Portfolio Investment

Transcription

1 Portfolio Investment Robert A. Miller Tepper School of Business CMU Lecture 5 Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 1 / 22

2 Simplifying the framework for analysis We have already enriched our models of auctions beyond the point we can solve the most complicated of them. Competitive equilibrium is commonly assumed in portfolio analysis, widely used to finesse descriptions of institutional detail: 1 Markets are perfectly liquid. (Bid equals ask) 2 Individual order size does not affect prices. (Price taking behavior) 3 There are neither unfilled orders nor excess inventories. (Markets clear) Note the definition does not answer: 1 How is the competitive equilibrium price formed? 2 Why does every trader believe he or she cannot influence price? 3 Does a competitive equilibrium price exist? Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 2 / 22

3 Price taking investors The first two conditions of competitive equilibrium imply that investors are price takers, and are most plausible for small investors. Large trades typically affect the spread, and therefore block traders should also account for the effects of their trading activity on prices they pay. About half the trading in financial securities of publicly listed firms are large trades (over $200,000), many executing off the limit order books. We now derive the fundamental equation that characterizes the trade-off at the individual level between current and future consumption in terms of asset returns, a personalized mean variance frontier for small investors. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 3 / 22

4 Assets Suppose there are J financial securities, and let q t 1,j denote the amount of the j th security owned at the beginning of period t. Let π tj denote the physical firm growth underlying the j th security revealed at the beginning of period t and applied to the securities carried from the previous period. Perhaps the simplest example is a tree, from which you can cut its branches for heat and eat its fruit, or let the tree grow more and plant its fruit. For expositional simplicity, assume markets are perfectly liquid, and let p tj denote the price of (buying or selling) the j th security in period t, again in consumption units. Note that p tj 1 since individuals can consume (liquidate) the asset. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 4 / 22

5 Budget constraint Apart from receiving investment income, the individual agent also receives wages at the beginning of each period t, denoted by w t, also measured in consumption units. Finally we let m t denote the money (liquid non-interest bearing assets) held over from one period to the next. To summarize, upon entering the current period, the individual agent receives a return on assets she holds, receives a wage, trades, consumes, and carries some assets and cash over to the next period. Her budget constraint is: c t w t + (m t 1 m t ) + J p tj (π tj q t 1,j q tj ) Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 5 / 22

6 Maximization problem Let E t [ ] denote the expectations operator based on information at time t, to indicate that random variables signifying events in the future are integrated with the respect to a probability distribution that conditions on all relevant current information. Given her endowment (m t 1, q t 1,1,... q t 1,J ) at each time t before death at T the consumer investor makes portfolio decisions to sequentially maximize: [ T ] u (c t ) + E t β s t u (c s ) s=t+1 and making her portfolio choices (m s, q s1,..., q sj ) for all s {t,..., T } subject to the sequence of all the future budget constraints. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 6 / 22

7 Non-satiation in consumption Since u (c t ) is strictly increasing, all wealth is consumed and therefore: c t = w t + (m t 1 m t ) + J p tj (π tj q t 1,j q tj ) Define disposable wealth at the beginning of period t as: W t w t + m t 1 + J p tj π tj q t 1,j Thus remaining lifetime utility for period t onwards is: ( ) u W t m t p tj q tj +E t T s=t+1 J w s + (m s 1 m s ) β s t u + J p sj (π sj q s 1,j q sj ) Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 7 / 22

8 Rebalancing Let us suppose one asset does extraordinarily well, and the others do terribly, and you end up with the same wealth as you did in the previous period. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 8 / 22

9 The first order condition for assets that are held Conditional only on disposable wealth W t and the stochastic properties of future wages w s, a consumer investor makes the same choices regardless of his initial portfolio. When economic conditions change suddenly, consumer investors can adjust the portfolios instantaneously. Other adjustments, like career and housing moves, occur less frequently, because human capital cannot be traded, and housing is highly differentiated. The interior first order condition requires that for each k {1,..., J}: ) p tk u (W t m t p tj q tj = E t J p t+1,k π t+1,k βu w t+1 + (m t m t+1 ) + J p t+1,j (π t+1,j q tj q t+1,j ) Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 9 / 22

