Downside risk-neutral probabilities

Abstract

We show that there exists a probability measure under which the CAPM formula for expected returns holds for general utility functions and probability distributions. This probability measure, the “downside risk-neutral” measure, is adjusted to incorporate the effects of downside risk and higher degree risks. It thus belongs to the same family as the risk-neutral measure, which is also a risk-adjusted measure. Using risk preference theory, we interpret this change in probability measure in terms of risk substitution.

Appendices

Appendix A: Additional proofs

The expression$1 - A(c_0) \mathbb {E}^{\Lambda _3} \left[ \tilde{c}_1-c_0\right]$in Eq. (12)

We show that this expression is strictly positive.

First, consider the states s such that $u^{(1)}(c_0) + u^{(2)}(c_0) (c_s-c_0)>0$ and, therefore, $\eta _{3,s}>0$ (given Eq. (3) and $u^{(1)}>0$), which implies $\lambda _{3,s} \ge 0$. Then we have $\lambda _{3,s} \left[ u^{(1)}(c_0) + u^{(2)}(c_0) (c_s-c_0) \right] \ge 0$. Dividing each term by $u^{(1)}(c_0)$, which is strictly positive, and summing over these states s, we have

$$\begin{aligned} \sum _{s|\eta _{3,s}>0} \lambda _{3,s} \left[ 1 - A(c_0) (c_s-c_0) \right] \ge 0. \end{aligned}$$

(17)

Second, consider the states s such that $u^{(1)}(c_0) + u^{(2)}(c_0) (c_s-c_0)<0$, and therefore $\eta _{3,s}<0$ (given Eq. (3) and $u^{(1)}>0$), which implies $\lambda _{3,s} \le 0$. Then we have $\lambda _{3,s} \left[ u^{(1)}(c_0) + u^{(2)}(c_0) (c_s-c_0) \right] \ge 0$. Dividing each term by $u^{(1)}(c_0)$, which is strictly positive, and summing over these states s, we have

$$\begin{aligned} \sum _{s|\eta _{3,s}<0} \lambda _{3,s} \left[ 1 - A(c_0) (c_s-c_0) \right] \ge 0. \end{aligned}$$

(18)

Third, because $\lambda _{3,s} \left[ 1 - A(c_0) (c_s-c_0)\right] \ge 0$ for any s and $1 - A(c_0) (c_s-c_0) \ne 0$ (otherwise $\eta _{3,s}$ would not exist), both expressions on the left-hand sides of (17) and (18) will be equal to zero only if $\lambda _{3,s}=0$ for all s, which would imply $\sum _s\lambda _{3,s} \ne 1$, a contradiction. Therefore, at least one of the expressions on the left-hand sides of (17) and (18) is strictly positive, while the other is nonnegative. Adding up these two sums yields

$$\begin{aligned} \sum _{s} \lambda _{3,s} \left[ 1 - A(c_0) (c_s-c_0) \right] =1 - A(c_0) \mathbb {E}^{\Lambda _3} \left[ \tilde{c}_1-c_0 \right] > 0. \end{aligned}$$

$\square$

Proof of Proposition 3

Let $\tilde{m} \equiv \frac{1}{R_\mathrm{f}} \frac{ f(c_0,\tilde{c}_1)}{\mathbb {E} ^{\Lambda _3}[f(c_0,\tilde{c}_1)]}$. The pricing kernel $\tilde{m}$ in Proposition 2 is linear in $\tilde{c}_1$. Since in addition $\tilde{c}_1$ is linear in $\tilde{R}^w$ (see Eq. (1)), we have $\tilde{m} = a + b \tilde{R}^w$, and

$$\begin{aligned} a \equiv \frac{1}{R_\mathrm{f}} \frac{1+A(c_0)c_0}{1-A(c_0)(\mathbb {E} ^{\Lambda _3}[\tilde{c}_1]-c_0)}, \quad b \equiv \frac{1}{R_\mathrm{f}} \frac{-A(c_0)[w_0-c_0]}{1-A(c_0)(\mathbb {E} ^{\Lambda _3}[\tilde{c}_1]-c_0)}. \end{aligned}$$

