Economics Note: The Distribution of Leveraged ETF Returns

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Division of  Economic and Risk Analysis 1 
 
Economics Note: The Distribution of Leveraged ETF Returns 
 
Division  of  Economic and Risk Analysis
1
               November 2019 
 
I. Introduction 
 
Leveraged exchange-traded funds  (“LETFs”) seek  to  generate  returns  that  are  equal  to  a  multiple,  
inverse, or inverse  multiple  of the return on  a particular index or  benchmark over a short  period of  time, 
typically   one   trading  day.
2
 These  funds   allow  investors   to  obtain  levered  (or   inverse)  exposure    t o   a n  
underlying  asset  class  without  using  brokerage  margin  accounts  or  engaging  in  more  complex  trading  
strategies  using  futures  or  options.  Therefore,  LETFs  can  offer  a  cost-effective, easily  accessible,  and 
generally  liquid   tool  to   increase  or   decrease  such   exposure.   At  the  same  time,  LETFs   employ   dynamic  
trading strategies to achieve their objectives  and, when  held  over  longer periods,  can have  returns with 
complex properties, similar  to those of  options. As  a result, investor  protection concerns regarding these 
funds have been  raised.
3
  
Investors  who  hold  LETFs  over longer  holding  periods  can experience returns that deviate significantly 
from  an LETF’s target multiple due to compounding.
4
 We  analyze the performance of LETFs  by: (a) deriving 
the  theoretical  distribution   of  the  investment  returns  for  a  buy-and-hold   strategy  in  an  LETF;  and  (b)  
estimating the empirical distribution  of  investment returns  for a hypothetical  LETF by  repeatedly sampling 
historical  S&P  500  Index  returns.  While  the  theoretical  approach  allows   us   to  study   the  entire   return  
distribution   of  an  LETF  and  the  effect  of  different  values  for  leverage  and  investment  horizon  on  that  
distribution, such  an analysis  is only  possible  under a set of simplifying  assumptions  about the underlying 
index  return  distribution.  Conversely,  while  the  empirical  analysis  allows  us  to  rely  on  the  historical  
distribution   of  the  underlying  index  returns,  and  thus  can  capture  more  complex  properties  such  as  “fat  
                                        
                    
 
1
 The Staff of the Division of Economic and Risk Analysis of the U.S. Sec ur i ties a nd Exc hange Commission c omp osed  
thi s  note. The Commission has expressed no view regarding the analysis, fi nd ings, o r  c o nc lusion s c o ntain ed  h er ein. 
Nor has the Commission approved or disapproved its content. 
2
 For example, an ETF employing leverage to double the daily return of the S&P 500 Index would aim to return an 
i nves tor 2% on a  da y the S&P 500 i ncreases  by 1%. Similarly, a n i nver se ETF empl oying lever age to r etur n twi ce the 
inverse of the S&P 500 would aim to return to an investor 2% on a day when the S&P 500 declines by 1%. 
3
 See,  e.g.,  SEC  Chairman Jay Clayton’s public statement on “Taking Significant Steps to Modernize our Regulatory 
Framework” (Sept. 26, 2019), available at 
https://www.sec.gov/news/public-statement/clayton-20 19-09-26-three-
r u l ema k ings. 
4
 Bec a us e of potential c onfusion a mong i nvestors a bout the per for mance obj ectives of LETFs  a nd i nverse LETFs , 
the  SEC a nd FI NRA i s sued a n “I nvestor Al ert” a dvising i nvestors tha t “bec ause l ever aged a nd i nver se ETFs  r eset 
each day, their performance can quickly diverge from the performance of the underlying index or benchmark. In 
other words, it is possible that you could suffer significant losses even if the long-term per formance of the i ndex 
showed a gain.” See “Lever a ged a nd I nverse ETFs : Spec ialized Pr oducts wi th Extr a Ri sks for  Buy-and-Ho ld 
Investors.” (Aug. 1, 2009) Available at: https ://www.sec.gov/investor/pubs/lever agedetfs-alert.htm
. 

