# 金融代写|投资组合代写Investment Portfolio代考|NBA5120

## 金融代写|投资组合代写Investment Portfolio代考|RISK-NEUTRAL PROBABILITIES

Another way to represent prices is via a risk-neutral probability. To distinguish between a risk-neutral probability and the probability under which we have been taking expectations, it is common to call the latter (whether objective or subjective) the physical probability. ${ }^5$ A risk-neutral probability is defined from the physical probability and a strictly positive SDF. Assume there is a strictly positive $\operatorname{SDF} \tilde{m}$, and, for each event $A$, let $1_A$ denote the indicator function of $A$, that is, for each state of the world $\omega, 1_A(\omega)=1$ if $\omega \in A$ and $1_A(\omega)=0$ if $\omega \notin A$.
Suppose first that there is a risk-free asset. For each event $A$, define
$$\mathbb{Q}(A)=R_f E\left[\tilde{m} 1_A\right] .$$
Then, $\mathbb{Q}$ is the risk-neutral probability associated to $\tilde{m}$. In a finite-state world, (3.21) means that the risk-neutral probability of each state is the product of $R_f$, the physical probability of the state, and the value of $\tilde{m}$ in the state (it is also the product of $R_f$ with the state price). In general, $\mathbb{Q}$ is a probability: $\mathbb{Q}(A) \geq 0, \mathbb{Q}(\Omega)=1$, where $\Omega$ is the set of all states of the world, and if $A_1, A_2, \ldots$ is a sequence of disjoint events, then $\mathbb{Q}\left(\cup A_i\right)=\sum \mathbb{Q}\left(A_i\right)$. As with any probability, there is an expectation operator associated with $\mathbb{Q}$. Denote it by $E^$. The definition of $\mathbb{Q}$ can be restated as $$\mathrm{E}^\left[1_A\right]=R_f \mathrm{E}\left[\tilde{m} 1_A\right],$$
because the expectation of an indicator function is the probability of the event. More generally, the definition of $\mathbb{Q}$ implies that
$$\mathrm{E}^[\tilde{x}]=R_f \mathrm{E}[\tilde{m} \tilde{x}]$$ for every $\tilde{x}$ for which the expectation $\mathrm{E}[\tilde{m} \tilde{x}]$ exists. Because the price of any payoff $\tilde{x}$ is $\mathrm{E}[\tilde{m} \tilde{x}]$, equation (3.22) implies that the price of any payoff $\tilde{x}$ is $$\frac{1}{R_f} \mathrm{E}^[\tilde{x}]$$

## 金融代写|投资组合代写Investment Portfolio代考|HANSEN-JAGANNATHAN BOUNDS

This section derives lower bounds on the standard deviations of SDFs due to Hansen and Jagannathan (1991). The Hansen-Jagannathan bounds have real economic significance. As discussed previously, an asset pricing model is a specification of an SDF $\tilde{m}$. A model can be rejected by the Hansen-Jagannathan bound if $\tilde{m}$ is not sufficiently variable. An illustration of the economic significance of the Hansen-Jagannathan bound with a risk-free asset is given in Exercise 7.2.
Continue to assume that asset payoffs have finite variances and the law of one price holds. Therefore, an SDF exists.
Hansen-Jagannathan Bound with a Risk-Free Asset
Assume there is a risk-free asset. Then, (3.12) holds for any SDF $\tilde{m}$, which we repeat here:
$$\mathrm{E}[\tilde{R}]-R_f=-R_f \operatorname{cov}(\tilde{m}, \tilde{R}) .$$
Letting $\operatorname{corr}(\tilde{m}, \tilde{R})$ denote the correlation of $\tilde{m}$ with $\tilde{R}$, we can write this as
$$\operatorname{corr}(\tilde{m}, \tilde{R}) \times \operatorname{stdcv}(\tilde{m})=\frac{\mathrm{E}[\tilde{R}]-R_f}{R_f \operatorname{stdev}(\tilde{R})} .$$

Because the correlation is between $-1$ and 1 , this implies
$$\operatorname{stdev}(\tilde{m}) \geq \frac{\left|\mathrm{E}[\tilde{R}]-R_f\right|}{R_f \operatorname{stdev}(\tilde{R})} .$$
Recalling that $1 / R_f=\mathrm{E}[\tilde{m}]$, we can rewrite this as
$$\frac{\operatorname{stdev}(\tilde{m})}{\mathrm{E}[\tilde{m}]} \geq \frac{\left|\mathrm{E}[\tilde{R}]-R_f\right|}{\operatorname{stdev}(\tilde{R})} .$$
The Sharpe ratio of a risky asset with return $\tilde{R}$ is the ratio of the risk premium $\mathrm{E}[\tilde{R}]-R_f$ to the risk $\operatorname{stdev}(\tilde{R})$. Thus, the ratio on the right-hand side of (3.35) is the absolute value of the Sharpe ratio of the return $\tilde{R}$. Hence, the ratio of the standard deviation of any SDF to its mean must be at least as large as the maximum absolute Sharpe ratio of all returns. This is one version of the Hansen-Jagannathan (1991) bounds.

## 金融代写|投资组合代写Investment Portfolio代考|RISK-NEUTRAL PROBABILITIES

$$\mathbb{Q}(A)=R_f E\left[\tilde{m} 1_A\right] .$$

$\mathbb{Q}(A) \geq 0, \mathbb{Q}(\Omega)=1$ ， 在哪里 $\Omega$ 是世界上所有状态的集合，如果 $A_1, A_2, \ldots$ 是一系列不相交的事件， 那么 $\mathbb{Q}\left(\cup A_i\right)=\sum \mathbb{Q}\left(A_i\right)$. 与任何概率一样，有一个期望算子与 $\mathbb{Q}$. 表示为 $\mathrm{E}^{\wedge}$. 的定义 $\mathbb{Q}$ 可以重述为
$$\mathrm{E}^{\left[1_A\right]}=R_f \mathrm{E}\left[\tilde{m} 1_A\right]$$

$$\left.\mathrm{E}^{[} \tilde{x}\right]=R_f \mathrm{E}[\tilde{m} \tilde{x}]$$

$$\left.\frac{1}{R_f} \mathrm{E}^{[} \tilde{x}\right]$$

## 金融代写|投资组合代写Investment Portfolio代考|HANSEN-JAGANNATHAN BOUNDS

Hansen-Jagannathan 与无风险资产绑定

$$\mathrm{E}[\tilde{R}]-R_f=-R_f \operatorname{cov}(\tilde{m}, \tilde{R}) .$$

$$\operatorname{corr}(\tilde{m}, \tilde{R}) \times \operatorname{stdcv}(\tilde{m})=\frac{\mathrm{E}[\tilde{R}]-R_f}{R_f \operatorname{stdev}(\tilde{R})} .$$

$$\operatorname{stdev}(\tilde{m}) \geq \frac{\left|\mathrm{E}[\tilde{R}]-R_f\right|}{R_f \operatorname{stdev}(\tilde{R})} .$$

$$\frac{\operatorname{stdev}(\tilde{m})}{\mathrm{E}[\tilde{m}]} \geq \frac{\left|\mathrm{E}[\tilde{R}]-R_f\right|}{\operatorname{stdev}(\tilde{R})} .$$

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