Helloworld!

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Abstract This paper responds to three questions not well answered for many students of finance: (1) How do investors choose where to aim on an efficient frontier of trade-offs between portfolio expected return and risk? (2) What do investors do when return variance is inadequate as a summary of risk? (3) How can investors take into account the need to avoid interim shortfalls on the way to a more distant goal? To have great impact, the answers must be simple enough to be carried in the heads of practical investors. Since vivid examples help, the last part of this paper examines five cases: the option income fund, extreme CPPI (Constant Proportion Portfolio Insurance), the Long-Term Capital Management case, the Enron case and split capital trust funds. In each case, the discretionary wealth hypothesis is used, a reformulation of what are sometimes known as fractional Kelly strategies, to pinpoint the easily remedied blind spots that led to unexpected disappointments.

Keywords: risk management, discretionary wealth, Kelly rule, logarithmic utility, asset allocation, shortfall, downside risk

Introduction

The proper balance of pursuit of return versus diversification and reserves is the essence of risk management – for investors as well as biological species. Those who practise it well benefit from superior long-term returns. This paper responds to three questions not well answered for many students of finance:

1 How do investors choose where to aim on an efficient frontier of trade-offs between portfolio expected return and risk?

2 What do investors do when return variance is inadequate as a summary of risk?

3 How can investors take into account the need to avoid interim shortfalls on the way to a more distant goal?

To have great impact, the answers must be simple enough to be carried in the heads of practical investors. Since vivid examples help, the last part of this paper examines five cases: the option income fund, extreme CPPI (Constant Proportion Portfolio Insurance), the Long-Term Capital Management (LTCM) case, the Enron case and split capital trust funds. Each case uses the approach of Wilcox (2003), a reformulation of what are sometimes known as fractional Kelly strategies, to pinpoint the easily remedied blind spots that led to unexpected disappointments. Because this approach may be unfamiliar, its origins are first examined as a reaction to a more narrow view of risk management often taught to investment professionals.

How we got here

As early as the 1700s, Daniel Bernoulli suggested making decisions about risky outcomes based on their expected ‘utility’; for utility’s measure, he proposed the natural logarithm of wealth. Most economists, however, have regarded this function as essentially arbitrary, choosing to explain risk aversion with a broader class of utility functions that were curved so as to produce declining marginal utility with increases in wealth. The stronger the curvature, the stronger is the implied risk aversion.

Shortly after World War II, von Neumann and Morgenstern proposed a set of axioms defining rational utility functions for choices involving what they called lotteries. They took as given the decision-maker’s utility curve implied by his subjective preferences, so long as it is ‘rational’, without questioning whether it served specified objective goals.

In the 1950s, Harry Markowitz introduced his approach to constructing an efficient frontier between (1) investment portfolio expected single-period return and (2) its risk measured as return variance. For theoretical justification, he appealed to the von Neumann utility theory, at least for those typical utility functions that could be approximated with quadratic curves. In the 1960s, Sharpe, Lintner, Tobin and others developed the capital asset model (CAPM) for stock market pricing, in turn based on the Markowitz mean-variance model for the representative investor. In the 1970s, Black, Scholes and Merton introduced option pricing models founded on arbitrage with continuous, frictionless dynamic hedging, for which return variance was a sufficient measure of risk. The trinity of mean-variance optimisation, CAPM and the Black-Scholes option pricing model still make up the core of formal education for many quantitatively oriented investment professionals. This core offers valuable insights. It leaves serious omissions, however, with regard to how one should seek long-term investment growth without risking financial embarrassment.

In parallel, there has been a persistent thread of modelling activity that seeks more comprehensive answers to risk management. John Kelly (1956), a Bell Telephone researcher, wrote a single paper that was to instigate a continuing stream of research on the periphery of finance. Titled ‘A New Interpretation of Information Rate’, it showed that the maximum expected rate of growth of a gambler’s capital was obtained by maximising the expected log return of capital at each bet. (Log return is simply ln(1 + r), where r is the conventional fractional return.) The paper showed what proportion of the gambler’s stake to hold back in reserve at each bet. This rule, derived for maximising expected exponential growth rates, was mathematically equivalent to Bernoulli’s specification of utility as the natural logarithm of wealth. Consequently, some subjective utility functions were seen as better than others from the viewpoint of objective growth criteria.