10 Consumption and asset choice Substituting the definition of consumption back into the first order condition we obtain the interior solution for assets that are held over to the next period: p tk u (c t ) = E t [ pt+1,k π t+1,k βu (c t+1 ) ] Since u (c) is a concave increasing function it follows that if no units of the k th asset are held, that is q tk = 0 then: p tk u (c t ) > E t [ pt+1,k π t+1,k βu (c t+1 ) ] This equation shows the condition under which the distribution of returns on the k th asset are too low to warrant keeping any units. If p tk > 1 they are sold, and if p tk = 1 they are liquidated. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 10 / 22

11 The fundamental equation of portfolio choice Rearranging the first order condition yields: [( ) ] pt+1,k π 1 = t+1,k E t β u (c t+1 ) p tk u E t [r t+1,k MRS t+1 ] (c t ) where: MRS t+1 β u (c t+1 ) u (c t ) is the marginal rate of substitution between c t and c t+1 and: ( ) pt+1,k π r t+1,k t+1,k is the real return on the k th security. In words, the return on an asset return is discounted by the marginal rate of substitution between current and future consumption. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 11 / 22 p tk

12 The constant relative risk aversion case When u (c t ) = ct 1 α / (1 α) the equation yields: [ ( ) α ] ct βe t r t+1,k = 1 c t+1 For example, in a world where there is no uncertainty, and r t+1 is the risk free rate, this equation simplifies to: βr t+1 ( ct c t+1 ) α = 1 And specializing to u (c t ) = ln (c t ) we obtain: [ ( )] ct βe t r t+1,k = 1 c t+1 Finally if u (c t ) = c t the expected return is equated with the interest rate for all interior choices: βe t [r t+1,k ] = 1 Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 12 / 22

13 Risk correction Recall from the definition of a covariance: cov (r t+1,k, MRS t+1 ) = E t [r t+1,k MRS t+1 ] E t [r t+1,k ] E t [MRS t+1 ] = 1 E t [r t+1,k ] E t [MRS t+1 ] = 1 E t [r t+1,k ] /r t+1 where the second line uses the fundamental equation of portfolio choice, and the third the definition of the risk free rate. Rearranging this equation gives the risk correction for the k th asset: E t [r t+1,k ] r t+1 = r t+1 cov (r t+1,k, MRS t+1 ) When c t+1 and r t+1,k tend to move in opposite directions (like insurance payouts), then cov (r t+1,k, MRS t+1 ) is positive and the investor accepts a mean return less than the risk free rate. When they mainly move together (like the market portfolio and wages in many occupations), then cov (r t+1,k, MRS t+1 ) is negative and he requires more. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 13 / 22

14 Market clearing Asset Pricing The third defining characteristic of competitive equilibrium, market clearing, closes the model. Let {1,..., N} denote the population of individual agents, and write r t,j q (n) t 1,j for the amount of the j th asset owned by agent n in period t. If p tj > 1, then the j th asset is not consumed by anybody in period t so market clearing means: N n=1 ( r t,j q (n) t 1,j q(n) tj ) = 0 If the j th asset is consumed by anybody in period t, then p tj = 1. All other assets, collectively denoted by A, are priced at one and are perfect substitutes in consumption. Market clearing only requires: N n=1 ( c (n) t w (n) t + m (n) t m (n) t 1 ) = N n=1 j A ( ) r t,j q (n) t 1,j q(n) tj Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 14 / 22

15 Mutual funds Financial intermediation If trading was a costless exercise, then investing through a mutual fund would be free. Consider investing through a personally tailored mutual fund which comprises the n th investor s portfolio: Its real return is defined by: π t+1,s ( J s (n) J t = p tj q (n) tj p t+1,j q (n) tj π t+1,j ) /s (n) t The investor could economize on his transactions by purchasing shares in a mutual fund that replicate the financial portfolio he would have purchased himself. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 15 / 22

