


































http://journals.sfu.ca/abr  ADVANCES IN BUSINESS RESEARCH 
2014, Volume 5, pages 33-49 

 
 

33 

 

Long-Run Risk of Dynamic Asset Allocation Strategies  

Thomas S. Howe 
Illinois State University 

Ralph A. Pope 
California State University, Sacramento  

This study uses empirical resampling to examine the risk of three of Perold and Sharpe’s (1988) 
dynamic asset allocation strategies – buy and hold, constant mix, and constant proportion 
portfolio insurance (CPPI). Generally we find that the lower the floor percentage the greater 
the risk. However, which strategy has the most (or least) risk depends on how risk is measured. 
Finally, despite the positive floor buy and hold places on portfolio value, buy and hold is less 
risky than constant mix only in a few cases. 

Keywords: portfolio management; risk measurement 

 Introduction 

 When viewed in a one-period context under a traditional mean-variance framework common 
stocks are on average much riskier than Treasury bills. However, proponents of the concept of time 
diversification [for example, Reichenstein (1986), Levy (1978)] argue that over long horizons common 
stocks as a whole may not be riskier than Treasury bills if certain shortfall risk measures are considered 
to be the relevant risk measures. Levy points out that over every 25-year period beginning with January 
1926 or later the rate of return on the stock market as a whole has exceeded that of Treasury bills.  
 Butler and Domian (1991, 1992) point out that Levy’s finding may be biased in favor of 
common stocks, since the 25-year periods overlapped, and therefore were not independent. The 25-
year periods examined were January 1926 through December 1950, February 1926 through January 
1951, and so on. To overcome this weakness, they use empirical resampling (also known as historical 
simulation or bootstrapping) to generate independent series of returns. Although their results vary 
somewhat depending on the sub-period used in their resampling, the benefits of time diversification 
are apparent, especially over long investment horizons. 
 The above studies focus on long-run returns of individual asset classes rather than portfolios 
composed of more than one asset class; thus, they do not examine the effect of diversification between 
asset classes. When one diversifies, one has rebalancing decisions to make—decisions that can have a 
considerable effect on the risk and return, especially over the long-run. Perold and Sharpe (1988) 
present three dynamic strategies which combine investment in stocks with investment in Treasury 
bills to reduce the risk of equity portfolios—buy-and-hold (BH), constant mix (CM), and constant 
proportion portfolio insurance (CPPI). Dichtl and Drobetz (2010) discuss the attractiveness of such 
strategies to both institutional and private investors and examine the performance of the CPPI 
strategy. 
 Perold and Sharpe discuss the payoffs and exposures of these strategies. For example, they 
point out that during a strictly rising or strictly falling market BH dominates CM while during an 
oscillating market CM dominates. However, they do not examine the empirical long-run performance 
or risk. While Dichtl and Drobetz examine the long-run performance of the CPPI strategy, they use 
a period of only 22 years; given the volatility of the U.S. equity market, one may consider this to be 
too short a period. 

http://journals.sfu.ca/abr


LONG-RUN RISK OF DYNAMIC ASSET ALLOCATION STRATEGIES 

34 

 

 This study differs from the above-mentioned studies in that it (1) examines a far greater variety 
of risk measures than they do; and (2) focuses entirely on the risk of strategies rather than both risk 
and return. Although variance is a traditional measure of risk and is a key part of Markowitz portfolio 
theory and the capital asset pricing model, it is not clear that variance is always the most useful risk 
measure. Tversky (1990) points out that losses loom much larger than gains in people’s minds. Also, 
the consequences of falling short of a given return target could determine the appropriate measure of 
risk. For example, if failure is defined as earning less than the Treasury bill rate of return, the 
probability of earning less than what one could have by investing in Treasury bills could be a useful 
measure of risk (Balzer (1994) and Sortino and Price (1994)). On the other hand, as Balzer (1994) 
points out, shortfall probability is an incomplete risk measure because it fails to consider the amount 
of the shortfall. If the consequences of falling short of the return that could have been earned from 
investing in Treasury bills is more severe the farther one’s return is below that of Treasury bills, 
expected shortfall or lower partial variance measures which consider the frequency and magnitude of 
deviations below what could have been earned by investing in Treasury bills may be relevant. Finally, 
value at risk is a commonly-used risk management tool. If value at risk is considered to be the 
appropriate risk measure, specified lower percentiles of wealth relatives, such as the first or fifth 
percentiles, may be relevant. Still, if the conclusions as to which strategy is riskiest and the effect of 
floor percentage on risk are the same regardless of risk measure the choice of risk measure is 
unimportant.  
 The questions examined in this study are: (1) which strategy has the most (or the least) risk; 
(2) what impact the floor percentage has on the risk; (3) whether the answers to the first two questions 
are the same for all risk measures; and (4) what effect, if any, the choice of investment horizon has on 
the answers to the first three questions. Of these, the third question is the one of most interest. 
 This study uses empirical resampling (historical simulation) as used in Butler and Domian 
(1991, 1992) and Howe (1999) to examine the risk of the BH, CM, and CPPI strategies. The study 
uses monthly Morningstar/Ibbotson SBBI data going back to 1926 as the basis of the simulation. 
Thus, although this study uses fewer data points than Dichtl and Drobetz’s 22 years of daily returns, 
the sample period is much longer. 
 This study focuses on an investment horizon of 10 years. However, to examine whether the 
conclusions regarding the risks of the strategies relative to each other and the relationship between 
risk and floor percentage depend on the investment horizon it also examines investment horizons of 
20 and 30 years. This study uses CPPI floor percentages ranging from 10 percent to 95 percent. For 
comparison, this study also examines the long-run risk and ending wealth of the BH and CM strategies 
using equity positions of 10 through 95 percent. In addition to variance, this study uses a number of 
downside risk measures: value at risk, conditional value at risk, and lower partial moments. This study 
examines three types of lower partial moments: shortfall probability, expected shortfall, and lower 
partial variance. Lower partial moments require that the minimum acceptable wealth relative be 
specified. This study uses three different minimum acceptable wealth relatives: the mean wealth 
relative, the wealth relative one would have earned by investing entirely in Treasury bills, and a wealth 
relative of 1.0, which corresponds to a return of zero. 
 This study finds that generally the lower the floor percentage the greater the risk. However, 
which strategy is the riskiest (or the least risky) depends on how risk is measured. For risk measured 
based on deviations from the mean or deviations from the risk-free rate the CPPI strategy is the 
riskiest and the constant mix strategy is the least risky, with the buy and hold strategy being only 
slightly riskier than the constant mix strategy. For the other risk measures examined in this study the 
findings as to how the risks of the strategies compare with each other are very mixed. Finally, while 
the buy and hold strategy places a positive floor on value of the portfolio while the constant mix 
strategy does not, this does not make the buy and hold strategy less risky than the constant mix strategy 



