Portfolio Management Asset Selection Models

INTRODUCTION

To attract assets, managers must stand out in the crowd.  The challenge is that, in today’s increasingly competitive marketplace, differentiation is both more important and more difficult than ever before.

Bank,  an institution that deals in money and its substitutes and provides other money-related services. In its role as a financial intermediary, a bank accepts deposits and makes loans. It derives a profit from the difference between the costs (including interest payments) of attracting and servicing deposits and the income it receives through interest charged to borrowers or earned through securities. Many banks provide related services such as financial management and products such as mutual funds and credit cards. Some bank liabilities also serve as money—that is, as generally accepted means of payment and exchange.

The main objective of asset selection for any portfolio manager is to select stocks, which can form a portfolio having higher expected return for given risk levels. The level of risk associated with the assets, stocks in our case, becomes a major deciding factor when choosing stocks from the ever increasing number of stocks in the financial market. Markowitz (1952), used stock return variance as a measure of risk and the prime deciding factor in stock selection in his pioneering portfolio theory. Jack Treynor (1961, 1962), William Sharpe (1964), John Lintner (1965) and Jan Mossin (1966) independently, proposed Capital Asset Pricing Theory, (CAPM), to quantify the relationship between beta of an asset and its corresponding return, which in potential applications, given appropriate assumptions, simplified the Markowitz portfolio theory by reducing the number of parameters required for asset selection.

Literature Review

The use of a single factor risk metric as in the CAPM oversimplifies a complex market. Eugene Fama and Kenneth French developed the Fama-French three factor model, which described “value” and “size” to be the most significant factors, outside of market risk, for explaining the realized returns of publicly traded stocks. The three factors, beta, SMB (for size effect), HML (for value), as proposed by Fama-French gives the projected return of a stock as a combination of these three factors. The natural approach to quantify the model is to apply OLS regression, which assumes a linear relationship across the mean of the distribution, and thus doesn’t quantify or assess the lower and upper tails of the return distribution which may play a major part when it comes to the efficient quantification of risk. A new and more robust alternative to OLS is Quantile Regression developed by Koenker and Basset (1978), which gives the capability of modelling the conditional quantiles across the distribution. Modelling the whole distribution becomes important when the return distribution becomes skewed due to adverse market conditions like the recent Global Financial Crisis, and the incapability of OLS to quantify the lower tails of the distribution can lead to wrong asset selection which could lead to greater loss.

Data Envelopment Analysis (DEA), Charnes et.al. (1978) and Banker et.al. (1984) is a powerful technique adopted from the operational research area. DEA is used for evaluating and comparing performances of organizational units in multi-attribute and multidimensional environment by determining the relative efficiency of a productive unit by considering its closeness to an efficiency frontier. In this paper we use the Fama-French factor model coefficients as calculated from Quantile Regression as an input to DEA for asset selection process. We show by a comparative analysis of asset selected by means of OLS and assets selected by application of Quantile Regression that the assets selected by the latter give better returns when combined in a equally weighted portfolio. We also show that the assets selected by application of Quantile Regression not only give better returns in normal market conditions but also in conditions of extreme financial distress.

THE FAMA-FRENCH THREE FACTOR MODEL

Jack Treynor (1961, 1962), William Sharpe (1964), John Lintner (1965) and Jan Mossin (1966) independently, proposed Capital Asset Pricing Theory, (CAPM), to quantify the relationship between the beta of an asset and its corresponding return. CAPM stands on the broad assumption that, that only one risk factor is common to a broad-based market portfolio, which is beta. (As derived from sweeping assumptions about common expectations, frictionless markets, etc). Modelling of the CAPM using OLS assumes that the relationship between return and beta is linear, as given in equation:

rA = rf + βA(rM − rF ) + α + eb#%l!^+a?

whereb#%l!^+a?

rA is the return of the asset

rM is the return of the market

rF is the risk free rate of return

α is the intercept of regression

e is the standard error of regression;

Fama and French (1992,1993) extended the basic CAPM to include size and book-to-market effects as explanatory factors in explaining the cross-section of stock returns. SMB (Small minus Big) gives the size premium which is the additional return received by investors from investing in companies having a low market capitalization. HML (High minus Low), gives the value premium which is the return provided to investors for investing in companies having high book-to-market values.

MB is a factor measuring "size risk", which comes from the view that, small companies (companies with low market capitalization), are expected to be relatively more sensitive to various risk factors, which is a result of their undiversified nature and their inability to absorb negative financial events. HML, on the other hand is a factor which proposes association of higher risk with “value” stocks (high B/M values) as compared to “growth” stocks (low B/M values). This is intuitively justified as firms or companies ought to attain a minimum size in order to enter an Initial Public Offering (IPO).

The three factor Fama-French model is written as:

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Stock exchange,  also called stock market, or (in continental Europe)bourse, organized market for the sale and purchase of securities such as shares, stocks, and bonds.

