Jump-diffusion international asset allocation

Abstract

We examine international asset allocation with jump-diffusion assets in the presence of risky deviations of exchanges rates from purchasing power parity when investors consume both traded and nontraded goods. We show that adding new jump risks to existing diffusion assets does not alter investors’ original optimal portfolios of diffusion assets, as long as diffusion-risk premia remain unchanged. We also show that hedge portfolios against purchasing power parity deviations are integral parts of optimal portfolios for investors from different countries, and they can be constructed by using foreign and domestic inflation-indexed bonds. Moreover, country-specific demand for risky assets can arise from nontraded-good-specific inflation-rate-differential risks.

Appendix

1.1 Proof of Theorem 1

Recall $\varphi _c = S_AP_c = e^DP_c^*$ and $\varphi _n = S_AP_n = e^D P_c^* (P_n/P_c)$. Let $(G_c, G_n)$ be the inverse of $(U_{C_{Rc}},U_{C_{Rn}})$ with respect to $(C_{Rc},C_{Rn})$. Then the solution to (17) and (18) is given by

$$\begin{aligned} C_c(t)= & {} \varphi _c(t) G_c(\varphi _c(t)\Psi (t),\varphi _n(t)\Psi (t),t), \end{aligned}$$

(31)

$$\begin{aligned} C_n(t)= & {} \varphi _n(t) G_n(\varphi _c(t)\Psi (t),\varphi _n(t)\Psi (t),t). \end{aligned}$$

(32)

And the optimal consumption budget is

$$\begin{aligned}&E^{Q} \left[ \int _0^T e^{-\int _0^tr(u)du} (C_c + C_n) dt \right] \\&\quad = \frac{1}{q} E \left[ \int _0^T q e^{-\int _0^tr(u)du} \xi (t) (C_c + C_n) dt \right] \\&\quad = \frac{1}{q} E \left[ \int _0^T \Psi (t)\left\{ \varphi _c(t) G_c + \varphi _n(t) G_n \right\} dt \right] \end{aligned}$$

Thus, by the Bayes rule

$$\begin{aligned}&E^{Q} \left[ \left. \int _t^T e^{-\int _0^sr(u)du} (C_c + C_n) ds \quad \right| \, {\mathcal {F}}_t \right] \\&\quad = \frac{1}{q\xi (t)} E \left[ \left. \int _t^T q^l e^{-\int _0^sr(u)du} \xi (s) (C_c + C_n) ds \, \right| \, {\mathcal {F}}_t \right] \\&\quad = e^{-\int _0^tr(u)du}\frac{1}{\Psi (t)} E \left[ \left. \int _t^T \Psi (s)\left\{ \varphi _c(s) G_c + \varphi _n(s)G_n \right\} \, ds \, \right| \, {\mathcal {F}}_t \right] \\&\quad = e^{-\int _0^tr(u)du}\frac{1}{\Psi (t)} E \left[ \left. \int _t^T \Psi (s)e^{D(s)}P_c^*(s) \left\{ G_c + \frac{P_n(s)}{P_c(s)} G_n \right\} \, ds \, \right| \, {\mathcal {F}}_t \right] \\&\quad = e^{-\int _0^tr(u)du}F(\Psi (t), D(t), P_c^*(t), P_c(t), P_n(t), X(t),t) = e^{-\int _0^tr(u)du}F(t). \end{aligned}$$

For the last equality/definition, we have utilized the joint Markovian assumption. Then $F(0) = W(0)$, $F(T) = 0$, and one can modify the standard procedure of [10] for jumps to show the optimal portfolio policy as follows:¹⁵

$$\begin{aligned}&W(t)\left( \alpha _A^{\top }\phi _A + \alpha _J^{\top }\phi _J\right) = (\Delta F_1(t),\ldots ,\Delta F_L(t)) \end{aligned}$$

(33)

$$\begin{aligned}&W(t) \alpha _A^{\top } \sigma _A = - F_{\Psi } \Psi \nu ^{\top } + F_{D} \sigma _{D}^{\top } + F_{P_c^*} P_c^*(t) \sigma _{p_c^*}^{\top } + F_{P_c} P_c(t) \sigma _{p_c}^{\top } \nonumber \\&\quad \qquad \qquad \qquad +\, F_{P_n} P_n(t) \sigma _{p_n}^{\top } + F_X^{\top } \sigma _X. \end{aligned}$$

(34)

Since $F > 0$ by the assumption of an interior solution, Eq. (34) can be rewritten as in Eq. (21).