Howe & Pope 

35 
 

unless one views risk as a particular low percentile of the ending wealth distribution, and even then 
only if the investment horizon is sufficiently short.  
 

 Survey of Relevant Literature 

The relevant literature addresses long-run assets class returns, portfolio insurance, and risk 
measurement. 
 
Long-Run Asset Class Return Studies 

One type of study examining long-run returns is a Monte Carlo simulation study, in which the 
return distributions (along with the autocorrelations and cross-correlations) of the various asset classes 
are assumed. Examples of such studies are Leibowitz and Langetieg (1989) and Shoven and Sialm 
(1998). Monte Carlo simulation studies involve sampling from assumed distributions, often normal or 
lognormal distributions with specified means and variances. In addition to avoiding the problem of a 
small number of independent multi-year periods, simulation studies allow one to vary the expected 
returns and standard deviations (and other parameters if the assumed distribution is non-normal). This 
enables one to examine the sensitivity of the results to the expected returns, variances, and so on. A 
disadvantage of the Monte Carlo simulation study is its sensitivity to the choice of return-generating 
process. Given the finding that return distributions tend to have fatter tails than a normal or lognormal 
distribution, using a normal or lognormal distribution is likely to underestimate the frequency and 
magnitude of extremely low returns, thus making common stocks look less risky than they really are 
(Lucas and Klaassen, 1998). 
 A second type of study involves assuming return-generating distributions and analytically 
deriving the future wealth or return distributions. Examples include Reichenstein (1986), Ho, 
Milevsky, and Robinson (1994) and McCabe (1999). While there is less of a problem of sampling error 
than in the Monte Carlo simulations, the results of this type of study are still sensitive to the choice of 
return-generating process. 
 A third type of long-run investment performance study is the empirical resampling (also 
known as bootstrapping or historical simulation) study. Depending on the extent to which past stock 
and Treasury bill market performances repeat themselves, this is potentially more realistic than the 
Monte Carlo simulation study. Empirical resampling differs from Monte Carlo simulation by using 
the observed distribution of monthly returns rather than an assumed distribution. Butler and Domian's 
initial study (1991) uses empirical resampling to examine the performance of lump-sum investments 
in common stocks and Treasury bonds. Their second study (1992) examines Treasury bills, corporate 
bonds, and low-capitalization common stocks as well. In addition, their second study considers the 
case of making deposits into a retirement fund, where an equal amount (in real terms) is invested each 
month. Both of their studies find that in the vast majority of cases common stocks earn more than 
bonds over periods of 20 years or longer. 
 
Portfolio Insurance Studies 

The major motive behind portfolio insurance is to minimize the chance of the large losses 
possible from investing in common stocks. Perold and Sharpe (1988) present four portfolio insurance 
strategies: buy-and-hold, constant mix, constant proportion portfolio insurance (CPPI), and option-
based portfolio insurance. Two decades later, Dichtl and Drobetz (2010) point out the continuing 
popularity of CPPI strategies, despite criticism from the academic community. 
 
 
 



LONG-RUN RISK OF DYNAMIC ASSET ALLOCATION STRATEGIES 

36 

 

Risk Measurement 
 Variance and standard deviation have long been presented in textbooks as the most 
commonly-used measures of total risk. However, they are not necessarily the best measures. Balzer 
(1994) lists several undesirable events that could imply risk measures other than variance are relevant. 
This list, which he points out is non-exhaustive, is dominated by shortfall situations such as negative 
returns or returns less than the risk-free rate. As further support for shortfall-based risk measures, 
Tversky (1990) points out that losses loom much larger than gains in people’s minds. 
 If investors are more concerned with downside variability than upside volatility, this would 
suggest that one should consider skewness when examining risk. Models which consider skewness 
include a three-moment CAPM (Kraus and Litzenberger 1976) and the mean-lower partial moment 
CAPM (Bawa and Lindenberg 1977). 
 

Methodology 

 
This section describes the historical simulation methodology used in this study as well as 

dynamic asset allocation strategies and expectations. 
  

