Systems Approach to Investment Management: Does Constant Proportion Investment beat Volatility Pumping?
By Dhruv Sharma
Dhruv Sharma is a credit risk manager. Prior to
working their Mr. Sharma worked at the largest mortgage company in the
Keywords: asset allocation, investment, system, risk/reward, simulation
Introduction
The aim of this study is to compare 3 investment
formulas: dollar cost averaging, value averaging, and constant proportion or
volatility pumping investing. By analyzing these investment schemes as
systems, new insights are found via monte carlo
simulation. The performance of these systems is assessed for a widely
applicable base case scenario with 2 widely used assets: S&P500 large cap
stocks, and small cap stocks. While dollar cost averaging and constant
proportion have extensive literature, value averaging has been studied less[1]. The results of the study show that
in contrast to past studies, by
Overview of 3 investment Formulas
Dollar Cost Averaging
Dollar cost averaging is the most well known system for investing money over time. Using a dollar cost averaging approach an investor puts in a fixed amount of money regularly in asset classes of their choosing at each appropriate unit of time. The advantages of dollar cost averaging are that it is easy to follow, systematic, and allows an investor to buy assets at different costs over time. The downturn in the market in 2008 clearly shows the value of dollar cost averaging, as people who may have put in a large lump sum investment in early 2008 will have experience considerable loss in a short period of time.
Value Averaging
Mark Edleson invented Value Averaging while at Morgan Stanly. In his own words, “Value averaging requires a target value for you to achieve at each point in time (Edleson, 88). With Value Averaging the emphasis is “on resulting value instead of an investment cost. Instead of a ‘fixed dollar’ rule as with dollar cost averaging…the rule under value averaging is to make the value of your stock holding go up by 100 or some other amount each month (39).
With Value Averaging an investor must determine the target end goal of the investment and set a value path, which indicates how much, the investor’s portfolio should be worth at each time t. If the value of the asset is greater than the expected value per the value path the investor must invest more capital to make the portfolio value equal to the value path. Likewise if the value of the assets is above the value path the investor must sell enough assets to again balance the portfolio towards the predetermined value path.
The calculation for Value Path is as follows:
C =investment contribution
Vt=Value Path at time t
N=investment horizon
r=expected rate of return
Vt=C *t (1+R)^t where t=time (73).
Although Value Averaging has had less research on it directly, it is a variation on the fixed amount investing studied by Sid Browne (Browne, 1998). The only difference between fixed amount investment and value averaging is that value averaging involves increased the fixed value targeted at each time period by a growth factor weighted by time.
Constant Proportion Investing aka 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)”
Given this a rebalancing period of 1 year seems safe and is simple to implement from a practical matter. For the purpose of this study an annual rebalancing frequency will be used.
The desirable properties of value averaging are that it is goal focused and ideal for investment managers who must show regular returns to investors over time. This is intuitive and straightforward. Value Averaging is similar to volatility pumping or constant proportion investment in that in both techniques rebalancing is done toward a preset goal. In Value Averaging the rebalance to a desired portfolio value while in constant proportion investing the rebalance is to a desired proportion of value of assets. A key difference between the techniques is that constant proportion investing is selffinancing while value averaging can be selffinancing depending on asset values but it may require additional investment to meet the value target. If the market has huge declines then making the value path rebalance may require a lot of additional capital, which an investor may not have without borrowing. If there are multiperiod declines then the investor could conceivably run out of money and be in debt. In contrast constant proportion can always be selffinanced without need for additional capital unless an asset loses all value and becomes insolvent. Although constant proportion investment is selffinancing one can investment additional capital at regular intervals like in dollar cost averaging just as long as the additional capital in combined in such a way that the desired proportions of the asset make up of the portfolio are achieved.
For the purpose of this investigation we use returns from index funds as they are free from credit risks such as firm level bankruptcy or insolvency. It is worth noting that passive investment does not work well with aggressive asset allocations. For example buying and holding a company that is likely to go bankrupt will end up in losses which can eat up any excess (abnormal) returns from other risky assets in one’s portfolio. Buy and hold does not go well with aggressive positions in highly volatile assets with credit risk. This is consistent with Perold and Sharpe’s research on dynamic asset allocation which showed buy and hold to be superior 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). One important point to note here is that although lump sum buy and hold of the present value of the capital an individual will earn and save would yield the best results, most individual investors end up implementing buy and hold through dollar cost averaging as they invest via 401k or IRA accounts over time[2]. This protects them from the risk of starting lump sum investing at the top of a market bubble and is also more consistent with their earnings flow.
A repeatable simple investment system to manage one’s portfolio is important. The role of discipline in investment is important for success. An ideal investment approach would allow investors the ability to safely contribute to their investment plan over time, remain within desired risk preferences, maximize return within given assets in the portfolio, and be diversified over time. This approach must meet the needs of the person making the investment. A systematic approach is our best defense against mistakes and costly behavioral biases.
The 3 systems of investment we are analyzing like all systems have the following components:
Components of Systems of Investment
System of Investment 
Dollar Cost Averaging 
Value Averaging 
Constant Proportion 
Inputs: 
expected value of investment throughout investment period 
expected value of investment throughout investment period 
expected value of investment throughout investment period 
Decision Variables: 
Amount,asset allocation 
Amount, target growth rate, asset allocation 
Proportion, asset allocation 
Output: 
terminal portfolio value 
cash, terminal portfolio value 
terminal portfolio value 
Focus: 
Averaging cost over time, systematic 
focused on maintaining and growth portfolio value over time to meet end goal. 
focused on maintaining proportions and increasing terminal wealth 
Approach to Evaluating Performance of Investment Systems
The question of what index of performance to use for comparison is a critical issue for comparing investment systems[3].
Pitfalls of IRR
IRR is a popular but flawed metric due to the fact that it is erroneously thought to be easy to use, easy to calculate, and does not require the need to specify a discount rate (Gibson, 1991). The following simple example highlights the known issue with IRR:
Cash flows for 2 different mutually exclusive investments 





