EMF EXECUTIVE MASTER IN FINANCE TRASFORMA LE SFIDE IN GRANDI OPPORTUNITÀ SDA Bocconi I Executive Master in Finance EMF Prof. Massimo Guidolin ACADEMIC DIRECTOR ANDREA BELTRATTI Short vs. PROGRAM Long-Term DIRECTORand ALESSIA Tactical BEZZECCHI vs. Strategic Asset Allocation MILANO I ITALY
PLAN OF THE LECTURE GENERALITIES: PREFERENCE-BASED PORTFOLIO SELECTION LONG RUN ASSET ALLOCATION AND DYNAMIC PROGRAMMING THE CONTRARIAN NATURE OF REBALANCING PREDICTABILITY AND TIME-VARYING INVESTMENT OPPORTUNITIES HEDGING DEMANDS: LARGE OR SMALL? REBALANCING IS SHORT VOLATILITY THE REBALANCING PREMIUM SDA Bocconi I Executive Master in Finance EMF 2
INTRODUCTION By definition, asset allocation lies at the heart of buy-side asset management and plays a crucial role However, its importance is even more pronounced when the goals of portfolio managers have a long-horizon nature o This occurs both because asset allocation becomes more complex and because it interacts with two technical aspects: 1. Asset/liability management (ALM) 2. The costs and benefits of rebalancing According to experts (e.g., Andrew Ang, in his 2014 book), «the foundation of long-term investing is to rebalance to fixed asset positions, which are determined in a one-period portfolio choice problem where the asset weights reflect the investor s attitude toward risk. Rebalancing is counter-cyclical strategy: it buys low and sells high Rebalancing goes against investors behavioral instincts and is also a short volatility strategy SDA Bocconi I Executive Master in Finance EMF 3
DYNAMIC PORTFOLIO CHOICE In a dynamic long-term ptf. choice problem, the horizon T is long (say, 10 years) but the investor is aware of being able to change her portfolio at (equally spaced) B rebalancing points (say, every year) o The length of a period between rebalancing points is then T/(B+1) o For instance, T = 10 years and T/(B + 1) = 10/10 = 1 year Ptf. weights can change in response to time-varying investment opportunities through economic recessions or expansions and in response to the final term t + T approaching (retirement, say) o Same goes for her liabilities, that will be addressed again later Assume the investor s utility function and risk aversion remain constant over time and this is understood by the investor The utility function (of final wealth) of an investor is a device that turns all possible wealth levels that can be achieved into an index of satisfaction or, if you will, happiness o Utility fnct. U(W t+t ) converts future wealth W t+t into real numbers SDA Bocconi I Executive Master in Finance EMF 4
DIGRESSION: UTILITY FUNCTIONS AND RISK AVERSION SDA Bocconi I Executive Master in Finance EMF 5
DIGRESSION: UTILITY FUNCTIONS AND RISK AVERSION The expected utility theorem obtains under a set of axioms that can be summarized as (rather strong) requirements of rationality of the decision-maker o An introductory summary can be found at http://didattica.unibocconi.it/ mypage/doc.php?iddoc=23431&idute=135242&idr=14063&tipo=m&lingua=it a in lecture 7 o Not surprisingly many of these axioms are violated in reality In the framework, the more risk averse the asset owner, the more these bad times hurt and the more she prefers sure things o With mean-variance utility, asset owners care only about means (which they like), and variances (which they dislike) o Mean-variance utility defines bad times as low means and high variances: Ptf. returns o MV utility is closely related to power/crra utility because we can consider expected utility using CRRA utility and MV to be approximately the same: SDA Bocconi I Executive Master in Finance EMF 6
DIGRESSION: UTILITY FUNCTIONS AND RISK AVERSION o Utility functions are concave, so their second derivatives are negative and the second term in the equation is negative An investor maximizing expected utility is approximately the same as an investor maximizing mean for a given level of variance o However, often other things also matter and unfortunately meanvariance utility does not allow us to capture them Other things such as skewness, kurtosis, jumps, regimes, etc. o Mean-variance utility does not assume returns are normal o Often people confuse using mean variance utility with assuming normally distributed returns o This error occurs because, with normal distributions, there are only two parameters the mean and the variance (they are sufficient statistics) o Levy and Markowitz (1979) showed that using mean-variance utility is often a good approximation with non-normal returns SDA Bocconi I Executive Master in Finance EMF 7