In most countries the stock exchange has two important functions. As a ready market for securities, it ensures their liquidity and thus encourages people to channel savings into corporate investment. As a pricing mechanism, it allocates capital among firms by determining prices that reflect the true investment value of a company’s stock. (Ideally, this price represents the present value of the stream of expected income per share.)

Membership requirements of stock exchanges vary among countries, mainly with respect to the number of members, the degree of bank participation, the rigour of the eligibility requirements, and the level of government involvement. Trading is done in various ways: it may occur on a continuous auction basis, involve brokers buying from and selling to dealers in certain types of stock, or be conducted through specialists in a particular stock.

Project valuation and selection has attracted a substantial amount of attention among researchers and practitioners over the past few decades. Several methods has been suggested for this purpose, including discounted cash flow analysis (DCF, see e.g. Brealey and Myers 2000), project portfolio optimization (see Luenberger 1998), and options pricing analysis, where the focus has been on the recognition of the managerial flexibility embedded in the project (Dixit and Pindyck 1994, Trigeorgis 1996). In DCF analysis, it is proposed that a project’s cash flows should be discounted at the rate of return of an asset that is equivalent in risk to the project, so that the opportunity costs of alternative investment opportunities are properly accounted for. However, while it may be straightforward to use DCF analysis when opportunity costs are solely imposed by securities, it is less obvious how the discount rate should be adjusted to account for further opportunity costs imposed by alternative project opportunities in the firm’s portfolio. b#%l!^+a?Options pricing analysis, on the other hand, requires replication of the project’s cash flows with financial instruments, which may be difficult in practice.

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Mixed Asset Portfolio Selection

In a MAPS problem, available investment opportunities are divided into two categories: securities, which can be bought and sold in any quantities

projects, lumpy all-or-nothing type investments. From a technical point of view, the main difference between these two types of investments is that the projects’ decision variables are binary, while those of the securities are continuous. Another difference is that the cost, or price, of securities is determined by a market equilibrium model, such as the CAPM, while a project’s investment cost is an endogenous property of the project.

Portfolio selection models can be formulated either in terms of rates of return and portfolio weights, like in Markowitz-type formulations, or by using a budget constraint, expressing the initial wealth level, and maximizing the investor’s terminal wealth level. When properly applied, both approaches yield identical results. We use the second approach with MAPS, because it is more suitable to project portfolio selection. We first formulate single-period MAPS models, where the investments are made at time 0 and the objective at time 1 is optimized. These models will allow us to generate several insights and show how MAPS is related to Markowitz’s (1952) model and the CAPM. We then develop the multi-period MAPS model based on Contingent Portfolio Programming.

Early portfolio selection formulations were bi-criteria decision problems minimizing risk while setting a target for expectation. Later, the mean-variance model was formulated in terms of expected utility theory (EUT) using a quadratic utility function. However, there are no similar utility functions for most other risk measures, including the widely used absolute deviation (Konno and Yamazaki 1991). Therefore, we distinguish between two classes of portfolio selection models:

(1) preference functional models, such as the expected utility model

(2) bi-criteria optimization models or mean-risk models.

LIABILITY AND RISK MANAGEMENT

The traditional asset-management approach to banking is based on the assumption that a bank’s liabilities are both relatively stable and unmarketable. Historically, each bank relied on a market for its deposit IOUs that was influenced by the bank’s location, meaning that any changes in the extent of the market (and hence in the total amount of resources available to fund the bank’s loans and investments) were beyond a bank’s immediate control. In the 1960s and ’70s, however, this assumption was abandoned. The change occurred first in the United States, where rising interest rates, together with regulations limiting the interest rates banks could pay, made it increasingly difficult for banks to attract and maintain deposits. Consequently, bankers devised a variety of alternative devices for acquiring funds, including repurchase agreements, which involve the selling of securities on the condition that buyers agree to repurchase them at a stated date in the future, and negotiablecertificates of deposit (CDs), which can be traded in a secondary market. Having discovered new ways to acquire funds, banks no longer waited for funds to arrive through the normal course of business. The new approaches enabled banks to manage the liability as well as the asset side of their balance sheets. Such active purchasing and selling of funds by banks, known as liability management, allows bankers to exploit profitable lending opportunities without being limited by a lack of funds for loans. Once liability management became an established practice in the United States, it quickly spread to Canada and the United Kingdom and eventually to banking systems worldwide.

A more recent approach to bank management synthesizes the asset- and liability-management approaches. Known as risk management, this approach essentially treats banks as bundles of risks; the primary challenge for bank managers is to establish acceptable degrees of risk exposure. This means bank managers must calculate a b#%l!^+a?reasonably reliable measure of their bank’s overall exposure to various risks and then adjust the bank’s portfolio to achieve both an acceptable overall risk level and the greatest b#%l!^+a?shareholder value consistent with that level.

