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A Better Performance Measure

2011 August 28

I have been convinced for a long time that the inappropriate way investment performance is measured is part of the reason that many investors,  even large pension funds, are shortchanged relative to what they should have achieved.  The following are ideas on performance measures customized to the risk management needs of the investor.  They are easy enough for any investor who can use a spreadsheet to implement.  This version is pretax, meant for those whose funds are mostly in accounts like IRAs and 401(k) plans.

Who Benefits From Investment Performance Measurement?

Disproportionately, it is the lucky.  The reason is that most investors invest in particular securities and investment managers based on excellent past returns, and there is overwhelming evidence that this leads to disappointment.  Not only are past returns not very predictive of future returns, but returns unadjusted for the risk control needs of the particular investor are generally misleading.

Investment consultants benefit, because they generally provide measurements for use by institutional investors, and because they are  often able to sell consulting services to find replacement investment managers based on, among other things, these measurements.

Investors themselves benefit very little.  Investors in mutual funds do worse on average than the mutual funds, because they buy and sell at the wrong times, and even if they stay fully invested, the average mutual fund doesn’t do as well as an index of its universe – such as the S&P 500.  A study of large pension funds (Goyal and Wahal, Journal of Finance, 2008) found that the investment managers fired by pension funds actually did better in succeeding years, on average, than did their replacement hires.  Though one can certainly argue that somehow investors would have done even worse without  performance measurements, in fact investors as a whole tend to select new investments from those vehicles that have had much better than average measured performance in the past, and this appears not to do them any good.

Motivating Ourselves and Our Brokers, Advisers, and Money Managers

There is a cycle in hiring and firing either our agents or our own investment strategies.  We pick people and strategies, and we fire them, and start over.  Presumably we do this in large part based on their performance records.  It doesn’t work very well for two reasons.  First, it is hard to distinguish skill from luck in investing.  Second, we don’t try very hard to devise performance measures that would serve as interim motivators in an investment world that is so unpredictable.  In general, we want to motivate for better long-term compound return and that turns out to be highly dependent on risk and on our ability to sustain risk without dropping out.  (Again, taxes are important, too.)  What we are suggesting here is that we pay more attention to this middle step, the motivation that comes from a performance measure more aligned with out interests.


Good Motivating Criteria


Our performance measure should be congruent with investor benefit.  It should take into account risk as well as return, because risks taken generalize much better to the future than do particular returns, and too much risk will cause compound return over longer periods to tend to be less than they could otherwise be.   Too much risk also means that we are likely to have to drop out of the game prematurely and never realize those favorable stock market returns that the textbooks tell us will eventually arrive.  And it should also be easy to measure without knowing a lot of statistics.


Before introducing our new risk measure, we need to make clear the impact of leverage on your personal ability to bear risk.  Your ability to bear investment risk is a function of how much you can afford to lose relative to the size of your investments. You can think of the ratio of your investments to the amount you can afford to lose, that is, your discretionary wealth, as a kind of financial leverage.   In the exhibit below we see a very simple balance sheet, with assets on the left and liabilities and something like net worth on the right.  The liability can include both a debt you have now and the present value  of what you will need to finance your retirement.  The investment portfolio here is about 2.8 times as large as what you can barely afford to lose.  That is, if your investments fall by 10%, what you can afford to lose will fall by 28%.  Even moderate fluctuation in investment returns will result in a rather significant fluctuation in the discretionary wealth buffer that stands between your investments and what you will need to pay off your debts and finance your retirement.
 Simple Balance Sheet

Of course, in a more realistic balance sheet, you may have other assets beyond investments as well as the present value of future savings or an inheritance that will also affect what you can afford to lose without damaging your ability to pay off your future obligations.  But the point is still the same.  Your leverage determines your ability to bear risk.

How does this kind of personal leverage affect your results at different levels of risk bearing?  The exhibit below illustrates for an investor with a fairly typical leverage ratio of 2.8.  It shows expected annualized returns over many periods for different risk-taking assumptions as they apply to both investments and discretionary wealth for this investor, given some realistic assumptions about the pre-tax returns and volatility of a mixture of index cash, bonds, and stocks today.

The general shape of expected long-term annualized returns as a function of risk-taking is curved.  Intermediate levels of risk do better than either very little risk or very great risk.  Look first at the flatter (blue) line.  It indicates that if there were no leverage — that is, if the investor could afford to keep playing the game even if nearly all his or her money were lost —  the optimum equity allocation is just slightly over 100%.  That is, in this example, one could invest on margin and do a little better.  The line is relatively flat, so that very little long-run result difference is expected among the various equity allocations between 80% and 120%.  Now look at the more sharply curved (red) line.  It shows the expected annualized return on the smaller base of discretionary wealth.  For low levels of risk, the leverage results in a high rate of return, but the curve peaks at around a 35% stock allocation and then drops sharply to negative rxpected annualized return by about a 120% leverage — that is, a 20% margin account, all in stocks.  Note that the chart generously assumes that the cost of borrowing on margin is no higher than the interest return on the fixed income portion of the portfolio.  In reality, margin borrowing is more expensive and would result in a faster drop off.

