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Chapter 1 Introduction Capital Markets, Consumption, and Investment

By Loretta Sims,2014-08-11 22:50
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Chapter 1 Introduction Capital Markets, Consumption, and Investment

Topic 1: Finance Theory Under Certainty (One period model) Copeland, Weston, Shastri Chapter 1

    How do capital markets benefit society? Consider a one-period economy. Without capital markets and without production, individuals are constrained to either (1) consume their particular initial endowments, y, y, or (2) store 01

    all of, or a portion of, their time 0 endowment (at a zero interest rate) for consumption at time 1.

    Where current endowments are low, but where future endowments will be high, this could cause hardship. Capital markets allow these individuals to borrow against their anticipated future income (endowments). They borrow from other individuals who lend excess current income (endowments) at positive interest rates. Why does the interest rate have to be positive?

    The investment/consumption decision depends on the individual's subjective preference for consumption across the different time periods, opportunities for investment in productive opportunities, and the market interest rate, r.

    To analyze this decision, we will consider Irving Fisher's Theory of the Real Interest Rate (1930).

    What we get from the analysis:

    1) How capital markets lead to an efficient allocation of resources to investment projects;

    2) A foundation for net present value rule;

    3) Fisher Separation Theorem.

    Assumptions:

    1) One period world (today and end of the period).

    2) Certainty

    3) Perfect capital markets (i.e., no taxes, transaction costs, etc,)

    4) Initial wealth endowment (y and y) 01

    5) Individual preferences for consumption today (t = 0) vs. consumption at the end of the period (t = 1) are a

    function of their utility function, U(C, C) and associated indifference curves 01

    C0The slope of an individual’s indifference curve at a particular point (C, C) is denoted as . MRS01C1

    1~?~?UCCUCC(,)(,)C01010?(?(MRSWe will see later that . Further assume that individuals C?(?(1CC?)?)01

    ~?~?U(C,C)U(C,C)0101?(?(prefer more consumption to less, and , and the marginal utility of 00?(?(CC01?)?)

    22~?~?U(C,C)U(C,C)0101?(?(consumption is negative, and . We will talk more about utility 0022?(?(CC01?)?)

    theory in chapter three. What do these utility functions look like graphically?

    6) Market determined interest rate for borrowing and lending, r. Let T(C,C) be the transformation function 01

    relating t = 0 and t = +1 consumption opportunities. The lower the consumption this period, the higher the

    investment (at interest rate r) and, therefore, the higher the consumption next period. The marginal rate of

    C0transformation at a particular point, for example at (C, C), is denoted as . With borrowing and 01MRTC1

    C0lending at a fixed interest rate r, , for all possible C,C. MRT(1r)01C1

    Later:

    7) Firms with productive capacity, defined by the marginal rate of transformation and initial capital. Let

    T(P,P) be the transformation function that relates dividend payments (production) today with dividend 01

    payments (production) next period. The lower the dividend this period, the higher the t = 0 investment.

    This results in a higher dividend payment (production) next period. First, a graphical presentation

    In each case, note how the individual’s budget constraint (i.e., opportunity set) expands (or shrinks). Also, note how the individual’s utility from consumption changes as the budget constraint changes.

    Case 1: Individual endowed y, y, no storage, no capital markets, no production. Note the amount of “utility” 01

    the individual has at this point.

    Case 2: Individual endowed y, y, storage, no capital markets, no production. The tangency point of the 01

    individual’s indifference curve with the “storage” line indicates preferred location for individual.

    A. Who would want to store?

    B. How much better off is this individual?

    C. Who wouldn’t want to store?

    Case 3: Individual endowed y, y, storage, capital markets, no production. The “capital market line” reflects the 01

    interest rate and has a constant slope of -(1+r).

    A. What do the X and Y intercepts signify?

    B. Let’s graphically identify an individual’s preferred location on the capital market line. As before,

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    notice that the individual’s indifference curve is tangent to the capital market line at this preferred

    location. What does this mean? That is, at what rate is the individual indifferent to exchanging time 0

    and time 1 dollars at this point?

    C. Where would two different individuals, with difference time preferences, locate on the capital market

    line?

    D. Important point: at their respective tangency points, what rate are these individuals indifferent to

    exchanging time 0 and time 1 dollars?