For Part (ii), note that $G_c$ and $G_n$ are functions of $(D^l,Y_c^l,Y_n^l)$. Since $(D^l, Y_c^l,Y_n^l, X)$ are jointly Markov, one can alternatively write F as follows:

$$\begin{aligned}&e^{-\int _0^tr(u)du}F(.,t) \\&\quad = e^{-\int _0^tr(u)du}\frac{1}{\Psi (t)} E \left[ \left. \int _t^T \Psi (s)e^{D(s)}P_c^*(s) \left\{ G_c + \frac{P_n(s)}{P_c(s)} G_n \right\} \, ds \, \right| \, {\mathcal {F}}_t \right] \\&\quad = e^{-\int _0^tr(u)du}\frac{1}{\Psi (t)} E \left[ \int _t^T \left. \left\{ e^{D(s)}Y_c(s) G_c + e^{D(s)}Y_n(s) G_n \right\} \, ds \, \right| \, D(t), Y_c(t), Y_n(t), X(t) \right] \\&\quad = e^{-\int _0^tr(u)du}\frac{1}{\Psi (t)} H(D(t), Y_c(t), Y_n(t),X(t),t). \end{aligned}$$

Thus, we must have

$$\begin{aligned} F(\Psi (t), D(t), P_c^*(t), P_c(t), P_n(t), X(t),t) \equiv \frac{1}{\Psi } H(D(t), Y_c(t), Y_n(t), X(t),t). \end{aligned}$$

Therefore,

$$\begin{aligned} F_{\Psi }= & {} - \frac{1}{\Psi ^2}H + \frac{1}{\Psi }H_{Y_c}P_c^* + \frac{1}{\Psi } H_{Y_n}\frac{P_c^*P_n}{P_c}\\ F_{P_c^*}= & {} H_{Y_c} + H_{Y_n}\frac{P_n}{P_c}\\ F_{P_c}= & {} - H_{Y_n}\frac{P_c^*P_n}{P_c^2}\\ F_{P_n}= & {} H_{Y_n}\frac{P_c^*}{P_c}. \end{aligned}$$

Thus,

$$\begin{aligned} F_{\Psi }\Psi = - F + F_{P_c^*}P_c^* \end{aligned}$$

and

$$\begin{aligned} P_c F_{P_c} + P_n F_{P_n} = 0. \end{aligned}$$

Therefore, (34) becomes

$$\begin{aligned} W(t) \alpha _A^{\top } \sigma _A = -F_{\Psi } \Psi \nu ^{\top } + \left( F+ F_{\Psi } \Psi \right) \sigma _{p_c^*}^{\top } + F_D\sigma _{D}^{\top } + F_{P_n} P_n(t) \left( \sigma _{p_n}^{\top } - \sigma _{p_c}^{\top }\right) + F_X^{\top }\sigma _X. \end{aligned}$$

(35)

Thus, we have Eq. (23).

To prove Part (iii), let $\bar{Y}_c^l(t) := e^{D(t)} Y_c^l(t)$ and $\bar{Y}_n^l(t) := e^{D(t)} Y_n^l(t)$. When volatilities, drifts and jump rates of traded and nontraded prices, exchange rates and asset prices are all constant over time, $(\bar{Y}_c^l,\bar{Y}_n^l)$ are Markov. Thus since $G_c$ and $G_n$ are functions of $(\bar{Y}_c^l,\bar{Y}_n^l)$, F can be rewritten as follows:

$$\begin{aligned}&e^{-rt}F(\Psi (t), D(t), P_c^*(t), P_c(t), P_n(t),t) \\&\quad = e^{-rt}\frac{1}{\Psi (t)} E \left[ \left. \int _t^T \left\{ \bar{Y}_c(s) G_c + \bar{Y}_n(s) G_n \right\} \, ds \, \right| \, \bar{Y}_c(t), \bar{Y}_n(t) \right] \end{aligned}$$