Time Period 
0 
1 
2 
3 
4 
5 
IRR 
Modified IRR (funding and interim rate both =0) 

Investment 1 
100 
100 
100 
100 
100 
1000 
24% 
15% 

Investment 2 
100 
100 
100 
100 
100 
600 
30% 
12% 












In IRR interim positive cashflows are assumed to be reinvested at the IRR rate which can lead to biased and inaccurate results. Especially when comparing investment systems as systems that do not have positive interim cash flows will appear to be at a disadvantage like dollar cost or constant proportion when compared to a system like value averaging which can result in positive interim cash flows[4]. Unfortunately all prior studies of value averaging, asserting its dominance over dollar cost investing, have been based on IRR[5].
Further Pitfalls of Cashflow and Compounding in comparing System Performance
Given the power of compounding over long periods of time it is no wonder that lump sum investing appears to dominate other investment schemes. If rate of return is used as a metric for example then the first 10 years of a person’s invested capital would, in a diversified equity fund, over time have the highest return on investment, barring an economic catastrophe. The funds invested later in the investment horizon would not have as high a rate of return as they would not have had sufficient time to compound. When investing over long periods in a diversified market index fund the compounding effect can have an impact on the overall rate of return based on cashflows. A system like value averaging where the amount of capital invested increases over time will appear to have a lower rate of return on the entire capital invested as it is more weighted toward investments in the future. Using NPV would show that even if the rate of return of Value Averaging is lower it will result in more wealth creation. The pitfall with NPV in this scenario is that Value Averaging returns greater NPV because it invests more money in the market or assets than alternative strategies and as a result more money is invested in the market and has the benefit of compounding.
A Fair Comparison
A correction to IRR that makes it less misleading is to use the modified IRR in which the one sets the reinvestment rate for earlier cashflows to an appropriate like the risk free rate or 0 if the cash will not be reinvested.
A correction to the NPV issue of a system investing more money than other systems to be compounded over long periods would be to compute the expected value of investment in value averaging and adjusting the amount invested in other schemes to be use a higher cost basis for investment comparison. This requires a simulation on VA to compute the expected value of capital invested over time first and then adjusting inputs to dollar cost or constant proportion investing,
As simpler more elegant solution to the pitfalls discussed above is to use the geometric rate of return computed between the final value of the portfolio at the end of investment horizon from the total cost incurred by the investment system.
In this study we will use a variant of geometric return called the continuously compounded rate of return which is the natural log of (terminal wealth generated by the scheme/total investment)/number of periods (Benninga,148). This is less computationally intensive to compute than the geometric return as it involves division rather than taking a root of the return by the number of investment periods (148).
As value averaging can produce large positive cash flows along the way the assumption here is that those cashflows are reinvested at a risk free rate of 0 in the modified IRR calculation. These positive cashflows are counted in the terminal value of the portfolio at the end of the investment horizon.
Scenario
The
The model assumes 5K are invested at yearly intervals in a tax deferred Roth IRA.
The simulation is run 1000 times[6]. Transactions costs are assumed to be minimal and the risk free rate is taken as 0 for modified/adjusted internal rate of return.
Per best practice as pointed out by Wilcox it is assumed that all portfolios are based on discretionary wealth and do not represent wealth the individual investor needs until the end of the investment horizon. This ensures the investor is not ruined, as the wealth invested is not needed for consumption.
The 2 assets classes studied are large cap US stocks and small cap stocks. The mean annual return for large cap US stocks of 12.7% and standard deviation of 20% was used while a mean of 17.7% with a standard deviation of 34.4% is used (Ross, 354). These means/deviations are based on the annual returns from 19261998 computed by Ibbotson associates (354).
.
Results of Simulation
In analyzing returns based on the 3 investment schemes we can see that using traditional IRR replicates the dominance of value averaging over dollar cost averaging.
Traditional IRRcashflows as they occur
IRR 