   
Division of  Economic and Risk Analysis 2 
 
tails”, it does  not allow us  to analyze how the properties of  LETF returns vary with an LETF’s leverage and 
the investment horizon as precisely. 
Our  analysis  of  the  theoretical  distribution   of  long-term   LETF  returns  shows   that,  under   certain  
simplifying   assumptions,   the  likelihood   of   experiencing  losses   from  a   long-term  investment  in   an  LETF  
increases with leverage, while the magnitude of  potential gains, when  they do  occur, also  increases with 
leverage. The returns to holding an option have similar characteristics.  Our subsequent empirical analysis 
using randomly sampled S&P 500 Index returns  suggests that the derived  theoretical distribution  provides 
a good approximation to levered ETF returns on the S&P 500 Index.  
 
II. Summary  of  the  Exchange-Traded  Funds Industry 
 
The ETF industry  has experienced extensive  growth since the first  U.S. ETF began trading in 1993. Since 
2007, the average growth rate in the number  of  ETFs has  been 10% annually  and total net assets of  ETFs 
have increased 15% annually; as of September 2019 there were 1,910 ETFs  with  total net assets  of  $3,081 
billion.
 
LETFs started trading in  2003 and, as of  September 2019, there were 164 LETFs  with $33.9 billion 
in total net assets (comprising  approximately 1% of  all ETF AUM).
5
  
 
III. Theoretical  and Empirical Return Distributions  for  LETFs 
Theoretical  Distribution  of LETF Returns 
Using simplifying  assumptions,  we  derive the return from  a buy-and-hold  investment  in an  LETF as a 
function  of   the  leverage  multiple  and  holding  period.
6
  These  simplifying   assumptions  allow  us   to  
understand  the  first-order   effects  of   investing  in   an  LETF  that  is   rebalanced  to  achieve  a  constant  daily  
leverage multiple. We assume daily  log-returns 푟푟
푡푡+1
 of the underlying  index between period t and period  
t+1 are independent  and  identically distributed  normal variables: 
 
                                                             
5
 Form N-CEN da ta . 
6
 Our assumptions are that (i) daily returns for an underlying index are independent and identically distributed log-
normal (there are no jumps and no serial correlation) and (ii) tha t there a re no transaction costs that would i mpact 
the  performance of an LETF because of daily rebalancing. Furthermore, we use a standard approximation to get 
the  log return of the levered portfolio. This approximation holds perfectly in continuous time and reasonably well 
over s hort ti me horizons. While we could analyze how LETF returns vary wi th underlying asset vol atility, varying 
the  volatility of the underlying asset also typically changes its expected return (e.g., bonds are less volatile than 
s toc ks  but ea r n lower expected returns).   Therefore, we focus on how the leverage multiple and holding period 
a ffec t LETF r etur ns  bec ause thes e fac tor s a re i ndependent of the under lying a sset bei ng s tudied. 

   
Division of  Economic and Risk Analysis 3 
 
This assumption implies  that  the log k-period return from period t to period t+k on an l-times  levered asset 
is  also approximately normal: 
 
Because  many  LETFs  use  the  S&P  500  Index  as  an underlying  index,  we calibrate the  parameters of 
our  theoretical  model  to  match  the  annual  mean  and  standard  deviation  of  S&P  500  Index  returns. This 
allows  us  to analyze the effects of  different values  for  the leverage (l)  and  holding  period  (k) parameters 
on  the  distribution   of  LETF  payoffs.
7
 Figures  1  and  2  graph  the  distribution   of  gross  returns—the payoff 
from  a  one-dollar   initial  investment—for  various   levered  and  inverse-levered  ETF  structures over  a six-
month horizon.
8
 As  leverage increases, the probability  mass of  the return  distribution shifts to the left. At 
the same  time, the mass  allocated to  the right  tail of  the distribution  also  increases. These changes  reflect 
                                                             
7
 Based on S&P 500 index historic al r etur ns, we a s sume a n a verage a nnual return of 6% and an annual volatility of 
15% for the underlying index. 
8
 More precisely, the figures show probability density functions that can be used to evaluate the probability that an 
outcome (x axis) falls wi thin a particular range of values. This probability is given by the integral of the density (y 
a xi s) over  that r a nge.   W hen the y axis has values larger than 100%, this does not mean the associated outcome 
has a probability of more than 100% of happening.   Rather, the area under the curve indicates the probability 
associated with a set of outcomes, and the total area under a given curve is always equal to 1. 
Figure 1: Theoretical distributions (density functions) of gross returns (the payoff from a $1 initial 
i nves tment) over  a  6-month investment horizon for the underlying index (Black) a n d LETFs  wi th 
l ever a ge mul tiples of plus two (Blue), pl us thr ee (Green), and plus four (Red), r espectively. 
 