In a remarkable chapter on long-term returns, Markowitz (1959) showed, using a Taylor series expansion, that the mean single-period fractional return less half its variance gives a fair approximation of the continuously compounded or log rate of return, thereby giving new meaning to his mean-variance criterion. But he did not dwell on why conservative investors might be more averse to variance or on the possibility that, in some cases, higher terms in the Taylor series expansion involving return skewness and kurtosis (fat-tails) might also be relevant.

Those who over the intervening years have tried to insert optimal growth criteria into the mainstream financial teaching include, among others, Latane, Thorp, Hakansson and Ziemba. One difficulty was that, for diversified portfolios of common stocks, the Markowitz mean-variance optimisation closely approximated the results of maximising expected log return, so that there seemed relatively little practical motivation to switch to a new paradigm. A second was theoretical – the expected log return criterion led to more risk taking than most investors seemed to want. Though it could account for moderate risk aversion, it could not account for strong risk aversion. A third was related to the sociology of science – a great many finance papers had already been written based on a utility theory that admitted a wide range of subjective utility functions.

Hakansson (1971) attempted to appease mainstream utility theorists by proposing a decision model incorporating both the expected log return and its variance. Merton and Samuelson (1974) rebutted strongly that it was a fallacy to suggest that this formulation was capable of representing the class of power utility functions suggested, except of course for logarithmic utility.

Yet the need for something more comprehensive than mean-variance optimisation based on subjective risk preferences has grown steadily more apparent as more sophisticated option instruments and hedging techniques have multiplied. Ziemba and others (see MacLean and Ziemba, 1999) have continued to promote the concept of fractional Kelly strategies – strategies that mix an expected growth rate criterion with one of a number of safety-promoting criteria without reference to utility theory. It is fair to say that, although these mixed strategies would tend to produce good results, they have not enjoyed widespread adoption. The barriers are not just an intellectual desire for a connection to utility theory. To capture the imagination of practical investors, one needs simplicity. To incorporate this into the education system for financial professionals, one needs the tightest possible theoretical cohesion so as to recruit academic thought.

There is a large literature that attempts to deal with one or more of the ‘missing’ issues: multi-period investment problems, skewed and fat-tailed return distributions, and the specification of more useful risk preferences. While respecting this great body of work, this paper argues for a particular fractional Kelly strategy, the discretionary wealth hypothesis, which is extremely comprehensive in scope, simply related to utility theory, and easy to implement. It is based on the utility measure ln(w – c), where w represents wealth, and c represents the current shortfall constraint. There is, however, much of interest to be found in a broad array of recent work on growth-optimal strategies. See Cover (1991), Browne (1999), Kritzman and Rich (2002), Leland (1999), Michaud (2003), Stutzer (2000) and Wilcox (2003) for related reading.

The model

Volatility reduces compound returns

Many investors do not realise that risk interferes with the compounding of returns. What happens if a dollar is alternately doubled and halved each period for ten periods? The mean single period return is 25 per cent. But the ending wealth is the same as the starting wealth, for a 0 per cent return. Now consider what happens if for each period a fair coin is flipped to determine the outcome, with ‘heads’ giving a return of 100 per cent and ‘tails’ a return of -50 per cent. The median result is still an equal number of heads and tails, producing a 0 per cent return at the end of 10 periods. The expected wealth is far higher – it is 1.25^sup 10^, or over $9, the same as if a constant return of 25 per cent per period were compounded.

Holding expected single-period return constant, one sees that the addition of riskiness in these returns has no effect on mean wealth achieved, but an enormous effect on median wealth achieved. In this case, the ending wealth distribution is so skewed that only a small proportion (less than 20 per cent) of the possible sequences of ten coin flips will produce wealth as great as the mean. The tiny percentage of outcomes with nine or ten heads, achieving wealth of $256 and $1024, disproportionately affect the averages, but has little relevance to the typical result.