DC 
VA 
CP 
1 asset (large cap); 20 year 
mean 
10.62% 
11.98% 
10.63% 
Stdev 
4.17% 
4.31% 
4.17% 

2 assets; 50% large cap /50% small cap allocation; 20 Year 
mean 
12.50% 
14.07% 
12.58% 
Stdev 
4.42% 
4.77% 
3.93% 

2 assets; 70 % large cap/30 allocation; 20 Year 
mean 
12.05% 
13.24% 
12.01% 
Stdev 
3.93% 
4.04% 
3.65% 

2 assets; 30 % large cap/70 allocation; 20 Year 
mean 
13.20% 
15.56% 
13.38% 
Stdev 
5.01% 
5.23% 
4.71% 
Using the adjusted or modified IRR reveals that IRR is not superior to dollar cost averaging. Both dollar cost averaging and constant proportion investing dominate value averaging in terms of mean return.
Adjusted IRRall positive cash flows counted at end of investment term
IRRadjusted (positive cashflows at end; interim cash flows not reinvested at IRR) 

DC 
VA 
CP 
1 asset; 20 year 
mean 
11.14% 
10.22% 
11.15% 
Stdev 
4.16% 
3.20% 
4.16% 

2 assets; 50% large cap /50% small cap allocation; 20 Year 
mean 
12.61% 
11.80% 
12.68% 
Stdev 
4.49% 
3.72% 
3.91% 

2 assets; 70 % large cap/30 allocation; 20 Year 
mean 
12.38% 
11.58% 
12.25% 
Stdev 
3.96% 
3.35% 
3.72% 

2 assets; 30 % large cap/70 allocation; 20 Year 
mean 
13.08% 
12.11% 
13.30% 
Stdev 
4.99% 
3.97% 
4.63% 
Using traditional IRR makes it appear that Value Averaging performs better than dollar cost averaging and constant proportion investing. This advantage is due to the fact that VA results in some positive cashflows earlier than VA or Constant Proportion schemes. As positive cash flows from VA may be used up in subsequent years the positive cashflows should only be realized at the end of the investment term. This is equivalent to using modified IRR with a 0 rate of reinvestment of interim cash flows. This is important as positive cash flows in value averaging can get used for subsequent period short falls.
Examining the continuously compounded rate of return of the total return from the investment scheme based on the total amount invested in the investment yields similar results to the modified IRR.
Rate of Return –
Continuously Compounded return 

DC 
VA 
CP 
1 asset (large cap); 20 year 
Mean 
6.52% 
4.41% 
6.52% 
Stdev 
3.08% 
2.26% 
3.08% 

2 assets; 50% large cap /50% small cap allocation; 20 Year 
Mean 
8.05% 
5.09% 
8.10% 
Stdev 
3.06% 
2.42% 
2.69% 

2 assets; 70 % large cap/30 allocation; 20 Year 
Mean 
7.62% 
5.06% 
7.59% 
Stdev 
2.63% 
2.01% 
2.43% 