Historical Simulation 
 This study uses the empirical resampling methodology of Butler and Domian (1991, 1992) to 
simulate Treasury bill and common stock returns. The steps in the empirical resampling procedure are 
as follows: 
1. Randomly select, with replacement, 600 months from the sample period, in this study January 

1926 through December 2009. 
2. Obtain the return for each of these months for each asset category 
3. Use the returns from step 2 to calculate wealth relatives for 600 months. 
4. Repeat the first three steps 9,999 more times to generate distributions of the wealth relatives. 
Thus, the simulation uses 10,000 series of 600 simulated monthly returns. To better identify the effects 
of the choice of strategy, floor percentage, and risk measure on the results, the same 10,000 series are 
used for each strategy, floor percentage, and risk measure. 

Although this procedure generates wealth relatives for periods as long as 50 years, the study 
will focus on the results for 10-, 20-, and 30-year periods. The common stock and Treasury bill returns 
used in this study are the monthly large-company common stock and 30-day Treasury bill returns 
from Morningstar/Ibbotson Associates. 
 
Dynamic Asset Allocation Strategies 
 The constant mix strategy specifies an initial common stock/Treasury bill allocation and 
rebalances the portfolio each period that the common stock and Treasury bill returns were not equal. 
Thus, the portfolio return in any given month i, Rip, is: 
 

R)x1(RxR iTBCSiCSCSip    (1) 

 
where 
xCS = weight of the portfolio invested in the common stock index 
RiCS = return on the common stock index in period i 
RiTB = return on Treasury bills in period i 
 
This study implements the constant mix strategy by creating a file of returns for each month from 



Howe & Pope 

37 
 

January 1926 through December 2009 for each weight of common stocks used in this study. In 
addition to examining the performance of 100 percent stock and 100 percent Treasury bill portfolios, 
this study examines the performance of portfolios containing the following proportions of common 
stock: 0.05, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, and 0.90, which imply floor proportions of 
0.95, 0.90, 0.80, 0.70, 0.60, 0.50, 0.40, 0.30, 0.20, and 0.10, respectively. Thus, this study creates ten 
such return files to sample from in the simulation. 
 The buy and hold strategy assumes a specified initial weight of common stocks but no 
rebalancing. Therefore, the wealth relative, that is, the ending wealth per dollar initially invested, after 
n periods, Wnp, is: 
 

W)x1(WxW nTBCSnCSCSnp        (2)  

 
where 
xCS = weight of the portfolio initially invested in the common stock index 
WnCS = wealth relative on the common stock index in period n 
WnTB = wealth relative on Treasury bills in period n. 
 
This study uses the same initial weights for the buy and hold strategy as for the constant mix strategy. 

For the CPPI strategy, the dollar amount allocated to common stocks as of period n, DnCS, is 
calculated as 

)]R1(FW[mD iTB

n

1i
0npnCS  


  (3) 

where 
m = CPPI multiplier 
F0 = initial floor value of the portfolio 
 
The multiplier for a typical CPPI strategy is greater than 1. The greater the multiplier or the lower the 
floor the more aggressive the strategy. If the common stock index drops by a proportion greater than 
1/m in one period the strategy fails in that the value of the portfolio drops below the floor. In such a 
case, this study assumes the entire portfolio is invested in Treasury bills. In this study leveraging is not 
allowed; therefore, in this study, DnCS is the lesser of Wnp or the result from equation (3). 
 The implementation of CPPI strategies in this study involves a simulation with the following 
steps for each period t: 
 
1. As of the beginning of period t calculate Wt−1p and DtCS. 
2. Calculate the weights of common stocks and Treasury bills as DtCS/Wt−1p and 1−DtCS/Wt−1p, 

respectively. 
3. Applying these weights to the returns on common stocks (xtcs) and Treasury bills (1−xtCS), 

respectively, for period t, calculate the return on the portfolio, Rtp, as 
  Rtp = xtcsRtCS + (1−xtCS)RtTB  
4. Calculate the wealth relative for the end of period t as Wt−1p (1+Rtp). 
 
This study uses CPPI multipliers of 1.5 and 2.0 and floor proportions of 0.10, 0.20, 0.30, 0.40, 0.50, 
0.60, 0.70, 0.80, 0.90, and 0.95. 
 This study applies various risk measures to the cross sectional distribution of 10-, 20-, and 30-
year wealth relatives. Other than mean absolute deviation and value at risk, all of the risk measures 
this study examines are generalizations of: 



LONG-RUN RISK OF DYNAMIC ASSET ALLOCATION STRATEGIES 

38 

 

 

  


10000

1j
jB

K
N/)WW(LS  (4) 

where 
L    = a dummy variable equal to 1 for all observations included in calculating the risk measure 
          and equal to zero for all other observations.  For observations below the benchmark L 
          equals 1. For downside risk measures, L equals 0 for all observations above the 
          benchmark while for risk measures which consider all observations L equals 1 for all 
          observations above the benchmark. 
WB = benchmark wealth relative. For calculations of the variance and standard deviation WB is 
          the cross-sectional mean of the wealth relatives.In addition to the mean, this study uses 
          benchmark wealth relatives equal to 1.0 (representing a geometric mean return of zero) 
          and that earned by a 100 percent investment in T-bills.  
K    = a parameter equal to 2 for variance-based measures, 1 for expected shortfall measures, and 
          0 for shortfall probabilities 
N    = the number of degrees of freedom. If K equals 0 or 1, N equals the number of 
          observations.  If K equals 2, N equals the number of observations minus 1. 
 
