Lecture 3: Separation Fund Theorem

1. Lecture 3: Separation Fund Theorem

2. Separation Fund Theorem Contents • Recommended reading • Investment choices under certainty • The investment schedule • Indifference curves • Optimal consumption decision • The money market line • Investing and financing decisions • Separation of investment and financing decisions • A formal solution • An example

3. Separation Fund Theorem Recommended reading • Bodie, Z. Kane, A. and Marcus, A. (2014). Investments. McGraw-Hill.10th, ed. • McMillan, Pinto, Pirie, Gerhard, “Investments: Principles of Portfolio and Equity Analysis”, pp. 200-206 • Levy and Sarnat, ‘Portfolio and Investment Selection: Theory and Practice’, Chapter 3

4. Separation Fund Theorem Investment choices under certainty The investment schedule Suppose there is an investor with an initial endowment equal to w0=1.000$ who should make a decision about the portion of her wealth that will be consumed now and the portion that will be invested in productive assets.   Investment opportunity schedule – production opportunity set: Assume that the available options for this representative investor consist of four 1-year investments, the NCFs of which are shown below (in a descending order of profitability (IRR)): t0 t1 a) (100) 200 b) (100) 150 c) (500) 600 d) (300) 315

5. Separation Fund Theorem Investment choices under certainty The investment schedule The investment opportunity set for this representative individual investor ($1,000 initial endowment) can be illustrated in a two dimensional graph the horizontal axis of which represents the current consumption while the vertical one represents the future consumption. t0 t1 Investment 1: invests 100$, consumes 900$ consumes 200$ Investment 1,2: invests 200$, consumes 800$ consumes 350$ Investment 1,2,3: invests 700$, consumes 300$ consumes 950$ Investment 1,2,3,4: invests 1000$, consumes 0$ consumes 1265$

6. Separation Fund Theorem Investment choices under certainty The investment schedule Figure 1 Investment/Opportunity set

7. Separation Fund Theorem Investment choices under certainty The investment schedule The discrete attainable points of this figure could be connected to form the investment opportunity curve. This is tantamount to assuming that the investment projects are infinitely divisible, that is they can be broken down into very small components so that the investment alternatives can be represented by a continuous curve instead of by a series of discrete points. Figure 2 Investment Opportunity curve The slope of a line tangent to the investment opportunity curve is the rate at which a dollar of consumption foregone today is transformed by productive investment into a dollar of consumption tomorrow. It is the marginal rate of transformation (MRT) offered by the production/investment opportunity set.

8. Separation Fund Theorem Investment choices under certainty Indifference curves Consider a rational individual who investigates all alternatives in order to make a decision about current and future consumption (c0,c1) which will maximize his satisfaction. Figure 3 Indifference Curves

9. Separation Fund Theorem Investment choices under certainty Indifference curves One possible combination is M. Whenever a combination such as M is replaced by an alternative one located in the direction of the arrow marked a, there is an increment in current consumption without any penalty in the second period’s consumption. In the case that M is replaced by another alternative located in the direction of the arrow marked b, there is a reduction in the second period’s consumption without any compensating benefit in the current consumption. Since, any movement in the direction of arrow b reduces the investor’s satisfaction while any movement in the direction of arrow a, increases it, a point can be found between a and b (for example N) at which the individual’s satisfaction is neither increased nor decreased.

10. Separation Fund Theorem Investment choices under certainty Indifference curves Figure 4 Indifference Curves At this point N, the impact of the increase in c0 on the individual’s satisfaction is fully offset by the decrease in c1. So the investor is indifferent to the choice between the two consumption combinations represented by M and N. This is generalized in order to account for all possible combinations leading to the curve of the following figure, the indifference curve:

11. Separation Fund Theorem Investment choices under certainty Indifference curves Figure 5 Indifference Curves • The indifference curve declines from left to right which indicates that the rational investor must be compensated by an increase in future consumption when his current consumption is reduced. The slope of the indifference curve at each point is the marginal rate of substitution, MRS. • The indifference curve is convex, since each additional decrease in current consumption requires increasingly larger increment of future consumption, if the individual is to remain indifferent to the change. The indifference curves of a single individual cannot intersect.

12. Separation Fund Theorem Investment choices under certainty Indifference curves The individual would prefer a combination which would allow him to reach the highest feasible indifference curve. For example, the following figure represents the individual’s indifference map on an opportunity set of alternative investment opportunities (a, b, d, f and g). The individual would prefer to reach indifference curve I5, but such a combination is not attainable. The optimal choice is to reach the indifference curve I3, as no other existing choice will permit him to reach a higher level of satisfaction, and as a result, a-investment constitutes his optimal choice. Figure 6 Indifference Curves

13. Separation Fund Theorem Investment choices under certainty Indifference curves The utility function of an individual characterizes his preferences. Every individual is assumed to prefer more consumption to less. In other words, the marginal utility of consumption is always positive. Also, we assume that the marginal utility of consumption is decreasing. The total utility curve shows the utility of consumption at the beginning of the period, assuming that the second-period consumption is held constant. Figure 7 Utility function

14. Separation Fund Theorem Investment choices under certainty Indifference curves If we consider the second period’s consumption and the corresponding utility, the three-dimension result provides a description of trade-offs between current and future consumption. Figure 8 Utility function • The dashed lines represent contours along the utility surface where various combinations of current and future consumption provide the same total utility. Since, all points along the same contour (i.e. points A & B) have equal total utility; the individual will be indifferent with respect to them. Therefore, the contours are called indifference curves.