Define

$$\begin{aligned} \bar{H}(\bar{Y}_c^l(t), \bar{Y}_n^l(t),t) := E \left[ \left. \int _t^T \left\{ \bar{Y}_c^l(s) G_c + \bar{Y}_n^l(s) G_n \right\} \, ds \, \right| \, \bar{Y}_c^l(t), \bar{Y}_n^l(t) \right] . \end{aligned}$$

Then we must have

$$\begin{aligned} F(\Psi (t), D(t), P_c(t), P_n(t),t) \equiv \frac{1}{\Psi (t)} \bar{H}(\bar{Y}_c(t), \bar{Y}_n(t),t). \end{aligned}$$

Therefore,

$$\begin{aligned} F_{\Psi }= & {} - \frac{1}{\Psi ^2} \bar{H} + \frac{1}{\Psi }\bar{H}_{\bar{Y}_c}e^DP_c^* + \frac{1}{\Psi } \bar{H}_{\bar{Y}_n}\frac{e^DP_c^*P_n}{P_c}\\ F_{P_c^*}= & {} \bar{H}_{\bar{Y}_c}e^D + \bar{H}_{Y_n}\frac{e^DP_n}{P_c}\\ F_{D}= & {} \bar{H}_{\bar{Y}_c}e^DP_c^* + \bar{H}_{Y_n}\frac{e^DP_c^*P_n}{P_c} = P_c^* F_{P_c^*}\\ F_{P_c}= & {} - \bar{H}_{Y_n}\frac{e^DP_c^*P_n}{P_c^2}\\ F_{P_n}= & {} \bar{H}_{Y_n}\frac{e^DP_c^*}{P_c}. \end{aligned}$$

Thus,

$$\begin{aligned} F_{\Psi }\Psi= & {} - F + F_{P_c^*}P_c^*\\ 0= & {} P_c F_{P_c} + P_n F_{P_n}\\ F_{D}= & {} F_{P_c^*}P_c^*, \end{aligned}$$

and (34) becomes

$$\begin{aligned} W(t) \alpha _A^{\top } \sigma _A = -F_{\Psi } \Psi \nu ^{\top } + (F + F_{\Psi } \Psi ) (\sigma _{p_c^*}^{\top } + \sigma _{D}^{\top }) + F_{P_n} P_n(t)( \sigma _{p_n}^{\top } - \sigma _{p_c}^{\top }). \end{aligned}$$

Therefore, we have Eq. (24).

1.2 Proof of Proposition 1

Recall

$$\begin{aligned} C_{Rc}(t) = \frac{C_c(t)}{e^{D(t)}P_c^*(t)}, \quad \text{ and } \quad C_{Rn}(t) =\frac{C_n(t)}{e^{D(t)}(P_c^*(t)/P_c(t))P_n(t)}. \end{aligned}$$

Let $\bar{Y}_c(t) = e^{D(t)}P_c^*(t)\Psi (t)$ and $\bar{Y}_n(t) = e^{D(t)}(P_c^*(t)/P_c(t))P_n(t)\Psi (t)$, as defined in the proof of Part (ii) of Theorem 1. Then the FOCs are

$$\begin{aligned}&\kappa a_c \left( C_{Rc} \right) ^{a_c-1}\left( C_{Rn} \right) ^{a_n} = \bar{Y}_c,\\&\kappa a_n \left( C_{Rc} \right) ^{a_c}\left( C_{Rn} \right) ^{a_n-1} = \bar{Y}_n. \end{aligned}$$

Thus the above two imply that

$$\begin{aligned} C_{Rc} = \left( \frac{a_c}{a_n}\right) C_{Rn}\frac{\bar{Y}_n}{\bar{Y}_c}. \end{aligned}$$

