An hour's thinking about how to manage risk more effectively will offer far greater typical returns than an hour spent searching for the next super stock. That is what I concluded after researching "Better Risk Management." The purpose of this essay is to make good on that promise. The first part deals only with issues that are relevant to all investors. The second part focuses on the delegation of investment authority to active managers seeking to improve on an index benchmark.

- Maximizing expected compound return of discretionary wealth. Why is this a suitable framework for investment policy? How much risk aversion is optimal in the deriva-tion of asset-allocation benchmarks? Are we missing something in evaluating the benefits of index funds? How should we react to major bull and bear markets?
- Better ways to motivate active managers who depart from index benchmarks. How should differences from benchmark per-formance be measured? What is left out of conventional risk budgeting? How can we better understand and avoid undesirable closet indexing on the one hand and unnecessary risks on the other?

**Part I: How Managing Risk ****Can Improve Return**

A good essay has a personal voice. This one reflects my personal experience as a pro-fessional money manager trying to make sense of the schizophrenic attitudes we all have toward risk. We love what it can do for us and hate the experience. In dealing with this paradox, I've found it helpful to establish sim-ple models that seem to capture the essence of the problem and then apply them to con-crete practical examples. **Describing Return Risks: **When we invest, we do not know in advance the profits or losses we will realize as a percentage of our initial capital. We use the concept of probability to express the relative frequency of each potential outcome return. Very high and very low returns will have low probabili-ties, with middling returns being more fre-quent. The central tendency of the return probability distribution may be summarized as the mean return. In many practical situa-tions, we also care about another measure of central tendency, the 50th percentile, or medi-an return. The most conventional measure of return dispersion is the statistical variance, the square of the standard deviation.

For most purposes, these three measures, mean, median and variance, are sufficient to derive very good investment policies. Using additional higher order statistical measures such as skewness or kurtosis as inputs to pol-icy may be relevant when either the investor is extremely conservative or the underlying port-folio is very risky, but for brevity we will not pur-sue such cases here. We also will not discuss the periodically popular semivariance, judging it an unnecessary complication.

**Compounding Risky Returns: **When we invest over multiple periods, the returns expe-rienced in each period affect the capital avail-able at the start of the next period. In the pres-ence of risk, this positive-feedback loop does unexpected things to the return distribution of the final results, as compared with the return distribution of any single period.

Even if the single-period return distribution is symmetric, with a mean equal to its medi-an, the probability distribution of the wealth at the end of many periods will be very uneven. Its mean will exceed its median, often by a great deal. Think of it as the Bill Gates Effect, when one has unusually good returns many periods in a row and compounds the result to astonishingly high wealth. An average includ-ing Bill Gates gives a distorted picture of what is likely to happen to individuals starting their economic careers.

Such winning streaks are hard to plan for. For active managers of other people's money, it is likely to be more important to be in the top half of the manager universe with high probability after five years than to have a small chance of an extremely high return and a large chance at a below-average return. The same also is true for individual investors. Most investors need to know with some con-fidence what they can count on for planning purposes. I believe what we want our invest-ment policy to achieve is best described by our median ending wealth, rather than by our mean ending wealth.

If we wanted only to improve our mean multi-period ending wealth, we would need only to improve the mean return each period. Risk in the form of dispersion of outcomes would be irrelevant to the calculation. But it is a mathematical fact that median, or typical, multi-period results are affected by both mean and variance of the probability distri-bution for single-period returns. The greater the single-period variance, the lower the typical ending wealth.

How can we relate final median results to single-period characteristics so that they can be better managed? The logarithmic return is defined as the natural logarithm of the sum of one plus a hundredth of the percentage return. If the number of time periods is large, the mean logarithmic single-period return determines not only the mean compound return, but also the median ending wealth. (Confused? The mean compound return and the compound return necessary to reach the mean ending wealth sound alike, but they are different concepts.)

The mean log return is approximated as a simple quantity-the single-period expected return *(*E) less half its variance *(*V), or *E-*V/2. For example, if the average stock market return net of inflation were 6%, and annual standard deviation of return were 20%, our formula would give us 0.06-(0.20) 2 /2, or 0.04 for mean log return, and about 4% annualized compound real return, assuming no taxes.

(This idea of maximizing expected log returns leads to strategies governing trade size variously known in high-risk trading and gambling circles as the Kelly rule, which has been around since its discovery in the 1950s. Below, I will add an important twist to the Kelly rule that makes it practical for even very conservative investors.)