 Normally x% value at risk is defined as the loss which is exceeded (100-x)% of the time, and 
a positive number for value at risk corresponds to a loss. In this study, however, the wealth relative 
that corresponds to this implies a positive return rather than a negative return in many cases. To avoid 
the possible confusion from negative value at risk numbers, this study reports the (100-x)th percentiles. 
As presented by Jorion (2011), the (100-x)% conditional value at risk equals the expected loss below 
the xth percentile of ending wealth conditional on ending wealth being below the xth percentile of 
ending wealth. Thus, (100- x)% conditional value at risk can be calculated from equation (4) where: L 
equals 1 if the wealth relative is less than the xth percentile wealth relative and 0 otherwise; WB is xth 
percentile wealth relative; K equals 1; and N equals the number of observations less than the xth 
percentile wealth relative.  
 
Expectations 

One would not expect the ending wealth relatives to be normally distributed. For one thing, 
the distribution of common stock returns has been found to be fatter-tailed than the normal 
distribution. Also, even if one-period returns were normally distributed, the ending wealth relatives 
would not be normally distributed, since the wealth relatives are products of weighted sums of 
lognormal random variables. Even if the ending wealth relatives could be assumed to be normally 
distributed, including only the observations below the threshold leaves a distribution that is non-
normal; for these reasons, standard significance tests such as the F-test for equality of variances are 
not appropriate. In addition, the same 10,000 series of stock and Treasury bill returns are used for 
each strategy, floor percentage, and risk measure. Therefore, the results for the different strategies, 
floor percentages, and risk measures are not independent of each other; this would also invalidate the 
standard statistical tests. 

Even if valid statistical tests were available, the method used to calculate the wealth relatives 
would make the tests largely meaningless. Every set of 10,000 calculated wealth relatives is based on 
the same 10,000 sets of common stock and Treasury bill returns; the only things that differ when the 
strategy or floor percentage is changed are the weights of common stock and Treasury bills. Thus, this 
study is best viewed as a demonstration rather than a traditional hypothesis-based empirical study. In 
addition, historical simulation, which this study employs, assumes that the distributions of Treasury 



Howe & Pope 

39 
 

bill and common stock returns over the 1926-2009 period are typical of the Treasury bill and common 
stock markets. 
 Even though the simulated results generated in this study do not lend themselves to statistical 
testing, we can develop a priori expectations of some of the results. Because on average common stocks 
earn a higher rate of return than Treasury bills, one would expect the weight of common stocks in a 
buy and hold portfolio to generally increase over time. Because common stocks have a higher variance 
of return than Treasury bills, this implies that the variance and mean absolute deviation of the buy and 
hold strategy are higher than those of the constant mix strategy for any given initial weight of common 
stocks in the portfolio greater than 0 but less than 1. However, because the constant mix strategy 
involves buying stock as the value of the portfolio drops, the value of the constant mix portfolio could 
theoretically drop to practically 0. 
 On the other hand, the initial investment in T-bills places a nonzero lower limit on the possible 
value of the portfolio in the buy and hold strategy. Therefore, it is conceivable that some of the lower 
percentiles of the ending wealth distribution could be higher using the buy and hold strategy than the 
constant mix strategy, given the same initial common stock / Treasury bill weights for the buy and 
hold strategy as for the constant mix strategy. Because the greater the percentage of stock the greater 
the variation in wealth relative, the greater the percentage invested in common stocks the greater the 
conditional value at risk. This implies that the conditional value at risk for the buy and hold strategy 
is higher than that of the constant mix strategy  
 The buy and hold strategy is a special case of the CPPI strategy, in which the multiplier equals 
1 and the floor equals the initial investment in T-bills. Because for any given floor a higher CPPI 
multiplier implies a higher proportion invested in common stocks, one would expect the average 
variance of ending wealth for the CPPI strategy with a multiplier of 1.5 to be higher than those for 
the buy and hold strategy but lower than those for a CPPI strategy with a multiplier of 2.0. 
 

Results 
 

 Table 1 presents the mean-based risk measures, in this study the mean absolute deviation, 
variance, and semi-variance of the wealth relatives, for each strategy for the 120-month investment 
horizon. Throughout this study the results for the 240- and 360-month investment horizons are 
presented only if they differ qualitatively from those for the 120-month investment horizon. Also, the 
results for the CPPI strategy with a multiplier of 2.0 are qualitatively identical but more pronounced 
than those for the CPPI strategy with a multiplier of 1.5 and are therefore not presented. The floor is 
the percentage invested in Treasury bills for the constant mix and buy and hold strategies and is the 
floor initially calculated by equation (3) for the CPPI strategies. Initial floors of 0 and 100 percent 
correspond to 100 percent allocations in common stocks and Treasury bills, respectively.  
 As expected, for the most part, as the floor decreases the mean absolute deviation and variance 
of the wealth relative increase. There appears to a minor exception to this for CPPI strategies with a 
floor percentage of approximately 1−1/CPPI multiplier. The reason for this is unclear. 
 As one would expect given the difference in the proportion invested in common stocks and 
the risk of common stocks relative to Treasury bills, the constant mix strategy consistently has the 
lowest mean absolute deviation and variance of wealth relative and the CPPI strategy with a multiplier 
greater than 1 has the highest mean absolute deviation and variance of wealth relative. 