By substituting the above back into the FOCs, we have

$$\begin{aligned} C_{Rc}= & {} \left( \kappa a_c\right) ^{\frac{1}{1-a}} \left( \frac{a_n}{a_c}\right) ^{\frac{a_n}{1-a}} \bar{Y}_c^{\frac{a_n-1}{1-a}} \bar{Y}_n^{\frac{-a_n}{1-a}}, \\ C_{Rn}= & {} \left( \kappa a_n\right) ^{\frac{1}{1-a}} \left( \frac{a_c}{a_n}\right) ^{\frac{a_c}{1-a}} \bar{Y}_c^{\frac{-a_c}{1-a}} \bar{Y}_n^{\frac{a_c -1}{1-a}}. \end{aligned}$$

Note that both $C_{Rc}$ and $C_{Rn}$ are lognormally distributed with jumps because $P_c^*$, $P_c^l$, and $P_n^l$ are lognormally distributed, and $\Psi$ and $e^D$ are lognormally distributed with jumps.¹⁶ Define $\eta _c(t) = \bar{Y}_c(t)C_{Rc}(t)$ and $\eta _n(t) = \bar{Y}_n(t)C_{Rn}(t)$. Then

$$\begin{aligned} \eta _c(t)= & {} \left( \kappa a_c\right) ^{\frac{1}{1-a}} \left( \frac{a_n}{a_c}\right) ^{\frac{a_n}{1-a}} \left( \bar{Y}_c(t)\right) ^{\frac{- a_c}{1-a}} \left( \bar{Y}_n(t)\right) ^{\frac{-a_n}{1-a}},\\ \eta _n(t)= & {} \left( \kappa a_n\right) ^{\frac{1}{1-a}} \left( \frac{a_c}{a_n}\right) ^{\frac{a_c}{1-a}} \left( \bar{Y}_c(t)\right) ^{\frac{-a_c}{1-a}} \left( \bar{Y}_n(t)\right) ^{\frac{-a_n}{1-a}}. \end{aligned}$$

Since $\bar{Y}_c$ and $\bar{Y}_n$ are lognormally distributed with jumps, so are $\eta _c$ and $\eta _n$. Thus dynamics of $\eta _c(t)$ and $\eta _n(t)$ can be expressed as in the following forms:

$$\begin{aligned} \frac{d\eta _c(t)}{\eta _c(t-)}= & {} \mu _{\eta c} dt + \sigma _{\eta c}^{\top }dz(t) + \phi _{\eta c}^{\top }dN(t) \\ \frac{d\eta _n(t)}{\eta _n(t-)}= & {} \mu _{\eta n} dt + \sigma _{\eta n}^{\top }dz(t) + \phi _{\eta n}^{\top }dN(t). \end{aligned}$$

where $\mu$’s, $\sigma$’s and $\phi$’s for $\eta _c$ and $\eta _n$are all constant. In fact, since $\eta _c(t) = (a_c/a_n)\eta _n(t)$, the dynamics of both $\eta _c$ and $\eta _n$ are identical to each other except their initial values. That is, $\mu _{\eta c} = \mu _{\eta n}$, $\sigma _{\eta c}= \sigma _{\eta n}$ and $\phi _{\eta c} = \phi _{\eta n}$. Therefore,

$$\begin{aligned} K_c:= & {} E_t\left[ \int _t^T \eta _c(s) ds \right] = E\left[ \left. \int _t^T \eta _c(s) ds \, \right| \, \bar{Y}_c(t), \bar{Y}_n(t) \, \right] \\= & {} \eta _c(t) E_t\left[ \int _t^T \frac{\eta _c(s)}{\eta _c(t)} ds \right] \\= & {} \frac{1}{\mu _{\eta c} + \phi _{\eta c}^{\top }\lambda } \eta _c(t) \left( e^{\left( \mu _{\eta c} + \phi _{\eta c}^{\top }\lambda \right) (T-t)} -1 \right) ,\\ K_n:= & {} E_t\left[ \int _t^T \eta _n(s) ds \right] = \frac{a_n}{a_c}K_c. \end{aligned}$$

If $\mu _{\eta c}$ is zero, then $K_c$ and $K_n$ become the limits of the above quantities as $\mu _{\eta c}$ approaches zero.