**An Unheralded Advantage For Index ****Funds: **The *E-*V/2 relationship to long-term results gives us an interesting clue as to why broad-based index funds with many stocks do so well against many active managers with more concentrated portfolios. It is not just a matter of low fees and trading costs. By diver-sifying specific risks, the index portfolio with more stocks has lower return variance, and thus, other things equal, a higher median long-term wealth. Consider the expectation you have for the skill of an active manager drawn at random. You should expect return in excess of index return approximately equal to fees, netting zero. The mean single-period percentage returns may be identical to that of the index fund and so will be the mean termi-nal wealth. But the index fund will have lower risk and thus higher median terminal wealth. Most of the time, you will do better with the index fund.

**Discretionary Wealth Vs. Total Wealth: **

Now let us address the frequent real-world case that the investor is more averse to risk than seems, at first glance, to be implied sim-ply by subtracting only half the return variance from our objective to maximize expected compound return. The logarithmic concept, or Kelly-type rules, can be universally applied. To do so, you need to assimilate the concept of discretionary wealth.

Suppose you, as assistant treasurer of the XYZ Corp, are managing a $1 billion pension fund currently invested 50% in stock-index funds and the rest in near-cash securities. (To keep the example simple, we exclude risky bonds.) If you were to lose even a quarter of the portfolio, the fund would be unable to meet its obliga-tions and catastrophe would ensue. In other words, no matter how it is invested, 75% of your portfolio is reserved against obligations. Your discretionary wealth is only 25% of the total portfolio wealth. Are you taking too much risk? Or not enough?

To answer this, first consider how you should think about the money that is invest-ed in stocks, even though it is not part of your discretionary wealth. In effect, it is money borrowed from your reserve! It is financial leverage on your discretionary wealth. In this case, you are using twice as much capital on risky assets as you have in discretionary wealth. What multiple would be most advantageous? Less, or more?

To figure out how good your current 50% allocation to stocks is, note that expected return on discretionary wealth depends on the leverage multiple implied by your total stock exposure, as well as on expected return and variance for a pure stock holding. Single-peri-od mean return on discretionary wealth will go up in proportion to the leverage multiple m, but its variance, which reduces typical results, will go up as the leverage multiple *m squared.*

The mean log return for discretionary wealth invested in risky assets is *m*E-*m*2 V/2. How does this apply to your pension fund situation? The expected compound return for discretionary wealth will be 2(.06)-2 2 (0.2 2 /2 or 0.12-0.08 or about 4%. You can expect your fund's discretionary wealth to tend to grow nicely. Note in contrast that if you were to decide to invest all your money in stocks, but could only afford to lose 25%, the leverage multiple would be 4. This would give a much different picture of mean log return. Then you would have 4(0.06)-42 (0.2 )/2, or about negative 8%. The nega-tive expected log return is a warning that the most typical result is that your 25% cushion would erode rather than increase.

**Optimal Leverage: **Now, what would be the optimal percentage of XYZ's pension fund allocation to stocks? Too little leverage, and you are not taking sufficient advantage of opportunities. Too much, and occasional big losses will set back your discretionary capital base and disrupt the compounding process. The best leverage is one that maximizes expected compound return. It is approximate-ly the ratio of expected return to variance, or *E/*V. It should be kept in mind that this assumes you really know *E *and V. Since both these are estimates, and since a negative expected return can result if you underesti-mate the risk relative to the return, the wise practitioner will make sure to estimate them both conservatively. Nevertheless, under-standing that there is an optimum leverage and that it depends on these two factors is a giant step forward in the formulation of a prop-er asset allocation plan for you at XYZ Corp.

Taking our example further, in your case, you have estimated *E *and *V *such that optimal leverage is 0.06/0.04 = 1.5. Consequently, you should allocate 1.5 times your 25% dis-cretionary wealth, or 37.5% of your fund's portfolio, to stocks. You should reduce your stock holdings, because your discretionary wealth is a bit overleveraged at 2 times. When you do so, your expected compound return will be increased. Your new expected com-pound return will be about 1.5(0.06)-1.5 2(0.22 /2, or 4.5%. You will have increased the typical growth rate of your fund by more than 10% of the base growth rate.

**Dynamic Policy: **Suppose you now have followed the optimal plan above, putting 37.5% of your fund in stocks. In the next year, your stock holdings went up by 25%, while cash was flat, after disbursements for current obligations. Should XYZ Corp. stick with the old asset-allocation plan? The answer will depend on what has happened to your reserve needs. Fortunately, in this case, they are unchanged in dollar terms.

