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deterministic function of the labor income volatility. ˜T is the time of retirement.
I also take the retirement into account, as it is an important factor to the household’s long-term financial plan. I assume that the breadwinner retires at a pre-specified date on the account of survival to the date. After the breadwinner’s retirement, the household receive a retirement income that is proportional to the income at the time of retirement, that is,
yt= ΥyT˜, T ≤ t ≤ T .˜
2.4 Household’s decision
In the economy, the household optimizes consumption, investment and insurance problem to maximize the its expected utility. The consumption plan ct denotes the time t consumption rate of the good, and the investment strategy π(t) = (πBt, πSt) denotes the amount of wealth invested in the bond and the stock. The household’s financial wealth at time t is denoted by Xt, and evolves in the following manner:
dWt= rtWt+ πtTΣ (rt, t) λ − ct+ yt− Ix+t dt + πtTΣ (rt, t) dZt,
where t ∈ [0, min (Tx, T )]. I assume that the household’s horizon T is shorter than the maturity of the bond, i.e. T ≤ ˆT . This ensures that the bond is a long-lived asset from the household’s viewpoint. If the breadwinner dies at time t, 0 < t ≤ T , then the rest of the family will receives the insurance benefit Iθx+t
x+t and financial wealth Wt. Therefore, the total legacy of the household leaves to its posterity is denoted by
Lt= Wt+Ix+t θx+t
.
The objective function of our optimization problem is in line with Huang and Milevsky (2008) and Huang, Milevsky, and Wang (2008), which are the first to separate the role of breadwinner from the traditional representative household. The household now faces a two-stage optimization problem conditional on the breadwinner’s survival. If the breadwinner is alive, then the household continues to receive the labor (or retirement) income, and make consumption-investment-insurance decisions. If the breadwinner dies in the last period, then the rest of the family receives a death benefit from insurance and use it to support the remaining life.
Conditional on the breadwinner’s death, the household faces a different optimization problem
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that does not involve labor income and insurance. This setup provides an incentive for the household to purchase life insurance throughout the life-cycle period to hedge the mortality risk. The indirect utility function for this optimization problem is denoted by
V (W, r, y, t) = sup
where the conditional expectation is measured given the values of W, r, y, at time t and given the strategy (ct, πt, Ix+t). Atis the set of all such triples of strategy (ct, πt, Iz+t). I supposed the household’s risk preference is of Constant Relative Risk Aversion (CRRA), i.e., U (X) = X1−γ1−γ with γ > 1 is the relative risk aversion parameter. Therefore, the indirect utility function satisfies
The Hamilton-Jacobi-Bellman (HJB) equation associated with this dynamic optimization problem is the value function for the household when there is no labor income and no life insurance and Ψ satisfies the following HJB equation:
δΨ = sup
I conclude this section with the solution of Ψ.
Proposition 1
Ψ (W, r, t) = 1
1 − γg (r, t)γW1−γ, (3)
where
Z T
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Proof. The proof of Proposition 1 is in Appendix A.The terminal condition of the HJB Equation (1) is
V (W, r, y, Tx) = Ψ (W, r, Tx; T ) .
The first-older conditions of the HJB Equation (1) are
ct= (VW(Wt, rt, yt, t))−1/γ, (4)
The first term of the right hand side (RHS) in Equation (5) is the myopic portfolio, the second term of the RHS is the income rate hedging portfolio, and the third term of the RHS is the interest rate hedging portfolio.
3 Solution of optimal strategies
I derive the analytical solution of consumption, investment, and insurance from first-order conditions in this section. I assume there is no short sale constraint and income risk is fully hedgable with investment assets following Munk and Sorensen (2010). Since personal income is hedgable, it can be replicated by financial assets, thus allows us to derive the analytical solution for expected labor income. The expected labor income represents the human capital in this model. It is both a vital part in the optimal strategies and informative for the household’s life-cycle wealth pattern. I solve the expected labor income now.
At time t, the expected value of the labor income from time t to T can written as
Ht≡ H (yt, rt, t) = EtQ
where Q is the equivalent martingale measure. In this situation, the household can get the amount Htfor selling his labor income. For the household, the optimal problem become to the
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financial wealth of Xt+ Ht and no labor income. The explicit solution of the labor income is in the Proposition 2.
Proposition 2. The labor income is given by
Ht=
where ˜T is the time to retirement, and s−tpx+t is the probability of the breadwinner surviving from age x + t to age x + s,
Proof: The proof is similar to Theorem 2.1.1 in Kraft and Munk (2011). The only difference is that I consider the mortality in the model.
Proposition 2 shows the labor income depends on the income rate y, the interest rate r, and the survival probability s−tpx+t. Based on Bodie, Merton and Samuelson (1992), the optimal problem can be converted to the case where the household has the initial financial wealth Wt+ Ht and no labor income. In the sense of Ingersoll (1987), with Vasicek (1977) interest rate model, and power utility function, I can write the indirect utility function J as the following equation
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I summarize the solution of value function in Proposition 3. Using Proposition 3, I can find the optimal consumption, investment and insurance strategy.
Proposition 3. The indirect utility function is given by
V (W, r, y, t) = 1
1 − γgl(r, t)γ(W + H (y, r, t))1−γ, (7) where the function gl(r, t) is defined by
gl(r, T ) =
Using first order conditions and the above proposition, the optimal consumption is
ct= Wt+ H (yt, rt, t) gl(rt, t)
and the optimal investments in the bond and the stock are
πBt = 1 and the insurance demand is
Iz+t = θx+tγ−1γ (µx+t)γ1 (Wt+ H (yt, rt, t)) − θx+tWt.
Proof: By substituting Equation (7) into the HJB Equation (1) and using the Feynman–Kac formula, I can verify that the function gl(r, t) satisfies Equation (8). The superscript l in
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the function gl(r, t) indicates the force of mortality with loading θx+t is included in the value function V . If there is no loading, i.e., θx+t= µx+t, then gl(r, t) = g(r, t). In Proposition 3, the function N (s − t) represents the utility loss the household suffered from the mortality risk plus the extra loading premium on the life insurance.
The optimal investment strategy on stock and bond can be separated into three compo-nents, namely the myopic portfolio, the interest rate hedging portfolio, and the income hedging portfolio, each serves an independent purpose. The myopic portfolio is the mean-variance opti-mal portfolio. The interest rate hedging portfolio comes from the household’s demand to hedge against fluctuation in future investment opportunities. Finally, the demand of income hedging portfolio is the result of income shock and the risk in present value of future income. Here, income shock is spanned by the stock and bond, so the household can hedge it by investing in the both financial assets. The risk of present value of future income is an interest rate risk in essence, because it reflects the volatility of future discount factor. The income hedging portfolio includes both stock and bond, and the relative holdings on them is related to the size of the income volatility and the correlation structure between the income and risky assets.
4 Numerical example
In this section, I illustrate the household’s behavior on investment and insurance demand with a numerical example. I first discuss the effect of mortality to the household’s wealth. Then I compare the investment strategies of the investor who has insurance demand to those who does not. The parameters of the financial asset, labor income and mortality model follows from Munk and Sorensen (2010) and Pirvu and Zhang (2012). Table 2 presents the parameters.
[Insert Table 2 about here.]