Question

Apparently, Simon Hoyle's article did not mention what would happen to the risk if an investor decided to buy more than one share. Explain how adding new shares to a portfolio can affect the risk and return of that portfolio. You should use the concepts of correlation coefficient and the standard deviation in your explanations.

Answer 1

Risk of a portfolio is often regarded as the standard deviation of that portfolio.

Portfolio risks can be calculated, like calculating the risk of single investments, by taking the standard deviation of the variance of actual returns of the portfolio over time. This variability of returns is commensurate with the portfolio's risk, and this risk can be quantified by calculating the standard deviation of this variability. Standard deviation, as applied to investment returns, is a quantitative statistical measure of the variation of specific returns to the average of those returns. One standard deviation is equal to the average deviation of the sample.

Standard Deviation Formula for Portfolio Returns

s = Standard Deviation rk = Specific Return rexpected = Expected Return n = Number of Returns (sample size) n – 1 = number of degrees of freedom, which, in statistics, is used for small sample sizes

Therefore, the more the standard deviation, the higher the risk.

Covariance is a statistical measure of how 1 investment moves in relation to another. If 2 investments tend to be up or down during the same time periods, then they have positive covariance. If the highs and lows of 1 investment move in perfect coincidence to that of another investment, then the 2 investments have perfect positive covariance. If 1 investment tends to be up while the other is down, then they have negative covariance. If the high of 1 investment coincides with the low of the other, then the 2 investments have perfect negative covariance. The risk of a portfolio composed of these assets can be reduced to zero. If there is no discernible pattern to the up and down cycles of 1 investment compared to another, then the 2 investments have no covariance.

A perfect positive correlation means that the correlation coefficient is exactly 1. This implies that as one security moves, either up or down, the other security moves in lockstep, in the same direction. A perfect negative correlation means that two assets move in opposite directions, while a zero correlation implies no relationship at all.

For example, large-cap mutual funds generally have a high positive correlation to the Standard and Poor's (S&P) 500 Index - very close to 1. Small-cap stocks have a positive correlation to that same index, but it is not as high - generally around 0.8.

However, put option prices and their underlying stock prices will tend to have a negative correlation. As the stock price increases, the put option prices go down. This is a direct and high-magnitude negative correlation.

The coefficient of correlation rxy between two variables x and y, for the bivariate dataset (xi,yi) where i = 1,2,3…..N; is given by –

r(x,y)=cov(x,y)/σx*σy

where,

⇒ cov(x,y): the covariance between x and y