    E. What is the amount of lending / borrowing at time 0 for these two individuals?

    F. How is consumption at time 1 affected by the decision to borrow or lend at time 0?

    G. In sum, how does the existence of capital markets (i.e., the ability to borrow / lend at some positive

    interest rate r) benefit individuals? Is anyone worse off with the introduction of capital markets? Is

    everyone better off?

    Case 4: Individual (100% owner of a firm) endowed with y, y, with productive investment opportunities, but 01

    no capital markets. The curved line is called the “production possibility frontier” or “investment opportunity schedule. Let’s assume that projects are infinitely divisible and independent, ordered from best (highest return) to worst (lowest return), resulting in smooth curved line. The initial endowment is the lower right point of the curve. Movement up and to the left is investment in the firm’s projects. How is this case different (the same as) case 3?

    A. How do the different owners time preferences affect the investment decisions of their two firms?

    B. What is the marginal rate of production at the owners’ preferred locations? What do we call the

    marginal rate of production in an undergraduate finance class?

    C. At their respective tangency points, what rate are these individuals indifferent to exchanging time 0

    and time 1 dollars?

    Case 5: Individual (100% owner of a firm) endowed y, y, with productive investment opportunities, and 01

    capital markets.

    A. How much better off is the consumer in case 5 than in case 4?

    B. Examine the graph. Be able to point out:

    Initial endowment (y, y) 01

    Dividend payments by the firm (P, P) 01

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    Investment by the firm

    Rate of return on the last “dollar” invested

    PV and FV of consumer’s wealth at initial endowment

    PV and FV of dividend payments

    NPV of investment

    Preferred location by consumer on the capital market line (and PV and FV of this preferred location)

    Amount of borrowing or lending at time 0 (difference between C and P) and associated adjustment to 00

    time 1 consumption (difference between C and P) 11

    Some valuable finance concepts from Case 5

    1) Fisher separation all investors in the firm are indifferent to exchanging time 0 and time 1 dollars at the market

    interest rate. Thus, they can delegate the investment decisions to firm managers who can make all individuals

    (investors) best off if they use the market interest rate to make investment decisions (i.e., investment doesn’t

    depend on the composition / preferences of the owners). The firm’s managers do this by:

    2) Investing in all projects in which have an IRR > r, or, equivalently, investing in all projects with a positive NPV. 3) Investment policy sets the amount of time 0 and time 1 dividends, which determines investor wealth. 4) Is investor wealth affected by dividend policy? For instance, say the firm wants (for some reason) to pay more

    dividends at time 0. How should they do this?

    Now, a mathematical presentation: Start with an initial endowment of y, y. Let’s allow for borrowing and 01

    lending at interest rate r and no production (case 3 above). To determine the individual's choice between consumption this period versus next period, the following problem needs to be solved:

    maxU(C,C)s.t.T(C,C)00101 1 C,C01

    To solve this constrained maximization problem, the following lagrangian function is formed:

    LU(C,C)T(C,C) 2 0101

    Taking the partial derivative of L w.r.t. C, C and λ and setting each equation equal to 0 (to find the first order 01

    conditions for a maximum value) yields:

    U(C,C)T(C,C)L01010 3 CCC000

    4

    U(C,C)T(C,C)L01010 4 CCC111

    L 5 T(C,C)001

    Therefore, the solution to the consumers choice problem will be defined by equations (3) (5). From these

    equations:

    U(C,C)U(C,C)T(C,C)T(C,C)01010101 6 CCCC0101

    In the context of our problem, an indifference curve is the combination of consumption possibilities at time 0 and 1 for which a consumer is indifferent. Mathematically, it is defined by the collection of points satisfying the following differential equation:

    dU(,) = + = 0 7 CCUdCUdC0101CC01

    This implies that

    UdC1C0 8 = -dCU0C1

    So, the LHS of equation (6) is the equation for the slope of the individual’s indifference curve. (Note, I will continue

    to use the short-hand designation of the partial from this point forward.) In a similar manner, the right hand side of equation (6) is the slope of the opportunity set.

    Equation (6) says that the consumer derives the maximum utility when he/she is at a point on his/her indifference curve that has a slope equal to the slope of the opportunity set (or budget constraint).

