A New Proposal for Log Optimal Wealth Creation for Low Income Families and Other Public financing

By Dhruv Sharma




            This article aims to propose the creation of a marketplace mechanism that can create significant wealth for the poor and low income citizens of a nation. The innovative approach described here is to create legislation that allows for the creation of a new product for investment targeted to low-income families.  The distinguishing feature of this product is that it creates wealth without imposing negative externalities on the wealthy or future generations as traditional policy tools like taxes do.




Recent research on ‘Encouraging Retirement Savings by Low-Income Families’, by Haisley and Lowenstein, shows that although low income families save disproportionately less and discusses novel approaches to increase low income savings ( 2009). 


Haisley and Lowenstein’s behavioral research shows that behavioral finance can be applied to increase savings via innovative techniques such as creating lottery like mechanisms for savings which appeal to low income families. 


This finding is important as “lotteries that offer a small chance of a high reward may be particularly useful for increasing motivation in low income populations” (Haisley, 2008).

If this is true then it can be further posited that low income families will find the chance of accumulating large wealth to be even more appealing gambles. 


Problem of Savings Disparity for Low Income Families

Low income families are not able to save for various reasons such as lack of free cash flows of savings due to debt aka less money to save, along with “no tax incentive for asset accumulation” (Haisley, 2009).  From my personal experience low income families tend to save via homeownership which has been a traditional source of wealth creation as opposed to stock market or 401k plans.  In general “Wealth accumulation for low-income and minority households, although low, experiences a major increase through home ownership (Boehm, Schlottman, 2005).  Despite this as income and home prices are correlated there is greater risk in recessionary periods of time for families to lose income and face a decling home vale (Caplin, Chan, Tracey, and Freeman, 1997).  This creates the scenario that when local job markets experience shock the value of the savings of home owners may dissipate just when they may need them to smooth out economic crisis.  Low income families also have low savings due to high debt (Warren, Sullivan, Westbrook, 2000).  If income is the primary source of savings then the 3 decades of real income stagnation for low income Americans is a growing societal problem (Lie, 2009).   Given this problem, it is no surprise that wealthy humanitarians like Bill Gates Sr. and Warren Buffet are calling for higher taxes on the wealthy (Krangel, 2009). 


Alternative Solutions to Taxation: Log Optimal System

While the Huey Long approach of taxing the Rockefellers of the world is not un-appealing other options which might also be optimal should be examined.  In this regard we can view the problem of low savings due to income and wealth disparity and also the underlying issue of “real income growth” disparity (Gordon, 2007). This disparity might contribute to the reason why lotteries are appealing to low income individuals along with behavioral biases.  If we reframe the partiality of low income households to gambling like systems as desire for rapid wealth accumulation then it would seem sensible to proscribe log-utility to low income households.  This is the utility for people who “want to become millionaires” (Thorp, 1971).  Recent research has shown that people who consume compounding debt, which grows exponentially, also fail to see benefits of compounding savings due to exponential bias (Stango & Zinman, 2007). 


If lack of capital, no real wealth growth through income, desire for rapid capital accumulation evidenced by lotto fever and tendency of falling victim to exponential growth bias are symptoms of the problem, then what is the optimal  “financial medicine”, as Zinman would say, to address this issue? 


One promising solution is to offer low income households a log optimal investment vehicle which will grow capital faster than any other competing investment approach and will accumulate wealth in minimal time (compared to other approaches) (Breiman, 1961).


Background of Log-Utility and Its Properties


The Log optimal portfolio or growth optimal investment has its root in information theory in the work of J.R. Kelly who pioneered the Kelly Criterion (Kelly, 1956). Kelly’s formula allowed one to compute the fraction of one’s wealth one should wager in a favorable gambling scenario.  Breiman in the 1960’s proved mathematically desirable properties of the Kelly rule or log optimal investment (Sewell, 2009).  Thorp popularized the use of Kelly criterion for portfolio management in various articles and by application in real life (Poundstone, 2005).   If the optimal fraction f is wagered according to the maximization of the expected value of the Logarithm of X (aka E(LogX)where X is the capital invested then the investor never over invests.  By using fractional, smaller wager than optimal f, an investor can assure that financial ruin will not occur as not all of the investors money is invested in risky assets.


Attractive properties of the Kelly or Log Optimal portfolio summarized by Ziemba are as follows:

  • “Maximizing log optimal growth maximizes the rate of rate growth asymptotically”
  • “Time to reach a pre-assigned investment goal is minimized with a strategy maximizing the log of capital”
  • “The log optimal bettor never risks ruin.”
  • “The E(LogX) bettor has an optimal myopic policy”

(Ziemba & Ziemba, 2007).