Although variance is a traditional measure of risk and is a key part of Markowitz portfolio 
theory and the capital asset pricing model, it is not clear that variance is always the most useful risk 
measure. Given evidence such as Tversky (1990) that losses loom much larger than gains in people’s  



LONG-RUN RISK OF DYNAMIC ASSET ALLOCATION STRATEGIES 

40 

 

Table 1.  
Mean-Based Risk Measures — 120 Month Investment Horizon 

    
Floor % MAD Variance Semivariance 

   
 Buy and Hold 

0 1.4603 4.1618 1.1241 
10 1.3143 3.3708 0.9105 
20 1.1682 2.6632 0.7195 
30 1.0222 2.0389 0.5509 
40 0.8763 1.4979 0.4049 
50 0.7303 1.0403 0.2813 
60 0.5844 0.6659 0.1803 
70 0.4386 0.3748 0.1017 
80 0.2931 0.1671 0.0456 
90 0.1485 0.0427 0.0120 
95 0.0785 0.0117 0.0036 
100 0.0312 0.0015 0.0007 

    
 Constant Mix 

0 1.4603 4.1618 1.1241 
10 1.2219 2.7974 0.8131 
20 1.0097 1.8415 0.5742 
30 0.8212 1.1794 0.3933 
40 0.6541 0.7277 0.2588 
50 0.5067 0.4262 0.1612 
60 0.3769 0.2311 0.0927 
70 0.2632 0.1110 0.0471 
80 0.1644 0.0428 0.0192 
90 0.0806 0.0103 0.0049 
95 0.0471 0.0035 0.0017 
100 0.0312 0.0015 0.0007 

    
CPPI 1.5 

0 1.4603 4.1618 1.1241 
10 1.4603 4.1618 1.1241 
20 1.4605 4.1631 1.1249 
30 1.4627 4.1732 1.1307 
40 1.4699 4.2006 1.1404 
50 1.4691 4.1804 1.1170 
60 1.4032 3.8140 0.9791 
70 1.2476 3.0740 0.7307 
80 0.9789 2.0083 0.4187 
90 0.5662 0.8079 0.1333 
95 0.3077 0.2848 0.0393 
100 0.0312 0.0015 0.0007 

 
  



Howe & Pope 

41 
 

minds, downside risk measures may be more useful that variance or mean absolute deviation. 
 Table 1 presents one such measure–the semi-variance of the wealth relatives. Although semi-
variance considers only variations below the mean while variance and mean absolute deviation 
consider all variations from the mean, whether below or above the mean, the semi-variances show 
exactly the same pattern as the mean absolute deviation and the variance. This suggests that even if 
downside deviations weigh considerably more on people’s minds than upside deviations do, this does 
not qualitatively change the risk of the strategies relative to each other or the relationship between 
floor percentage and risk. The finding that the semi-variances are less than half of the corresponding 
variances, especially for the CPPI strategy, is consistent with positive skewness in the distributions of 
the wealth relatives. 
 Table 2 presents three shortfall risk measures for which the benchmark is the wealth relative 
of a 100 percent investment in Treasury bills. The measures are the shortfall probability, the expected 
shortfall, and the lower partial variance relative to Treasury bills.  
 For the buy and hold strategy it can be shown that if the initial investment in common stocks 
is greater than zero the probability of the wealth relative being less than that from a 100 percent 
investment in Treasury bills is independent of the initial investment in common stocks and, therefore, 
the floor percentage. When the magnitude of the shortfall in considered, as in the expected shortfall 
and the lower partial variance, the risk increases as the floor percentage decreases. For the constant 
mix strategy, the increase in risk as the floor percentage decreases is apparent for all three risk 
measures. 

For the CPPI strategy, the relationship between floor percentage and risk measure varies 
somewhat depending on the risk measure. For floor percentages of greater than 90 percent the 
probability of a wealth relative less than that from Treasury bills increases as the floor percentage 
decreases. However, for floor percentages less than 90 percent the probability of a wealth relative less 
than that from Treasury bills declines as the floor percentage decreases. This reversal is less 
pronounced and occurs at a lower floor percentage for the expected shortfall than for the shortfall 
probability. While there is a slight reversal in the lower partial variance, it is much less pronounced 
and occurs at a lower floor percentage than the reversal in expected shortfall. This finding is not 
surprising, considering that the amount of the shortfall affects the lower partial variance proportionally 
more than it affects the expected shortfall, and the amount of the shortfall has no effect on the 
shortfall probability. 
 For all three risk measures, the constant mix strategy is the least risky and the CPPI strategy is 
the riskiest for all floor percentages between 10 and 95 percent. However, the buy and hold strategy 
is only slightly riskier than the constant mix strategy. 
 Table 3 presents the same three risk measures as Table 2 except that the benchmark return is 
zero. For floor percentages of 80 percent or higher all three risk measures were zero for all strategies; 
therefore, results for floor percentages greater than 80 percent are not presented. In general the 
relationships between floor percentage and risk measure are consistent with those in Table 2, although 
the decline in the shortfall probability for the CPPI strategy as the floor percentage decreases is much 
less pronounced and begins at a much lower floor percentage than it does in Table 2. Unlike the cases 
shown in Tables 1 and 2, the buy and hold strategy generally appears less risky than the constant mix. 
This is consistent with the hypothesis that the buy and hold strategy places a positive lower limit on 
the value of the portfolio while the constant mix strategy implies that one sells T-bills and buys stocks 
as the value of the portfolio drops and therefore has a theoretical lower limit of zero. However, as 
Table 4 shows, this effect disappears when the investment horizon is lengthened to 240 months. 
 