Please work through the following example yourself. You may find it challenges your pre-vious assumptions. You wisely continue to project *E *and *V *near their long-term averages at 0.06 and 0.04, giving a continued optimum multiple of 1.5. But your fund's wealth has gone up from $1 billion to $1.094 billion. Your required reserve is now 750/1,094=68.6% of your fund's portfolio. Your discretionary wealth ratio consequently has risen from 25% to 31.4%. Ergo, your new optimum equity posi-tion is 1.5*31.4%, or 47.1% of $1,094 million, amounting to $515 million!

Market action has increased your stock position only from $375 million to $469 mil-lion. Consequently, you need to take $46 mil-lion from your fixed-income position and put it in stocks. It is a good thing you have been studying multi-period investment policy and risk management, or you might have been tempted to let your positions ride, or worse, to have rebalanced back to 37.5% equity.

If the optimum leverage is greater than one, and if the reserve against shortfall is unchanged in dollar terms, then a gain should be followed by further investment. On the contrary, a major loss should be fol-lowed by pulling in your horns so that you do not become overleveraged. This is the lesson ignored during the famous hedg-fund disaster experienced by Long Term Capital Management.

Note that this riding with a winner and retreating if one loses is the opposite of con-ventional rebalancing. Buying more after gains and selling more after losses is called for in the XYZ Corp. pension fund's case for two interacting reasons. First, expected return is higher than its variance, resulting in an opti-mal leverage greater than one. Second, reserve obligations did not rise as fast as dis-cretionary wealth. On the other hand, in many other practical cases, working out the math would lead you to a policy of rebalancing rather than exploitation.

Each case can be analyzed on its own merits. What is always advantageous, however, is to recognize that the proper mix of stocks and bonds in a portfolio depends on optimal leverage multiplied by the discretionary wealth left after a calculated reserve against disaster.

**Portfolio Insurance:**In the case of the XYZ Corp. pension fund, the dynamic action called for is in the same direction as would result under Constant Proportion Portfolio Insurance (CPPI). However, the position shifts are far less because they are based on calculated optimal leverage rather than on the larger multiplier necessary to give a dramatic optionlike effect. Conventional CPPI plans call for a multiplier of five or more times a wealth cushion. This creates an overleveraged policy with negative expected log return on the cushion. Such plans are likely to lead to getting stuck disappointingly near the postulated floor.

**Part II: Better Ways To Measure And Motivate Active Managers**

As assistant treasurer of XYZ Corp., you now are given the assignment of designing a strategy for active management of the stock portion of your pension-fund portfolio. You are convinced that active managers, properly selected and motivated, will produce results better than can be gotten with your current index-fund investment.

Your investment consultant agrees. She advises beginning with a representative index benchmark that could be achieved passively through your index fund strategy. Then, she proposes helping you select one manager each for three different investment styles: a top-down strategist, a qualitative bottom-up stock picker and a quantitative manager. Each manager will be measured quarterly, based on value added above the benchmark. She also recommends that you monitor for each manager an information ratio of average value added divided by average tracking error relative to the benchmark. She suggests a cutoff information ratio of 0.5 be applied in the manager search, but with a higher goal. She comments that very frequent measurements would not be meaningful because the market is so unpredictable. Consequently, she recommends establishing an evaluation period of three years, after which the managers will be reviewed for retention.

Curious as to how well this advice fits with your recent mastery of multi-period investing, you decide to partition total expected log return. You are going to add the active return results to the benchmark results. You remember from college statistics that the variance of a sum is the sum of the two variances plus twice their covariance. Working out the math, you discover that there are actually going to be three, not two, parts to your expected compound return and the resulting median ending wealth.

Total E-V/2 equals:

- E-V/2 for the benchmark index;
- Plus active value added -- (tracking error squared)/2;
- Minus the covariance between the benchmark return and the value added.

That is, we get an expected compound return for the benchmark, plus an independent expected compound return for active management, less an unexpected drag caused by covariance in risks. Note that in this third part, no division by two appears because we already have divided it into twice the covariance.

The first thing you have discovered is that the consultant, perhaps to simplify things for you, has left out something very important. There is no remaining measurement or incentive for the active manager to try to reduce or even limit covariance between value added and the benchmark return. So long as the manager stays within a tracking error limit, he or she is just as happy to buy high-risk stocks as low-risk stocks! In fact, he is usually happier -- for two reasons. First, the manager may think buying higher-risk stocks is an easy way to get higher returns. (Think back to the times you have discovered emerging market or high-beta small stocks in supposedly conservative U.S. institutional portfolios.) Second, more insidious, the manager may be doing portfolio construction using a Markowitz optimizer but substituting tracking error for total risk.