Recall $F(t,.) = \frac{1}{\Psi (t)}(K_c + K_n)$, i.e., the current level of wealth is the same as the present value of future consumption. Since $K_c$ and $K_n$ are positive almost surely for $t < T$, we have $F > 0$ almost surely. However,

$$\begin{aligned} (K_c)_{Y_c} = \frac{a_c}{a -1} \frac{1}{\mu _{\eta c} + \phi _{\eta c}^{\top }\lambda } (\kappa a_c)^{\frac{1}{1-a}} \left( \frac{a_c}{a_n} \right) ^{\frac{a_n}{a - 1}} (\bar{Y}_c)^{\frac{1 - a_n}{a -1}} (\bar{Y}_n)^{\frac{a_n}{a -1}} \left( e^{\left( \mu _{\eta c} + \phi _{\eta c}^{\top }\lambda \right) (T-t)} - 1\right) < 0. \end{aligned}$$

Since

$$\begin{aligned} (K_c)_{\bar{Y}_c}= & {} \frac{a_c}{a - 1} \frac{K_c}{\bar{Y}_c}, \qquad (K_n)_{\bar{Y}_c} = \frac{a_n}{a_c} (K_c)_{\bar{Y}_c},\\ (K_c)_{\bar{Y}_n}= & {} \frac{a_n}{a - 1} \frac{K_c}{\bar{Y}_n}, \quad \text{ and } \quad (K_n)_{\bar{Y}_n} = \frac{a_n}{a_c} (K_c)_{\bar{Y}_n}, \end{aligned}$$

we have $(K_n)_{\bar{Y}_c}, \, (K_c)_{\bar{Y}_n}, \, (K_n)_{\bar{Y}_n} \, < 0$. Also since $F = \frac{1}{\Psi }(K_c + K_n) = \frac{1}{\Psi } K_c \frac{a}{a_c}$,

$$\begin{aligned}&\frac{F_{P_n}P_n}{F} = \frac{P_n}{F} \frac{1}{\Psi } \frac{a}{a_c}(K_c)_{\bar{Y}_n}\frac{\partial \bar{Y}_n}{\partial P_n} = - \frac{a_n}{1 - a } < 0\\&\frac{F_{P_c}P_c}{F} = \frac{P_c}{F} \frac{1}{\Psi } \frac{a}{a_c}(K_c)_{\bar{Y}_n}\frac{\partial \bar{Y}_n}{\partial P_c} = \frac{a_n}{1 - a} > 0 \\&\frac{F_{P_c^*}P_c^*}{F} = \frac{P_c^*}{F} \frac{1}{\Psi } \frac{a}{a_c}\left( (K_c)_{\bar{Y}_c}\frac{\partial \bar{Y}_c}{\partial P_c^*} + (K_c)_{\bar{Y}_n}\frac{\partial \bar{Y}_n}{\partial P_c^*} \right) = - \frac{a}{1 - a} < 0 \\&\gamma = - \frac{\Psi F_{\Psi }}{F} = - \frac{\Psi }{F} \frac{a}{a_c} \left\{ - \frac{1}{\Psi ^2} K_c + \frac{1}{\Psi } \left( (K_c)_{\bar{Y}_c}\frac{\partial \bar{Y}_c}{\partial \Psi } + (K_c)_{\bar{Y}_n}\frac{\partial \bar{Y}_n}{\partial \Psi } \right) \right\} = \frac{1}{1 - a} > 1 \\&\frac{F_D}{F} = \frac{1}{\Psi F}\frac{a}{a_c} \left( (K_c)_{\bar{Y}_c}\frac{\partial \bar{Y}_c}{\partial D} + (K_c)_{\bar{Y}_n}\frac{\partial \bar{Y}_n}{\partial D} \right) = - \frac{a}{1-a} < 0. \end{aligned}$$