   
Division of  Economic and Risk Analysis 4 
 
the increased skewness
9
  of  the  return  distribution: negative returns become more likely, while positive 
returns, when  they do  occur, tend  to be larger in magnitude. While we show  results for a fixed investment 
horizon  and  various  leverage  multiples,  fixing  leverage and varying the investment  horizon has a similar 
effect: as the investment horizon  increases, so does the skewness  of LETF returns.  
Empirical  Distribution  of  LETF Returns 
The  theoretical  analysis  above  relies  on  simplifying   assumptions   to  derive  the  return distribution  of  
an  investment  in  an  LETF  over  longer  holding   periods.  To  relax  these  assumptions,   we  simulate  LETF  
returns  based  on  randomly  sampled  historical  S&P  500  Index returns  and   analyze  their  distribution   for  
various  leverage multiples.  
Specifically, using daily return data for the S&P 500 Index from January 1964 to July 2017, we simulate 
100,000 daily index  return  price  paths  over  investment  horizons  ranging  from one  month  to  one  year.
10
 
Then,  for  each  simulated  return  price  path,  we  compute  the  holding  period  return  of  the  index  and the 
implied holding  period return for daily-rebalanced LETFs  with leverage ratios from minus  four  to plus  four.  
Figures 3 and 4 below are the empirical analogs of Figures 1 and 2 above,  and  are qualitatively similar 
to  the  theoretical  results  in  the  previous  section:  as  the  magnitude  of  the  leverage  ratio  increases,  
                                                             
9
 Skewness is a measure of asymmetry in a statistical distribution reflecting the degree to which the distribution 
c ur ve i s  skewed to the l eft or right of the mean of a variable.   
10
 W e gener a te pr i c e paths by randomly s ampling daily S&P 500 I nd ex  r eturns with r eplacement. This method 
c a ptur es c er tain fea tures of the historical return distribution (e.g., non-normality), but does not capture any ti me-
s er i es c or relation i n returns. 
Figure 2:  Theoretical  distributions  (density functions) of gross returns  (the  payoff  from  a  $1  initial  
i nves tment) over  a  6-month investment horizon for the underlying index (Black) and inverse LETFs  with 
l ever a ge mul tiples  of mi nus one (Blue), minus two (Green), minus three (Purple), and minus four (Red),  
r es pec ti vely. 

   
Division of  Economic and Risk Analysis 5 
 
negative returns  become more likely,  while  positive  returns, when they  do  occur,  tend  to  be  larger  in  
magnitude. In Figure 5, we fix the leverage ratio to plus  four  and show  the effect of  holding  an LETF over 
holding  periods ranging from  one month to a year. These empirical distributions  show  that increasing the 
amount  of  time  an  LETF  is   held  has  an  effect  that  is  similar  to  increasing  the  magnitude  of  the  leverage  
ratio in Figures 3 and  4, consistent with our theoretical results. 
 
Investor  Preferences  for  LETFs 
Risk-averse  investors  generally  prefer  higher  positive  skewness   in  returns  and  higher  expected  
returns,  while  having  an  aversion   to  volatility  and  other  even   moments,  such   as  variance and kurtosis.  
However,  since  the leverage multiple and the investment   horizon   both  increase  skewness   while  
simultaneously   decreasing  the  Sharpe  ratio  of   an LETF investment, it is not a  priori clear that investors 
ought  to  disfavor  LETFs  only   because  they  do  not  deliver  the  leveraged  multiple  of  a  correspondingly  
levered  but   non-rebalanced  investment.
11
 In  other  words,   investors   might  prefer  the   higher  skewness  
provided  by holding  an LETF over longer horizons  even if it means they expect lower risk-adjusted  returns  
and do  not necessarily receive the LETF’s daily  leverage multiple. 
 
