The securities market economy -- theory Abstracting again to the two- period analysis - - but to different states of payoff

The securities market We again assume an initial period, o, and some date in the future, 1 We again assume an initial period, o, and some date in the future, 1 This is a two-period analysis as an abstraction from the securities market to point out some basic foundations, now that we have laid out basic intertemporal utility issues, expected utility, and risk aversion This is a two-period analysis as an abstraction from the securities market to point out some basic foundations, now that we have laid out basic intertemporal utility issues, expected utility, and risk aversion

Again, consumption at date o is Co Again, consumption at date o is Co Consumption at date 1 is C1 Consumption at date 1 is C1 But C1 can come in different states, S But C1 can come in different states, S C1 is actually an S-dimensional vector C1 is actually an S-dimensional vector C1 = (C11, C12,...., C1s) for states S = 1,..., s states of the world C1 = (C11, C12,...., C1s) for states S = 1,..., s states of the world

C1s can be Cs if there is no confusion that we are in the future date 1 C1s can be Cs if there is no confusion that we are in the future date 1 Co and C1 (or C1s) may be restricted to be positive Co and C1 (or C1s) may be restricted to be positive We will assume there is utility over this intertemporal consumption as U(Co, C1), like, for example U = Ln(Co, C1) as we have seen before We will assume there is utility over this intertemporal consumption as U(Co, C1), like, for example U = Ln(Co, C1) as we have seen before

Utility and Consumption U = Ln(Co, C1) restricts Co, C1 > 0 U = Ln(Co, C1) restricts Co, C1 > 0 We assume there are I agents in the economy, i = 1, 2,..., I We assume there are I agents in the economy, i = 1, 2,..., I Utility of the ith agent is given by U i (Co, C1) --- which gives a consumption plan for the agent Utility of the ith agent is given by U i (Co, C1) --- which gives a consumption plan for the agent

The agents endowment Agent i’s endowment of wealth or previous payoff from securities that adds to wealth is given by Wo i at date o -- Wo could mean i starts with some portfolio of securities Agent i’s endowment of wealth or previous payoff from securities that adds to wealth is given by Wo i at date o -- Wo could mean i starts with some portfolio of securities The agents endowment of wealth or previous payoff from securities at date 1 is W1 i The agents endowment of wealth or previous payoff from securities at date 1 is W1 i

Properties of the utility function U is increasing at date o if U is increasing at date o if U(Co΄, C1) ≥ U(Co, C1) when Co΄≥Co for every C1 U(Co΄, C1) ≥ U(Co, C1) when Co΄≥Co for every C1 U is increasing at date 1 when U is increasing at date 1 when U(Co, C1΄) ≥ U(Co, C1), for C1΄ ≥ C1 for every Co U(Co, C1΄) ≥ U(Co, C1), for C1΄ ≥ C1 for every Co

U is strictly increasing for U(Co΄, C1) > U(Co, C1) for Co΄> Co for every C1 U is strictly increasing for U(Co΄, C1) > U(Co, C1) for Co΄> Co for every C1 U(Co, C1΄) > U(Co, C1) for every C1΄> C1 for every Co U(Co, C1΄) > U(Co, C1) for every C1΄> C1 for every Co If U is strictly increasing over all dates, Co, C1, then Utility is strictly increasing If U is strictly increasing over all dates, Co, C1, then Utility is strictly increasing

The agents problem in this securities market Let P = price Let P = price Let h = holding, or holdings, like a portfolio of securities (h is not payoff or payment as we discuss later in connection with expected utility) Let h = holding, or holdings, like a portfolio of securities (h is not payoff or payment as we discuss later in connection with expected utility) Let X be the payoff matrix from the holdings of securities Let X be the payoff matrix from the holdings of securities

The agent’s problem is to: The agent’s problem is to: Max Co,C1,h U(Co, C1) Max Co,C1,h U(Co, C1) subject to: subject to: Co  Wo – Ph Co  Wo – Ph C1  W1 + hX C1  W1 + hX and this is the two-period securities problem and this is the two-period securities problem

Now let’s go back to the agent’s problem Max Co,C1,h U(Co, C1) Max Co,C1,h U(Co, C1) subject to: subject to: Co  Wo – Ph Co  Wo – Ph C1  W1 + hX C1  W1 + hX Notice, Wo – Ph is exactly how much we can consume in period 0, or this is just wealth minus how much we are investing Notice also, that C1 is just added wealth + what we get from our holdings as a return

Now we have an optimization problem to maximize U(Co, C1) subject to two constraints Now we have an optimization problem to maximize U(Co, C1) subject to two constraints One constraint says that current consumption is less than or equal to the initial period wealth endowment minus how much the agent pays for the portfolio One constraint says that current consumption is less than or equal to the initial period wealth endowment minus how much the agent pays for the portfolio

