIN-VARIANCE (RISK) PORTFOLIO 𝜎𝐵 2 − 𝑟𝐴,𝐵 𝜎𝐴 𝜎𝐵 𝑊𝐴 = 2 𝜎𝐴 + 𝜎𝐵 2 − 2𝑟𝐴,𝐵 𝜎𝐴 𝜎𝐵

𝑊𝐵 = 1 − 𝑊𝐴

RISK & RETURN IN A THREE OR MORE SECURITY Portfolio’s expected return: 𝑅𝑝 = ෍(𝑊𝑗 𝑥 𝑅𝑗 )

Portfolio’s standard deviation: 𝜎𝑝 =

(𝑊𝐴 2 𝜎𝐴 2 + 𝑊𝐵 2 𝜎𝐵 2 + 𝑊𝐶 2 𝜎𝐶 2 + 2𝑊𝐴 𝑊𝐵 𝑟𝐴,𝐵 𝜎𝐴 𝜎𝐵 +

2𝑊𝐵 𝑊𝐶 𝑟𝐵,𝐶 𝜎𝐵 𝜎𝐶 + 2𝑊𝐴 𝑊𝐶 𝑟𝐴,𝐶 𝜎𝐴 𝜎𝐶 )

RISK & RETURN IN THE N-SECURITY PORTFOLIO • Most real-world portfolio-analysis problems involve portfolios larger than three stocks. • To assess the risk of portfolio containing more securities demand for the analysis of the correlation or covariance exists between each pair of the securities. • Example in case of four-portfolio securities; A, B, C and D, the possible two-stock combinations are AB, AC, AD, BC, BD, and CD. Here 6 correlations or covariance have to be determined. • According to Markowitz, the number of correlation or covariance can be calculated as follows: Number of COV or Corr =

(𝑁2 −𝑁) 2

THE EFFICIENT FRONTIER & INVESTOR UTILITY • The efficient frontier concept was introduce by Markowitz in 1952 and is a cornerstone of modern portfolio theory. • All portfolio on the efficient frontier is the best portfolio; so then we will discard portfolio below the minimum-variance portfolio. • The slope of the efficient frontier curve decreases steadily as we move upward. • This implies that adding equal increments of risk as we move up the efficient frontier gives the diminishing increments of expected return.

COMBINING A RISK-FREE ASSET WITH A PORTFOLIO • Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets.

SHARPE INDEX MODEL • William Sharpe has developed a simplified variant of the Markowitz model. • According to Sharpe, the fluctuations in the value of a stock relative to that of another, do not depend primarily upon the characteristics of those two securities alone. • Both are more apt to reflect a broader influence that might be described as general business conditions. • Sharpe reduced the number of covariance estimates needed under Markowitz techniques: Markowitz COV =

(𝑁2 −𝑁) 2

Sharpe COV = N

SHARPE INDEX MODEL • According to Sharpe, the index coefficient refer to only N measures of each security as it relates to the index. • In other words, the return on any stock depends upon some constant (α), plus some coefficient (β), times the value of a stock index I, plus a random component (e).

RISK & RETURN MEASUREMENT UNDER SHARPE MODEL Single-Index Model (Single Investment)

Multi-Index Model (Portfolio)

Expected return: E(Ri) = α + βiRm

Expected return: Rp= αp + βpRm Where; αp=Ʃ Wiαi βp=Ʃ Wiβi

Risk: The variance of return on any asset is the total risk. Total risk = systematic risk + unsystematic risk 𝜎𝑖2 = 𝛽𝑖 2 𝜎𝑚 2 + 𝜎𝑖𝑒 2 Where; 𝜎𝑖2 = 𝑎𝑠𝑠𝑒𝑡 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝛽𝑖 = Beta 𝜎𝑚 2 = market variance 𝜎𝑖𝑒 2 = asset i residual variance

Risk of a portfolio: 𝜎𝑝 2 = 𝛽𝑝 2 𝜎𝑚 2 + 𝜎𝑖𝑒𝑝 2 Where; 𝜎𝑖𝑒𝑝 2 =portfolio’s residual variance 𝜎𝑖𝑒𝑝 2 = Ʃ 𝑤𝑖 2 𝜎𝑖𝑒 2

COVij=βiβj 𝜎𝑚 2

𝜎𝑝 =

𝜎𝑝 2