Focus on logarithmic returns

Compounding returns is mathematically equivalent to summing the logarithmic returns each period, and taking the anti-logarithm of the result. Suppose each return is risky. Then, by the Central Limit Theorem of statistics, if the successive log returns are independent and identically distributed with a finite variance, their sum will tend toward a normal distribution as the number of periods increase. The normal distribution is symmetric; consequently, its mean and median are equal. Thus if the mean log return is maximised, the median log return also tends to be maximised. Since taking the antilog does not disturb rank order, one also tends to maximise median wealth achieved by the compounding process.

This conclusion – that maximising expected log returns each period will maximise median wealth attained after a sufficient number of periods – does not depend on the returns of individual periods having a normal distribution. It does not even depend on the Central Limit Theorem. If the sum of log returns tends toward any symmetric distribution, one thereby maximises median wealth. Unless the single-period return distributions are very unusual, the convergence is surprisingly fast, so that this approach to estimating median results is usually an excellent approximation within a horizon of ten periods or less.

Kelly’s rule applied to coin flipping (Google if needed)

Kelly’s rule applied to stock portfolios (Google if needed)

More generally, investors confront single-period return distributions that are effectively continuous, described by estimated statistical parameters such as mean, variance, skewness and kurtosis. How then can expected log returns be calculated? This can be approximated more and more closely using a Taylor series expansion of ln(1 + r), and taking the expected value of each term in the series one wishes to retain. There is a deep relationship between the Taylor series expansion of ln(1 + r) and the successive statistical moments about its mean value.

Note that for successively independent security returns, both E and V are proportional to the time interval between successive periods. The higher-order statistical moments involving return skewness and kurtosis (fat-tails), however, are functions of V raised to higher powers. Consequently, one strategy for taming the downside risk not captured by variance is to increase the frequency of decision making, thereby reducing the impact of these high-order terms on expected log return and median ending results.

If a portfolio is well behaved, one can estimate the expected log return with a simplified approximation of Equation (2): E – V/2. This is a good way to get a first intuitive feel for the process, but it will be optimistic if risky returns have significant negative skew or excess kurtosis (fat-tails). It will be pessimistic if there is significant positive skew, as when stocks are combined with a put option on the market.

Cash reserves versus borrowing

Suppose the Kelly rule investor wishes to determine directly how much wealth to put into his or her stock portfolio, with the rest left in a cash reserve earning a real return of 0 per cent. Assume that the investor can also borrow at a real rate of 0 per cent. Instead of maximising the expected ln(1 + [function of] × r), one maximises expected ln(1 + L × r), where L may be termed leverage, taking a value either greater or less than 1.

To give a concrete example, suppose you are responsible for an untaxed portfolio of stocks with an expected real return of 6 per cent and an annual standard deviation of return of 20 per cent. A first approximation of expected log return is L × E – L^sup 2^ × V/2. This expression is maximised when L = E/V. In this case, the optimum leverage is 0.06/0.04 or 1.5. Assuming one was confident of one’s estimates and borrowing at 0 per cent real interest were possible, to maximise expected growth one should borrow to finance a third of one’s investments in the stock market.

Putting aside for the moment that one may find this uncomfortably aggressive, let us see what happens if one uses not just a simplified version of the first two terms, but the first four terms of Equation (3), so as to take into better account the downside risk of occasional market crashes. Suppose one has estimated the skewness of the portfolio return at -0.9 and kurtosis at 4. These characteristics are not far from those of the S&P500 index for annual returns since 1926. Using whatever computation tools are available (Excels ‘solver’ works fine, but one can get there by hand trial and error), one finds that the best L in this case is about 1.22. That is, one should buy on margin (remember, the assumed real interest rate is 0 per cent) but less than half as aggressively as if one had not looked beyond variance as the full measure of risk.

One might wonder how a criterion based on expected log return could lead to taking any chance of losing everything, as one does whenever leverage L is not less than 1. This is the result of maximising only the first four terms, and not the full Taylor infinite series. Whenever one does so, one allows for the possibility of catastrophic events, but these are very unlikely to occur in a single lifetime. This is an important and usually appropriate modification of the Kelly rule. The 50-year probability of a fatal car accident for an individual in the US is on the order of 1/200, yet one rationally continues to drive. If normal lifetimes were 1000 years, this might be a poor decision.