 



LONG-RUN RISK OF DYNAMIC ASSET ALLOCATION STRATEGIES 

42 

 

Table 2. 

Shortfall Risk Relative to Risk-Free Rate — 120 Month Investment Horizon 

    

Floor % Prob. < RF Exp. Shortfall vs RF LPV vs RF 

    

Buy and Hold 

 
0 0.1757 0.0635 0.0345 

10 0.1757 0.0571 0.0280 

20 0.1757 0.0508 0.0221 

30 0.1757 0.0444 0.0169 

40 0.1757 0.0381 0.0124 

50 0.1757 0.0317 0.0086 

60 0.1757 0.0254 0.0055 

70 0.1757 0.0190 0.0031 

80 0.1757 0.0127 0.0014 

90 0.1757 0.0063 0.0003 

95 0.1757 0.0032 0.0001 

    

Constant Mix 

0 0.1757 0.0635 0.0345 

10 0.1681 0.0550 0.0276 

20 0.1588 0.0471 0.0216 

30 0.1513 0.0397 0.0164 

40 0.1443 0.0328 0.0119 

50 0.1362 0.0263 0.0082 

60 0.1295 0.0203 0.0052 

70 0.1226 0.0146 0.0029 

80 0.1164 0.0094 0.0013 

90 0.1100 0.0045 0.0003 

95 0.1078 0.0022 0.0001 

    

CPPI 1.5 

0 0.1757 0.0635 0.0345 

10 0.1757 0.0635 0.0346 

20 0.1757 0.0637 0.0354 

30 0.1773 0.0659 0.0349 

40 0.1882 0.0709 0.0289 

50 0.2171 0.0737 0.0206 

60 0.2498 0.0674 0.0133 

70 0.2658 0.0535 0.0075 

80 0.2717 0.0365 0.0034 

90 0.2731 0.0186 0.0009 

95 0.2729 0.0096 0.0002 

 
 



Howe & Pope 

43 
 

  
Table 3.   
Shortfall Risk Relative to Return of Zero — 120 Month Investment Horizon 

    

Floor % Prob. (Loss) Expected  Loss LPV vs 0 

 

Buy and Hold 

0 0.0625 0.0137 0.0046 

10 0.0534 0.0099 0.0028 

20 0.0444 0.0064 0.0015 

30 0.0310 0.0036 0.0007 

40 0.0196 0.0016 0.0002 

50 0.0079 0.0004 0.0000 

60 0.0008 0.0000 0.0000 

70 0.0000 0.0000 0.0000 

80 0.0000 0.0000 0.0000 

    

Constant Mix 

0 0.0625 0.0137 0.0046 

10 0.0519 0.0101 0.0031 

20 0.0415 0.0069 0.0019 

30 0.0297 0.0044 0.0010 

40 0.0205 0.0024 0.0005 

50 0.0117 0.0011 0.0002 

60 0.0045 0.0003 0.0000 

70 0.0007 0.0000 0.0000 

80 0.0000 0.0000 0.0000 

    

CPPI 1.5 

0 0.0625 0.0137 0.0046 

10 0.0625 0.0137 0.0046 

20 0.0629 0.0140 0.0048 

30 0.0674 0.0150 0.0047 

40 0.0788 0.0142 0.0034 

50 0.0778 0.0090 0.0014 

60 0.0438 0.0025 0.0002 

70 0.0019 0.0000 0.0000 

80 0.0000 0.0000 0.0000 

 
 
 
 
 
 
 
 



LONG-RUN RISK OF DYNAMIC ASSET ALLOCATION STRATEGIES 

44 

 

Table 4.   
Shortfall Risk Relative to Return of Zero — 240 Month Investment Horizon 

    

Floor % Prob (Loss) Expected  Loss LPV vs 0 

   

 Buy and Hold 

0 0.0155 0.0035 0.0013 

10 0.0102 0.0019 0.0006 

20 0.0055 0.0008 0.0002 

30 0.0027 0.0002 0.0000 

40 0.0002 0.0000 0.0000 

50 0.0000 0.0000 0.0000 

60 0.0000 0.0000 0.0000 

70 0.0000 0.0000 0.0000 

    

Constant Mix 

0 0.0126 0.0028 0.0010 

10 0.0101 0.0021 0.0008 

20 0.0059 0.0013 0.0004 

30 0.0036 0.0007 0.0002 

40 0.0019 0.0003 0.0001 

50 0.0008 0.0001 0.0000 

60 0.0001 0.0000 0.0000 

70 0.0000 0.0000 0.0000 

    

CPPI 1.5 

0 0.0155 0.0035 0.0013 

10 0.0155 0.0035 0.0014 

20 0.0175 0.0041 0.0014 

30 0.0208 0.0037 0.0009 

40 0.0164 0.0014 0.0002 

50 0.0006 0.0000 0.0000 

60 0.0000 0.0000 0.0000 

70 0.0000 0.0000 0.0000 

 
 
 Table 5 presents the fifth and first percentiles of ending wealth and the conditional value at 
risk associated with them. As expected based on the proportion of common stock in the portfolio, 
the lower the floor percentage the lower the first and fifth percentiles of ending wealth for the buy 
and hold and constant mix strategies. For the CPPI strategy this relationship reverses slightly for low 
floor percentages, a finding consistent with the CPPI findings shown on Tables 1 through 3.  