 
                                                             
11
 A  ma thema ti cal a nalysis of how these moments  var y wi th a n LETF’s  l everage mul tiple a nd the i nvestment 
horizon is available in a separate technical appendix. 
Figure 3: Empi rical distributions (density functions) of gross returns (the payoff from a $1 initial 
i nves tment) over a 6-month investment horizon for the underlying index (Black) an d L E T F s  wi th  
l ever a ge mul tiples of pl us two (Blue), pl us thr ee (Green), and plus four (Red), r espectively. 

   
Division of  Economic and Risk Analysis 6 
 
IV. Similarities  Between  LETFs  and  Options 
 
As discussed  above, both the leverage and the holding period of  an LETF increase the skewness  of the 
payoff  distribution  while  decreasing the expected payoff  per unit  of  volatility. Similarly,  just  as a long-term 
investment in  an LETF becomes  more likely  to pay off  only  a fraction of the initial investment  as the daily 
leverage multiple increases, the probability of  an option  paying off nothing at maturity increases with the 
degree to which the option  is  out-of-the-money. 
For  example,  a  call  option’ s  payoff is  similar  to  that  of  the  underlying  stock  when  the option  is  deep 
in the money.
 
However, as the strike price increases, a call option  investment becomes more levered, the 
mass of  its return distribution  shifts  to the left, and the probability  of rare large payoffs  in the right tail of 
this  distribution   increases.
12 
This  is   similar  to the effect of  increasing an LETF’s leverage multiple or the 
period  over  which  the  LETF  is  held.  As  the  option  becomes  more  out of the money, it has a payoff  that is 
zero most  of  the time, resulting  in  an investment  return of -100%.
13
 However, in  the unlikely  event of  a 
                                                             
12
 Thes e s i milarities are wi th res pect to call options. An analysis with respect to put options would be identical, but 
the  c ompa r ative s tatic s ar e r ever sed. 
13
 Note  that some unlevered investments in equity or debt markets, such as distressed firm debt or unprofitable 
growth s tocks, can also exhibit payoff distributions tha t a re s imilar to out-of-the-money options. 
Figure 4: Empi rical distributions (density functions) of gross returns (the payoff from a $1 initial 
i nves tment) over a  6-month investment horizon for the underlying index ( Bl a ck) a nd  i nver s e   
LETFs  wi th l ever age mul tiples of mi nus one (Blue),  mi n us  two   (Green), mi nus thr ee (P u rple),  
and minus four (Red), r espectively.
 

   
Division of  Economic and Risk Analysis 7 
 
non-zero payoff, the investor is likely  to receive a large positive return on the investor’s initial investment. 
As  discussed   above,  under  certain  assumptions,   investors  may  value  the  positively   skewed  payoff   
characteristics of  LETFs  and  may  value  these  same  characteristics  in  options,  despite  the  fact  they  may 
require investors  to accept a high likelihood  of low payoffs. 
Figure 6 shows  how the empirical distribution  of the holding period  returns on an S&P 500 call option 
changes with the strike price of the call option. As with the LETF distributions  in Figures 1 and 3, increasing 
a  call  option’s  leverage  by  increasing  its  strike  price  shifts   probability  mass   to  the  left,  increasing  the  
likelihood   of   negative  payoffs.   At  the  same  time,  the  probability  of  rare  but  extremely  high  payoffs  
increases with the call option’s leverage. Comparing Figure 6 to Figures 1 and 3, the effect of  leverage on 
call  options   manifests  more  suddenly,   which  is   to  be   expected  given   the  non-linearity  in  their payoffs.  
Finally,  it  is  worth noting  that an option  contract can be  replicated by  a dynamic  trading strategy in the 
underlying   asset  or   index,  while  LETFs  are  also   dynamic  trading  strategies  in  the  underlying  index   or  
benchmark.  
 
 
 
 
Figure  5:   Empirical  distributions  (density  function)  of  gross  returns  (the   payoff  from  a  $1   initial  
i nves tment) for  a  four ti mes l ever ed LETF for investment horizons of one month (Black) , th r ee mo nths  
(Blue) , s i x  mo n ths (Green), and one yea r  (Red), r es pec tivel y.
  

