Portfolios of Two Perfectly Positively Correlated or Perfectly Negatively Correlated Assets

Perfect Positive Correlation. Exhibit 4.4 shows the plotted means and standard deviations obtainable from portfolios of two perfectly positively correlated stocks. Points A and B on the line, designated, respectively, as "100% in stock 1" and "100% in stock 2," correspond to the mean and standard deviation pairings achieved when 100 percent of an investor's wealth is held in one of the two investments. Bold segment AB graphs the means and standard deviations achieved from portfolios with positive weights on the two perfectly correlated stocks. Moving up the line from point A toward point B

Part II Valuing Financial Assets

Exhibit 4.4 Mean-Standard Deviation Diagram: Portfolios of Two Perfectly Positively Correlated Stocks

Mean Short in

Mean Short in

Perfectly Negatively Correlated Stocks

places more weight on the investment with the higher expected return (stock 2). Above point B, the portfolio is selling short stock 1 and going "extra long" (weight exceeds 1) in stock 2. Below point A, the portfolio is selling short stock 2.

Exhibit 4.4 demonstrates that it is possible to eliminate risk—that is, to achieve zero variance—with a portfolio of two perfectly positively correlated stocks.9 To do this, it is necessary to be long in one investment and short in the other in proportions that place the portfolio at point C.

The graph of portfolios of two perfectly positively correlated, but risky, investments has the same shape (a pair of straight lines) as the graph for a portfolio of a riskless and a risky investment. This should not be surprising because a portfolio of the two perfectly correlated investments is itself a riskless asset. The feasible means and standard deviations generated from portfolios of, say, the riskless combination of stocks 1 and 2 on the one hand, and stock 2 on the other, should be identical to those generated by portfolios of stocks 1 and 2 themselves. Also, when the portfolio weights on both investments are positive, the mean and the standard deviation of a portfolio of two perfectly correlated investments are portfolio-weighted averages of the means and standard deviations of the individual assets. Consistent with Result 4.6, such portfolios are on a line connecting the two investments whenever the weighted average of the standard deviations is positive.

Perfect Negative Correlation. For similar reasons, a pair of perfectly negatively correlated risky assets graphs as a pair of straight lines. In contrast to a pair of perfectly positively correlated investments, when two investments are perfectly negatively correlated, the investor eliminates variance by being long in both investments.

9This also was shown in Example 4.13.

Chapter 4 Portfolio Tools 119

Example 4.16: Forming a Riskless Portfolio from Two Perfectly Negatively Correlated Securities

Over extremely short time intervals, the return of IBM is perfectly negatively correlated with the return of a put option on the company. The standard deviation of the return on IBM stock is 18 percent per year, while the standard deviation of the return on the option is 54 percent per year. What portfolio weights on IBM stock and its put option create a riskless investment over short time intervals?

Answer: The variance of the portfolio, using equation (4.9b), is

.182x2 + .542(1 - x)2 - 2x(1 - x)(.18)(.54) = [.18x - .54(1 - x)]2

This expression is 0 when x = .75. Thus, the weight on IBM stock is 3/4 and the weight on the put option is 1/4.

The Feasible Means and Standard Deviations from Portfolios of Other Pairs of Assets

We noted in the last subsection that with perfect positive correlation, the standard deviation of a portfolio with positive weights on both stocks equals the portfolio-weighted average of the two standard deviations. Now, consider the case where risky investments have less than perfect correlation (p < 1). Result 4.2 states that the lower the correlation, the lower the portfolio variance. Therefore, the standard deviation of a portfolio with positive weights on both stocks is less than the portfolio-weighted average of the two standard deviations, which gives the curvature to the left shown in Exhibit 4.5. The

Exhibit 4.5 Mean Standard Deviation Diagram: Portfolios of Two Risky Securities with Arbitrary Correlation, p

Mean Return

100% in stock 2

100% in stock 2

Portfolio Correlation
100% in stock 1

Standard Deviation of Return ctp

Part II Valuing Financial Assets degree to which this curvature occurs depends on the correlation between the returns. Consistent with Result 4.2, the smaller the correlation, p, the more distended the curvature. The ultimate in curvature is the pair of lines generated with perfect negative correlation, p = -1, which is the smallest correlation possible.10

The combination of the two stocks that has minimum variance is a portfolio with positive weights on both stocks only if the correlation between their returns is not too large. For sufficiently large correlations, the minimum variance portfolio, the portfolio of risky investments with the lowest variance, requires a long position in one investment and a short position in the other.11

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Responses

  • asmarina
    What is the variance of perfectly positively correlated stocks?
    8 years ago
  • linda
    What is a positive standard deviation for investor?
    7 years ago
  • alanna
    What is negative correlated with ibm?
    7 years ago
  • heli
    What happens to standard deviation of portfolio if correlation is perfect positive?
    7 years ago
  • sebastian
    What are positive correlated assets?
    7 years ago
  • semhar
    What is the standard deviation of positively correlated stocks?
    7 years ago
  • wanda
    What is the standard deviation of perfectly correlated stocks?
    7 years ago
  • sven
    Are stocks with perfect correlation risky?
    7 years ago
  • CALVIN FULLER
    What is the minimum variance portfolio of two perfectly negatively correlated assets?
    7 years ago
  • robert
    Are perfectly positively correlated the range?
    6 years ago
  • Monika
    What is the volatility of a perfectly correlated portfolio?
    6 years ago
  • LAILA
    What investments are negatively correlated?
    6 years ago
  • TAPIO
    Do postively correlated assets eliminate risk?
    6 years ago
  • russell sledge
    When the correlation of assets if perfectly positive?
    6 years ago
  • gino
    What is a perfectly positively correlated stock?
    6 years ago
  • DONNINO MANNA
    WHAT IF RETURN PERFECTLY COORELATED?
    6 years ago
  • senay
    Are WAG and VAR positively or negatively correlated?
    6 years ago
  • Artemisia
    What are two companies that are postively correlated?
    6 years ago
  • sanna pusa
    When two stocks are perfectly correlated what happens to the portfolios standard deviation?
    6 years ago
  • myrtle
    What will happen to portfolio risk with perfectly positive correlation?
    6 years ago
  • LAILA
    Does standard deviation work when stocks are not perfectly positively correlated?
    5 years ago
  • Armas
    What do we say two asset are perfectly correlated?
    4 years ago
  • salla
    When two stocks are perfectly negative correlated, what is the risk of the portfolio?
    3 years ago
  • pirjo ruoho
    What this means is that if assets A and B are postively correlated , variance is zero short postion?
    2 years ago
  • Bernd
    What would be the point of investing in two perfectly negatively correlated stocks?
    6 months ago
  • Cupido
    How to tell if two statedependent assets are perfectly positively correlated?
    30 days ago

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