The fifth percentiles of ending wealth are only slightly lower for the buy and hold strategy than 
for the constant mix strategy while the first percentiles of ending wealth are in almost all cases higher 
for the buy and hold strategy than for the constant mix strategy. Furthermore, both the 95% and 99% 
conditional values at risk are lower for the buy and hold strategy than for the constant mix strategy, 
with the 95% conditional value at risk enough lower for the buy and hold strategy to more than make  



Howe & Pope 

45 
 

Table 5.  
Lower Percentiles and Conditional Value at Risk — 120 Month Investment Horizon 

     
Floor % 5th Percentile 95% CVAR 1st Percentile 99% CVAR 

     
 Buy and Hold 

0 0.9257 0.1918 0.6002 0.0989 
10 0.9767 0.1827 0.6818 0.0879 
20 1.0289 0.1733 0.7639 0.0773 
30 1.0807 0.1560 0.8465 0.0674 
40 1.1304 0.1385 0.9296 0.0582 
50 1.1809 0.1189 1.0120 0.0490 
60 1.2306 0.0815 1.0978 0.0439 
70 1.2805 0.0637 1.1762 0.0329 
80 1.3281 0.0456 1.2542 0.0247 
90 1.3694 0.0277 1.3254 0.0195 
95 1.3852 0.0221 1.3491 0.0143 
100 1.3701 0.0015 1.3457 0.0012 

     
 Constant Mix 

0 0.9257 0.1918 0.6002 0.0989 
10 0.9842 0.1856 0.6676 0.0995 
20 1.0439 0.1786 0.7390 0.0984 
30 1.1015 0.1676 0.8137 0.0947 
40 1.1589 0.1553 0.8918 0.0885 
50 1.2120 0.1378 0.9752 0.0821 
60 1.2629 0.1178 1.0603 0.0719 
70 1.3106 0.0950 1.1497 0.0608 
80 1.3515 0.0672 1.2383 0.0451 
90 1.3841 0.0371 1.3218 0.0260 
95 1.3919 0.0233 1.3538 0.0173 
100 1.3701 0.0015 1.3457 0.0012 

     
 CPPI 1.5 

0 0.9257 0.1918 0.6002 0.0989 
10 0.9257 0.1918 0.6002 0.0990 
20 0.9229 0.1932 0.5968 0.1037 
30 0.8935 0.1765 0.6011 0.0585 
40 0.8694 0.1142 0.6803 0.0370 
50 0.9223 0.0804 0.7915 0.0284 
60 1.0128 0.0622 0.9146 0.0247 
70 1.1142 0.0487 1.0362 0.0201 
80 1.2168 0.0370 1.1577 0.0185 
90 1.3140 0.0255 1.2725 0.0164 
95 1.3584 0.0221 1.3222 0.0143 
100 1.3701 0.0015 1.3457 0.0012 

  



LONG-RUN RISK OF DYNAMIC ASSET ALLOCATION STRATEGIES 

46 

 

up for the fifth percentile of ending wealth being slightly lower for the buy and hold strategy than for 
the constant mix strategy. Combined, this implies that the average ending wealth of the bottom 1% 
and 5% of the wealth relatives is higher for the buy and hold strategy than for the constant mix strategy 
for almost all floor percentages examined in this study. This is consistent with the hypothesis that the 
buy and hold strategy places a positive lower limit on the value of the portfolio while the constant mix 
strategy does not. However, as Table 6 shows, this does not show up in the 240-month results. This 
suggests that floor on the portfolio value provided by the buy and hold strategy provides less downside 
protection the longer the investment horizon. 
 

  Summary and Conclusion 
 

This study has focused on the following questions: (1) of the buy and hold, constant mix, and 
CPPI strategies, which strategy has the most (or the least) risk; (2) what impact the floor percentage 
has on the risk; (3) whether the answers to the first two questions are the same for all risk measures. 
In addition, this study has examined what effect, if any, the choice of investment horizon has on the 
answers to the first three questions. The study makes no attempt to determine which strategy is 
optimal; this would depend on not only on how the investor perceives risk, but also on the investor’s 
risk tolerance and return target.  

This study generally finds the CPPI strategy to be the riskiest and the constant mix strategy to 
be least risky of the three strategies. Consistent with expectations, the strategies which involve the 
greatest investment in common stocks showed the greatest variance of ending wealth. Also generally 
consistent with expectations, as the floor percentage decreases the mean absolute deviation and 
variance of ending wealth increased in most cases, the exception being the constant proportion 
portfolio insurance strategy with a floor percentage below approximately 1-1/CPPI Multiplier. 
 Even though variance is the most traditional risk measure used in this study, the other risk 
measures based on deviations from the mean—semi-variance and mean absolute deviation—yield the 
same conclusions regarding the risk of the strategies relative to each other and regarding the 
relationship between risk and the floor percentage. In addition, the risk measures based on deviations 
below the risk-free rate yielded the same conclusions. 
 If this were the case for all risk measures, the question of what risk measure to use would be 
largely moot. However, it is not the case. For the other risk measures examined in this study no strategy 
was consistently the most—or the least—risky. This raises the question of which risk measure is the 
best to use. There is no clear answer to this question. Traditional finance theory suggests that variance 
is the best risk measure. However, evidence that investors prefer positive skewness (Kraus and 
Litzenberger 1976) and that deviations below the mean weigh two or more times as heavily on 
investors’ minds than deviations of equal magnitude above the mean (Tversky 1990) suggests that 
downside risk measures are more appropriate than risk measures which consider upside and downside 
deviations equally.  