Contemporary banks face a wide variety of risks. In addition to liquidity risk, they include credit risk (the risk that borrowers will fail to repay their loans on schedule), interest-rate risk (the risk that market interest rates will rise relative to rates being earned on outstanding long-term loans), market risk (the risk of suffering losses in connection with asset and liability trading), foreign-exchange risk (the risk of a foreign currency in which loans have been made being devalued during the loans’ duration), and sovereign risk (the risk that a government will default on its debt). The risk-management approach differs from earlier approaches to bank management in advocating not simply the avoidance of risk but the optimization of it—a strategy that is accomplished by mixing and matching various risky assets, including investment instruments traditionally shunned by bankers, such as forward and futures contracts, options, and other so-called “derivatives” (securities whose value derives from that of other, underlying assets). Despite the level of risk associated with them, derivatives can be used to hedge losses on other risky assets. For example, a bank manager may wish to protect his bank against a possible fall in the value of its bond holdings if interest rates rise during the following three months. In this case he can purchase a three-month forward contract—that is, by selling the bonds for delivery in three months’ time—or, alternatively, take a short position—a promise to sell a particular amount at a specific price—in bond futures. If interest rates do happen to rise during that period, profits from the forward contract or short futures position should completely offset the loss in the capital value of the bonds. The goal is not to change the expected portfolio return but rather to reduce the variance of the return, thereby keeping the actual return closer to its expected value.

The risk-management approach relies upon techniques, such as value at risk, or VAR (which measures the maximum likely loss on a portfolio during the next 100 days or so), that quantify overall risk exposure. One shortcoming of such risk measures is that they generally fail to consider high-impact low-probability events, such as the bombing of the Central Bank of Sri Lanka in 1996 or the September 11 attacks in 2001. Another is that poorly selected or poorly monitored hedge investments can become significant liabilities in themselves, as occurred when the U.S. bank JPMorgan Chase lost more than $3 billion in trades of credit-based derivatives in 2012. For these reasons, traditional bank management tools, including reliance upon bank capital, must continue to play a role in risk management.

Mean-variance optimization

(MVO) refers to a mathematical process that calculates the security or asset class weights that provide a portfolio with the maximum expected return for a given level of risk; or, conversely, the minimum risk for a given expected return. The inputs needed to conduct MVO are security expected returns, expected standard deviations, and expected cross-security correlations. For his work in developing this process, Harry Markowitz was awarded a share of the 1990 Nobel Prize in Economics.

When first developed, mean-variance optimization was applied only to portfolios of individual stocks. Today, this technique is applied with increasing frequency on an asset class level. This trend is appropriate for two reasons. First, the inputs required by the Markowitz model are more difficult to estimate for individual securities than they are for asset classes. Second, the range of asset classes available to investors is now much larger, especially given the increased willingness of U.S. investors to consider global investing.

Institutional investors are not the only ones to benefit from this development. Retail brokerage houses have traditionally only provided stock selection advice to their individual clients. However, with increasing frequency they are suggesting a greater degree of passive security selection, and instead are providing asset allocation recommendations to their investors. This is accomplished by using optimization to create allocations that provide their individual accounts with greater expected return, less risk, or both. In addition, sophisticated techniques derived from utility theory and behavioral economics can be employed to develop questionnaires that more accurately gauge an individual’s risk preferences.

Optimization has also found a home with pension funds who consider not just the b#%l!^+a?assets themselves when choosing investment mixes, but the fund liabilities and the interaction between the two. The resulting allocations maximize the expected fund surplus (assets minus liabilities) for a given level of risk. b#%l!^+a?

The consequence of mean-variance optimization is a set of asset class weights that can be used as a longterm guide for investing. This is often described as the portfolio’s strategic asset allocation plan. The portfolio weights should be updated occasionally to reflect changes in estimates of the long-term parameters or different needs of the portfolio. However, these changes will likely result in small revisions in the portfolio composition.

Dynamic asset allocation refers to strategies that continually adjust a portfolio's allocation in response to changing market conditions. The most popular use of these strategies is portfolio insurance. Broadly speaking, portfolio insurance is any strategy that attempts to remove the downside risk faced by a portfolio. A popular means of implementing portfolio insurance is to engage in a series of transactions that give the portfolio the return distribution of a call option.

Option replication is based upon the work of Fischer Black and Myron Scholes who showed that under certain assumptions the payoff of an option can be duplicated through a continuously-revised combination of the underlying asset and a risk-free bond. Hayne Leland and Mark Rubenstein extended this insight by showing that a dynamic strategy that increased (decreased) the stock allocation of a portfolio in rising (falling) markets and reinvested the remaining portion in cash would replicate the payoffs to a call option on an index of stocks.

Through the mid-1980s, the popularity of portfolio insurance programs soared. It has been alleged that the procyclical nature of these strategies contributed to greater market volatility, particularly during the stock market crash of October 19, 1987. Moreover, portfolio insurance proved to be unsuccessful in totally eliminating losses on the day of the crash. Consequently, the use and viability of portfolio insurance is controversial. Nevertheless, portfolio insurance continues to play a significant role in the world of asset allocation today.