Volatility Pumping for excess Growth of Capital

An application of log optimal portfolios is volatility pumping.  This is done by investing a fixed fraction of one’s wealth in risky assets using optimal fractions based on maximizing the log return.  Thus log optimal portfolios are involving maintaining constant proportions and can be referred to as constant proportion investing or volatility pumping.  Constant Proportion investing requires an investor to determine what proportion of capital to allocate to each chosen asset class.  The investor then rebalances the assets to the preset proportions as asset prices fluctuate.  In addition to determining proportions for each asset class in one’s portfolio the investor must also determine how often to rebalance towards the proportions.   Volatility pumping works by reducing the magnitude of variance of returns which reduce returns over time (Luenberger, 1998).  As shown by Luenberger, Volatility pumping does not add much value in a one asset portfolio as this increases risk without much increase in return.  For volatility pumping to be effective at least 2 risky assets are needed.


One concern regarding rebalancing frequency is transaction costs.  Recently Luenberger and Kuhn have shown that “ if the length of rebalancing interval is on the order of 1 year…[then] the loss incurred by infrequent rebalancing is surprisingly small (Kuhn & Luenberger, 38).  Kuhn and Luenberger have shown that “continuous rebalancing only slightly outperforms discrete rebalancing if there are no transaction costs and if the rebalancing intervals are shorter than about one year (1)”


Opportunity of Economic Crisis: uncertainty and excess volatility

Luenberger points out to take advantage of volatility pumping one must have multiple assets with large volatility.  Given the current economic crisis the use of a log optimal volatility based portfolio takes advantage of the crisis to grow capital aggressively. Evstignev & Schenk-Hoppe, in their appropriately called paper: “From Rags to Riches: On Constant Proportions Investment Strategies” showed that if prices are volatile and assuming small transaction costs, constant proportion strategies yield a strictly positive exponential rate of growth (2001).  Perold and Sharpe’s research on dynamic asset allocation showed buy and hold to be superior to constant proportion investing only in markets trending upward while constant proportion referred to as constant mix tended to be better with greater fluctuations in the market (Perold and Sharpe, 1998)   Volatility pumping is most advantageous when markets are in turmoil and excess volatility exists such as in the current credit crisis.  As noted in the last major economic crisis,“stock return variability[is]...unusually high during depression  and is difficult to explain” (Schwert, 1989). 


Universal Portfolios for Low Income Families


An extension of log optimal portfolio concept is that of “Universal portfolios” by Thomas Cover.  Interestingly Cover worked for California State Lottery from 1986 to 1994 (Levy, 2000).    Prof. Cover’s Universal Portfolios are based on data-compression algorithms to create the so-called "universal portfolio." [where] each day the stock proportions in the universal portfolio are readjusted to track the best performing constant proportions.  The universal portfolio performs as well as the best strategies that keeping a constant proportion of wealth in each stock would have performed in hindsight” (2000).  Universal portfolios are not one constant proportion strategy but instead consist of investing “uniformly in all constant rebalanced portfolio strategies” (2000).   Yale’s Andrew Barron and Wei Qiu have recently developed algorithms that work on subsets of stocks making the approach easier to implement (Barron and Qiu,2007). 


The power of Universal portfolios is shown by the following example from Barron

Looking at the stock market in recent years, there are selections of stocks for which if the portfolio is rebalanced monthly to a constant proportion then over about a decade, e.g. from Jan 1992 through Dec 2002, one's wealth would have multiplied 5462 fold!  Moreover, investing in only 4 particular stocks in this way would have had a wealth factor of over 4000.  That is, $1000 invested in that selection of stocks would have become over $4 million.  Other strategies with non-constant portfolios would have had even higher return (Barron, 2003).


Role for Public Policy in the creation of Growth Optimal Portfolios for Low Income Households


Given the powerful nature of log optimal portfolios and universal portfolios which are resilient to volatility and catastrophic events, these techniques are an appropriate tool to wealth creation for low income households.  If governments can create incentives and subsidies for the creation of such products to be sold to low income families using behavioral engineered products such as the matched savings plans and lottery based investment schemes then low income families will be willing to participate in a gamble that is in their favor and has the potential to aggressively grow capital for savings purposes. 


Subsidies needed to make Universal Portfolios for Low Income Households a reality:

  • Transaction cost free trading for investment plans for low income households
  •             This could be implemented via tax credits to exchanges.
  • Government could provide a large enough starting capital base relative to which the transaction costs would be small through aggressive initial matching
  • Like the plans discussed by Haisley and Lowenstein these investment schemes would be matched savings plans where government would match savings with an appropriate matching incentive.
  • The owner of this option own the terminal wealth generated by the investment scheme.