DYNAMIC ASSET ALLOCATION MODEL o A risk-averse investor aims at avoiding risks o One can show that the definition of risk aversion is equivalent to the mathematical definition of concavity of U(W t+t ) As a side detail, a concave function implies that U (W t+t ) < 0 o Returning to the dynamic ptf. problem, at the beginning of each period t, the investor chooses a set of portfolio weights, x t o Asset returns are realized at the end of the period t+1, and the portfolio weights chosen at time t, x t, with investor s wealth at the end of the period, given by here r p,t+1 is the return on the ptf. invested according to x t weights o For instance, in the case of T = 5 and 4 rebalancing points, we have: SDA Bocconi I Executive Master in Finance EMF 8
DYNAMIC ASSET ALLOCATION MODEL o At the beginning of each period investor chooses portfolio weights, x t o The sequence of weights, {x t }, is called a dynamic trading strategy The investor wishes to maximize expected power utility of end of period wealth at time T by choosing a dynamic series of ptf. weights: subject to constraints = (power utility) ( is the coefficient of relative risk aversion) o Some examples of constraints are that an investor may not be able to short (this is a positivity constraint so x t 0) o May not be able to lever (so the ptf. weight is bounded, 0 x t 1) o Can only sell a portion of her ptf. each period (a turnover constraint) o Note that uncertainty is dealt with by maximizing the current expectation of utility of future, terminal wealth Power utility is characterized by constant relative risk aversion In practice, it means that the percentage weight assigned to different assets shall not depend on how wealthy you are, W t SDA Bocconi I Executive Master in Finance EMF 9
DYNAMIC ASSET ALLOCATION MODEL Although the portfolio weights x t+τ are, of course, only implemented at time t + τ, the complete set of weights {x t } from t to T 1 is chosen at time t, the beginning of the problem E.g., x t could take on two values at time t: hold 50% in equities if we are in a recession and 70% if we are in a bull market o The optimal dynamic trading strategy is completely known from the beginning, even though it changes through time o As asset returns change, the strategy optimally responds, and as utility and liabilities change, the strategy optimally responds o The rate of response to news, will be a function of, which measures how averse to risk an investor is: is expected return and σ is volatility and x* is risky investment This solution is the classical mean-variance one However, one can prove that also power utility is locally meanvariance so that this formula provides in general a good approximation o However, x* just concerns a one-period problem, say equity weight SDA Bocconi I Executive Master in Finance EMF 10
DYNAMIC PROGRAMMING SOLUTION (HINTS) The dynamic problem is an optimal control problem, solved by dynamic programming o Because they are technical, details are in an Appendix o Long-horizon wealth is a product of one-period wealth: o Apply CRRA expected utility to long-horizon wealth, under some conditions, we have a series of one-period CRRA utility problems: The portfolio returns, r p,t+1 depend on the portfolio weights chosen at the beginning of the period, x t o Since U(W t ) appears outside the expectation, initial wealth doesn t matter o This is called the wealth homogeneity property of power utility o We can re-write this equation as: o Dynamic programming implies that optimization problem need to be solved backwards, according to Bellman s principle SDA Bocconi I Executive Master in Finance EMF 11
DYNAMIC PROGRAMMING SOLUTION (HINTS) SDA Bocconi I Executive Master in Finance EMF 12
IMPLICATION 1: BUY AND HOLD IS NOT OPTIMAL Dynamic programming turns the long-horizon problem into a series of one-period problems Dynamic portfolio choice over long horizons is first and foremost about solving one-period problems and viewing their solution this way demolishes two widely held misconceptions: 1. Buy and Hold Is Not Optimal In a buy-and-hold problem, the investor chooses portfolio weights at the beginning of the period and holds the assets without rebalancing over the entire long-horizon problem o The buy-and-hold problem treats the long-horizon problem as a single, static problem o Buy-and-hold strategies are nested by the dynamic portfolios but they are dominated by the optimal strategy that trades every period Long-horizon investing is not to buy and hold; long-horizon investing is a continual process of buying and selling SDA Bocconi I Executive Master in Finance EMF 13