Tactical asset allocation (also known as market timing or active asset allocation) is the process of diverging from the strategic asset allocation when an investor’s short-term forecasts deviate from the longterm forecasts used to formulate the strategic allocation. If the investor can make accurate short-term forecasts, tactical asset allocation has the potential to enhance returns. In practice, tactical asset allocation (TAA) models tend to recommend contrarian trades, that is, they recommend purchasing (selling) an asset as its current market value drops (rises).6 When viewed in this light, TAA becomes the mirror image of portfolio insurance. In other words, tactical asset allocators are the investors providing portfolio insurance.

One consequence of TAA is that by overweighting certain assets during certain times and underweighting others, the portfolio is riskier because of its reduced diversification. Therefore, the strategy would need to generate abovemarket returns as compensation for this added risk. Whether or not tactical asset allocators have achieved this is a matter of continuing study. It is certain, however, that because the potential returns from successful TAA would be large, researchers will continue their investigations, and investors will continue to listen to their findings.

Entry, branching, and financial-services restrictions

Historically, many countries restricted entry into the banking business by granting special charters to select firms. While the practice of granting charters has become obsolete, many countries effectively limit or prevent foreign banks or subsidiaries from entering their banking markets and thereby insulate their domestic banking industries from foreign competition.

In the United States through much of the 20th century, a combination of federal and state b#%l!^+a?regulations, such as the Banking Act of 1933, also known as the Glass-Steagall Act, b#%l!^+a?prohibited interstate banking, prevented banks from trading in securities and insurance, and established the Federal Deposit Insurance Corporation (FDIC). Although the intent of the Depression-era legislation was the prevention of banking collapses, in many cases states prohibited statewide branch banking owing to the political influence of small-town bankers interested in limiting their competitors by creating geographic monopolies. Eventually competition from nonbank financial services firms, such as investment companies, loosened the banks’ hold on their local markets.

In large cities and small towns alike, securities firms and insurance companies began marketing a range of liquid financial instruments, some of which could serve as checking accounts. Rapid changes in financial structure and the increasingly competitive supply of financial services led to the passage of the Depository Institutions Deregulation and Monetary Control Act in 1980. Its principal objectives were to improve monetary control and equalize its cost among depository institutions, to remove impediments to competition for funds by depository institutions while allowing the small saver a market rate of return, and to expand the availability of financial services to the public and reduce competitive inequalities between the financial institutions offering them.

In 1994 interstate branch banking became legal in the United States through the passage of the Riegle-Neal Interstate Banking and Branching Efficiency Act. Finally, in 1999 the Financial Services Modernization Act, also known as the Gramm-Leach-Bliley Act, repealed provisions of the Glass-Steagall Act that had prevented banks, securities firms, and insurance companies from entering each other’s markets, allowing for a series of mergers that created the country’s first “megabanks.”

RATIONALE FOR DEPOSIT INSURANCE

Most countries require banks to participate in a federal insurance program intended to protect bank deposit holders from losses that could occur in the event of a bank failure. Although bank deposit insurance is primarily viewed as a means of protecting individual (and especially small) bank depositors, its more subtle purpose is one of protecting entire national banking and payments systems by preventing costly bank runs and panics.

In a theoretical scenario, adverse news or rumours concerning an individual bank or small group of banks could prompt holders of uninsured deposits to withdraw all their holdings. This immediately affects the banks directly concerned, but large-scale withdrawals may prompt a run on other banks as well, especially when depositors lack information on the soundness of their own bank’s investments. This can lead them to withdraw money from healthy banks merely through a suspicion that their banks might be as troubled as the ones that are failing. Bank runs can thereby spread by contagion and, in the worst-case scenario, generate a banking panic, with depositors converting all of their deposits into cash. Furthermore, because the actual cash reserves held by any bank amount to only a fraction of its immediately withdrawable (e.g., “demand” or “sight”) deposits, a generalized banking panic will ultimately result not only in massive depositor losses but also in the wholesale collapse of the banking system, with all the disruption of payments and credit flows any such collapse must entail.

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The benefits of diversification as measured by the rules of mean-variance portfolio theory have increased in recent years, but the level of diversification in investor portfolios has not. It remains much below the optimal level currently prescribed by mean-variance optimization, which exceeds 300 stocks.

The view of investors in behavioral portfolio theory is different from the view of investors in mean-variance portfolio theory. Whereas "mean-variance investors" consider their portfolios as a whole and are always risk averse, "behavioral investors" do not consider their portfolios as a whole and are not always risk averse. In the simple version of behavioral portfolio theory, investors divide their money into two layers of a portfolio pyramid, a downsideprotection layer designed to protect them from poverty and an upside-potentialayer designed to make them rich. In the complete version of the theory, investors divide their money into many layerseach of which corresponds to a goal or aspiration.