    Example: Assume that y and y are $500 and $1650 respectively. Let U(C,C) = ln[(C)(C)] = ln(C) + 0101010

    ln(C). Therefore, the consumer derives 13.6231 utiles from the endowed consumption pattern. Notice there is no 1

    time preference.

    U U What is and ? CC00

    Note these are also the first and second derivatives with respect to C. 1

    Let r = 10%. (Note that the present value (PV) of the endowed consumption bundle is $2000.) This implies that the outer boundary of the opportunity set (i.e., the equation for the capital market line) is defined by the following equation:

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    C = 22001.1C 9 10

    Therefore, write T(C,C) as: 01

    T(C,C) = 22001.1CC 10 0101

    Equation (2) gives the form of the lagrangian function.

    Lln(C)ln(C)(22001.1CC) 11 0101

    Equations (3), (4), and (5), given above, give the necessary first order conditions for a maximum. What are the equations for the F.O.C.s?

    L 12 C0

    L 13 C1

    L 14

    Now, let’s solve for the maximum

    *What is ? C0

    *What is ?. C1

    *C0The slope of the consumer's indifference curve at the optimum, should be equal to the slope of the budget *MRSC1

    C0constraint, , i.e., capital market line (= -1.1). Using equation (6): MRTC1

    UdC1C0 15 = -dCU0C1

    **Also, = $1000 and = $1100 results in 13.9108 utiles. Testing other possible combinations: C = $990 and C = CC0101

    $1111 13.9107 utiles; C = $1010 and C = $1089 13.9107 utiles. Notice that the PV of the preferred 01

    consumption pattern is still $2000 [also for ($990, $1111) and ($1010, $1089)]. Therefore, reallocation of consumption along the capital market line does not increase (or decrease) the wealth of the consumer.

    As discussed above, while ($1000, $1100) solved the maximization problem for one particular consumer, it is likely that other consumers will prefer another allocation across the two time periods. For example, starting with

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initial endowments of ($500, $1650), compare two additional individuals. Assume further that for individual M, the

    MRS at this point is -1.02, and the MRS for individual P is -1.20.

    To add some concreteness to the example, assume that individual M, is indifferent to giving up $1.00 t = 0

    consumption to get $1.02 in t = +1 at the initial endowment of ($500, 1650). This would imply a subjective interest rate r = 2%. Notice that the subjective interest rate is higher for individual P, r = 20%. Mp

    Starting at the initial endowment, and faced with an opportunity to invest or borrow at a one period interest rate of r = 10%, what would individuals M and P do?

    In other words, t = 0 consumption is so valuable to P that he/she is willing to borrow (against t = +1

    consumption) at an interest rate of up to 20% to increase t = 0 consumption. On the other hand, since r > r, M M

    would want to lend money to increase future consumption. From equation (6) we know that individual M will

    continue to lend until he/she gets to a point where his/her MRS is equal to -1.1. The same can be said about

    individual P.

    Therefore, individual preferences for consumption today (individual P) and for next period (individual M), do

    not affect the result that at their preferred allocation, all consumers are indifferent to trading t = 0 and t = +1 dollars

    **CC00at the market interest rate. In other words, = for all consumers. MRS**MRTCC11

    Example with production and no capital markets (Case 4): As before, assume U(C,C) = ln(C) + ln(C). 0101

    3Assume PPF is given by the following equation: P = $5000 0.00004 (P) + 1650 for value of P between $0 and 100

    $500. (In this case, the initial endowment for the firm is $500 at time 0, $1650 at time 1. Investing all $500 brings in

    $6500 at time 1.) Dividend payments (and consumption, C and C) at time 0 and time 1 equal cash flows from the 01

    production function (P and P). With no capital markets (and no storage, to makes things simple), consumer (owner) 01

    needs to consume dividend payments. Find the optimum in same manner as above.

    **** = = $346.391496 and = = $4987.50 CCPP1001

    The slope of PPF at this point is the slope of the indifference curve for the consumer/owner = -($4987.50/$346.391496) = -14.3984. Thus, the last “dollar invested had an IRR of 1339.84%. As mentioned, owner

    M will direct the firm to invest differently than owner P. This implies that managers must direct

    investment/dividend policy based on the desires of its owners. This could create a problem if 50% of the firm is owned by M and 50% by P.