Equation (3) teaches three fundamental ideas about leverage:

1 You can have too much of it. Even with costless borrowing, there is an optimum leverage beyond which expected log return, and therefore median ending wealth, will fall off sharply. Expected log return may be negative even if single period average percentage return is positive. Negative expected log return implies eventual failure with certainty.

2 The first critical relationship is E/V, the ratio of expected return to its variance. (Note that this is neither the Sharpe ratio nor the information ratio. They have their place, but not here.) If E/V is less than one, the optimal leverage for growth will tend to be less than one – absent portfolio insurance, one must hold cash reserves. If greater than one, there may be a reason to borrow to finance a higher rate of growth. (This has obvious implications for corporate strategy.)

3 Increased leverage has a different effect on each term of the Taylor series. It increases single period return only linearly – but the impact of risk as variance varies as leverage squared – the impact of any skewness varies as leverage cubed – and the impact of fat-tails varies as leverage raised to the fourth power. Lower leverage whenever there is a potential for negative skew or fat-tailed returns is very important in avoiding long-run catastrophe. High leverage in their presence is a red flag that may easily signal negative expected log return, and thus eventual ruin with certainty.

The discretionary wealth hypothesis

The discretionary wealth hypothesis supposes that the best, most robust investment objective for most purposes is to aim for maximum expected log return of discretionary wealth, that wealth in excess of an amount that would trigger an intermediate shortfall and stop the sequence. In poker, the shortfall analogy would be losing one’s stake and having to leave the table. That is, Equation (3) applies not to total wealth but to that fraction D that is discretionary. This is a form of fractional Kelly strategy – with the innovation that the safety objective is implicit in the choice of numeraire rather than in a separate objective. That is, higher risk aversion is represented not by greater curvature than inherent in ln(wealth), but by increasing the apparent logarithmic distance between wealths. This is accomplished by measuring distance against a smaller discretionary wealth.

With the modification of the truncated four-term Taylor series representation of expected log return, the discretionary wealth hypothesis is equivalent to supposing that, at any point in time, the investor’s utility is very nearly of the form ln(w – c), where w is wealth and w – c is discretionary wealth. To be clear, suppose that if one were to lose half one’s invested wealth, traumatic events such as a divorce, losing one’s job, or facing an unacceptable retirement would be triggered. The discretionary wealth fraction D is therefore 50 per cent. If the entire amount is invested in stocks, one’s leverage L is 2. This might be termed implicit leverage, because one is not borrowing externally, but against one’s own shortfall constraint. If stocks decline by 10 per cent, return on discretionary wealth is -20 per cent. If, as estimated earlier, optimal leverage were 1.22, then one should reduce stock allocation to 1.22 × D or 61 per cent.

The specification of a shortfall constraint is often much easier, and never harder, than to specify directly the Markowitz mean-variance risk aversion parameter. If the E – V/2 approximation of expected log return is satisfactory, then the discretionary wealth hypothesis applied to conventional Markowitz mean-variance optimisation consists of maximising E – V/(2D). Even when it is not, it may be useful to do mean-variance optimisation as a security-diversifying step before applying Equation (3) to the whole to take into account market factor skewness and kurtosis.

It can be seen from Equation (3) that this approach means that the investor’s separate risk aversions to variance, negative skew and kurtosis “will vary over time, generating trading. A complete mathematical analysis would take into account trading costs.

A complete practical investment consideration would also take into account whether investors as a whole had shared in one’s recent losses or gains. If so, return expectations might well vary. One now understands why surviving commodity futures traders, who usually have very high leverage and whose losses are shared only by part of the market, are so quick to cut their losses. In the stock market, in contrast, if one has low implicit leverage, it may make sense to let generally experienced bear market losses ride or even to buy more when the market has dropped substantially. One is also led to understand that investors with high implicit leverage should be natural buyers of positive skew through portfolio insurance, while those with deeper pockets should be natural sellers of portfolio insurance. Finally, referring to earlier material, one sees why an initial loss leading to high implicit leverage can accentuate the later dangers of negative skew and fat-tailed returns, leading to a death spiral if risk positions are not promptly pruned.