15. Separation Fund Theorem Investment choices under certainty Optimal consumption decision • Now we are in a position to combine the concepts of an investment opportunity curve and an indifference map in order to determine the investor’s optimal investment policy. • The following figure, superimposes the indifference curves of a hypothetical individual confronted by the investment opportunities. This figure shows that the cash flow patterns (c0*, c1*) denoted by c* permit the individual to reach her highest indifference curve (I2). This occurs at the point of tangency between the investment opportunity curve and an indifference curve, i.e. MRS = MRT (necessary but not sufficient condition). Figure 9 Utility function

16. Separation Fund Theorem Investment choices under certainty Optimal consumption decision The optimal consumption combination also dictates the optimal investment policy: • Point c* can be attained by consuming c0* in the current period and investing the amount w0 - c0* = i0 in order to provide a cash flow in the second period which is sufficient to support a consumption of c1*. • It is obvious from the analysis that two investors with different indifference curves will choose different investment-consumption combinations in spite of the fact that both have the same initial endowment w0 and face the same opportunity set. We turn now to see how this analysis and conclusions are changed when investors are allowed to borrow and lend at the riskless interest rate r.

17. Separation Fund Theorem Investment choices under certainty The money market line • Assuming that the investor can borrow and lend at interest rate r, he may borrow c1/(1+r) in this period and increase his current consumption to c0+c1/(1+r). In the second period he has to repay the loan plus interest, [c1/(1+r)] (1+r) = c1, which is exactly the amount available to him, and his consumption in the second period will be zero. t0 t1 + c1/(1+r) - [c1/(1+r)] * (1+r) = - c1 • The present value of his consumption mix in this case is given by PV = c0 + c1/(1+r), where PV is the present value of consumption, or the maximum that the investor can consume in the first period.

18. Separation Fund Theorem Investment choices under certainty The money market line The consumption in the first period as a function of the current consumption is given below by the equation: c1 = PV(1+r) - c0(1+r) • which is a straight line (with intercept PV(1+r) and slope -(1+r)), indicating that for a given PV of consumption infinitely many combinations of consumptions bundle (c0,c1) exist, all lying on the same straight line characterized by this PV. • By borrowing and lending at the rate r, the investor can move along this straight line as he wishes, adjusting the consumption bundle without changing the PV level; hence the name money market line. • If the investor has a higher initial consumption bundle, he can move on a superior money market line (PV2). On the other hand if his initial consumption bundle is lower, then he should move to an inferior money market line (PV3). By considering alternative values of PV we derive a family of parallel straight lines each with the property that all consumption combinations on a given line represent the same PV; hence the name iso-PV lines or equal-PV lines.

19. Separation Fund Theorem Investment choices under certainty The money market line Figure 10iso-PV lines • Which PV line will the investor desire to reach? • Obviously, he would prefer the highest line. But not all of these lines represent attainable cash flow combinations. • The feasibility of an iso-PV line depends on the individual’s initial endowment, w0, as well as the available investment opportunities (the investment opportunity curve).

20. Separation Fund Theorem Investment choices under certainty The money market line Figure 11iso-PV lines and investment opportunity curve • More specifically, the individual would prefer to reach the line PV4. However, none of the combinations (c0, c1) which lie on this line are attainable because this line lies to the right of the investment curve. • So, his choice would be the PV3 since this line includes the feasible combinations of c0* and c1* that correspond to the highest feasible money market line. This choice, is achieved if the individual’s investment has an initial capital outlay of I0=w0-c0*.

21. Separation Fund Theorem Investment choices under certainty Investing and financing decisions • Will an individual who participate into a financial market, i.e. who can borrow and lend at interest rate r, necessarily choose the cash flow (consumption) combination denoted by point c*? • The answer of this question is conditional on the fact that the investor’s preferences should be satisfied by considering both the investing and the financing opportunities of the individual. There are two cases for the optimum choice of the investor: Case 1: lending Figure 12 Lending money

22. Separation Fund Theorem Investment choices under certainty Investing and financing decisions Case 1: lending • The slope of the investment opportunity curve at c* is equal with the slope of the market line curve which is -(1+r). • In this case the individual invests the amount I0 (i.e. w0-c0*) at t0 and reaches the point c*, where the point of tangency between the highest attainable market line and the opportunity curve are the same. • But the indifference curve which passes through point c*(I1) lies below I2, which is also attainable if financial alternatives are taken into account. • Given his tastes, the individual can reach a higher level of satisfaction by lending the amount (c0*-c0**) at the interest rate r. • This is indicated by a movement along the market line to c** at which point the market line is tangent to indifference curve I2. • The individual prefers lending to investing beyond point c* because the interest that he will receive in the second period is greater than the investment’s cash flows of the second period. • The effective rate of interest on the financial transaction r is greater than the rate or return on productive investment beyond this point, as a comparison of the slopes of the investment curve and the PV line clearly shows.