The view of portfolios as pyramids of assets is part of common investment advice. For example, consider the investment pyramid presented in Figure 1 that the mutual fund b#%l!^+a?company Putnam Investments (available at www.putnam.com) prescribes for its investors. In the Putnam pyramid, income funds are placed at the bottom because they are designed to provide a regular stream of income and growth funds are placed at the top of the pyramid because they are intended to help build the value of the investment over time.

Increasing Risk and

Potential Reward

The pyramid structure of behavioral portfolios is also reflected in the upside-potential and downside-protection layers of "core and satellite" portfolios. Pietranico and Riepe described Charles Schwab and Company's version of core and satellite (called "Core and Explore") as comprising a well-diversified Core to serve as the "foundation" layer of the portfolio and a lessdiversified Explore layer to seek returns that are higher than the overall market-which layer entails greater risk.

One might argue that, although the portfolios are described as layered pyramids, which is consistent with behavioral portfolio theory, investors actually consider them as a whole, which is consistent with mean-variance portfolio theory. But such an argument is not supported by the evidence.

Investors consider the individual stocks they hold as part of the upside-potential layer of their portfolios and are willing to forgo the benefits of diversification in an attempt to reach their aspirations. The desire of investors to attain their upside-potential aspirations leads them to take higher risks in these layers than they take in the downside-protection layers. For example, their aspirations lead investors to buy aggressive growth funds, individual stocks, and call options, all of which have positive expected returns accompanying their high risks. Moreover, at the extreme, the desire of investors to reach their aspirations leads them to buy lottery tickets and participate in other gambles that have negative expected returns.

Investors do not gamble because they seek risk. Rather, they gamble because they badly want to reach their aspirations. Some gamblers, thinking that they have positive expected returns, misjudge the odds of their gambles, but other gamblers know the odds and gamble nevertheless because gambles with negative expected returns offer them the b#%l!^+a?only chance to reach their aspirations.

Mean-Variance Diversification

In mean-variance portfolio theory, the optimal level of diversification is determined by marginal analysis; that is, diversification should be increased as long as its marginal benefits exceed its marginal costs. The benefits of diversification in mean-variance portfolio theory are in the reduction of risk; the costs are transaction and holding costs. Risk is measured by the standard deviation of portfolio returns.

Although lack of diversification is a puzzle to mean-variance portfolio theorists, it is a main feature of behavioral portfolio theory. Polkovnichenko found in simulations of behavioral portfolio theory that optimal behavioral portfolios include an allocation of 15-50 percent to a single stock.

Investors want more than protection from poverty; they want riches as well. They construct their portfolios as layered pyramids with bonds in the bottom layer for protection from poverty, stock mutual funds in the middle layer for moderate riches, and individual stocks and lottery tickets in the top layer for great riches. b#%l!^+a?

People who hold undiversified portfolios, like people who buy lottery tickets, are gambling; they are accepting high risks without compensation in the form of high expected returns. Although gambling behavior is usually recognized as inconsistent with mean-variance portfolio theory, it is often dismissed as no more than a minor irritant to the theory-something people do for "entertainment" with minor amounts of "play money."

But gambling behavior should be considered a major puzzle to mean-variance portfolio theory because it consumes major amounts of investor money. Goetzmann and Kumar found that, on average, the value of investors' undiversified stock portfolios was 79 percent of their annual income; Polkovnichenko found that, on average, the value of investors' undiversified stock portfolios was 15- 33 percent of households' total financial wealth; and Moskowitz and Vissing-Jorgensen found that equity in privately held companies was more than 70 percent in the portfolios of entrepreneurs, even though the average return on private equity in these undiversified portfolios was no higher than the average return of a diversified portfolio of public equity.

Gamblers are often derided as mathematically challenged risk seekers who are overly optimistic about their odds and overly eager to take risks. The same might be said about undiversified investors. But what is motivating the behavior of both gamblers and undiversified investors is aspirations, not cognitive errors in mathematics or risk seeking. As Friedman and Savage wrote, "Men will and do take great risks to distinguish themselves, even when they know what the risks are". People gamble and hold undiversified portfolios because these activities are often their only ways to move from working class to middle class or from middle class to upper class. Gambling in America reported that when gamblers and nongamblers were asked to rate their need for "chances to get ahead" on a scale from a low of 1 to a high of 8, the mean score of the gamblers was 5.35 whereas the mean score for the nongamblers was 4.69.

Undiversified portfolios offer people with high aspirations a better chance to get ahead than do diversified portfolios.

Investors with undiversified portfolios may indeed have a higher tolerance for risk than those with diversified portfolios. Gentry and Hubbard compared the composition of portfolios of entrepreneurs with those of nonentrepreneurs and found the investments of entrepreneurs outside their enterprises to be no more conservative than the overall portfolios of nonentrepreneurs. This finding may indicate that entrepreneurs have a higher tolerance for risk, but it may mean that entrepreneurs and other undiversified investors allocate more to the upside-potential layers of their portfolios because they already have substantial downsideprotection layers.