Cases

Option income funds1

For a time in the US in the 1980s, option income mutual funds were extremely popular. Distributed through brokers, they owned either stocks or bonds and wrote call options on them; premiums were then distributed as ‘income’. These funds caused intense investor disappointment and withdrawals after net fresh money stopped flowing in and gradually declining opportunities for safely writing calls caused successive cuts in income distribution rates.

As securities went up, they were called away, leaving increasing concentrations of securities whose market values were below cost. Because interest rates were generally falling, this hit the stock-based income funds first. Calls on the far-underwater stocks could not be written, for fear that they would be called and losses would have to be recognised on the underlying stock that would offset the ability to distribute the proceeds from the sales of call options. As distributions began to fall, it was necessary to meet them with further stock sales, but these had to be drawn from those that were not yet deeply underwater, to avoid recognising more losses that would interfere with current distributions. Thus the problem got worse ever more quickly, and the funds entered a kind of death spiral. Bond-based option-income funds eventually went through a somewhat similar cycle, though it took longer to develop.

What can be said about option income funds from the viewpoint of risk management by shareholders? Suppose that fees and brokerage costs had been minimal, and that distributions had been held to average total return levels. In this more ideal world, should retiree shareholders have bought, as their primary holding, option-income funds as compared with balanced funds of similar mean expected return and risk? Using Equation (3), the answer is clearly negative. Most retirees, having insufficient income from other sources, in effect have high income needs from their portfolio, and therefore low discretionary wealth fractions, implying high implicit leverage. This makes them unusually badly affected by the negative return skew induced by the fund’s strategy.

What can be said from the viewpoint of the mutual fund management companies’ corporate risk management? It is clear that the fund designers had no idea that they were setting up a vicious circle (cash flow positive feedback once the decline started) that would result in extremely fat-tailed results – catastrophe. That is, the fund structure set up an artificially induced kurtosis. This self-induced return kurtosis problem was to reveal itself again in the LTCM and Enron cases.

Extreme CPPI2

Black and Perold (1992) responded to the desire for protection against downside risk with a simple scheme for producing portfolio insurance. The investor’s equity allocation is periodically reset at the lesser of 100 per cent or a fixed multiple of the difference between current wealth and a floor based on a percentage of original wealth. For example, the multiple might be five and the floor might be set at 80 per cent of original wealth, resulting in an initial equity allocation of 100 per cent. The investor participated fully if the market went up. If the stock market went down, the portfolio became more and more conservative, asymptotically approaching 100 per cent cash equivalents when 20 per cent of original wealth were lost.

Long Term Capital Management3

The case of the LTCM hedge fund’s collapse, more than any other example, demonstrates the blind spots in too narrow a core academic view of risk management.

LTCM began with leverage (actual outside borrowings) of L = 30 times. One wonders whether ratios of expected mean return to variance, E/V, were available to justify such aggressiveness, even ignoring the possibility of negative skew or fat-tailed return distributions that could be taken into account applying Equation (3). In particular, it is often forgotten that relative portfolio variance is far more sensitive to errors in return correlation in a long-short portfolio than it would be in an all-long portfolio. There is no way, however, for an outsider to plumb the mysteries here. What can be assessed, however, is management’s reaction to the dynamics of the situation.

LTCM was initially so successful that it attracted both additional contributions of funds and competition. One would usually suppose, other things equal, that expected return E would decline and that, consequently, optimal and actual leverage L would decline. Instead, as profits became more difficult, leverage was increased.

Again, a dynamic implication of Equation (3) in LTCM’s situation is that a loss should be followed by a reduction in the aggressiveness of the portfolio. That is, the risky assets should be reduced faster than their decline in market value. This would keep leverage on the reduced discretionary wealth from increasing. Instead, after setbacks, leverage was allowed to increase to 60 times.

Finally, LTCM became so large and its strategies so well known that it set up the conditions for a death spiral if it should get into trouble: others knew of its troubles and took advantage in trading as it finally tried to reduce its positions, causing further losses – self-induced return kurtosis and negative skew.

Enron4

Putting aside the well-known legal and ethical problems of Enron management, much of its problem was readily apparent as poor risk management.