Equation (11) can thus be rewritten as $P_j=\mathbb {E} ^{\Lambda _3}[\tilde{m} \tilde{x}^j]$. Dividing both sides by $P_j$ gives $1=\mathbb {E} ^{\Lambda _3}[\tilde{m} \tilde{R}_j]$, where $\tilde{R}_j \equiv \frac{\tilde{x}^j}{P_j}$ is the gross return of asset j. Rewrite this equation as

$$\begin{aligned} \frac{1}{\mathbb {E} ^{\Lambda _3}[\tilde{m}]} = \mathbb {E} ^{\Lambda _3}[\tilde{R}_j] + \frac{\mathrm{cov}^{\Lambda _3}(\tilde{m},\tilde{R}_j)}{\mathbb {E} ^{\Lambda _3}[\tilde{m}]}. \end{aligned}$$

Using $\mathbb {E} ^{\Lambda _3}[\tilde{m}] = \frac{1}{R_\mathrm{f}}$, and substituting for $\tilde{m}$, the equation above rewrites as

$$\begin{aligned} \mathbb {E} ^{\Lambda _3}[\tilde{R}_j]-R_\mathrm{f} =- \frac{b \, \mathrm{cov}^{\Lambda _3}( \tilde{R}^w,\tilde{R}_j)}{\mathbb {E} ^{\Lambda _3}[\tilde{m}]}. \end{aligned}$$

(19)

In particular, if the asset considered is the wealth portfolio with gross return $\tilde{R}^w$, we have

$$\begin{aligned} \mathbb {E} ^{\Lambda _3}[\tilde{R}^w]-R_\mathrm{f} = - \frac{b \, \mathrm{var}^{\Lambda _3}(\tilde{R}^w)}{\mathbb {E} ^{\Lambda _3}[\tilde{m}]}. \end{aligned}$$

(20)

Equating Eqs. (19) and (20),

$$\begin{aligned} \mathbb {E} ^{\Lambda _3}[\tilde{R}_j]-R_\mathrm{f} = \frac{\mathrm{cov}^{\Lambda _3}( \tilde{R}^w,\tilde{R}_j)}{\mathrm{var}^{\Lambda _3}(\tilde{R}^w)} \left[ \mathbb {E} ^{\Lambda _3}[\tilde{R}^w]-R_\mathrm{f} \right] . \end{aligned}$$

$\square$

Appendix B: ith-degree risk-neutral probabilities

We start by introducing the family of ith-degree risk-neutral probabilities. Assume that u is of class $C^i$, for an integer $i \ge 2$. For given i and s, let $\eta _{i,s}$ be defined implicitly as

$$\begin{aligned} u^{(1)}(c_s) \equiv \eta _{i,s} \sum _{k=0}^{i-2} \frac{1}{k!} u^{(k+1)}(c_0) (c_s-c_0)^{k}. \end{aligned}$$

(21)

The sum on the right-hand side of Eq. (21) is a Taylor expansion of marginal utility of order $i-2$ at $c_0$, which only reflects the preferences for risks of degree $i-1$ and lower. The left-hand side, the marginal utility of consumption, can be viewed as a Taylor series of marginal utility at $c_0$, which reflects the preferences for all degrees of risks. We henceforth consider economies such that $\eta _{i,s}$ exists for all s ($\eta _{i,s}$ exists generically—except for $\sum _{k=0}^{i-2} \frac{1}{k!} u^{(k+1)}(c_0) (c_s-c_0)^{k}=0$).