For example, Heaton and Lucas pointed out that the entrepreneurs' enterprises bring them substantial income, which may fill the entrepreneurs' downside-protection layer. Similarly, Gambling in America reported that gamblers were "more likely to have their future secured by social security and pension plans than non-gamblers and hold 60 b#%l!^+a?percent more assets". And Harrah's Profile of the American Casino Gambler reported that gamblers have higher incomes, on average, than the general population and that when gamblers are away from casinos, they are more risk averse than the general population. For example, 50 percent of gamblers save in retirement plans, compared with 40 percent of the general population, and 61 percent of gamblers always or almost always pay off their credit cards in full every month, compared with 52 percent of the general population.

Investors with undiversified portfolios may be overestimating the expected returns of their undiversified portfolios or underestimating the risks of their own portfolios. Fisher and Statman found that investors are overly optimistic about their returns. The authors reported, based on Gallup surveys, that investors expect higher returns from their own portfolios than from the stock market as a whole. Similarly, Benartzi found that participants in 401(k) programs concentrate their portfolios in the stocks of their employers, thus overestimating the expected returns from these stocks and underestimating their risk. In particular, Benartzi found that only 16.4 percent of respondents in his survey realized that a portfolio concentrated in company stock is b#%l!^+a?riskier than a portfolio diversified into the overall stock market.

The description of individual stocks as individual casino bets or lottery tickets does not imply that investors choose their stocks randomly. Rather, in an echo of the common recommendation to "invest in what you know," investors tend to choose stocks of companies they are familiar with. Indeed, many casino bettors and lottery buyers follow the same recommendation by betting on favorite colors at the roulette table or choosing lottery tickets with numbers that correspond to their children's birthdays.

There is some evidence that familiarity does increase the odds of winning. For example, Massa and Simonov and Bodnaruk found that investors had higher returns with stocks selected from their geographical vicinity than with other stocks. However, the overall evidence of investor ability to pick good stocks is discouraging. Benartzi found that high allocations to company stocks increase portfolio risk but that extra risk is not rewarded by extra return; company stocks do not perform better, on average, than other stocks. Barber and Odean found that, on average, returns going to investors who pick individual stocks and trade them infrequently trail the market. Those who pick individual stocks and trade frequently trail the market by even more.

Skill in stock selection can overcome the disadvantage of limited diversification. For example, recall that the gross benefit from increasing diversification from 20 stocks to 3,444 stocks is 0.88 percent if the correlation is 0.08 and the equity premium is 3.44 percent. The net benefit of increasing diversification when the 0.06 percent net cost of the Vanguard Total Market Fund is subtracted is 0.82 percent. So, investors who can beat the market by more than 0.82 percentage points a year overcome the disadvantage of 20-stock diversification.

In summary, in behavioral theory, investors do not diversify fully because diversified portfolios leave them with too little hope of reaching their upside aspirations. Behavioral investors place great importance on the upside-potential layers in their portfolios, but at the same time, they do not neglect the downside-protection layers. Indeed, behavioral investors establish downside-protection layers in their portfolios while holding only an undiversified handful of stocks in the upside-potential layers.

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Beyond Single-Period Investment Theory

Modern portfolio theory, as introduced by Markowitz and developed by many others, provides practical techniques for analysing – and optimising – the trade off between risk and reward. This body of work is applied in a number of contexts and in a number of ways (for example, Sharpe (1964), Ross (1976), Rom & Ferguson (1994)). A common strand is that these are single-period theories: there is a single time horizon, risk and return estimates are made over that horizon and these enable a choice of investment strategy to be made.

An equally important but less widely understood stream of work within the financial academic literature examines the investment problem over multiple periods. A challenge – and a key attraction – of these papers is that the objectives of the investor come much closer to the surface. For a real- world investor – a long-term institution such as a pension plan or an insurance company, or an individual seeking to manage their b#%l!^+a?savings so as to smooth their lifetime income – the multi-period experience is much more relevant, and the investment strategy can and should evolve through time in sympathy with their objectives.

Modern portfolio theory provides clear and useful guidance on the construction of risk-adjusted performance: Sharpe ratios, Sortino ratios and the like are heavily used across the capital markets industry. In contrast, for a specific investor in the real world, it is the moneyweighted returns that are important: not these ratios that depend upon time-weighted returns generated by a notional investment.

Policy Evolution Over Time

The usual output from Modern Portfolio Theory is an efficient frontier of investment strategies: there is a hedge portfolio, a growth portfolio, and the choice of a particular point on this frontier (the amount invested in the growth portfolio) is a matter of investor judgement. This choice might typically be expressed in terms of either a return target, risk target or risk aversion.

Dynamic strategies offer the opportunity to manage risk across time, but this also leads to a sudden explosion in the number of possible strategies. Fortunately, this is not a new problem in financial theory: quite the reverse. The dynamic investment problem for individuals has been the subject of a huge volume of financial theory, starting, in its modern form, with Merton (1969) and Merton (1971). While the initial model economies had constant interest rates, more recent work has moved this into more complex (stochastic interest rate) models, where the risks of assets and liabilities can be more fairly treated together.