First, it was heavily leveraged, in large part through off-balance sheet partnerships. It is not clear, except after the fact, whether there was a good basis in E/V for this leverage, but high leverage is a precondition for some problems noted below.

Secondly, in addition to short-term trading, it engaged in long-term supply contracts. Equation (3) indicates that the combination of long intervals between trading opportunities and high leverage dramatically heightens the impact of any negative skew or fat-tailed return distributions.

Thirdly, an obvious risk management failure was to increase this leverage further when additional partnerships appear to have been created to hide losses from GAAP accounting of corporate profits. As with LTCM, one can see a pattern of letting losses run rather than confronting them.

Fourthly, perhaps most critical, returns to the limited partners in off-balance sheet partnerships were supported by guarantees involving Enron stock and cash. Once partnership losses started, this made it still more difficult for Enron to report profit per share, further reducing both its creditworthiness and its stock price, making trading more difficult, requiring more depreciated stock for guarantees, and setting up further partnership losses that required still more Enron stock issuance. Again, a vicious circle was constructed that led to extreme self-induced return kurtosis. In this case, since guarantees were not reversed if there were healthy partnership profits, it also led to severe negative skew on corporate return. Management essentially sold a put on itself.

Split capital trust funds5

The investment trust industry in the UK has spawned a large number of so-called split capital trust funds. In recent years, many have severely disappointed investors. These trusts offer to the retail public several classes of securities. There are the Zero Dividend Preference Shares, which have a fixed schedule payout, so long as any capital remains, until a liquidation date. There are Income Shares, which collect all dividends and which are second in line for return of their capital at the liquidation date. (Actually, there may be two classes of Income Shares, one of which is ‘Ordinary’ Income Shares.) Finally, there are Capital Shares, last in line, that receive any remaining capital at liquidation time. That is not all. Many of these trusts are geared – that is they can borrow. Finally, they may invest in other Split Capital Trust Funds of a similar nature, creating cross holdings.

Can one determine from all this the investment characteristics of the Capital Shares? Because of cross holdings, there can be multiple layers of fees and expenses, reducing expected returns, but let us concentrate just on return variance. The UK Financial Service Authority (2002) issued a policy statement giving some indications of leverage for funds in different categories based on price movements from 31st March, 1999, to 31st March, 2002. During this period, the FTSE All-Share Index was down 11.7 per cent. Splits, presumably Capital Shares, with no cross holdings were down 39.1 per cent, those with cross holdings less than 20 per cent were down 82.2 per cent, and those with cross holdings over 70 per cent were down 98.0 per cent. Making allowances for interest and fees, it would appear that without cross holdings the effective leverage L was over three, while with cross holdings one might easily get leverage of eight times and, since one really cannot lose more than 100 per cent, leverages of considerably more than ten are quite plausible.

Conclusion

The big mistakes in risk management are straightforward, if sometimes painful to fix. One cannot expect a trouble-free investment world. But with better risk management education, one can expect a smoother growth path for investors and investment managers alike. This will come only through widespread use of robust mental shortcuts, abbreviations of more elaborate models. The discretionary wealth hypothesis offers a comprehensive risk management model encompassing both downside risk and interim shortfall constraints based on ln(w – c) utility. It is a fractional Kelly rule, but without complication. Through the device of Taylor series approximations, it yields mental shortcuts such as the LE – L^sup 2^ V/2 for quick assessment of expected log return, and E/V for optimal leverage on discretionary wealth. Where more careful estimates are required, adding terms for return skew and kurtosis (fat-tails) takes into account lower-probability extreme events. This approach also yields practical management insights quantifying intuition – such as (1) the benefit of shortening cycle times between decisions and reducing otherwise justified leverage when facing unusual negative skew or fat-tailed return distributions, (2) avoiding positive feedback structure that can set up a vicious circle when one gets into trouble, and (3) telling investors under what circumstances they should reduce their bets after a loss.

Charles Darwin is quoted as saying ‘The evolution of the human race will not be accomplished in the ten thousand years of tame animals, but in the million years of wild animals, because man is and will always be a -wild animal.’

The best response may be a quote from W Edwards Deming: ‘Learning is not compulsory – neither is survival.’