IMPLICATION 2: LONG-TERM INVESTING IS SHORT o There is much confusion in practice about this issue. In 2011, the World Economic Forum, for example, defined long-term investing as investing with the expectation of holding an asset for an indefinite period of time by an investor with the capability to do so. o A popular finance best-seller, Siegel s Stocks for the Long Run (1994): Siegel makes a case for sticking to a long-run allocation to equities o Hold on: if the allocation is constant, it is maintained by a constant rebalancing rule investors increase (decrease) their share of equities after equities have done poorly (well) to maintain the share o This re-affirms that long-run investors never buy and hold, they constantly trade The implication for the financial industry is clear: one needs professional advise and/or appropriate tools (funds, securities, derivatives, etc.) to wrap these complex dynamic strategies in 2. Long-Term Investing Is Short-Term Investing Long-run investors are no fundamentally different from myopic, short-term investors SDA Bocconi I Executive Master in Finance EMF 14
IMPLICATION 2: LONG-TERM INVESTING IS SHORT o Dynamic programming solves the long-horizon portfolio choice problem as a series of short-term investment problems o They do everything that short-run investors do, and they can do more because they have the advantage of a long horizon Where is the catch? The effect of the long horizon enters through the indirect utility in each one-period optimization problem Paradoxically, this is best seen just from the case in which there is no difference between long-run investors and short-run investors Suppose that returns are not predictable, or the investment opportunity set is independent and identically distributed (IID) o In every period asset returns are drawn from the same distribution that is independent of returns drawn in previous periods o Asset returns are like a series of coin flips, except coins can only land heads or tails, and returns can take on many different values o Unfortunately, the assumption of IID returns is realistic as far as mean returns are concerned, but it is not when applied to variances SDA Bocconi I Executive Master in Finance EMF 15
MYOPIC INVESTING UNDER CONSTANT OPPORTUNITIES o Returns are dependent because large squares and absolute values tend to be followed by other large values, like in volatility clustering Here use this case simply as a benchmark: under IID returns and a fixed risk-free rate, the dynamic ptf problem becomes a series of identical one-period problems If returns are not predictable, then the long-horizon weight is identical to the myopic ptf. weight: there is no difference btw. long-horizon investing and shorthorizon investing, all investors are short term The investment horizon is irrelevant The short-run weight is the myopic ptf. weight in All investors whether they have short or long horizons act like short-term investors in the IID world SDA Bocconi I Executive Master in Finance EMF 16
REBALANCING IS COUNTER-CYCLICAL (CONTRARIAN) o The optimal strategy is to manage ptf. risk and return in each period, treating each period s asset allocation as a myopic investment No confusion here: even under IID returns, the sequence of oneperiod problems defining a dynamic strategy never amounts to repeating the myopic strategy over time The dynamic problem is a series of one-period problems, and it involves rebalancing back to the same portfolio weight o The investor sells stocks when they have done well; conversely, if equity loses relative to other assets, equities have shrunk as a proportion of the total portfolio and more need to be purchased This rebalancing is irrelevant to a myopic investor, because the myopic investor is no longer investing after a single period Rebalancing is the most basic long-run investment strategy, and it is naturally counter-cyclical o It is a type of natural value-investing strategy o Value investing occurs when an investors buys cheap (presumably, under-priced) and sells expensive (over-priced) assets SDA Bocconi I Executive Master in Finance EMF 17
REBALANCING IS COUNTER-CYCLICAL (CONTRARIAN) o In his book, Andrew Ang shows that rebalancing, dynamic portfoliobased strategies have the power to easily outperform fix-mix strategies (initial 60-40) that are often the result of buy-and-hold SDA Bocconi I Executive Master in Finance EMF 18