   
Division of  Economic and Risk Analysis 8 
 
 
V. Conclusion 
 
The above analysis shows that the distribution  of  LETF returns becomes more  skewed as  their leverage 
multiple  increases: the likelihood  of  experiencing losses  from a  long-term investment in  an LETF  increases, 
while  the magnitude of  potential gains, when  they do  occur, also increases. These  features of  LETF returns 
are similar  to those  of options,  whose  skewness  increases with the extent to which in option  is  out of the 
money. Just  as investors  may  need a  higher level  of sophistication  to understand  the return characteristics 
of  options,   they  may also need  a  higher  level   of  sophistication   to  understand  the  returns  of  LETFs  over  
longer holding  periods. While a broker-dealer accepting a customer’s order for options  is subject to FINRA 
account  approval  and   due   diligence  requirements
14
,  similar requirements  for  transactions  in   LETFs  
currently do not exist. 
                                                             
14
 See, e.g.,  FINRA rule 2360(b)(16), (17) (requiring for options accounts, firm approval, diligence and 
r ec or dkeeping). 
Figure 6: Empi rical distributions (density function) of gross returns (the payoff from a $1 initial 
i nves tment) for  the underlying index ( Bl a ck) a n d 6-month call options on the S&P 500 with 
va r yi ng str ike pr ices.   The s trike pr ices c or respond to 25% (Blue), 75% (Green), 90% (P u rple),  
and 100% (Red) of the value of the S&P 500 at purchase.  The initial call price is obtained using 
the Bl a c k-Scholes formula assuming a risk-free i nteres t r ate of 5% and a n a nnual volatility of 
15%. 

Division of Economic and Risk Analysis 1 
 

Economics Note: The Distribution of Leveraged ETF Returns 
 

Division of Economic and Risk Analysis1             November 2019 

 

I. Introduction 
 

Leveraged exchange-traded funds (“LETFs”) seek to generate returns that are equal to a multiple, 
inverse, or inverse multiple of the return on a particular index or benchmark over a short period of time, 
typically one trading day. 2 These funds allow investors to obtain levered (or inverse) exposure to an 
underlying asset class without using brokerage margin accounts or engaging in more complex trading 
strategies using futures or options. Therefore, LETFs can offer a cost-effective, easily accessible, and 
generally liquid tool to increase or decrease such exposure. At the same time, LETFs employ dynamic 
trading strategies to achieve their objectives and, when held over longer periods, can have returns with 
complex properties, similar to those of options. As a result, investor protection concerns regarding these 
funds have been raised. 3  

Investors who hold LETFs over longer holding periods can experience returns that deviate significantly 
from an LETF’s target multiple due to compounding. 4 We analyze the performance of LETFs by: (a) deriving 
the theoretical distribution of the investment returns for a buy-and-hold strategy in an LETF; and (b) 
estimating the empirical distribution of investment returns for a hypothetical LETF by repeatedly sampling 
historical S&P 500 Index returns. While the theoretical approach allows us to study the entire return 
distribution of an LETF and the effect of different values for leverage and investment horizon on that 
distribution, such an analysis is only possible under a set of simplifying assumptions about the underlying  
index return distribution. Conversely, while the empirical analysis allows us to rely on the historical 
distribution of the underlying index returns, and thus can capture more complex properties such as “fat 
                                                             
1 The Staff of the Division of Economic and Risk Analysis of the U.S. Securities and Exchange Commission composed 
this note. The Commission has expressed no view regarding the analysis, findings, or conclusions contained herein. 
Nor has the Commission approved or disapproved its content. 
2 For example, an ETF employing leverage to double the daily return of the S&P 500 Index would aim to return an 
investor 2% on a day the S&P 500 increases by 1%. Similarly, an inverse ETF employing leverage to return twice the 
inverse of the S&P 500 would aim to return to an investor 2% on a day when the S&P 500 declines by 1%. 
3 See, e.g., SEC Chairman Jay Clayton’s public statement on “Taking Significant Steps to Modernize our Regulatory 
Framework” (Sept. 26, 2019), available at https://www.sec.gov/news/public-statement/clayton-2019-09-26-three-
rulemakings. 
4 Because of potential confusion among investors about the performance objectives of LETFs and inverse LETFs, 
the SEC and FINRA issued an “Investor Alert” advising investors that “because leveraged and inverse ETFs reset 
each day, their performance can quickly diverge from the performance of the underlying index or benchmark. In 
other words, it is possible that you could suffer significant losses even if the long-term performance of the index 
showed a gain.” See “Leveraged and Inverse ETFs: Specialized Products with Extra Risks for Buy-and-Hold 
Investors.” (Aug. 1, 2009) Available at: https://www.sec.gov/investor/pubs/leveragedetfs-alert.htm. 