Market Evolution over Time

We should emphasise that the evolutions in portfolio allocation that we discuss in this paper are triggered by the investor's objectives. Although the investor's preferences are assumed to stay constant, they say different things at different values of the portfolio. In contrast, it is very common for investors to change their portfolio allocation in response to changes in markets. The market opportunity set does change over time, and an informed investor should of course be wise to this and adjust accordingly.

Risk and Return Measures

Quantifying the Trade-off between Risk and Return

When choosing an investment strategy, investors select assets that are likely to meet their particular return goals while managing the likelihood and /or severity of returns falling short of these goals. Assessing the right trade-off between return and risk therefore requires careful thinking about: – Target outcome. The investor's goals should be clear. This may be expressed in terms of a benchmark or a return target, for example. Ideally these goals will be embedded in the wider situation facing the investor – the investor's asset allocation is likely to be only one of several levers that can be adjusted to meet these goals. This goal then helps determine the average return required of the investor's assets.

– Risk tolerance. A judgement of the extent to which the investor is prepared to tolerate the return falling short of this target return. In the simplest situation, this could be expressed as an aversion to volatility, although more explicitly focusing on the probability or severity of poor returns may be more appropriate.

With these in place, an investor will then seek to carry out a form of optimisation – finding a strategy which gives as good a trade off as possible between the average return and the variability relative to this average.

Mean-variance Approaches

The classic Markowitz approach involves trading the expected return outcome off against the volatility of returns around this outcome. This can be framed in multiple equivalent ways:

– maximise expected return for a given level of volatility; b#%l!^+a?

– minimise volatility for a given level of average return;

– maximise risk-adjusted return = return – aversion x variance.

These are equivalent: varying the level of volatility, return or aversion respectively in the three versions above sweeps out the same 'efficient frontier' of portfolios that are mean-variance efficient.

In a single-period situation, this can work very well. If we use one of the simplest models of asset price uncertainty, namely the assumption asset returns are distributed according to a multivariate normal distribution, we can readily create risk/return trade-offs for a variety of risk measures. For many risk measures, the joint-normality assumption means that these generally give identical frontier portfolios, so the simplicity of mean-variance optimisation makes it an attractive approach.

The situation in a multi-period situation is rather different. Jointlognormality is the natural simple choice in a multi-period framework. If we work in a continuous time dimension, then this results in multivariate Brownian motion with constant drift and covariance.

Efficient frontiers can be established for different rebalancing frequencies: from static portfolios, through portfolios that rebalance n times over a fixed period, through to a theoretical limit of continuous rebalancing.

The Sharpe ratio measure, based on standard deviation, may fail to adequately capture an investor's risk preferences once we move away from normally distributed outcomes. A metric which appears to adequately select between 'good' and 'bad' strategies in a simple situation may no longer be an adequate description of the investor's goals in situations that allow more complex outcomes.

Dynamic Asset Allocation Techniques

The Fokker-Planck equation allows us to manage some of this dimensionality. It allows the evolution of the probability density of the portfolio value to be modelled easily without resorting to Monte Carlo simulation. The Fokker-Planck equation is a PDE of a similar type to that which arises in option pricing. Whereas option prices satisfy a PDE – essentially the backward Kolmogorov equation – it turns out that the probability distribution of outcomes satisfies a forward Kolmogorov equation, also known as the Fokker-Planck equation. The coefficients in this equation depend on the asset strategy – solving this PDE approximately for a discrete version of the asset strategy thus provides an estimate of the probability distribution at the time horizon, and from this estimate the functional to be optimised can be readily derived. A nonlinear optimisation routine can then be applied to find the optimal asset strategy on a lattice, which should be close to the optimal asset strategy defined in a continuousspace domain.

A strategy for a dynamic optimisation problem consists of a set of asset weights w at each time and portfolio value. In principle, the weights could depend upon the path taken up to that time. However, as the objective functions we consider are defined in terms of statistics defined at the terminal time, the optimal weight at time t will depend only on this path via the portfolio value at time t. This assumes that the future opportunity set is homogeneous (but if it is not, the inclusion of state variables that determine the regime at time t allows the process to work) and that transaction costs are negligible. This is necessary to allow the tree to recombine.

If:

(i) the weights are functions of time t and portfolio value Y alone

(ii) the vector of drifts of the various assets modelled is constant or dependent on time only'

(iii) the variance covariance matrix of the various assets is either constant or dependent on time only, then the drift and volatility of the strategy depend only on time and portfolio value.