FIGHTING BEHAVIORAL DRIFTS: ROLE OF PLANNING o Figures look impressive but, in practice, rebalancing is hard: it leads to buying assets that have lost value and selling those that have risen o This goes against human nature: investors tend to be very reluctant to invest in assets that have experienced large losses; they are reluctant to relinquish positions that have done extremely well How many institutions take capital away from traders because they have been successful and give it to colleagues who have underperformed? o Yet good financial advising on long run asset management, when grounded in financial theory ought to fight behavioral trends The principal reason for articulating long-term investment policy explicitly is to enable the client and the ptf. manager to protect the portfolio from ad hoc revisions of sound long-term policy Key to help them stick to long-term policy when short-term exigencies are most distressing and the policy is in doubt o To the point of writing so-called Ulysses contracts In reality, some evidence (e.g., on Swedish households accounts) that there is a considerable amount of rebalancing occurring SDA Bocconi I Executive Master in Finance EMF 19
SMART (CONTINGENT) REBALANCING o But this evidence is reversed in other studies, e.g., on US data that seem to express considerable inertia Oddly, some financial institutions fail to stand out as much better than individual households in their use of rebalancing Rebalancing is so crucial in optimizing long-run investment strategies that recently it has been made smart : it is triggered by information arrival, not the passage of calendar time This is called contingent rebalancing, rather than calendar rebalancing and its goal is to save transaction costs Optimal rebalancing strategies trade off the utility losses of moving away from optimal weights versus transaction costs from rebalancing o If the benefits of rebalancing outweigh the cost of doing so, then it is an optimal time to rebalance, and rebalancing becomes contingent Rebalancing bands are often used, set around optimal targets rebalance rebalance SDA Bocconi I Executive Master in Finance EMF 20
SMART (CONTINGENT) REBALANCING o The bands are a function of transaction costs, liquidity, asset volatility, and minimum transaction sizes o When transaction costs are large or asset volatility is high, the bands are wider: the bands expand at an approximate rate of the cube root of the transaction costs For example, for a 5% transaction cost, the bands are approximately (0.05/0.001) 1/3 3.7 times larger than a 0.1% transaction cost Other authors suggest rebalancing to the edge of the band Whether you rebalance to the target or to the edge depends on whether the transaction costs are fixed (rebalance to target) or proportional like brokerage fees and taxes (rebalance to the edge) There are even more sophisticated rebalancing strategies with two bands surrounding the target weight There is no trade if the ptf. lies within the outer band, but, if the ptf. breeches the outer band, then the investor rebalances back to the inner band SDA Bocconi I Executive Master in Finance EMF 21
SMART (CONTINGENT) REBALANCING When returns are predictable, then rebalancing in long-term strategies gives additional value and opportunities o Equivalently, the IID constant investment opportunity case gives the lower bound to the economic value of rebalancing When expected returns and volatilities change over time, the optimal shortrun weight changes, i.e., it depends on conditional moments (forecasts) SDA Bocconi I Executive Master in Finance EMF 22
TIME-VARYING INVESTMENT OPPORTUNITIES Under time-varying, predictable returns, the optimal long-run strategy comprises the time-varying short-run strategy plus a hedging portfolio: Horizon The hedge ptf. has a complex mathematical expression but economically it provides insurance against changes in the investment opportunity set o It will include long, large positions in assets that pay out above their average when investment opportunities deteriorates Such as protective puts, short calls, credit default swaps, some utility stocks, etc. o The hedging demand, for an investor with log-utility (CRRA utility with γ = 1, U(W) = lnw) is 0: intuitively, a log-investor maximizes log returns, and long-horizon log returns are sums of 1-period returns o Because the ptf. weight is freely chosen each period, the sum is maximized by maximizing each individual term in the sum SDA Bocconi I Executive Master in Finance EMF 23