https://www.sec.gov/news/public-statement/clayton-2019-09-26-three-rulemakings
https://www.sec.gov/news/public-statement/clayton-2019-09-26-three-rulemakings
https://www.sec.gov/investor/pubs/leveragedetfs-alert.htm


   

Division of Economic and Risk Analysis 2 
 

tails”, it does not allow us to analyze how the properties of LETF returns vary with an LETF’s leverage and 
the investment horizon as precisely. 

Our analysis of the theoretical distribution of long-term LETF returns shows that, under certain 
simplifying assumptions, the likelihood of experiencing losses from a long-term investment in an LETF 
increases with leverage, while the magnitude of potential gains, when they do occur, also increases with 
leverage. The returns to holding an option have similar characteristics.  Our subsequent empirical analysis 
using randomly sampled S&P 500 Index returns suggests that the derived theoretical distribution provides 
a good approximation to levered ETF returns on the S&P 500 Index.  

 

II. Summary of the Exchange-Traded Funds Industry 
 

The ETF industry has experienced extensive growth since the first U.S. ETF began trading in 1993. Since 
2007, the average growth rate in the number of ETFs has been 10% annually and total net assets of ETFs 
have increased 15% annually; as of September 2019 there were 1,910 ETFs with total net assets of $3,081 
billion. LETFs started trading in 2003 and, as of September 2019, there were 164 LETFs with $33.9 billion 
in total net assets (comprising approximately 1% of all ETF AUM). 5  

 

III. Theoretical and Empirical Return Distributions for LETFs 

Theoretical Distribution of LETF Returns 
Using simplifying assumptions, we derive the return from a buy-and-hold investment in an LETF as a 

function of the leverage multiple and holding period. 6 These simplifying assumptions allow us to 
understand the first-order effects of investing in an LETF that is rebalanced to achieve a constant daily 
leverage multiple. We assume daily log-returns 𝑟𝑟𝑡𝑡+1 of the underlying index between period t and period 
t+1 are independent and identically distributed normal variables: 

 

                                                             
5 Form N-CEN data. 
6 Our assumptions are that (i) daily returns for an underlying index are independent and identically distributed log-
normal (there are no jumps and no serial correlation) and (ii) that there are no transaction costs that would impact 
the performance of an LETF because of daily rebalancing. Furthermore, we use a standard approximation to get 
the log return of the levered portfolio. This approximation holds perfectly in continuous time and reasonably well 
over short time horizons. While we could analyze how LETF returns vary with underlying asset volatility, varying 
the volatility of the underlying asset also typically changes its expected return (e.g., bonds are less volatile than 
stocks but earn lower expected returns).  Therefore, we focus on how the leverage multiple and holding period 
affect LETF returns because these factors are independent of the underlying asset being studied. 



   

Division of Economic and Risk Analysis 3 
 

This assumption implies that the log k-period return from period t to period t+k on an l-times levered asset 
is also approximately normal: 

 

Because many LETFs use the S&P 500 Index as an underlying index, we calibrate the parameters of 
our theoretical model to match the annual mean and standard deviation of S&P 500 Index returns. This 
allows us to analyze the effects of different values for the leverage (l) and holding period (k) parameters 
on the distribution of LETF payoffs. 7 Figures 1 and 2 graph the distribution of gross returns—the payoff 
from a one-dollar initial investment—for various levered and inverse-levered ETF structures over a six-
month horizon. 8 As leverage increases, the probability mass of the return distribution shifts to the left. At 
the same time, the mass allocated to the right tail of the distribution also increases. These changes reflect 

                                                             
7 Based on S&P 500 index historical returns, we assume an average annual return of 6% and an annual volatility of 
15% for the underlying index. 
8 More precisely, the figures show probability density functions that can be used to evaluate the probability that an 
outcome (x axis) falls within a particular range of values. This probability is given by the integral of the density (y 
axis) over that range.  When the y axis has values larger than 100%, this does not mean the associated outcome 
has a probability of more than 100% of happening.  Rather, the area under the curve indicates the probability 
associated with a set of outcomes, and the total area under a given curve is always equal to 1. 

