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

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

A second alternative to the expected utility theorem that is advocated by many is a group of criteria called safety-first models. The origin of these models is a belief that decision makers are unable, or unwilling, to go through the mathematics of the expected utility theorem but rather will use a simpler decision model that concentrates on bad outcomes. The name “safety first” comes about because of the emphasis each of the criteria places on limiting the risk of bad outcomes. Three different safety-first criteria have been put forth. The first, developed by Roy (1952), states that the best portfolio is the one that has the smallest probability of producing a return below some specified level. If $R_P$ is the return on the portfolio and $R_L$ is the level below which the investor does not wish returns to fall, Roy’s criterion is
$$\text { minimize Prob }\left(R_P<R_L\right)$$
If returns are normally distributed, then the optimum portfolio would be the one where $R_L$ was the maximum number of standard deviations away from the mean. For example, consider the three portfolios shown in Table 11.8. Assume $5 \%$ is the minimum return the investor desires. The investor wishes to minimize the chance of getting a return below $5 \%$. If the investor selects portfolio $\mathrm{A}$, then $5 \%$ is 1 standard deviation below the mean. The chance of getting a return below $5 \%$ is the probability of obtaining a return more than 1 standard deviation below the mean. If the investor selects investment $\mathrm{B}$, then $5 \%$ is 2.25 standard deviations below the mean. The probability of obtaining a return below $5 \%$ is the probability of obtaining a return more than 2.25 standard deviations below the mean. If he selects investment $\mathrm{C}$, the probability of obtaining a return below $5 \%$ is the probability of obtaining a return more than 1.5 standard deviations below the mean. Because the odds of obtaining a return more than 2.25 standard deviations below the mean are less than the odds of obtaining a return more than 1.5 or 1 standard deviation less than the mean, investment $\mathrm{B}$ is to be preferred.

As a second example, return to the problem shown in Table 11.1. Assume the investor wants to avoid negative outcomes. Portfolio $\mathrm{A}$ has a mean that is $7 / 10$ or 0.7 standard deviations above zero, $\mathrm{B}$ has a mean that is $8.5 / 15$ or 0.57 standard deviations above zero, and $\mathrm{C}$ has a mean that is $10 / 20$ or 0.5 standard deviations from zero. Thus $\mathrm{A}$ has the lowest probability of returns below zero and is preferred using Roy’s criterion.

## 金融代写|利率理论代写portfolio theory代考|MAXIMIZING THE GEOMETRIC MEAN RETURN

One alternative to utility theory is simply to select that portfolio that has the highest expected geometric mean return. Many researchers have put this forth as a universal criterion. That is, they advocate its use without qualifications as to the form of utility function or the characteristics of the probability distribution of security returns. The proponents of the geometric mean usually proceed with one of the following arguments. Consider an investor saving for some purpose in the future, for example, retirement in 20 years. One reasonable portfolio criterion for such an investor would be to select that portfolio that has the highest expected value of terminal wealth. Latane (1959) has shown that this is the portfolio with the highest geometric mean return. The proponents have also argued that the maximum geometric mean ${ }^4$

1. has the highest probability of reaching, or exceeding, any given wealth level in the shortest possible time ${ }^5$
2. has the highest probability of exceeding any given wealth level over any given period of time ${ }^6$

These characteristics of the maximum geometric mean portfolio are extremely appealing and have attracted many advocates. However, maximizing the geometric mean implicitly assumes a particular trade-off between the expected value of wealth and the occurrence of really bad outcomes. It is not clear that maximizing the geometric mean return is always appropriate.

Opponents quickly point out that, in general, maximizing the expected value of terminal wealth (or any of the other benefits discussed earlier) is not identical to maximizing the utility of terminal wealth. Because opponents accept the tenets of utility theory, and, in particular, the idea that investors should maximize the expected utility of terminal wealth, they reject the geometric mean return criteria.

In short, some researchers find the characteristics of the geometric mean return so appealing they accept it as a universal criterion. Others find any criterion that can be inconsistent with expected utility maximization unacceptable. Readers must judge for themselves which of these approaches is more appealing.

Having discussed the arguments in favor of and against the use of the geometric mean as a portfolio selection criterion, let us examine the definition of the geometric mean and some properties of portfolios that maximize the geometric mean criterion. The geometric mean is easy to define. Instead of adding together the observations to obtain the mean, we multiply them. If $R_{i j}$ is the $i$ th possible return on the $j$ th portfolio and each outcome is equally likely, then the geometric mean return on the portfolio $\left(\bar{R}{G j}\right)$ is $$\bar{R}{G j}=\left(1+R_{1 j}\right)^{1 / N}\left(1+R_{2 j}\right)^{1 / N} \ldots\left(1+R_{N j}\right)^{1 / N}-1.0$$
If the likelihood of each observation is different and $P_{i j}$ is the probability of the $i$ th outcome for portfolio $j$, then the geometric mean return is
$$\bar{R}{G j}=\left(1+R{1 j}\right)^{P_{1 j}}\left(1+R_{2 j}\right)^{P_{2 j}} \ldots\left(1+R_{N j}\right)^{P_{N j}}-1.0$$

# 利率理论代考

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

$$\operatorname{minimize} \operatorname{Prob}\left(R_P<R_L\right)$$

## 金融代写|利率理论代写portfolio theory代考|MAXIMIZING THE GEOMETRIC MEAN RETURN

1. 在尽可能短的时间内达到或超过任何给定财富水平的可能性最大 ${ }^5$
2. 在任何给定时间段内超过任何给定财富水平的可能性最大 ${ }^6$
最大几何平均数投资组合的这些特点极具吸引力，吸引了众多拥护者。然而，最大化几 何平均数隐含地假设了财富的预期价值与真正糟糕结果的发生之间的特定权衡。尚不清 楚最大化几何平均回报总是合适的。
反对者很快指出，一般来说，最大化终端财富的预期价值（或前面讨论的任何其他收
益) 并不等同于最大化终端财富的效用。因为反对者接受效用理论的原则，特别是投资 者应该最大化最终财富的预期效用的观点，他们拒绝几何平均回报标准。
简而言之，一些研究人员发现几何平均回报率的特征非常吸引人，他们接受它作为一个 普遍的标准。其他人发现任何可能与预期效用最大化不一致的标准都是不可接受的。读 者必须自己判断这些方法中哪种方法更有吸引力。
在讨论了支持和反对使用几何平均数作为投资组合选择标准的论点之后，让我们检查一 下几何平均数的定义和最大化几何平均数标准的投资组合的一些属性。几何平均数很容 易定义。我们不是将观察值加在一起以获得平均值，而是将它们相乘。如果 $R_{i j}$ 是个 $i$ th 可能的回报 $j$ 第一个投资组合和每个结果的可能性相同，然后是投资组合的几何平均回 报 $(\bar{R} G j)$ 是
$$\bar{R} G j=\left(1+R_{1 j}\right)^{1 / N}\left(1+R_{2 j}\right)^{1 / N} \ldots\left(1+R_{N j}\right)^{1 / N}-1.0$$
如果每个观察的可能性不同并且 $P_{i j}$ 是的概率 $i$ 投资组合的结果 $j$, 那么几何平均回报是
$$\bar{R} G j=(1+R 1 j)^{P_{1 j}}\left(1+R_{2 j}\right)^{P_{2 j}} \ldots\left(1+R_{N_j}\right)^{P_{N j}}-1.0$$

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