In the portfolio network in the figure above, investment arcs have multipliers that increase or decrease the flow on each arc. This kind of portfolio network with multipliers is called a generalized network. We now have all of the components necessary to model most portfolio situations. Multipliers increase or decrease the flow on an arc. The flow of money on an investment arc is multiplied by a gain or multiplier to increase […]

*Portfolio Networks with Income and Capital Gain*

or decrease the flow into a node. Multipliers have the units of total return. A multiple of 110 percent increase the flow on that arc by 10 percent. So 110 percent of the input money flow on the arc will flow out of the arc. Investments normally have income or dividends and capital gains or losses. So there can be a multiplier for the gain representing the increase in price. If the dividends are reinvested that will increase the flow. In this case the dividend yield can be included in the multiplier. In the portfolio network figure above there are multipliers that represent the capital gain on each investment arc. There are also profit boxes that represent the dividends or income on each investment arc. In addition there are bounds on the investment arcs that restrict investment in each investment or flow of money on each arc. Also note that we have added flow variables x1 to x4 representing the solution to the portfolio network. The variables x1 to x4 will give the optimal flow on each arc so x1 to x3 gives the amount we should invest in bonds, or invest in stock 1, or stock 2 to get the maximum money flow, x4 out of the portfolio network.

Portfolio Network with Total Return Multipliers

The portfolio network figure above shows the same portfolio network as in the previous figure but with reinvestment of income or dividends. The dividend yield is added to the gain multiplier to give the total return on that investment. The total return is used as a multiplier on that arc. This will be that most useful portfolio network since it simplifies the objective function to maximizing the flow out of the output node (node 1). The objective will be to maximize the flow x4 out of node 1. Using generalized portfolio networks with return multipliers the objective will always be to maximize the money coming out of the portfolio network.

*Investment Portfolio Network with Flow Weights and Margin Buying*

The above portfolio network figure shows an investment portfolio network with total return multipliers, m1 to mN. There are N possible investments. There are N arcs from node 0 to node 1. Investment weights are now the variables on each arc. The sum of the weights w1 to wN must equal 1. Each weight, w1 to wN gives the percent of the initial investment in the N investments. This is useful since the solution will apply to any size investment. This particular investment portfolio network also allows margin buying. In the margined portfolio network, wM gives the equity or money that the investor invests in the portfolio. This equity money is called the margin. In addition to the margin the investor can borrow money to add to his or her investment. The borrowed money requires payment of interest on the loan from node 1. By adding an investment arc from node 1 to node 0 with arrow pointing from the future to the present we can borrow money from the future portfolio. This arc from left to right is called a loan arc. The interest rate that must be paid on borrowed money in this portfolio network figure is iC . The weight, wB on the loan arc from node 1 to node 0 is the percent of the initial investment to be repaid at the end of the period. The Book gives a specific example of this portfolio network model. Notice that node 1 is at the end of period 1. If we are interested in a multiple period model there could also be a node 2, node 3, etc.