Figure 1: Theoretical distributions (density functions) of gross returns (the payoff from a $1 initial 
investment) over a 6-month investment horizon for the underlying index (Black) and LETFs with 
leverage multiples of plus two (Blue), plus three (Green), and plus four (Red), respectively.  



   

Division of Economic and Risk Analysis 4 
 

the increased skewness9 of the return distribution: negative returns become more likely, while positive 
returns, when they do occur, tend to be larger in magnitude. While we show results for a fixed investment 
horizon and various leverage multiples, fixing leverage and varying the investment horizon has a similar 
effect: as the investment horizon increases, so does the skewness of LETF returns.  

Empirical Distribution of LETF Returns 
The theoretical analysis above relies on simplifying assumptions to derive the return distribution of 

an investment in an LETF over longer holding periods. To relax these assumptions, we simulate LETF 
returns based on randomly sampled historical S&P 500 Index returns and analyze their distribution for 
various leverage multiples.  

Specifically, using daily return data for the S&P 500 Index from January 1964 to July 2017, we simulate 
100,000 daily index return price paths over investment horizons ranging from one month to one year.10 
Then, for each simulated return price path, we compute the holding period return of the index and the 
implied holding period return for daily-rebalanced LETFs with leverage ratios from minus four to plus four.  

Figures 3 and 4 below are the empirical analogs of Figures 1 and 2 above, and are qualitatively similar 
to the theoretical results in the previous section: as the magnitude of the leverage ratio increases, 

                                                             
9 Skewness is a measure of asymmetry in a statistical distribution reflecting the degree to which the distribution 
curve is skewed to the left or right of the mean of a variable.   
10 We generate price paths by randomly sampling daily S&P 500 Index returns with replacement. This method 
captures certain features of the historical return distribution (e.g., non-normality), but does not capture any time-
series correlation in returns. 

Figure 2: Theoretical distributions (density functions) of gross returns (the payoff from a $1 initial 
investment) over a 6-month investment horizon for the underlying index (Black) and inverse LETFs with 
leverage multiples of minus one (Blue), minus two (Green), minus three (Purple), and minus four (Red), 
respectively. 



   

Division of Economic and Risk Analysis 5 
 

negative returns become more likely, while positive returns, when they do occur, tend to be larger in 
magnitude. In Figure 5, we fix the leverage ratio to plus four and show the effect of holding an LETF over 
holding periods ranging from one month to a year. These empirical distributions show that increasing the 
amount of time an LETF is held has an effect that is similar to increasing the magnitude of the leverage 
ratio in Figures 3 and 4, consistent with our theoretical results.  

Investor Preferences for LETFs 
Risk-averse investors generally prefer higher positive skewness in returns and higher expected 

returns, while having an aversion to volatility and other even moments, such as variance and kurtosis.  
However, since the leverage multiple and the investment horizon both increase skewness while 
simultaneously decreasing the Sharpe ratio of an LETF investment, it is not a priori clear that investors 
ought to disfavor LETFs only because they do not deliver the leveraged multiple of a correspondingly 
levered but non-rebalanced investment. 11 In other words, investors might prefer the higher skewness 
provided by holding an LETF over longer horizons even if it means they expect lower risk-adjusted returns 
and do not necessarily receive the LETF’s daily leverage multiple. 

 

 

                                                             
11 A mathematical analysis of how these moments vary with an LETF’s leverage multiple and the investment 
horizon is available in a separate technical appendix. 

Figure 3: Empirical distributions (density functions) of gross returns (the payoff from a $1 initial 
investment) over a 6-month investment horizon for the underlying index (Black) and LETFs with 
leverage multiples of plus two (Blue), plus three (Green), and plus four (Red), respectively. 



   

Division of Economic and Risk Analysis 6 
 

IV. Similarities Between LETFs and Options 
 

As discussed above, both the leverage and the holding period of an LETF increase the skewness of the 
payoff distribution while decreasing the expected payoff per unit of volatility. Similarly, just as a long-term 
investment in an LETF becomes more likely to pay off only a fraction of the initial investment as the daily 
leverage multiple increases, the probability of an option paying off nothing at maturity increases with the 
degree to which the option is out-of-the-money. 

