Question

An individual stock has an annualized volatility of sigma. Consider a portfolio of G equally weighted positions.

a. What is the formula for the portfolio volatility assuming the stocks are uncorrelated?

b. what is the formula for the portfolio volatility assuming each pair of unique stocks have a positive correlation of p?

c. what happens to the formula as G --> +infinity?

Answer 1

Ans.

(a)Portfolio variance is a measurement of how the aggregate actual returns of a set of securities making up a portfolio fluctuate over time. This portfolio variance statistic is calculated using the standard deviations of each security in the portfolio as well as the correlations of each security pair in the portfolio.

The most important quality of portfolio variance is that its value is a weighted combination of the individual variances of each of the assets adjusted by their covariances. This means that the overall portfolio variance is lower than a simple weighted average of the individual variances of the stocks in the portfolio.

The equation for the portfolio variance of a two-asset portfolio, the simplest portfolio variance calculation, takes into account five variables:

w(1) = the portfolio weight of the first asset

w(2) = the portfolio weight of the second asset

o(1) = the standard deviation of the first asset

o(2) = the standard deviation of the second asset

Cov(1,2) = the covariance of the two assets, which can be sampled to: q(1,2)o(1)o(2), where q(1,2) is the correlation between the two assets

The formula for variance in a two-asset portfolio is:

Variance = (w(1)^2 x o(1)^2) + (w(2)^2 x o(2)^2) + (2 x (w(1)o(1)w(2)o(2)q(1,2))

For example, assume there is a portfolio that consists of two stocks. Stock A is worth $50,000 and has a standard deviation of 20%. Stock B is worth $100,000 and has a standard deviation of 10%. The correlation between the two stocks is 0.85. Given this, the portfolio weight of Stock A is 33.3% and 66.7% for Stock B. Plugging in this information into the formula, the variance is calculated to be:

Variance = (33.3%^2 x 20%^2) + (66.7%^2 x 10%^2) + (2 x 33.3% x 20% x 66.7% x 10% x 0.85) = 1.64%

Variance is not a particularly easy statistic to interpret on its own, so most analysts calculate the standard deviation, which is simply the square root of variance. In this example, the square root of 1.64% is 12.82%.

As the number of assets in the portfolio grows, the terms in the formula for variance increase exponentially. For example, a three-asset portfolio has six terms in the variance calculation, while a five-asset portfolio has 15.

Assumptions

The period of time that is long enough to provide a sufficient amount of data but short enough to yield a result that is representative of recent economic conditions. Generally, choosing a period between one month and one year is sufficient.

computing the average is equal to the sum of the stock price on each day the market is open divided by the number of trading days in the period you are evaluating.

volatility as just one of many tools for assessing the risk of your stock investments. For example, if the volatility of your stock portfolio is low for the period, future fluctuations of the stock price outside of the standard deviation can be the result of other economic factors that affect the price of all stock rather than the inherent risk of one particular stock.

(b) Correlation of p 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 correlation If the highs and lows of 1 investment move in perfect coincidence to that of another investment, then the 2 investments have perfect positive Correlation The most diversified portfolio consists of securities with the greatest negative correlation. A diversified portfolio can also be achieved by investing in uncorrelated assets, but there will be times when the investments will be both up or down, and thus, a portfolio of uncorrelated assets will have a greater degree of risk, but it is still significantly less than positively correlated investments. However, even positively correlated investments will be less risky than single assets or investments that are perfectly positively correlated. However, there is no reduction in risk by combining assets that are perfectly correlated.

(c) The curves approach infinity as growth approaches the discount rate. The mathematical result is quite simple, the perpetuity formula is just an infinite geometric sum which converges when the absolute value of the common ratio (1+g)/(1+r) is less than 1. This is true when g < r, but when g = r, one can see that the growth multiplier in each term in the sum cancels out to leave behind a 1 period discount factor. This leads to an infinite sum of a fixed dividend discounted back by one period (and hence the result of infinity). There is no really no financial interpretation to this value ,