2 assets; 30 % large cap/70 allocation; 20 Year 
Mean 
8.18% 
4.84% 
8.25% 
Stdev 
3.37% 
2.44% 
3.21% 
Although constant proportion returns in these runs not statistically significantly different from dollar cost averaging the standard deviation of the constant proportion returns are consistently lower than DCA for all scenarios with more than 1 asset. As Semivariance, as recommend by Leggio and Lien, which is variance below the mean asset return was also computed but gave similar results to standard deviation in most cases but in high volatility scenarios was lower for constant proportion investing (Leggio & Lien, 2003). As discussed earlier we can see that constant proportion or volatility pumping require >1 risky asset to have value and that the resulting value can be seen in lowered standard deviation. Also in the scenario where the majority of portfolio is invested in large cap stocks, dollar cost averaging appears to have a slightly higher rate of return than constant proportion investing. This highlights the fact that volatility pumping benefits from higher volatility assets and when their proportion is reduced it has less impact.
Value averaging has a much lower geometric return because VA invests more money later over time and money invested later in the investment term do not have time to compound.
The simulation shows that constant proportion investing dominates dollar cost averaging by reducing risk in terms of standard deviation for almost the same return. In addition both dollar cost averaging and constant proportion investing dominate value averaging in terms of mean rate of return. That said as value averaging can result in more money being invested in the assets it may have a higher net present value as it requires more capital to be invested on which the rate of return is earned. This can be corrected for by ensuring one invests the same amount in each system.
Extensions
A major advantage of building a monte carlo simulation for evaluating investment schemes is that it allows easy extension for what if analysis. We now will compare the methods in extreme scenarios to see which system works better in extreme scenarios.
The 2 extreme scenarios we will run the simulation are assets with negative returns and a scenario with extreme volatility (fat tails).
Also based on the simple intuition gained from studying the simple strategies discussed above an example using recent 2009 data with a hybrid strategy is studied.
Bizarre Scenario of Assets with Negative Return
Simulating scenarios via monte carlo simulation allows investors to test out the behavior of investment systems and develop an understanding of the complex behavior of systems and volatile assets. Recently research has claimed that any constant proportion investment fraction will dominate the return of any individual asset and that one can achieve a return >0 with assets with mean 0 returns. These scenarios will allow us to test out the assertion that constant proportion investing will provide a positive return even with assets with low or negative returns.
Continuously Compounded return 

DC 
VA 
CP 
Asset A 
Asset B 
Negative returns. 1%/2% , (same volatility) mean returns from assets with 50% allocation in each asset 
Mean 
1.03 
.75% 
1.03 
3.37% 
7.73% 
Stdev 
2.17% 
1.38% 
2.11% 
4.78% 
7.97% 

0 mean returns from assets (same volatility) with 50% allocation in each asset 
Mean 
.35% 
1.07% 
0.3% 
2.24% 
5.43% 
Stdev 
2.36% 
1.5% 
2.28% 
4.86% 
8.35% 

Normal returns per Ibbotson but with a standard deviation of 100% for both assets; 50% allocation each 
Mean 
3.1% 
4.12% 
5.9% 
20.2% 
11.5% 
Stdev 
8.95% 
3.34% 
8.74% 
23.17% 
21.95% 