TACTICAL ASSET ALLOCATION SDA Bocconi I Executive Master in Finance EMF 24
HEDGING DEMANDS Let s say something more on hedging demands two determinants: 1. Just like the myopic portfolio weight depends on the risk tolerance of an investor (i.e., 1/ ) so does the opportunistic portfolio; but here the investor s horizon plays a role 2. Hedging weights depend on asset-specific properties of how returns vary through time, i.e., whether they insure variation in investment opportunities directly or indirectly (via correlations) While the simple rebalancing strategy is counter-cyclical and has a value tilt, some hedging strategies optimal to investors are even more counter-cyclical and strongly value oriented There has been a debate in the academic literature on how large these hedging demand, long-run opportunistic effects really are o Campbell and Viceira (1999) estimate hedging demands to be very large that are easily double the average total demand for stocks; others estimate small hedging because predictability would be weak The point is that this component depends crucially on how one models predictability and hence on the statistical framework SDA Bocconi I Executive Master in Finance EMF 25
HEDGING DEMANDS AND PARAMETER UNCERTAINTY SDA Bocconi I Executive Master in Finance EMF 26
HEDGING DEMANDS AND PARAMETER UNCERTAINTY SDA Bocconi I Executive Master in Finance EMF 27
HEDGING DEMANDS AND PARAMETER UNCERTAINTY This occurs when the estimation uncertainty correlates with the investment opportunity set Long Run Dynamic Portfolio Under Parameter Uncertainty % allocation to stocks Static Myopic Demand When parameter uncertainty is ignored, the sign of hedging demands depends on level of the predictor (here dividend yield) Under parameter uncertainty hedging demands are almost certainly negative and large Rebalancing can also be interpreted as an option strategy and, in particular, a short (negative vega) volatility strategy SDA Bocconi I Executive Master in Finance EMF 28
REBALANCING AS A SHORT OPTION STRATEGY Suppose that a stock follows a binomial tree o Each period the stock can double, with prob. 0.5, or halve starting from an initial value of S = 1 o There are two periods, so there are three final nodes o At maturity, there are three potential payoffs of the stock have probabilities of 0.25, 0.5, and 0.25, respectively o In addition, the investor can hold a risk-free bond that pays 10% o Let us first consider a buy-andhold strategy that starts out with 60% equities and 40% in the risk-free asset o At the end of the 1 st period, the wealth of this investor can increase or decrease to which is shown by branching of the tree SDA Bocconi I Executive Master in Finance EMF 29
REBALANCING AS A SHORT OPTION STRATEGY o The optimal rebalanced strategy, which rebalances at time 1 back to 60% equities and 40% bonds gives the following tree o If we plot the payoffs of the buy-and-hold strategy vs. the rebalanced strategy as a function of the stock value at maturity time 2 we have: o The gains and losses on the buy-and-hold position are linear in the stock price o The payoffs of the rebalanced strategy are concave over the stock price This nonlinear pattern of the rebalancing strategy can be equivalently generated by short option positions SDA Bocconi I Executive Master in Finance EMF 30
REBALANCING AS A SHORT OPTION STRATEGY If we add a _ short European call (with strike $3.6760 maturing at time 2), _ a short European put (with strike $0.4660), _ the long bond position, and _ the buy-and-hold strategy, we get identical payoffs to the rebalancing strategy at time 2: A short volatility position that is financed by bonds together with the buy-and-hold strategy is identical to the rebalanced strategy o The rebalancing strategy is an active strategy that transfers payoffs from the extreme low and high realizations to middle stock realization SDA Bocconi I Executive Master in Finance EMF 31
REBALANCING AS A SHORT OPTION STRATEGY o Rebalancing does this by selling when stock prices are high and buying when stock prices are low o Short volatility positions do exactly the same o A short call option can be dynamically replicated by a short stock position and a long bond position: this buys equity when stock prices fall and sells equity when stock prices rise. o Likewise, a short put is also dynamically replicated by selling equity when prices rise and buying when prices fall o These are exactly the same actions as rebalancing. The benefit to rebalancing is investor specific: moving the payoffs from the extreme stock positions back to the center is optimal for the investor because it cuts back on risk o Because rebalancing is short volatility, it automatically earns the (negative) volatility risk premium; shorting negative premium > 0 o In moving the payoffs to the center, rebalancing increases the losses during extreme low markets, and underperforms the buy-and-hold strategy during extreme high markets and profits from reversals SDA Bocconi I Executive Master in Finance EMF 32