Expected Annualized Return


If the investor’s discretionary wealth were even lower than in this example, leverage would be still greater, and the discretionary wealth return would be higher at its maximum, since on a smaller base, but the optimium would be at a lower percentage allocation to equity.  This kind of exhibit makes clear the need for conservatism as the amount you can afford to lose, your discretionary wealth, shrinks relative to the investments you hold.

For those readers familiar with Markowitz mean-variance optimization, what we have done is to locate a customized tradeoff between risk and return based on the investor’s ability to bear risk, as shown in the exhibit below.  The advantage of this approach is that it is more objective and less likely to be far off the mark than asking yourself what your risk tolerance is, or inferring it from what you imagine you would feel under varying circumstances.



Efficient Frontier

The curves in the preceding example were constructed based on assumptions of return mean and variance, as well as their correlation, for fixed income and equity portions of a portfolio.  But that is an approximation of a more generalized form of this optimization — maximizing expected log leveraged return, or ln(1+Lr), where r is the portfolio return and L is the leverage multiple obtained from the balance sheet shown earlier.  This formula has the property of providing the best expected annualized rate of return. It takes into account not only mean and variance of return, but also the shape of the probability distribution — whether it is skewed or has “fat tails,” meaning non-normally high probabilities of very extreme outcomes.  Note that there is a disproportionate penalty attached to big losses.  This is because a significant loss interferes disproportionately with the ability to compound returns so as to produce a satisfactory long-term annualized return.  If you don’t believe this, consider a process in which you with equal probability gain 100% or lose 50%.  The expected return in a single period is +25%.  The expected long-term return over many periods is ZERO.  For the statistically minded, this function appropriately penalizes both negative skewness and high kurtosis, or “fat tails” of return probability distirbutions.  that is, risk reduces annualizes compound return from the expected return for a single period.  Log Leveraged Return

Now we can present our recommended measure of investment performance.  Average ln(1+Lr) gives us the growth rate of discretionary wealth.  Dividing this expression by leverage L scales the return down so that it is an appropriate risk-adjusted return for your investment portfolio.  You can think of it as approximately E – LV/2, where E is the expected single period return, V is the return variance, or the standard deviation squared, and L is the investor’s leverage relative to his or her discretionary wealth, or what can just barely be afforded as a loss.  The main difference is that the expected logarithmic leveraged return takes into account adjustments needed for the shape of the return distribution.

Now we have arisk-adjusted return measure appropriately customized to the needs of a particular investor with a leverage of 3 times.  That is, he could barely afford to lose a third of his investments without running into serious future problems.  The exhibit below shows the compounding effect of 4 hypothetical return histories.  They are constructed to begin and end at the same place, so as to illustrate the impact of taking the customized leverage of 3 times into the performance measure.  What we are trying to do is to measure performance in such a way that a sequence that would lead the investor to a greater chance of defaulting on their future obligations is penalized and a greater chance of enhancing their discretionary wealth so that they could increase their spending plans, is rewarded.

Each of the 4 sequences achieves a 7.3% return on investments.  but risk adjusted, the returns for the base “normal” case were reduced by volatility to 3.9%, a substantial reduction.  Even this case with returns generated from a normal distribution is heavily penalized by the apparent exceptional risk.  the higher variance sequence seems visually to be only a little more volatile, but of course when already high standard deviation is squared, apparently small differences loom much bigger, and the negative impact of this is multiplied here by 3 times!  Consequently, the customized risk adjusted return is only 2.8%.  Finally, to show the impact of changes in the shape of the return distribution when leverage is involved, but without affecting its mean or variance, sequences 3 and 4 were adjusted to produce a negatively skewed return distribution without excess kurtosis, and a stronger kurtosis (fat tails) return distribution without skew.  Each did better than the high variance case, but slightly worse than the “normal case,” producing risk-adjusted returns of 3.6%.


Hypothetical Return Histories

Well, you might say, such big differences reflect unusually high leverage or very highly volatile returns.  Actually, the return volatility is not particularly extreme for an investor holding individual stocks rather broadly diversified indexed funds in a balanced portfolio.  Well, then , what about the leverage?  Isn’t it rather high?  No, based on my experience, it is similar to that of many high net worth investors, and most investors probabily have higher leverages.

But it is true that the substantial reduction from unadjusted, unleveraged returns might give pause to the investor in any of these strategies.  That is the whole point.  Keeping track of your investments in this way reinforces two messages.  The first is to keep your risk-taking consistent with your ability to bear risk.  The second is to further motivate accumualting an adequate reserve of discretionary wealth so that you can afford to bear more risk and aspire to future increases in planned spending and gift-giving.  That is, it reinforces both good investing and good financial planning.

Jarrod Wilcox

August 28, 2011