Understanding levered funds' long term performance

Concept

Many investors expect levered index funds to deliver 2x the returns of the underlying index and are surprised to discover that this isn't always the case. The reason for this discrepancy, which is disclosed in the levered index funds' prospectus and website, is due to the effective holding period of the levered index fund being 1 day, while the average investor's holding period will be significantly different. The levered index fund will therefore rebalance its leverage to 2x at the end of every day, distorting the returns that an investor would realize were he or she to begin with 2x leverage and never rebalance. In periods of higher volatility this effect becomes more pronounced.
Underlying Index
Matching levered fund
Price
Daily Return
Price
Daily Return
Start
$100.0
$100.0
Day 1
103.0
3.00%
6.00%
106.0
Day 2
99.9
(3.01%)
(6.02%)
99.6
Day 3
103.9
4.00%
8.01%
107.6
Day 4
101.3
(2.50%)
(5.00%)
102.2
Day 5
105.1
3.75%
7.50%
109.9
Total Return
5.10%
9.88%
1.9x
<< levered index fund under performs
This example is taken from the ProShares ETF site: http://www.proshares.com/funds/performance/UnderstandingProSharesLongTermPerformance.html

Scenarios

Volatile Increasing
Volatile Decreasing
Underlying Index
Matching levered fund
Underlying Index
Matching levered fund
Price
Daily Return
Price
Daily Return
Price
Daily Return
Price
Daily Return
Start
$100.0
$100.0
$100.0
$100.0
Day 1
103.0
3.00%
6.00%
106.0
97.0
(3.00%)
(6.00%)
94.0
Day 2
99.9
(3.00%)
(6.00%)
99.6
99.9
3.00%
6.00%
99.6
Day 3
103.9
4.00%
8.00%
107.6
95.9
(4.00%)
(8.00%)
91.7
Day 4
101.3
(2.50%)
(5.00%)
102.2
98.3
2.50%
5.00%
96.3
Day 5
104.3
3.00%
6.00%
108.4
95.4
(3.00%)
(6.00%)
90.5
Day 6
101.2
(3.00%)
(6.00%)
101.9
98.2
3.00%
6.00%
95.9
Day 7
105.3
4.00%
8.00%
110.0
94.3
(4.00%)
(8.00%)
88.2
Day 8
102.6
(2.50%)
(5.00%)
104.5
96.7
2.50%
5.00%
92.6
Day 9
106.5
3.75%
7.50%
112.3
93.0
(3.75%)
(7.50%)
85.7
Day 10
105.4
(1.00%)
(2.00%)
110.1
94.0
1.00%
2.00%
87.4
Total Return
5.42%
10.10%
(6.04%)
(12.59%)
1.86x
<< under
2.08x
<< over
Non-Volatile Increasing
Non-Volatile Decreasing
Underlying Index
Matching levered fund
Underlying Index
Matching levered fund
Price
Daily Return
Price
Daily Return
Price
Daily Return
Price
Daily Return
Start
$100.0
$100.0
$100.0
$100.0
Day 1
100.5
0.50%
1.00%
101.0
99.5
(0.50%)
(1.00%)
99.0
Day 2
100.9
0.40%
0.80%
101.8
99.1
(0.40%)
(0.80%)
98.2
Day 3
101.2
0.30%
0.60%
102.4
98.8
(0.30%)
(0.60%)
97.6
Day 4
101.7
0.50%
1.00%
103.4
98.3
(0.50%)
(1.00%)
96.6
Day 5
102.1
0.40%
0.80%
104.3
97.9
(0.40%)
(0.80%)
95.9
Day 6
102.4
0.30%
0.60%
104.9
97.6
(0.30%)
(0.60%)
95.3
Day 7
102.9
0.50%
1.00%
105.9
97.1
(0.50%)
(1.00%)
94.3
Day 8
103.3
0.40%
0.80%
106.8
96.7
(0.40%)
(0.80%)
93.6
Day 9
103.7
0.30%
0.60%
107.4
96.5
(0.30%)
(0.60%)
93.0
Day 10
104.2
0.50%
1.00%
108.5
96.0
(0.50%)
(1.00%)
92.1
Total Return
4.18%
8.51%
(4.03%)
(7.91%)
2.04x
<< slightly over
1.96x
<< slightly under

NASDAQ

From
2008-01-01
To
2008-07-16
INDEX
ULTRA LONG
ULTRA SHORT
^IXIC
NASDAQ Composite
QLD
PROSHARES ULTRA Q
QID
PS UTLRSHRT QQQ
Date
Open
Close
Daily Ret
Total Ret
Open
Close
Daily Ret
Total Ret
Open
Close
Daily Ret
Total Ret
2008-Jul-16
2219.27
2284.85
2.96%
(12.52%)
70.19
72.97
3.96%
(23.87%)
46.48
44.52
(4.22%)
13.46%
2008-Jul-15
2197.18
2215.71
0.84%
(15.17%)
69.11
69.64
0.77%
(27.34%)
47.19
46.96
(0.49%)
19.67%
2008-Jul-14
2262.86
2212.87
(2.21%)
(15.28%)
72.8
69.94
(3.93%)
(27.03%)
44.79
46.55
3.93%
18.63%
2008-Jul-11
2233.38
2239.08
0.26%
(14.28%)
71.76
70.97
(1.10%)
(25.96%)
45.51
45.82
0.68%
16.77%
2008-Jul-10
2239.95
2257.85
0.80%
(13.56%)
71.82
73.23
1.96%
(23.60%)
45.46
44.76
(1.54%)
14.07%
2008-Jul-09
2290.63
2234.89
(2.43%)
(14.44%)
75.49
71.99
(4.64%)
(24.89%)
43.39
45.3
4.40%
15.44%
2008-Jul-08
2244.9
2294.44
2.21%
(12.16%)
72.4
74.84
3.37%
(21.92%)
45.38
43.71