To compute $\Delta F/F$, recall $\bar{Y}_n = \bar{Y}_c\left( \frac{P_n}{P_c}\right)$. Thus,

$$\begin{aligned} F= & {} \frac{1}{\Psi } \frac{a}{a_c} \frac{ e^{\left( \mu _{\eta c} + \phi _{\eta c}^{\top }\lambda \right) (T-t)} - 1}{\mu _{\eta c} + \phi _{\eta c}^{\top }\lambda } \eta _c(t) \\= & {} \frac{1}{\Psi } \frac{a}{a_c} \frac{ e^{\left( \mu _{\eta c} + \phi _{\eta c}^{\top }\lambda \right) (T-t)} -1 }{\mu _{\eta c} + \phi _{\eta c}^{\top }\lambda } \left( \kappa a_c\right) ^{\frac{1}{1-a}} \left( \frac{a_n}{a_c}\right) ^{\frac{a_n}{1-a}} \left( \bar{Y}_c(t)\right) ^{\frac{- a}{1-a}} \left( \frac{P_n}{P_c}\right) ^{\frac{-a_n}{1-a}}. \end{aligned}$$

However, since

$$\begin{aligned}&\frac{1}{\Psi } \bar{Y}_c^{-\frac{a}{1-a}} = \frac{1}{\Psi } \left( e^D P_c^* \Psi \right) ^{-\frac{a}{1-a}} = \Psi ^{-\frac{1}{1-a}} \left( e^D P_c^* \right) ^{-\frac{a}{1-a}},\\&F = \frac{a}{a_c} \frac{ e^{\left( \mu _{\eta c} + \phi _{\eta c}^{\top }\lambda \right) (T-t)} -1 }{\mu _{\eta c} + \phi _{\eta c}^{\top }\lambda } \left( \kappa a_c\right) ^{\frac{1}{1-a}} \left( \frac{a_n}{a_c}\right) ^{\frac{a_n}{1-a}} \left( \frac{P_n}{P_c}\right) ^{\frac{-a_n}{1-a}} \Psi ^{-\frac{1}{1-a}} \left( e^D P_c^* \right) ^{-\frac{a}{1-a}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Delta F_l(t)= & {} F_l(t) - F_l(t-) = F\left\{ e^{-\frac{a}{1-a}\phi _{D^l}}\left( \frac{\theta ^l}{\lambda ^l} \right) ^{-\frac{1}{1-a}} -1 \right\} ,\\ \Delta F_k(t)= & {} F_k(t) - F_k(t-) = F\left\{ \left( \frac{\theta ^k}{\lambda ^k} \right) ^{-\frac{1}{1-a}} -1 \right\} , \quad \text{ for } k \ne l. \end{aligned}$$

Next, to find the value function, note that

$$\begin{aligned} U(.,t) = \kappa \left( \frac{a_n}{a_c} \right) ^{a_n}\left( \frac{P_c(t)}{P_n(t)}\right) ^{a_n} C_{Rc}^a(t). \end{aligned}$$

Thus, optimal U is lognormally distributed with jumps. Let $\mu _v$ and $\phi _v$ be the drift rate and the L-vector of jump rates for the dynamics of the optimal U(., t). Then, since $\mu _v$ and $\phi _v$ are constant over time, the value function V is

$$\begin{aligned} V(t) = E\left[ \left. \int _t^T U(.,s) ds \right| U(.,t) \right] = \frac{1}{\mu _v + \phi _v^{\top }\lambda } U(.,t) \left( e^{\left( \mu _v + \phi _v^{\top }\lambda \right) (T-t)}- 1\right) . \end{aligned}$$

Therefore, by substitutions, the value function is computed to be (27). The above value function is well defined for $\mu _v + \phi _v^{\top }\lambda \ne 0$. If $\mu _v + \phi _v^{\top }\lambda = 0$, then the value function can be computed as the limit of the above quantity as $\mu _v + \phi _v^{\top }\lambda$ approaches zero. This completes the proof.