Consider the benchmark portfolio compared with one with the same expected return but lower total risk. The lower-risk portfolio clearly is better for the client. However, an optimizer fed tracking error never can choose it because any departure from the benchmark, even if it is in the right direction, creates tracking error that will be penalized.

The risk aversion implicit in our formula for expected compound return is the same no matter which of the three terms in which it appears-benchmark risk, tracking error squared or twice covariance between active and benchmark returns. Of course, using a factor of just one-half is appropriate only if you already have magnified your risk measure as you scale it to the smaller base of discretionary rather than total wealth. Whatever the number, it should be the same for all three terms-benchmark risk, tracking error squared risk and twice covariance risk. Just as it is a mistake in one direction to be uncaring of covariance, it is equally a mistake in the other direction to treat the active portfolio's tracking error as though it needed to be avoid-ed more than the risk in investing in a benchmark index fund.

Your consultant also has told you that you should look for a set of managers who each could deliver 1% added return per year, with a tracking error of 2%. That implies that you should only invest in managers with an opti-mal leverage of 0.01/ 0.0004 =25 times. But of course, you wouldn't apply anything like that much leverage. You only applied 1.5 times leverage in moving from discretionary wealth to your index fund in the past. The implicit risk aversion required of active man-agers is many times higher than for the benchmark.

It seems likely to you that this screening requirement will exclude many managers who have good capabilities for adding to expected compound return. It will spotlight instead those who happen to have pro-duced extremely consistent positive results over a history usually much too short from which to estimate their true information ratio with any confidence.

A bias toward over-optimistic cutoffs is not the only problem created by lack of analysis of the proper risk aversion. Whatever risk penal-ty should be applied to the total XYZ Corp. portfolio of long and short active positions, you should be somewhat more tolerant of tracking-error risk generated by an individual manager. That is because much of the squared tracking error of the individual manager will be diversified away through the assemblage of a stable of managers with three different investment styles, while the active return will remain.

As assistant treasurer of XYZ Corp., you wonder how the managers you hire will react to their situation. If their three-year perform-ance is poor, they face being fired. If they make bets big enough to maximize their contribution to your expected compound return, they increase that probability.

**A Caution On Information Ratios: **The Sharpe ratio, or information ratio as it is now usually termed, is a real advance over just looking at value added without considering variability in returns. However, note its formula in the context of maximizing expected com-pound return and thus median ending wealth. The information ratio is defined as *E/V *1/2 . This is not equivalent to *E-*V/2. When comparing two alternatives, the two criteria will not necessarily pick the same one as best, even if we ignore the previously noted issues of covari-ance and risk aversion.

Which is better, a 2% value added with 3% tracking error, or 1% value added with 1% tracking error? Unless you are very con-servative, you should prefer the 2% value added with a 3% tracking error because the increment to expected compound return is much greater.

Finally, investors tend to use information ratios as a convenient shortcut to scaling up return. I have heard a hedge-fund manager say that he can get three times the return if he uses three times the leverage. This is dangerously wrong. His clients' mean end-ing wealth out of the distribution of possibili-ties may be the result of three times the return. But their median ending wealth will fall short because the variance drag affect-ing typical results will go up as the leverage squared. At high enough risk levels, typical compound returns are not only less than optimal, they also are negative!

**Conclusion**

With very little mathematics, you can for mulate a successful multi-period investment policy based on expected logarithmic returns. I have adapted the Kelly rule used by gamblers and commodity traders to the world of conservative investing by refocus-ing it on discretionary wealth, rather than total wealth. In this framework, you can rec-ognize your implicit leverage and act accordingly to improve the average rate of compounding discretionary wealth under risk to better your median results.

Partitioning the *E-*V/2 formula into passive and active elements also allows you to identi-fy weaknesses in current approaches to dele-gating active investment responsibilities. Ignoring covariance between active value added and benchmark returns is the most obvious of these. Perhaps even more harmful is failing to motivate managers to fully exploit their skill through applying overly strict implic-it risk aversions to tracking error. The further by-product of reaching for the tail of return distributions necessary to meet the resulting search criteria biases selection toward man-agers with unsustainable good records who are especially likely to create disappointment. Finally, conventional practice relies too much on the information ratio as a sufficient meas-ure of contribution to the overall risk and return budget.