16 Mutual funds satisfy the fundamental equation too Financial intermediation Weighting the first order condition by q (n) tj and then summing over j gives: u (c t ) s (n) t = J p tj q (n) tj u (c t ) = J [ ] E t p t+1,j q (n) tj π t+1,j βu (c t+1 ) Dividing both sides of this equation by u (c t ) s (n) t yields the fundamental equation of portfolio choice for the mutual fund: 1 = E t J p t+1,jq (n) tj s (n) t π t+1,j β u (c t+1 ) u (c t ) E t [r t+1,s MRS t+1 ] There are no tax implications. Rather than buying a whole portfolio of stocks, the investor simply picks the mutual with a return that most suits his marginal rate of substitution at the same time he decides how much to consume in the current period. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 16 / 22

17 Why don t individuals form their own individual portfolios? Financial intermediation Since transaction fees per share decline with volume, and information about the risk characteristics of each stock is costly, investors with like minded strategies have an incentive to band together. Mutual funds also give investors a say on the board. Recall that with the same wealth and wage process (plus other demographics) investors with same utility function should hold the same portfolio. Thus mutual funds compete with each other by constructing portfolios that appeal to different niches in the market. This explains why the S & P composite of 1500 firms covers about 85 percent of the U.S. equity market, there are more than 8000 mutual funds to invest in. One final cautionary remark is that investing through an intermediary does carry an additional risk of fraud and incompetence (which can be diversified by investing in more than one fund). Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 17 / 22

18 Hedges Financial intermediation Companies cannot hedge risks or insure themselves against negative financial events they influence, because this would reduce their incentives to avoid the risk, creating a moral hazard problem. But why should companies hedge any risks, since individual investors and the financial retail sector can replicate every hedging strategy for themselves? Nevertheless many companies routinely hedge on input prices, and insure themselves against financial events they cannot control. For example Southwest Airlines buys oil futures. They fared well relative to airlines who do not hedge with the outbreak of the Iraq war, and with the devastation caused by hurricane Katrina. But Southwest lost when oil prices fell with the onset of the most recent recession. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 18 / 22

19 Why do companies hedge? Financial intermediation When Southwest buys oil on the futures market, are they simply smoothing their cost revenue stream and returns stream? In SCM (45-970) I argue that companies create value by concentrating on those activities which are more effi ciently achieved inside the firm and outsource those activities that are easily verified where it lacks a comparative advantage. For example Southwest manages the risks associated with those activities about which it has specialized knowledge, such as route and seasonal demand, relations with its unionized workforce, and aircraft maintenance. If there are transactions costs and information imperfections, hedging and insurance reduces distractions, helps solidify the company s mission and branding strategy, makes the company s performance more dependent on its core business, thus increasing its transparency and value as a financial security. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 19 / 22

20 Relaxing wealth maximization Summary Relaxing expected wealth maximization as a goal requires us to introduce new parameters defining what investors care about. Generally owners of financial securities care about the temporal properties of payouts (when dividends will be paid), and their risk properties (their correlation with future consumption). The degree of curvature in the utility function captures a person s willingness to gamble. We defined the subjective discount factor to measure intertemporal trade-offs. We discussed approaches and methods for empirically recovering risk attitude and time preference. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 20 / 22

21 Optimal portfolio choices Summary Explicitly solving for the bid ask spread in an limit order exchange is too diffi cult even with supercomputer help. Competitive equilibrium (pricing taking in perfectly liquid markets) greatly simplifies portfolio analysis. Optimal behavior requires the investor to discount, relative to the real interest rate, stocks with returns (like aggregate shocks to the economy) that are positively correlated with his own next period consumption choices. Similarly an investor should favor stocks over bonds when (like insurance) their returns help offset variation in his consumption next period. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 21 / 22

22 Financial intermediation Summary Market equilibrium does not require investors to hold the same portfolio, or even a mix of a small number of funds. However the optimal financial portfolio holdings of each investor is independent of their initial endowment unless there is a nontradeable component to their endowment that is correlated with a traded asset. Mutual funds and hedging by firms are two illustrations of the same phenomenon, financial intermediation. Financial intermediaries bundle securities for investors to tailor securities that suit the investor s attitudes towards current versus future consumption and certain versus risky consumption They also provide hedging insurance to investors and companies exposed to risks that are ancillary to their mission and tastes. Miller (Tepper School of Business CMU) Portfolio Investment Lecture 5 22 / 22