# 金融代写|利率理论代写portfolio theory代考|FINC6009

#### Doug I. Jones

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## 金融代写|利率理论代写portfolio theory代考|THE SHAPE OF THE PORTFOLIO POSSIBILITIES CURVE

Reexamine the earlier figures in this chapter and note that the portion of the portfolio possibility curve that lies above the minimum variance portfolio is concave, whereas that which lies below the minimum variance portfolio is convex. ${ }^{8}$ This is not due to the peculiarities of the examples we have chosen but rather is a general characteristic of all portfolio problems.

This can easily be demonstrated. Remember that the equations and diagrams we have developed are appropriate for all combinations of securities and portfolios. We now examine combinations of the minimum variance portfolio and an asset that has a higher retum and risk.
Figures $5.6 a, 5.6 b$, and $5.6 c$ represent three hypothesized shapes for combinations of Colonel Motors and the minimum variance portfolio. The shape depicted in $5.6 b$ cannot be possible because we have demonstrated that combinations of assets cannot have more risk than that found on a straight line connecting two assets (and that occurs only in the case of perfect positive correlation). But what about the shape presented in Figure $5.6 c$ ? Here all portfolios have less risk than the straight line connecting Colonel Motors and the minimum variance portfolio. However, this is impossible. Examine the portfolios labeled $U$ and $V$. These are simply combinations of the minimum variance portfolio and Colonel Motors. Since $U$ and $V$ are portfolios, all combinations of $U$ and $V$ must lie either on a straight line connecting $U$ and $V$ or above such a straight line. ${ }^{9}$ Hence $5.6 c$ is impossible, and the only legitimate shape is that shown in $5.6 a$, which is a concave curve. Analogous reasoning can be used to show that if we consider combinations of the minimum variance portfolio and a security or portfolio with higher variance and lower return, the curve must be convex, that is, it must look like Figure $5.7 a$ rather than $5.7 b$ or $5.7 c$.

## 金融代写|利率理论代写portfolio theory代考|The Efficient Frontier with No Short Sales

In theory we could plot all conceivable risky assets and combinations of risky assets in a diagram in return standard deviation space. We used the words “in theory” not because there is a problem in calculating the risk and return on a stock or portfolio but because there are an infinite number of possibilities that must be considered. Not only must all possible groupings of risky assets be considered but all groupings must be considered in all possible percentage compositions.

If we were to plot all possibilities in risk-return space, we would get a diagram like Figure 5.8. We have taken the liberty of representing combinations as a finite number of points in constructing the diagram. Let us examine the diagram and see if we can eliminate any part of it from consideration by the investor. In Chapter 4 we reasoned that an investor would prefer more return to less and would prefer less risk to more. Thus, if we could find a set of portfolios that

1. offered a bigger return for the same risk or
2. offered a lower risk for the same return

we would have identified all portfolios an investor could consider holding. All other portfolios could be ignored.

Let us take a look at Figure 5.8. Examine portfolios $A$ and $B$. Note that portfolio $B$ would be preferred by all investors to portfolio $A$ because it offers a higher return with the same level of risk. We can also see that portfolio $C$ would be preferable to portfolio $A$ because it offers less risk at the same level of return. Notice that, at this point in our analysis, we can find no portfolio that dominates portfolio $C$ or portfolio $B$. It should be obvious at this point that an efficient set of portfolios cannot include interior portfolios. We can reduce the possibility set even further. For any point in risk-return space, we want to move as far as possible in the direction of increasing return and as far as possible in the direction of decreasing risk. Examine point $D$, which is an exterior point. We can eliminate $D$ from further consideration given the existence of portfolio $E$, which has more return for the same risk. This is true for every other portfolio as we move up the outer shell from point $D$ to point $C$. Point $C$ cannot be eliminated because there is no portfolio that has less risk for the same return or more return for the same risk. But what is point $C$ ? It is the global minimum variance portfolio. ${ }^{10}$ Now examine point $F$. Point $F$ is on the outer shell, but point $E$ has less risk for the same return. As we move up the outer shell curve from point $F$, all portfolios are dominated until we come to portfolio $B$. Portfolio $B$ cannot be eliminated, for there is no portfolio that has the same return and less risk or the same risk and more return than point $B$. Point $B$ represents that portfolio (usually a single security) that offers the highest expected return of all portfolios. Thus the efficient set consists of the envelope curve of all portfolios that lie between the global minimum variance portfolio and the maximum return portfolio. This set of portfolios is called the efficient frontier.

# 利率理论代考

## 金融代写|利率理论代写portfolio theory代考|The Efficient Frontier with No Short Sales

1. 为相同的风险提供更大的回报或
2. 相同回报的风险更低

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