The second constraint indicates that future consumption has to be less than future period endowment plus the payoff of the holdings The second constraint indicates that future consumption has to be less than future period endowment plus the payoff of the holdings We also have a hidden constraint in here from our first discussions of this security world problem that Co, C1 ≥ 0 We also have a hidden constraint in here from our first discussions of this security world problem that Co, C1 ≥ 0

So, we now have what is called a Kuhn-Tucker optimization problem (as opposed to the Lagrangian problem we dealt with previously in our economic foundations discussions) So, we now have what is called a Kuhn-Tucker optimization problem (as opposed to the Lagrangian problem we dealt with previously in our economic foundations discussions) We now have inequalities to deal with We now have inequalities to deal with

The Kuhn-Tucker problem Max U(Co,C1) + λ[Co  Wo-Ph] +  [C1  W1 + hX], with two constraint multipliers, λ,  to deal with Max U(Co,C1) + λ[Co  Wo-Ph] +  [C1  W1 + hX], with two constraint multipliers, λ,  to deal with But we can solve these using some optimization algorithm ---- Excel, Matlab, etc But we can solve these using some optimization algorithm ---- Excel, Matlab, etc

The constrained optimization conditions (Kuhn-Tucker conditions) The constrained optimization conditions (Kuhn-Tucker conditions) (*) ∂ o U(Co,C1) – λ  0 and [∂ o U(Co,C1) – λ]Co = 0 [∂ o U(Co,C1) – λ]Co = 0 (**) ∂ S U(Co,C1) –  S  0 and [∂ S U(Co,C1) –  S ]C S = 0 [∂ S U(Co,C1) –  S ]C S = 0 (s, being some state of the payoffs ) (s, being some state of the payoffs ) λP = X  {Well, duh!!} λP = X  {Well, duh!!}

All that ∂ o U(Co,C1) means is that we have taken the derivative of the utility function with respect to Co All that ∂ o U(Co,C1) means is that we have taken the derivative of the utility function with respect to Co All that ∂ o U(Co,C1) – λ means is that we have taken the derivative of the whole constrained optimization problem with respect to Co --- similarly for ∂ S U(Co,C1) -  S All that ∂ o U(Co,C1) – λ means is that we have taken the derivative of the whole constrained optimization problem with respect to Co --- similarly for ∂ S U(Co,C1) -  S

The solution? If utility, U, is quasi concave, meaning U΄> 0 and U΄΄  0, the Kuhn-Tucker conditions derived from the constrained optimization are sufficient and necessary for the maximization (and we will not give the mathematics of that proof here) --- meaning we have at least quasi-concave utility If utility, U, is quasi concave, meaning U΄> 0 and U΄΄  0, the Kuhn-Tucker conditions derived from the constrained optimization are sufficient and necessary for the maximization (and we will not give the mathematics of that proof here) --- meaning we have at least quasi-concave utility

And, if the solution is interior (no border or axis solutions) and ∂ o U(Co,C1) > 0, ∂ S U(Co,C1) > 0, then inequalities in (*) and (**) are satisfied with equality And, if the solution is interior (no border or axis solutions) and ∂ o U(Co,C1) > 0, ∂ S U(Co,C1) > 0, then inequalities in (*) and (**) are satisfied with equality So λP = X  becomes, So λP = X  becomes, X(  /λ) = P = X(∂ S U(Co,C1) / ∂ o U(Co,C1) ) = X/r = JUST PRESENT X(  /λ) = P = X(∂ S U(Co,C1) / ∂ o U(Co,C1) ) = X/r = JUST PRESENT VALUE VALUE

So the price of security j ( the cost in units of date o consumption of a unit increase in the holding of the jth security) = the sum over states of its payoff in each state multiplied by the marginal rate of substitution between consumption in that state and consumption at date 0 So the price of security j ( the cost in units of date o consumption of a unit increase in the holding of the jth security) = the sum over states of its payoff in each state multiplied by the marginal rate of substitution between consumption in that state and consumption at date 0

Intertemporal marginal rate of substitution ∂ S U(Co,C1) / ∂ o U(Co,C1) is the marginal rate of substitution between consumption in that state s and consumption at date o ---- MRS Co,C1 ∂ S U(Co,C1) / ∂ o U(Co,C1) is the marginal rate of substitution between consumption in that state s and consumption at date o ---- MRS Co,C1 Notice, that P = X/r is just SIMPLE PRESENT VALUE Notice, that P = X/r is just SIMPLE PRESENT VALUE Present value P = X/r = (the earnings)/discounted Present value P = X/r = (the earnings)/discounted We have basic finance! We have basic finance!