For example, a call option’s payoff is similar to that of the underlying stock when the option is deep 
in the money. However, as the strike price increases, a call option investment becomes more levered, the 
mass of its return distribution shifts to the left, and the probability of rare large payoffs in the right tail of 
this distribution increases. 12 This is similar to the effect of increasing an LETF’s leverage multiple or the 
period over which the LETF is held. As the option becomes more out of the money, it has a payoff that is 
zero most of the time, resulting in an investment return of -100%. 13 However, in the unlikely event of a 
                                                             
12 These similarities are with respect to call options. An analysis with respect to put options would be identical, but 
the comparative statics are reversed. 
13 Note that some unlevered investments in equity or debt markets, such as distressed firm debt or unprofitable 
growth stocks, can also exhibit payoff distributions that are similar to out-of-the-money options. 

Figure 4: Empirical distributions (density functions) of gross returns (the payoff from a $1 initial 
investment) over a 6-month investment horizon for the underlying index (Black) and inverse 
LETFs with leverage multiples of minus one (Blue), minus two (Green), minus three (Purple), 
and minus four (Red), respectively. 



   

Division of Economic and Risk Analysis 7 
 

non-zero payoff, the investor is likely to receive a large positive return on the investor’s initial investment. 
As discussed above, under certain assumptions, investors may value the positively skewed payoff 
characteristics of LETFs and may value these same characteristics in options, despite the fact they may 
require investors to accept a high likelihood of low payoffs. 

Figure 6 shows how the empirical distribution of the holding period returns on an S&P 500 call option 
changes with the strike price of the call option. As with the LETF distributions in Figures 1 and 3, increasing 
a call option’s leverage by increasing its strike price shifts probability mass to the left, increasing the 
likelihood of negative payoffs. At the same time, the probability of rare but extremely high payoffs 
increases with the call option’s leverage. Comparing Figure 6 to Figures 1 and 3, the effect of leverage on 
call options manifests more suddenly, which is to be expected given the non-linearity in their payoffs.   
Finally, it is worth noting that an option contract can be replicated by a dynamic trading strategy in the 
underlying asset or index, while LETFs are also dynamic trading strategies in the underlying index or 
benchmark.  

 

 

 

 

Figure 5: Empirical distributions (density function) of gross returns (the payoff from a $1 initial 
investment) for a four times levered LETF for investment horizons of one month (Black), three months 
(Blue), six months (Green), and one year (Red), respectively.  



   

Division of Economic and Risk Analysis 8 
 

 

V. Conclusion 
 

The above analysis shows that the distribution of LETF returns becomes more skewed as their leverage 
multiple increases: the likelihood of experiencing losses from a long-term investment in an LETF increases, 
while the magnitude of potential gains, when they do occur, also increases. These features of LETF returns 
are similar to those of options, whose skewness increases with the extent to which in option is out of the 
money. Just as investors may need a higher level of sophistication to understand the return characteristics 
of options, they may also need a higher level of sophistication to understand the returns of LETFs over 
longer holding periods. While a broker-dealer accepting a customer’s order for options is subject to FINRA 
account approval and due diligence requirements14, similar requirements for transactions in LETFs 
currently do not exist. 

                                                             
14 See, e.g., FINRA rule 2360(b)(16), (17) (requiring for options accounts, firm approval, diligence and 
recordkeeping). 

Figure 6: Empirical distributions (density function) of gross returns (the payoff from a $1 initial 
investment) for the underlying index (Black) and 6-month call options on the S&P 500 with 
varying strike prices.  The strike prices correspond to 25% (Blue), 75% (Green), 90% (Purple), 
and 100% (Red) of the value of the S&P 500 at purchase.  The initial call price is obtained using 
the Black-Scholes formula assuming a risk-free interest rate of 5% and an annual volatility of 
15%. 


	Economics Note: The Distribution of Leveraged ETF Returns
	I. Introduction
	II. Summary of the Exchange-Traded Funds Industry
	III. Theoretical and Empirical Return Distributions for LETFs
	Theoretical Distribution of LETF Returns
	Empirical Distribution of LETF Returns
	Investor Preferences for LETFs

	IV. Similarities Between LETFs and Options
	V. Conclusion