Definition 2

Let $\nu _i \equiv \sum _{s=1}^S p_s \eta _{i,s}$, and

$$\begin{aligned} \lambda _{i,s} \equiv p_s \frac{ \eta _{i,s} }{\nu _i} = \frac{p_s \eta _{i,s}}{\sum _{s=1}^S p_s \eta _{i,s}}. \end{aligned}$$

(22)

The set $\{ \lambda _{i,s} \}$ is the set of ith-degree risk-neutral probabilities, and $\Lambda _i$ is the ith-degree risk-neutral probability measure.

The Radon–Nikodym derivative that yields ith-degree risk-neutral probabilities, $\frac{\mathrm{d}\lambda _{i,s}}{\mathrm{d}p_s}=\frac{\eta _{i,s}}{\nu _i}$, is at any point proportional to the ratio of marginal utility of consumption to a Taylor expansion of marginal utility of order $i-2$ at $c_0$. Setting $i=4$ yields outer risk-neutral probabilities.

The next Proposition establishes that the ith-degree risk-neutral probabilities not only coincide with physical probabilities in the limit, as $i \rightarrow \infty$, but also for any i such that $u^{(i)}=0$.

Proposition 4

The ith-degree risk-neutral probability measure, $\Lambda _{\infty ,s}$, coincides with the physical probability measure for $i \rightarrow \infty$. Moreover, for preferences such that $u^{(i)}=0$, $\Lambda _i$ coincides with the physical probability measure.

Proof

First, by analogy with Eq. (21), for any given s, let $\eta _{\infty ,s}$ be defined implicitly as ($\eta _{\infty ,s}$ exists generically—except for $\sum _{k=0}^{\infty } \frac{1}{k!} u^{(k+1)}(c_0) (c_s-c_0)^{k}=0$):

$$\begin{aligned} u^{(1)}(c_s) \equiv \eta _{\infty ,s} \sum _{k=0}^{\infty } \frac{1}{k!} u^{(k+1)}(c_0) (c_s-c_0)^{k}. \end{aligned}$$

(23)

Given that $u^{(1)}$ is analytic at $c_0$, the Taylor series of $u^{(1)}$ give

$$\begin{aligned} u^{(1)}(c_s) = \sum _{k=0}^{\infty } \frac{1}{k!} u^{(k+1)}(c_0) (c_s-c_0)^{k}. \end{aligned}$$

(24)

Comparing with (23), this implies that $\eta _{\infty ,s}=1$ for any s. Then Definition 2 implies that $\lambda _{\infty ,s}=p_s$ for any s.

Second, suppose that $u^{(i)}=0$, which implies that $u^{(j)}=0$ for $j \ge i$. Then, for any s, a Taylor expansion gives

$$\begin{aligned} u^{(1)}(c_s) = \sum _{k=0}^{i-2} \frac{1}{k!} u^{(k+1)}(c_0) (c_s-c_0)^{k}. \end{aligned}$$

(25)

Equation (21) then implies that $\eta _{i,s}=1$ for any s, and Definition 2 implies that $\lambda _{i,s} = p_s$ for any s. $\square$

We now show how to express the price of any asset with ith-degree risk-neutral probabilities. Let $C_{k+1}(c_0) \equiv \frac{ u^{(k+1)}(c_0)}{u^{(1)}(c_0)}$ be the coefficient of absolute preference for the $k+1$th-degree risk at $c_0$. A negative preference (i.e., an aversion) for the $k+1$th-degree risk implies that $C_{k+1}(c_0)<0$. The price $P_j$ of an asset with stochastic payoff $\tilde{x}^j$ may be decomposed in the following terms.