Tiny mean returns (1% and 2% with normal volatility (stdev=.2,stdev=.3 respectively) 
Mean 
0.43% 
1.46% 
0.54% 
1.17% 
2.97% 
Stdev 
2.43% 
1.55% 
2.31% 
4.63% 
7.74% 
In examining the extreme scenarios it appears that all 3 systems perform better than the individual asset returns. In particular value averaging dominates both dollar cost averaging and constant proportion investing for scenarios with negative returns. The reason for this appears to be the fact that whenever abnormal returns are generated above the value path those returns are removed from reinvestment. In contrast constant proportion always reinvests abnormal returns back in the assets. In normal conditions this helps compound growth over time but under extreme risk and unfavorable markets this aspect of constant proportion investing produces a negative effect. Although value averaging performs better with assets yielding negative returns, constant proportion investing performs better than value averaging in high volatility scenarios with positive mean. Constant proportion investing does not perform as well as value averaging in scenarios where the mean asset return is negative or 0. That said constant proportion performs better than dollar cost averaging and the individual asset returns. This is especially of interest as recent research has claimed that any constant proportion strategy will dominate the growth of weighted average return of individual assets in a market. Although the extreme scenario returns above support this, the base case simulation of returns using the base case mean and variation does not. The constant proportion returns while better than dollar cost and value averaging under normal market mean and variance, per Ibbotson, did not perform better than the individual asset returns as claimed by Dempster, Evstignev & SchenkHoppe (2007).[7]
An interesting extension of this research would be to combine the strengths of value averaging and constant proportion investing. This hybrid would be a constant proportion scheme but would check the value path against the overall value of the portfolio and would move abnormal returns of large amounts or some fraction of the abnormal returns into a risk free asset. This would then result in a technique that performs well in normal long term investment scenarios with extreme variance as well as strange unforeseen scenarios of long term negative market conditions.
Another important point to note is that although certain techniques dominate the dominance is probabilistic. This means in general one technique may dominate the other technique as reflected in the mean but in the 1000 simulations another technique may dominate 2030% of the time. Picking a system does not guarantee that it would have beaten other systems but one can pick the system more likely to dominate given that it has better odds to giving the investor higher returns.
Extension: Simple Hybrid Model
From our study of constant proportion or value averaging strategies we can see the benefits or rebalancing towards a proportion in normal markets and rebalancing toward preset targets are beneficial in extreme markets as shown by the performance of value averaging. A simple practical application of this is the following hybrid investment system comprising of the following trading rule:
If portfolio value >Threshold1 then maintain X percentage in cash;
If portfolio value < threshold1 but the market exceeded the threshold in the past then retain Y percentage in cash else 0% in cash.
This investment rule was derived from the successful performance of value averaging during extreme market scenarios above. The intuition behind the rule is determining how best to safeguard abnormal returns provided by a market which then underperforms. This scenario is not unlike the current economic outlook.
Asset Class: For this exercise daily returns were analyzed for 2 highly volatile assets for the period of Jan 2009 to July 2009.
Ticker 
mean return 
stdev 
fnm 
0.001767201 
0.082087668 
fnmq 
0.004951051 
0.120798123 
Competing Systems: This hybrid model was tested against the alternative strategy of always keeping a fixed proportion in cash.
Per recent research by Daryanani this example uses frequent and recent data.
Analysis of Simple Hybrid Investment Rule
The simple hybrid rule requires 3 parameter decision to be made in addition to the asset allocation decision. The 3 parameters which must be tuned and optimally set are:
The financial planner must run
It is important to note asset allocations are applied after cash proportion is maintained. For example if 10% in FNM and 90% in FNMQ is the portfolio allocation and 10% is the cash allocation then 10% and 90% asset allocations apply to the 90% of portfolio value after cash has been removed.
A graph of the pareto optimal risk reward trade offs of daily compounded return vs. standard deviation shows that the hybrid rule allows greater return to be achieved with lower risk given appropriate parameter settings.
One added strength of this new hybrid model is that it can easily be optimized via a linear program to set either the asset allocation parameters or the model decision parameters conditioned on either setting. Also both the asset allocation decisions and hybrid strategy parameters can jointly be optimized via a linear program in excel as well.
Conclusion
By carefully analyzing investment systems and uncertain
asset returns using a
Thus by using simulation within a normative model for evaluating investment schemes we have been able to discover the following insights:
Further extensions of this work would be test the hybrid model of using constant proportion with a value path estimate to detect when abnormal returns occur and to put them in a risk free asset. In addition research on combining extreme returns with positive means can be combined with constant proportion investing to yield superior returns[8]. For interested readers another extension would be augmenting the constant proportion strategy with adaptive or dynamic proportions[9].
