In this paper, we will investigate optimal investment and consumption strategies in the market with CRRA utility function. By using shadow price, which is the virtual price between the purchase and offer price, we can derive the optimal investment and consumption strategies. With KOSPI200 and house price index data, we obtained the optimal investment and consumption strategies and also checked the effect of transaction costs on two risky assets and consumption.
In this frictionless market, where there is one risk-free asset and one risky asset with CRRA utility function, Merton has shown that the optimal allocation strategy is to invest a constant portion of assets in the risky asset. In the transaction cost market, the optimal strategy is defined by two straight lines. Outside the no-transaction region, the optimal strategy for the investor is to adjust his portfolio to the nearest line so that the stock can be in the no-transaction region.
So, using the shadow price approach, he derived the optimal investment and consumption strategies by solving the optimization problem in the frictionless market. In this paper, we used the optimal investment and consumption strategies obtained in [1], applied the results to the real data. And in Section 3, we will derive the optimal investment and consumption strategies in the frictionless market with two risky assets and briefly review the paper [1].
And finally, the optimal strategies using KOSPI200 and house price index data will be shown.
The market
Utility function
Shadow price approach
But before we review the paper, we analyze the frictionless market with two risky assets.
Frictionless market with two risky assets
Then, The last two terms are zero and Hamilton-Jacobi-Bellman(HJB) equation is made as. In this subsection, we will see that the candidate solution we derived in section 3.1.1 is an optimal solution. Since π1s, π2s and Wtp are bounded (Wtp is bounded from Jensen's inequality), expectation of last two terms is zero.
Result of Choi [1]
In this section, we investigate the optimal trading of the asset with liquid, illiquid and consumption risk. We recall that from section 3.1.1, the optimal ratio of liquid, illiquid and consumption risk assetscM, π1M, πM2 in the frictionless market is given as.
Optimal strategy
From inference 4.1.3 we can see that the optimal consumption rate in the market with transaction costs is always greater than that without transaction costs. One possible assessment of this effect is that the existence of transaction costs makes investments less attractive and causes an increase in consumption. In the previous model we assumed an interest rate of zero, but since the interest rate in the real world is not zero, we will look at the interest rate in this section.
Data description
And from adaptive inflation expectation, the expected inflation rate is based on the lagged inflation rate. The transaction cost λ mainly consists of purchase tax, local education tax, special agricultural tax, real estate agent's commission and legal fees.
Results
In other words, the optimal trade in the illiquid asset is to maintain the proportion of investment in the illiquid asset as. In section 4, we have seen the optimal share of the liquid asset when the investor sells or buys the illiquid asset. Similarly, when the invested proportion of the illiquid asset reaches π1 = 0.51, the investor sells the illiquid asset and the proportion of the liquid asset is π¯2 = 0.519 at that time.
After the investor sells (or buys) an illiquid asset, the optimal share of the liquid asset is 51.876% (or 51.813%). In Section 4, we obtained the asymptotic expansion of the fraction of the optimal consumption rate for small transaction costs λ. Because the coefficient λ23 is the sensitivity of the increase in the consumption rate due to transaction costs is 0.000125, which is small compared to the case of an illiquid asset.
Remember also that −q(1−ρ2(1+q)2)σ12ζ2, the coefficient λ23 represents the sensitivity of the increase in the consumption rate generated by the transaction costs. To compare with the effect of transaction costs on consumption, where the market model only has the illiquid asset, we therefore consider. So in the market with a liquid asset, the effect of transaction costs on consumption is stronger.
Also, we can see that the incremental amount of the optimal consumption rate is much smaller than the case with a liquid asset. The optimal investment of the illiquid asset is to trade the proportion of the illiquid asset within a range (no transaction region). And when an investor sells (respectively buys) the illiquid asset, the optimal percentage of the liquid asset is 51.8% (respectively, 51.9%).
Finally, the optimal strategy for the consumption rate is to consume 0.12% of total wealth in a month. As the transaction costs change, we can see that the effect of the transaction costs is large on the illiquid asset. Compared to the model with only an illiquid asset, the effect of transaction costs on consumption is more noticeable.
The reason for this is that the existence of the liquid assets causes an increase in the volatility of the overall wealth process, and therefore more frequent trading of the illiquid assets. We have seen that the existence of liquid assets does indeed have an effect compared to the market having only illiquid assets.