Proposition 5

If u is of class $C^i$, for a given integer $i \ge 2$, then

$$\begin{aligned} P_j= & {} \frac{1}{R_\mathrm{f}} \frac{ \sum _{k=0}^{i-2} \frac{1}{k!} C_{k+1}(c_0) \mathbb {E}^{\Lambda _i} \left[ (\tilde{c}_1-c_0)^{k} \tilde{x}^j \right] }{ \sum _{k=0}^{i-2} \frac{1}{k!} C_{k+1}(c_0) \mathbb {E}^{\Lambda _i} \left[ (\tilde{c}_1-c_0)^{k} \right] } \end{aligned}$$

(26)

$$\begin{aligned}= & {} \frac{1}{R_\mathrm{f}} \mathbb {E}^{\Lambda _i} \left[ \frac{ f_i(c_0,\tilde{c}_1)}{ \mathbb {E}^{\Lambda _i}[f_i(c_0,\tilde{c}_1)] } \tilde{x}^j \right] , \end{aligned}$$

(27)

where $f_i(c_0,\tilde{c}_1) \equiv \sum _{k=0}^{i-2} \frac{1}{k!} C_{k+1}(c_0) (\tilde{c}_1-c_0)^{k}$.

Proof

Substituting $u^{(1)}(c_s)$ from (21) in Eq. (2), the asset price $P_j$ may be decomposed in the following terms:

$$\begin{aligned} P_j= & {} \sum _{s=1}^S p_s \left[ \beta \eta _{i,s} \sum _{k=0}^{i-2} \frac{1}{k!} \frac{ u^{(k+1)}(c_0)}{u^{(1)}(c_0)} (c_s-c_0)^{k} x_s^j \right] \nonumber \\= & {} \beta \nu _i \mathbb {E}^{\Lambda _i} \left[ \sum _{k=0}^{i-2} \frac{1}{k!} C_{k+1}(c_0) (\tilde{c}_1-c_0)^{k} \tilde{x}^j \right] \nonumber \\= & {} \beta \nu _i \left[ \sum _{k=0}^{i-2} \frac{1}{k!} C_{k+1}(c_0) \mathbb {E}^{\Lambda _i} \left[ (\tilde{c}_1-c_0)^{k} \tilde{x}^j \right] \right] . \end{aligned}$$

(28)

Given that Eq. (28) must hold for any asset, including the risk-free asset whose payoff is $x_s=1$ for all s and whose price is by definition of the gross risk-free rate $R_\mathrm{f}$ equal to $P_\mathrm{f}=\frac{1}{R_\mathrm{f}}$, we have

$$\begin{aligned} \frac{1}{R_\mathrm{f}}= \beta \nu _i \left[ \sum _{k=0}^{i-2} \frac{1}{k!} C_{k+1}(c_0) \mathbb {E}^{\Lambda _i} \left[ (\tilde{c}_1-c_0)^{k} \right] \right] . \end{aligned}$$

(29)

Substituting (29) in (28) gives Eq. (26). $\square$

By construction, for any $i \ge 2$, the asset price in Proposition 5 is the same as the one in the standard stochastic discount factor formula in Eq. (2).

For preferences such that $u^{(i)}=0$, $\Lambda _i$ coincides with the physical probability measure (see Proposition 4), and the price of any asset with stochastic payoff $\tilde{x}^j$ is

$$\begin{aligned} P_j = \frac{1}{R_\mathrm{f}} \frac{ \sum _{k=0}^{i-2} \frac{1}{k!} C_{k+1}(c_0) \mathbb {E} \left[ (\tilde{c}_1-c_0)^{k} \tilde{x}^j \right] }{ \sum _{k=0}^{i-2} \frac{1}{k!} C_{k+1}(c_0) \mathbb {E} \left[ (\tilde{c}_1-c_0)^{k} \right] } = \frac{1}{R_\mathrm{f}} \mathbb {E} \left[ \frac{ f_i(c_0,\tilde{c}_1)}{ \mathbb {E}[f_i(c_0,\tilde{c}_1)] } \tilde{x}^j \right] , \end{aligned}$$

(30)

where $f_i(c_0,\tilde{c}_1)$ is as in Proposition 5. For utility functions such as CRRA and CARA utility which are such that $u^{(i)} \ne 0$ for any $i \in (1,2,\ldots )$, the pricing kernel under physical probabilities can only be obtained in polynomial form as a Taylor series, with an infinite number of terms.