Notes
To ensure simulated prices do not become negative the max worse returns were capped at 99%.
VA is tantamount to fixed investment allocation studied by Sid Browne with the difference being the addition of a growth factor. Sid Browne showed that for 1 single asset case this approach results in a higher rate of return than constant proportion investing but that this difference can be considered an insurance premium due to the fact that any single asset can go bankrupt. If an investor only plans to use 1 asset then VA is beneficial. Most realistic scenarios, with the number of assets being >1, favor constant proportion investing.
Bibliography
Anderson, Keith P. and Brooks, Chris. (April 2005). ‘Extreme Returns from Extreme Value Stocks: Enhancing the Value Premium” Available at SSRN: http://ssrn.com/abstract=739667
Benninga, Simon (2002). Financial Modeling. MIT Press.
Browne, Sid (1998). “The Return on Investment from Proportional Portfolio Strategies”. Advanced Applications of Probability. 30, 216238.
Daryanani,. Gobind,. (2008). “Opportunistic Rebalancing: A. New. Paradigm for Wealth Managers.” Journal of Financial Planning.
Dempster,M; Evstignev, I.V; SchenkHoppe, K.R. (2007) “VolatilityInduced Financial Growth”. Quantitative Finance Vol. 7 No. 2.
Gibson, J.E. & Scherer, Bill (1991). How to do a Systems Analysis.
Leggio, K.L., Lien, D. (2003) Comparing Alternative Investment Strategies Using Risk Adjusted Performance Measures. Journal of Financial planning; 16,1.
Leong, K. (1998) Easy Simulation on Spreadsheets without Addins. Decision Line. March 1998. http://www.decisionsciences.org/decisionline/vol29/29_2/pom_29_2.pdf
Luenberger, D.G. (1998) Investment Science.
Kuhn, D. & Luenberger, D (2007). Analysis of Rebalancing Frequency in Log Optimal Selection. Submitted for publication.
Marshall, P.S. (2000) A Statistical Comparison of Value Averaging vs. Dollar Cost Averaging and Random Investment techniques. Journal of Financial and Strategic Decisions. Vol 13.
Perold, A.F.; Sharpe, W.F. (1995) Dynamic Strategies for Asset Allocation. Financial Analysts Journal. (Jan/Feb 1988):1627
Ross, S. ,Westerfield. R. & Jordan, B. (1997) Fundamentals of Corporate Finance.
McGraw Hill.
Wilcox, J (2004) Risk Management: Survival of the Fittest. Journal of Asset Management. Volume 5, Number 1, 1 June 2004 , pp. 1324
Appendix of Full results of Hybrid Strategy and Simple Constant Proportion Strategies with FNM/FNMQ JanJuly 2009
Results for Various Hybrid Strategies (note strategies where parameter Y is lower than X perform much worse in general and where Y is >=X perform better; This indicates that after periods of abnormal returns it is safer to keep more assets in cash even if the market appear undervalued if there is great volatility)
Red cells represent nonpareto optimal decision points.
Results from Always Investing Proportion in Cash and Using simple Constant Proportion Investing Alone
[1] For literature on dollar cost averaging see: Michael J. Brennan, Feifei Li, and Walt Torous, "Dollar Cost Averaging" (June 24, 2005). Finance. Paper 1705. http://repositories.cdlib.org/anderson/fin/1705
[2] Interestingly Gabay and Herlemont recently cite constant proportion beats buy and hold over a sufficiently long enough horizon in Gabay, D. & Herlemont, D. (2007)Benchmarking and Rebalancing. Conférence AFG sur le Benchmarking, 29 June 2007. http://www.yats.com/doc/optimalgrowthafgpaperen.pdf .
The long horizon can be longer than a normal life span as has been shown by Browne and Aucamp.
Aucamp,Donald
C. (1993) On the Extensive Number of Plays to Achieve
Browne, Sid.,(2000)
Can You Do Better Than Kelly In the Short Run? Chapter 12 (pages 215 231) in Finding the Edge: Mathematical Analysis of Casino
Games, edited by Vancura, 0., Endington, W., and
Cornelius, J. Reno: Institute for the Study of Gambling & Commercial
Gaming,
[3] See Leggio & Lien for other useful metrics like the Sortino ratio and Upside potential ratio.
[4] See for example the article entitled “The Trouble with IRR: Assumptions about reinvestment based on internal rate of return can lead to major capital budget distortions” in the McKinsey Quarterly October 20, 2004. Also http://www.pitt.edu/~sbm12/busfin1311/docs/capbud.ppt has details on Modified IRR.
[5] Both Marshall and Edleson use IRR to show Value Averaging’s superiority relative to dollar cost averaging.
[6] Wittwer,
J.W. (2004) “Summary Statistics for
[7] The large cap stock index had a mean continuous compounded return of 9.81% with standard deviation of 4.14% while the small cap stocks had a mean continuous compounded return of 12.03% with a sample standard deviation of 6.46%. Thus the mean returns of the individual assets were higher than the investment schemes but the risk was much higher as the standard deviation was higher. This is counter to SchenkHoppe’s work. The returns of the individual assets being lower than the constant proportion returns held true in the extreme scenarios as predicted by SchenkHoppe etal.
[8] See for example the work by Keith Anderson & Chris Brooks on “Extreme returns from extreme value stocks: Enhancing the Value Premium”. 2005.
[9] The easiest and simplest example of this is Flam, Sjur Didnk (2007). “Portfolio Management without Probabilities or Statistics”. In this example wealth is reallocated “towards assets that most recently generate better returns than the entire portfolio”. Source: http://ideas.repec.org/p/man/sespap/0508.html
For a more elaborate adaptive strategy see Thomas M. Cover. Universal Portfolios. Mathematical Finance, 1(1): 129, January 1991.