REBALANCING AS A SHORT OPTION STRATEGY o This is one reason that rebalancing performed well during 1926 1940 and 1990 2011 in the earlier Figures o When there are strong reversals from the steepest crashes, rebalancing works wonders o Conversely, if reversals do not occur, such as in permanent bull or permanent bear markets, then rebalancing will underperform the buy-and-hold strategy SDA Bocconi I Executive Master in Finance EMF 33
ASSET-LIABILITY MANAGEMENT Long-run ptf. strategies are also strongly affected by the need to manage liabilities When liabilities are introduced, the optimal portfolio strategy has four components: The liability hedging portfolio is the portfolio that best ensures the investor can meet those liabilities We solve for it by holding asset positions that produce the highest correlation with the liabilities Often however, simpler approaches are used to avoid to carry around separate hedging components, for assets and liabilities Their goal is remove (in a financial sense) the liabilities and hence operate on wealth only: 1. Cash flow matching or immunization, constructing a perfect match of liability outflows each period, through bonds SDA Bocconi I Executive Master in Finance EMF 34
ASSET-LIABILITY MANAGEMENT 2. Duration (more generally, asset-liability) matching, take positions in assets to perfectly neutralize the risk exposures (e.g., to the level factor in the case of duration, to level and slope in the case of duration+convexity, etc.) o E.g., if liabilities increase when credit spreads narrow, for example, as they do for pension funds, then the liability-hedging portfolio must hold large quantities of assets that are sensitive to credit risk o The Merton Samuelson advice of long-horizon asset allocation extended to liabilities is to meet the liabilities and then to invest the excess wealth over the present value of liabilities as before If assets are insufficient to meet current liabilities, then investors must face that default will happen in some states of the world Portfolios can be constructed to minimize this probability, but avoiding insolvency requires a different optimization than the maximization of utility To long-horizon investors, rebalancing yields a risk premium often called growth-optimal investing, volatility pumping, and Kelly rule SDA Bocconi I Executive Master in Finance EMF 35
THE REBALANCING PREMIUM Suppose that the price of each underlying asset is stationary; that is, the price of each asset tends to hover around a fixed range and never goes off to infinity By rebalancing to a fixed constant weight each period, an investor can generate wealth that increases over time, and any such rebalancing strategy will eventually beat the best buy-and-hold ptf. o Erb and Harvey (2006) call it turning water into wine Mathematically, this arises as a consequence of compounding, under rebalancing wealth is while under buy-and-hold Buy-and-hold o The compounding of products gives rise to many nonlinearities over time, which are called Jensen s terms, and the effect of the nonlinear terms increases over time o In a one-period setting, geometric and arithmetic returns are economically identical, thus, there is no rebalancing premium for a short-run investor SDA Bocconi I Executive Master in Finance EMF 36
THE REBALANCING PREMIUM o Over multiple periods, the difference between geometric and arithmetic returns is a function of asset volatility: the greater the volatility, the greater the rebalancing premium o For U.S. stocks, the rebalancing premium a long-run investor can earn is approximately 1% a year The rebalancing premium is too good to be true: Rebalancing is a short volatility strategy that does badly compared to buy and hold when asset prices permanently continue exploding to stratospheric levels or permanently implode to zero and disappear If there are assets that experience irreversible capital destruction, then rebalancing leads to buying more assets that eventually disappear this is wealth destruction o The rebalancing premium can only be collected for assets that will be around in the long run, so rebalance over broad asset classes or strategies: global equities, global sovereign bonds, real estate, etc., rather than individual stocks SDA Bocconi I Executive Master in Finance EMF 37