23. Separation Fund Theorem Investment choices under certainty Investing and financing decisions Case 2: Borrowing Figure 13 Borrowing money

24. Separation Fund Theorem Investment choices under certainty Investing and financing decisions Case 2: Borrowing • In this case the individual, initially chooses point c*, that is investment at t0 of w0-c0*, where the market line is tangent at c* with the investment productivity curve, which means that the point of tangency between the highest attainable market line and the opportunity curve are the same. • But the indifference curve which passes through point c*(I1) lies below I2, which is also attainable if financial alternatives are taken into account. Given his tastes, the individual can reach a higher level of satisfaction by borrowing the amount (c0**-c0*) at the interest rate r, after investing the amount w0-c0* at t0. • This is indicated by a movement along the market line to c** at which point the market line is tangent to indifference curve I2. • Should the point of tangency with the investment curve c** lie to the right of point c*, the individual would again invest up to point c* as before, but would borrow the amount (c0**-c0*) in order to increase his current consumption. • The effective rate of interest on the financial transaction r is lower than the rate or return on productive investment below the point c*, as a comparison of the slopes of the investment curve and the PV line clearly shows.

25. Separation Fund Theorem Separation of investment and financing decisions • The slope of the investment opportunity curve at any point measures the marginal rate of productivity (MRT) or the rate of return on the marginal amount invested. The slope of the indifference curve at each point measures the subjective rate of substitution (MRS) between present and future consumption. It is called subjective since it differs from one investor to another. The slope of the PV lines measures the price of money, i.e. the interest rate. • Equating the slope of the investment opportunity curve to the slope of the money market lines we find the optimal investment. Then, equating the slope of the indifference curves to the slope of the money market line through the optimal investment point, we find the optimal amount of borrowing or lending, i.e., the optimum financial decision.

26. Separation Fund Theorem Separation of investment and financing decisions • The striking feature of the analysis is that the optimal investment decision denoted by point c* does not depend on the shape of the indifference curves. Whether the individual desires to redistribute his consumption over time by either borrowing or lending, the investment decision remains the same. Thus, as long as the individual chooses investments so as to maximize PV of consumption, that is to reach the highest market line, he also ensures that he will be able to maximize his utility (satisfaction) by redistributing (if necessary) his consumption over time by means of borrowing or lending. • The independence of investment and financing decisions is called ‘separation’ and lies at the very heart of the modern theory of finance. It is the existence of an efficient capital market which permits the individual to reach physical investment decisions without explicitly considering the financing decisions, as long as the opportunity cost of using the capital resources is fully reflected in the evaluation of the economic investment opportunities.

27. Separation Fund Theorem A formal solution From the separation theorem it is obvious that in order to find the optimal investment-consumption decision we have to follow two steps: Step 1: • equate the slope on the investment opportunity curve to the slope of the money market line -(1+r): the solution of this equation yields the optimal investment in physical assets. • suppose that the investment schedule is given in the functional form by some function g relating the futures consumption to investment: c1 = g(w0-c0) = g(I) • to find the optimal investment I*, equate the slope of the investment opportunity curve to the slope of the money market line -(1+r): dc1/dc0 = g’(I) = -(1+r) • given a specific investment opportunity function, we can solve the above equation which yields the optimal investment in physical assets, I*. 

28. Separation Fund Theorem A formal solution Step 2: • equate the slope of the money market line through the optimal investment point from step 1 to the slope of the indifference curve • the solution yields the optimal financial activity, namely the optimal borrowing or lending, and leads to the optimal consumption bundle reached by combining investment in physical assets and financial activity • the investor’s utility U from any consumption bundle (c0,c1) is given by U=f(c0,c1), where U denotes utility and f is the functional form of the indifference curve • we seek the tangency point between the money market line through the optimal consumption point from step 1 and one of the individual’s indifference or utility curves • take a total differential of U to obtain:

29. Separation Fund Theorem A formal solution • since on a given indifference curve (the one that is tangent to the highest money market line) there is no change in the level of the utility between different consumption bundles, dU must be equal to zero, which implies that on a given indifference curve we must have, • But the LHS of this equation is the slope of the given indifference curve. The slope is along a given indifference curve U, since we measure the required change in c1, when we decrease c0, such that the investor’s utility does not change. • thus to solve for the optimal amount of borrowing or lending, use the following equation • the solution indicates whether the investor is a borrower or a lender

30. Separation Fund Theorem An example Suppose that there exist an investor with initial endowment equal to $ 20,000. The interest rate is 20%. The investment opportunity curve is determined by the following function: c1 = g(c0) = 240*(20,000-c0)^0.5. The average indifference curve is determined by the following equations: U(c0,c1) = f(c0,c1) = c0*c1. Questions • What is the optimum consumption decision? • What is the optimum investment decision? • What is the optimum financing decision?