We can also get NPV Notice also, that net present value is just NPV = -P + X/r, and if X/r > P, then NPV > 0, which is a concept with which we are all familiar Notice also, that net present value is just NPV = -P + X/r, and if X/r > P, then NPV > 0, which is a concept with which we are all familiar

What are we doing? Security j is identified by its payoff, x j Security j is identified by its payoff, x j Security j comes in payoffs of different states, s, at date 1, x js --- these payoffs are in terms of consumption good, C, since the payoff is converted to consumption Security j comes in payoffs of different states, s, at date 1, x js --- these payoffs are in terms of consumption good, C, since the payoff is converted to consumption

Securities are claims on cash flows coming from the production and sale of goods and services Securities are claims on cash flows coming from the production and sale of goods and services The payoffs x js can be positive, negative or equal to zero The payoffs x js can be positive, negative or equal to zero There are a finite number, j, of securities with payoffs x 1,.., x j There are a finite number, j, of securities with payoffs x 1,.., x j

Therefore, X is a j by s matrix of payoffs of all securities in the economy Therefore, X is a j by s matrix of payoffs of all securities in the economy Payoff matrix

The portfolio A portfolio is the holdings of J securities, j = 1, 2,..., J A portfolio is the holdings of J securities, j = 1, 2,..., J Holdings, h, can be positive, negative or equal to zero Holdings, h, can be positive, negative or equal to zero h > 0 is a long position in that particular set of securities h > 0 is a long position in that particular set of securities h < 0 is a short position h < 0 is a short position

So the portfolio is the J-dimensional vector, h, where h j is the holdings of security j So the portfolio is the J-dimensional vector, h, where h j is the holdings of security j The portfolio payoff is hX for all the holdings The portfolio payoff is hX for all the holdings The set of payoffs available via trades in securities markets is the Asset Span, but we will not worry too much about this definition The set of payoffs available via trades in securities markets is the Asset Span, but we will not worry too much about this definition

Price of the security P = the price of the security P = the price of the security P = (P 1, P 2,..., P J ) P = (P 1, P 2,..., P J ) Ph = the sum of the prices multiplied by the holdings over J prices and securities, and this is the price (or value) of the portfolio h at securities prices P Ph = the sum of the prices multiplied by the holdings over J prices and securities, and this is the price (or value) of the portfolio h at securities prices P

Return Return, r j, on security j is its payoff divided by its price --- x j / P j, for P j being nonzero Return, r j, on security j is its payoff divided by its price --- x j / P j, for P j being nonzero r j = x j /P j = gross return r j = x j /P j = gross return Net return is gross return - 1 Net return is gross return - 1

Example: S = 1, 2, three states S = 1, 2, three states x 1 = (1, 1, 1), x 2 = (1, 2, 2) --- two securities x 1 = (1, 1, 1), x 2 = (1, 2, 2) --- two securities Payoff matrix

Payoff states and risk Let’s look at security 1 over all its states, x 1 = (1, 1, 1) Let’s look at security 1 over all its states, x 1 = (1, 1, 1) The payoff is 1 in each state! The payoff is 1 in each state! Security 1 is risk free, that is the payoff is 1 in each state Security 1 is risk free, that is the payoff is 1 in each state E (x) = 1, VAR (x) = 0 =  2 E (x) = 1, VAR (x) = 0 =  2

Now look at security 2 Now look at security 2 x 2 = (1, 2, 2) x 2 = (1, 2, 2) Security 2 is risky, with a payoff of 1 in state 1, and then a payoff of 2 in states 2 and 3 Security 2 is risky, with a payoff of 1 in state 1, and then a payoff of 2 in states 2 and 3 E (x) = mean = 1.67, VAR (x) = 0.33 =  2 E (x) = mean = 1.67, VAR (x) = 0.33 =  2 St. dev (x) = √  2 = 0.58 St. dev (x) = √  2 = 0.58

The excel commands for mean and variance meanvariancest. dev

The excel formulae meanvariancest. dev 111=AVERAGE(C3:E3)=VAR(C3:E3) 122=AVERAGE(C5:E5)=VAR(C5:E5)=SQRT(G5) =STDEV(C5:E5)

We let P 1 = 0.8 and P 2 = 1.25 We let P 1 = 0.8 and P 2 = 1.25 Then, r 1 = X/P = (1/0.8, 1/0.8, 1/0.8), which is (1.25, 1.25, 1.25) Then, r 1 = X/P = (1/0.8, 1/0.8, 1/0.8), which is (1.25, 1.25, 1.25) And r 2 = X/P = (1/1.25, 2/1.25, 2/1.25) = (0.8, 1.6, 1.6) And r 2 = X/P = (1/1.25, 2/1.25, 2/1.25) = (0.8, 1.6, 1.6) Notice what these are?  C1/  Co – 1 = r, or  C1/  Co = 1 + r Notice what these are?  C1/  Co – 1 = r, or  C1/  Co = 1 + r