THE REBALANCING PREMIUM SDA Bocconi I Executive Master in Finance EMF 38
ON THE EFFECTS OF THE INVESTMENT HORIZON The fact that the long-run ptf. weight may be written as shows that the horizon T may enter through SAA as well as hedging demands o However, when T gets really large, SAA converges to a long-run, average asset allocation and T only enters the hedging component Therefore whether or not the weight on risky assets, x*, should decline or increase with age (i.e., increase or decrease with T) depends on the sign of the hedging demand! % allocation to stocks Long Run Dynamic Portfolio Under Parameter Uncertainty Static Myopic Demand SDA Bocconi I Executive Master in Finance EMF 39
ON THE EFFECTS OF THE INVESTMENT HORIZON Because there is debate as to the sign of hedging demands or even their optimality, no clear best practice in wealth management The issue is that there are multiple possible statistical predictability models o E.g., Barberis results are based on linear regressions in which past dividend yields (== trailing avg. of dividends divided by lagged price) forecasts future excess returns o In my own work, about 10 yrs ago, I have considered models with recurring regimes Also called Markov switching o The idea is that the meanvariance trade-off would change over time according some discrete state of the economy, like expansions vs. recessions, bull vs. bear markets, etc. Static Myopic Demand As you can see, size and sign of the hedging demand depends on the regime SDA Bocconi I Executive Master in Finance EMF 40
APPENDIX: DYNAMIC PROGRAMMING o Long-horizon wealth is a product of one-period wealth: o Apply CRRA expected utility to long-horizon wealth, under some conditions, we have a series of one-period CRRA utility problems: The portfolio returns, r p,t+1 depend on the portfolio weights chosen at the beginning of the period, x t o Since U(W t ) appears outside the expectation, initial wealth doesn t matter o This is called the wealth homogeneity property of power utility o Therefore we can always normalize W t so that U(W t ) = 1 and stop worrying about it o We can re-write this equation as: o Dynamic programming implies that optimization problem need to be solved backwards, according to Bellman s principle o Let s start at the end, btw. t+4 and t+5, where the investor chooses ptf. weights to maximize expected utility at the terminal horizon t+5 SDA Bocconi I Executive Master in Finance EMF 41
APPENDIX: DYNAMIC PROGRAMMING t+4 Prof. Bellman o In the green square, we have a static one-period problem, and, for CRRA utility without constraints, this is identical to the one-period mean-variance problem: t+4 SDA Bocconi I Executive Master in Finance EMF 42
APPENDIX: DYNAMIC PROGRAMMING o Because the problem is solved at t+4, the expectation may in principle depend on the information on the state of the economy or investment opportunities that will be available at that time o The maximum utility obtained at t + 4 is: where the portfolio return from t+4 to t+5,, is a function of the optimal portfolio weight chosen at t+4, r * pt,+5(w * t+4) o The maximum utility V t+4 is called the indirect utility o Next, we turn to the problem 2 periods before the end, in the blue box: at t+3, we solve for both the ptf. weights at t+3 and t+4, x* t+3 and x* t+4, o We already solved last period to find x* t+4, for any outcome at t+3 o Write the problem 2 periods before end as the problem from t+3 to t+4, plus the problem with known solution from t+4 to t+5: SDA Bocconi I Executive Master in Finance EMF 43
APPENDIX: DYNAMIC PROGRAMMING Prof. Bellman o This now leaves just one ptf. weight at t+3, x t+3, to solve SDA Bocconi I Executive Master in Finance EMF 44
APPENDIX: DYNAMIC PROGRAMMING o The problem is a standard single-period problem, except that it involves the indirect utility V t+4, and can be solved as: and this gives us o At this point, we apply the recursion to the t+2 problem having solved the t+3 and t+4 problems; again, the t+2 optimization is a 1-period problem o After solving the t+2 problem, we continue backward to t+1 and then finally to the beginning of the problem, time t. o Dynamic programming turns the long-horizon problem into a series of one-period problems SDA Bocconi I Executive Master in Finance EMF 45