The advantage of removing transaction costs for low income household investment products will allow investment algorithms to convert a few thousand dollars of savings into hundreds of thousands of dollars.  An alternative version of this scheme might be for the government to set aside money for each low income household and put in the investment scheme for low income households and subtract the initial capital from the terminal wealth after the investment scheme comes to maturity (which could be based on retirement age).  As the terminal wealth will be in excess of initial investment the tax on the excessive return could be recovered from the low income family wealth rather than by taxing other people.





Role for Private Enterprise


Investment banks and mutual funds or ETFs would provide the universal portfolio automated strategy as a product line.  For this product to work it would need to highlight the important properties of the product and market the product through appropriate distribution channels to low income households and automate the trading strategy via software.  This is important to shield the low income households form cognitive burden of investment and also from behavior biases shown to retard investment decision making.  The product would be simple for the purchaser as the low income investors are abstracted from the dealings of the funds, trading strategy, or the assets contained within it. 




Governments should create sufficient regulation and auditing of the log optimal universal portfolio schemes to ensure that appropriate investment fractions are used and no fraud is occurring.  As significant wealth will be created from the schemes it is important to sure the wealth makes it to the low income investment holders and that investment management is not charging excess fees.  Also there must be enough regulation to ensure too much leverage is not used in the investment schemes.





Giving low income families a subsidized engine for wealth creation will allow substantial value and social welfare to be created without taxation.  The aggressive growth of investment capital would serve the needs of low income that are lagging severely in real income growth, facing greater financial shocks, and are consequently unable to save for the long term.  Consistent with recent behavioral finance research low income households would be willing to invest in these lotteries like optimal gambling systems in a favorable game of savings.  Legislators in government, experts from Academia from Information Systems Sciences and Behavioral Finance experts in conjunction with private enterprise should work to create this product a reality.  This product would benefit to all countries with active stock exchanges and uses market volatility to help finance growth when and where it is most needed.  Once established log optimal portfolios free of transaction costs could become a source of public funds for various other public needs and are powerful alternative to taxes as sources of social welfare.







































Boehm, T.P., Schlottman, A. (2004) “Wealth Accumulation and Homeownership”

U.S. Department of Housing and Development Office of Policy Development and Research



Barron, Andrew (2003).  PORTFOLIO ESTIMATION FOR COMPOUNDING WEALTH. Retrieved May 3 2009 from .


Caplin, A., Chan,S., Tracy, J., & Freeman, C. (1997). Housing Partnerships

MIT Press.


Breiman, L., "Optimal Gambling Systems For Favorable Games," Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1, 65-78, 1961.


Collins, C. (2004) “The Economic Context: The Growing Disparities of Income and Wealth.” Retrieved from as of May 4 2009.


Evstigneev, I.V., Reiner, K. (2001) "From Rags to Riches: On Constant Proportions Investment Strategies" No iewwp089, IEW - Working Papers from Institute for Empirical Research in Economics.  Retrieved from as of May 1 2009.



Gordon, R.J. (2007) "Productivity Growth and the Distribution of Income: Results and Explanations" Presentation retrieved from  as of May 1 2009.


Haisley, Emily (2008) “The Appeal of Lotteries and their Use in Incentive Design”. Retreived from  May 2 2009.


Hailey, Emily; Lowenstein, George (2009) “Encouraging Saving by Low Income Families” U.S.-U.K. Conference on Behavioral Finance and Public Policy 2009.


Kelly, Jr, J. L., 1956. A New Interpretation of Information Rate. The Bell

System Technical Journal, 35(4), 917–926.


Krangel, Eric (2009) "Bill Gates' Dad: Raise Taxes On My Son!."   The business insider: Silicon Alley insider

Retrived from

 on May 2 2009.


Levy, Dawn. (2000)  "Universal portfolios take investors back to the future." News Service (4/12/00) Retrieved from as of May 4 2009.


Luenberer, D. (1998) Investment Science Oxford University Press.


Perold, André F., and William F. Sharpe. (1988) "Dynamic Strategies for Asset Allocation." Financial Analysts Journal. 44, no. 1 (January-February 1988): 16-27


Poundstone, W. (2005) Fortune’s Formula. Hill and Wang. New York.


Qiu, W., Barron, A. (2007) "Maximum wealth portfolios"

Retrieved from  May 4 2009


Sewell, Martin (2009) “Money Management”.  Retrieved from as of May 1 2009 .


Schwert, G.W. (1989) “Why Does Stock Market Volatility Change Over Time?” XLIV. No. 5. Journal of Finance


Thorp, E.O., (1972) "Portfolio Choice and the Kelly Criterion," Proceedings of the 1971 Business and Economics Section of the American Statistical Association


Zeimba, W., Ziemba, R. (2008) Scenarios for Risk Management and Global Investment Strategies. John Wiley and Sons. New York.


Zinman, J., Stango, Victor (2007) “Fuzzy Math in Household Finance: A Practical Guide” Retrieved from  as of May 3 2009.





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