Although shortfall probabilities and value at risk, which are measures of the mere probability 
of a sufficiently poor outcome, have been criticized for being too simplistic in that they ignore the 
severity of the shortfall, they have their uses. For example, using the probability of loss as a risk 
measure is consistent with an individual viewing risk as the possibility of losing money (Kaiser 1990). 
Also, value at risk is commonly used in financial institution risk management. Finally, investment 
policy statements may specify that the mere fact that a portfolio manager earns less than a specified 
rate of return over a specified period may be grounds for the portfolio manager being replaced (Trone, 
Allbright, and Taylor 1996). 

Even if the magnitude of the shortfall is considered, there is no theoretical reason that  



Howe & Pope 

47 
 

Table 6.  
Lower Percentiles and Conditional Value at Risk — 240 Month Investment Horizon 
    

Floor % 5th Percentile 95% CVAR 1st Percentile 99% CVAR 
    

             Buy and Hold 
0 1.5768 0.4172 0.8782 0.1989 
10 1.6244 0.3758 0.9924 0.1769 
20 1.6717 0.3343 1.1163 0.1651 
30 1.7190 0.2930 1.2284 0.1421 
40 1.7708 0.2564 1.3489 0.1282 
50 1.8153 0.2131 1.4617 0.1077 
60 1.8599 0.1708 1.5752 0.0894 
70 1.9077 0.1332 1.6863 0.0714 
80 1.9502 0.0942 1.7940 0.0573 
90 1.9863 0.0616 1.8851 0.0422 
95 1.9899 0.0468 1.9101 0.0277 
100 1.9232 0.0301 1.8746 0.0222 

    
            Constant Mix 

0 1.5768 0.4172 0.8782 0.1989 
10 1.6831 0.4137 0.9979 0.2118 
20 1.7788 0.4005 1.1220 0.2203 
30 1.8607 0.3757 1.2435 0.2182 
40 1.9327 0.3450 1.3658 0.2098 
50 1.9903 0.3055 1.4943 0.2010 
60 2.0264 0.2520 1.6114 0.1759 
70 2.0509 0.1964 1.7322 0.1512 
80 2.0593 0.1372 1.8394 0.1145 
90 2.0432 0.0757 1.9188 0.0634 
95 2.0150 0.0484 1.9355 0.0358 
100 1.9232 0.0301 1.8746 0.0222 

     
           CPPI 1.5 

0 1.5768 0.4172 0.8782 0.1989 
10 1.5768 0.4177 0.8749 0.1976 
20 1.5713 0.4404 0.8226 0.1711 
30 1.4872 0.4147 0.8244 0.0905 
40 1.3280 0.2483 0.9395 0.0551 
50 1.3364 0.1461 1.0996 0.0430 
60 1.4484 0.1056 1.2768 0.0415 
70 1.5901 0.0831 1.4541 0.0402 
80 1.7396 0.0685 1.6272 0.0386 
90 1.8790 0.0545 1.7880 0.0320 
95 1.9366 0.0468 1.8586 0.0265 
100 1.9232 0.0301 1.8746 0.0222 

 
 
 

 
 
 



LONG-RUN RISK OF DYNAMIC ASSET ALLOCATION STRATEGIES 

48 

 

investors’ utility is inversely proportional to the square, or any other given power, of the deviations 
below the target return. The square of these deviations may be easier to work with statistically, but 
this does not imply that investors really perceive risk that way. 
 Finally, although the buy and hold strategy guarantees a higher floor value than the constant 
mix strategy, for floor percentages of 90 percent or less this led to the buy and hold strategy having a 
higher first percentile of wealth relative over relatively short investment horizons such as ten years, 
but not over investment horizons of twenty or more years. Also, for floor percentages between 40 
and 70 percent the buy and hold strategy had a lower shortfall probability relative to a benchmark 
return of zero that the constant mix strategy did. All other downside risk measures were higher for 
the buy and hold strategy than for the constant mix strategy. Thus, while the buy and hold strategy 
theoretically has a floor greater than zero while the constant mix strategy does not, the effect of this 
floor appears to be beneficial only over relatively short investment horizons in the most extreme cases 
of poor performance.  

Because the study is a historical simulation, it assumes that the distributions of Treasury bill 
and common stock returns will continue to be similar to what they were over the sample period, 1926-
2009. This assumption limits the extent to which one can generalize the results. An extension to this 
study would be to change the average return or volatility of Treasury bills or common stocks. Finally, 
this study assumes the first order serial correlation is zero for both Treasury bills and common stocks. 
However, Butler and Domian (1991) suggest that this assumption has little quantitative and no 
qualitative effect on the results. 

 

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Thomas S. Howe received his Ph.D. degree in Business Administration (Finance) at Texas Tech 
University in 1986.  He is currently Professor of Finance at Illinois State University.  Dr. Howe has 
published 27 articles in journals, including the Journal of Economics and Finance, Financial 
Counseling and Planning, and the Quarterly Journal of Business and Economics.  
 
Ralph A. Pope received his D.B.A. degree in Finance at Mississippi State University in 1984.  He is 
currently Professor Emeritus at California State University, Sacramento.  Dr. Pope has published 46 
articles in peer-reviewed journals including the Journal of Economics and Finance, The Engineering 
Economist, and The Journal of Accountancy. 

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