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The Constant Proportion Portfolio Insurance (CPPI) has been first introduced in the context of capital protection techniques, by F. Black and R. Jones and F. Perold, as b#%l!^+a?a particular form of dynamic hedging. The investor invests in a portfolio for which he wants principal protection either of the whole initial amount or of at least a portion of it. The portfolio’s manager shifts asset allocation over the investment period between a risk-free investment and a collection of risky assets, having as a scope the preservation of a portion of money sufficient to acquire at maturity the protection requested by the investor. The position on the risk-less asset is equivalent to an ideal investment in a risk-less bond that matures at the protected amount. The position on the risky market is equivalent to a portfolio whose asset allocation is based entirely on risky assets, such as stocks, bonds, hedge funds and credit products. The amount of money that ideally has to be invested in the risk-less bond is called floor and depends strictly on the timing-to-maturity of the investment. The difference between the portfolio’s investment and the floor is called cushion.

In order to produce interesting performances, the cushion is invested in the risky market in a leveraged format, whose leverage originally, see for instance F. Perold and W. Sharpe and F. Perold and F. Black, has been taken as constant during the whole life of the product, thus motivating the name constant proportion. The higher the leverage is, the higher potential upside would be. However, high leverage increases the risk that the value of the risky exposure declines too fast for the portfolio’s manager to readjust the asset allocation. Indeed the portfolio, albeit with a small probability, could fall below the value of the floor, thus incurring in a possible loss for the investor. In such cases the portfolio’s manager has to choose among two alternatives: either he invests the floor in the risk-less bond that provides at maturity the granted capital or consumes small portions of the floor investing again in the risky market. In the first case the product ceases to exist and merely becomes a passive risk-free investment, thus motivating the name of closing-out effect. In the second case, the the portfolio’s manager, despite of its views, may incur in other losses that either compromise the possibility of having enough money to get at a certain point the protection, or erode completely the floor thus motivating a bankruptcy. This risk is commonly referred as gap risk.

To grasp the potential of the CPPI portfolio, it is worthy to analyze its advantages and disadvantages in comparison with classical capital protection methods, such as static portfolio hedging. Mainly static portfolio hedging such as Option Based Portfolio Insurance (OBPI), following R. Bookstaber and J. Langsam, initially invests a portion of the portfolio in the risk-free zero coupon bond in order to ensure for certain that the ensured amount is available at maturity. Therefore the asset allocation is established at the beginning and cannot be changed during the life of the product. Differently the CPPI portfolio’s asset allocation may be adjusted according to the performance of the risky market, thus given more flexibility to the portfolio’s manager. Moreover it is easy to note that the static portfolio hedging is equivalent to purchase a Vanilla Put option at the beginning of the financial horizon, having as underlying the portfolio’s value. Thus only the initial and terminal values of the risky asset determine the portfolio performance, regardless of the exact path the risky assets followed. In contrast the CPPI’s risky exposure grows and contracts in response to favorable and poor performance. Because the amount of the risky exposure is readjusted based on the previous performance, the overall portfolio performance is path-dependent.

The CPPI configures as an interesting alternative to static portfolio hedging, because of its flexibility that results from the dynamic asset allocation management. However the path-dependency complicates substantially the product’s structure, thus creating difficulties regarding the mathematical modellization that is required by a cautious risk analysis of the investment. While the case of the static portfolio hedging has been longly investigated, see for instance N.E. Karoui, M. Jeanblanc and V. Lacoste, the issue of the CPPI’s mathematical modellization seems to be completely open.

Given [0, T] financial horizon, the reference market consists in n−risky assets with prices {Sτ }τ∈[0,T] and a in a risk-less asset {S (0) τ }τ∈[0,T] , whose dynamics are specified by the equations:

dSτ = µSτ dτ + σSτ dWτ , τ ∈ (0, T]

S0 = s, a.s., b#%l!^+a?

And

dS(0) τ = rS(0) τ dτ, τ ∈ (0, T]

S0 0 = s0 , a.s.,

where W is a Brownian motion defined on the probability space (Ω, F,P) endowed with the filtration {Fτ }τ∈[0,T] . Here µ ∈ R is the assets’ expected return, σ ∈ R the volatility and r ∈ [0, 1] the yield interest rate. Coherently with the hypothesis of absence of arbitrage and completeness of the market, both diffusion constants are taken as bounded. The yield r has to be distinguished from the risk-free reference market’s spot interest rate d ∈ [0, 1] to which corresponds the money market account.

In cases where the risk-less asset coincides with the money market account, the identity r = d is true; in the other cases either r > d or r < d. The Constant Proportion Portfolio Insurance (CPPI) is a self-financed portfolio in which the portfolio’s asset allocation at any time is selected in order to ensure the purchase of risk-free bond that matures at the protected amount required by the investor. Denote by PT the granted amount at maturity and define the floor as the the present value of a zero coupon bond with maturity T and referring notional of PT :

Pτ = e-d(T –t)PT , τ ∈ [0, T].

Only a part of the available resources, called cushion, equal to the difference between the current portfolio’s wealth process {Vτ }τ∈[0,T] and the value of the floor, is invested in the risky assets:

Cτ = Vτ − Pτ , τ ∈ [t, T].

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