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    Example with production and borrowing and lending along the capital market line: Now, the owner directs

    management to invest in all projects that earn at least the market interest rate (10%). To solve, first find the tangency point of capital market line with the PPF (i.e., where the slope of the PPF is -1.1). This fixes dividend payments and the capital market line. Then, find the consumer’s preferred location on the capital market line. Solution:

    ** = $95.7427 and = $6614.8943 PP01

    **C = $3054.6415 and = $3360.1056 C01

    Some points of interest:

    *C01) What is ? MRS*C1

    Note - the same can be said for any other owner (i.e., at their preferred consumption point, the slope of their

    indifference curve for all owners is equal to the slope of the capital market line, i.e, -1.1.)

    2) Current value (PV) of firm (no investment in projects) = $500 + 1650 /1.1 = $2000

    3) Time one value of firm (maximum investment in capital markets, instead of projects) = $2200

    4) Time one value of firm (maximum investment in projects) = $6650

    5) Amount of optimal investment =

    6) PV of dividends (and PV of consumption) =

    7) FV of dividends (and FV of consumption) =

    8) NPV of total investment =

    9) Amount borrowed at time 0 =

    10) Amount paid back at time 1 =

    11) NPV of last “dollar” invested =

    12) IRR of last “dollar” invested =

    13) All owners of the firm are (weakly) better (i.e., have maximum utility) off if the firm invests optimally. So,

    they can delegate investment decisions to management (Fisher Separation).

    A word about the market interest rate: We have assumed that the market interest rate is given. However, the

    intuition behind the setting of the market interest rate can be determined by considering a world composed of two identical firms and two individuals (with P the owner of one firm and M the owner of the other). The market interest

    rate would be the interest rate that equates supply (money lent) with demand (money borrowed). Note that changes

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in the market rate affect investment in projects, and therefore dividend payment (so this is not a trivial problem).

    Asset valuation under certainty

    In the simple one-period world described above, the current value (V) of an asset that generates cash flow V at 0Ttime 1 (i.e., T = 1), is

    1VV(1r) 16 T0

    More precisely, this is the time 0 value of the asset, compounded once per period. We will derive this later. But first, we

    will derive the general formula for asset valuation (at time t, where t < T), using continuous compounding. Derivation of the formula for the value at time t of a single payment paid at time T (continuous compounding)

    The change in value of an asset, over some interval, is equal to:

    dVVrdt 17 tt

    1dVrdt 18 tVt

    Recall that d ln(x) / dx = 1/x. So, d ln(x) = 1/x dx.

    dln(V)rdt 19 t

    ?dln(V)?rdt 20 t

    ln(V)rtk 21 t

    rtkVe 22 t

    rtkVee 23 t

    kWrite equation for V and solve for e T

    rTkVee 24 T

    krTeVe 25 T

    Substitute back into equation (23):

    rtrTVVee 26 tT

    r(Tt)VVe 27 tT

    So, equation (27) is the time t value of a single payment of V, at interest rate r per period, continuous compounding. T

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    Example (1): What is the time 0 value of $1 received in five years (r = 10% per year)?

    Example (2): What is the time 2 value of $1 received in five years (r = 10% per year)? Derivation of the formula for the value at time t of an asset that continuously pays $C per period (continuous compounding)

    dVCdtVrdt 28 tt

    dVt 29 CVrtdt

    dVt 30 rVCtdt

    This is a first-order linear differential equation of the form:

    dy 31 p(x)yf(x)dx

    ?p(x)dxTo solve equation (31), use e as the integrating factor and multiply (31) by the integrating factor

    dy?p(x)dx?p(x)dx?p(x)dx 32 ep(x)eyef(x)dx

    ?p(x)dxeyThe left hand side of (32) is the derivative of the product of the integrating factor and the dependent variable: .

    So, write (32) as

    d?p(x)dx?p(x)dx;;eyef(x) 33 d(x)

    To solve, integrate both sides. Now, back to our formula:

    dVtrVC 34 tdt

    ?rdtrteeThe integrating factor is . Multiply by this integrating factor

    dVrtrtrtterVeCe 35 tdt

    drtrt;;VeCe 35 tdt

    Take an integral of both sides

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