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## 金融代写|利率理论代写portfolio theory代考|Case 1—Perfect Positive Correlation

Let the subscript $C$ stand for Colonel Motors and the subscript $S$ stand for Separated Edison. If the correlation coefficient is $+1$, then the equation for the risk on the portfolio, Equation (5.4), simplifies to
$$\sigma_{P}=\left[X_{C}^{2} \sigma_{C}^{2}+\left(1-X_{C}\right)^{2} \sigma_{S}^{2}+2 X_{C}\left(1-X_{C}\right) \sigma_{C} \sigma_{S}\right]^{1 / 2}$$
Note that the term in square brackets has the form $X^{2}+2 X Y+Y^{2}$ and thus can be written as
$$\left[X_{C} \sigma_{C}+\left(1-X_{C}\right) \sigma_{S}\right]^{2}$$
Because the standard deviation of the portfolio is equal to the positive square root of this expression, we know that
$$\sigma_{P}=X_{C} \sigma_{C}+\left(1-X_{C}\right) \sigma_{S}$$
while the expected return on the portfolio is
$$\bar{R}{P}=X{C} \bar{R}{C}+\left(1-X{C}\right) \bar{R}_{S}$$
Thus with the correlation coefficient equal to $+1$, both risk and return of the portfolio are simply linear combinations of the risk and return of each security. In footnote 3 we show that the form of these two equations means that all combinations of two securities that are perfectly correlated will lie on a straight line in risk and return space. ${ }^{3}$ We now illustrate that this is true for the stocks in our example. For the two stocks under study,$\bar{R}{P}=\frac{\sigma{P}-\sigma_{S}}{\sigma_{C}-\sigma_{S}} \bar{R}{C}+\left(1-\frac{\sigma{P}-\sigma_{S}}{\sigma_{C}-\sigma_{S}}\right) \bar{R}_{S}$

## 金融代写|利率理论代写portfolio theory代考|Case 2—Perfect Negative Correlation

We now examine the other extreme: two assets that move perfectly together but in exactly opposite directions. In this case the standard deviation of the portfolio is [from Equation (5.4) with $\rho=-1.0]$
$$\sigma_{P}=\left[X_{C}^{2} \sigma_{C}^{2}+\left(1-X_{C}\right)^{2} \sigma_{S}^{2}-2 X_{C}\left(1-X_{C}\right) \sigma_{C} \sigma_{S}\right]^{1 / 2}$$
Once again, the equation for standard deviation can be simplified. The term in the brackets is equivalent to either of the following two expressions:
$$\left[X_{C} \sigma_{C}-\left(1-X_{C}\right) \sigma_{S}\right]^{2}$$
or
$$\left[-X_{C} \sigma_{C}+\left(1-X_{C}\right) \sigma_{S}\right]^{2}$$
Thus $\sigma_{P}$ is either
$$\sigma_{P}=X_{C} \sigma_{C}-\left(1-X_{C}\right) \sigma_{S}$$
or
$$\sigma_{P}=-X_{C} \sigma_{C}+\left(1-X_{C}\right) \sigma_{S}$$
Because we took the square root to obtain an expression for $\sigma_{P}$ and because the square root of a negative number is imaginary, either of the preceding equations holds only when its right-hand side is positive. A further examination shows that the right-hand side of one equation is simply $-1$ times the other. Thus each equation is valid only when the righthand side is positive. Because one is always positive when the other is negative (except when both equations equal zero), there is a unique solution for the return and risk of any combination of securities $C$ and $S$. These equations are very similar to the ones we obtained when we had a correlation of $+1$. Each also plots as a straight line when $\sigma_{P}$ is plotted against $X_{C}$. Thus one would suspect that an examination of the return on the portfolio of two assets as a function of the standard deviation would yield two straight lines, one for each expression for $\sigma_{P}$. As we observe in a moment, this is, in fact, the case. ${ }^{5}$
The value of $\sigma_{P}$ for Equation (5.7) or (5.8) is always smaller than the value of $\sigma_{P}$ for the case where $\rho=+1$ [Equation (5.5)] for all values of $X_{C}$ between 0 and 1 . Thus the risk on a portfolio of assets is always smaller when the correlation coefficient is $-1$ than when it is $+1$. We can go one step further. If two securities are perfectly negatively correlated (i.e., they move in exactly opposite directions), it should always be possible to find some combination of these two securities that has zero risk. By setting either Equation (5.7) or (5.8) equal to 0 , we find that a portfolio with $X_{C}=\sigma_{S} /\left(\sigma_{S}+\sigma_{C}\right)$ will have zero risk. Because $\sigma_{S}>0$ and $\sigma_{S}+\sigma_{C}>\sigma_{S}$, this implies that $0<X_{C}<1$ or that the zero-risk portfolio will always involve positive investment in both securities.

# 利率理论代考

## 金融代写|利率理论代写portfolio theory代考|Case 1—Perfect Positive Correlation

$$\sigma_{P}=\left[X_{C}^{2} \sigma_{C}^{2}+\left(1-X_{C}\right)^{2} \sigma_{S}^{2}+2 X_{C}\left(1-X_{C}\right) \sigma_{C} \sigma_{S}\right]^{1 / 2}$$

$$\left[X_{C} \sigma_{C}+\left(1-X_{C}\right) \sigma_{S}\right]^{2}$$

$$\sigma_{P}=X_{C} \sigma_{C}+\left(1-X_{C}\right) \sigma_{S}$$

$$\bar{R} P=X C \bar{R} C+(1-X C) \bar{R}{S}$$ 因此相关系数等于 $+1$ ，投资组合的风险和回报都是每种证券的风险和回报的简单线性组 合。在脚注 3 中，我们表明这两个等式的形式意味着完全相关的两种证券的所有组合在风 险和回报空间中将位于一条直线上。 ${ }^{3}$ 我们现在说明这对于我们示例中的股票是正确的。对 于研究中的两只股票， $\bar{R} P=\frac{\sigma P-\sigma{S}}{\sigma_{C}-\sigma_{S}} \bar{R} C+\left(1-\frac{\sigma P-\sigma_{S}}{\sigma_{C}-\sigma_{S}}\right) \bar{R}_{S}$

## 金融代写|利率理论代写portfolio theory代考|Case 2—Perfect Negative Correlation

$$\sigma_{P}=\left[X_{C}^{2} \sigma_{C}^{2}+\left(1-X_{C}\right)^{2} \sigma_{S}^{2}-2 X_{C}\left(1-X_{C}\right) \sigma_{C} \sigma_{S}\right]^{1 / 2}$$

$$\left[X_{C} \sigma_{C}-\left(1-X_{C}\right) \sigma_{S}\right]^{2}$$

$$\left[-X_{C} \sigma_{C}+\left(1-X_{C}\right) \sigma_{S}\right]^{2}$$

$$\sigma_{P}=X_{C} \sigma_{C}-\left(1-X_{C}\right) \sigma_{S}$$

$$\sigma_{P}=-X_{C} \sigma_{C}+\left(1-X_{C}\right) \sigma_{S}$$

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