Why Leveraged ETFs and Inverse ETFs Suffer Time Decay?

There are lots of misinformation on the internet about leveraged and inverse ETFs. Most of them negative after investors observed the fact that the leveraged and inverse funds typically underperform x times the underlying index (x is the leverage ratio that can be -3, -2, -1, 2, or 3) after a period of time. 

 

A few examples here:

 

http://seekingalpha.com/article/1864191-what-you-need-to-know-about-the-decay-of-leveraged-etfs

https://www.tradeking.com/education/etfs/leveraged-and-inverse-etfs

http://www.nasdaq.com/article/leveraged-etfs-for-the-long-term-rockets-to-the-poorhouse-cm262071

 

The truth is:

 

1. these funds do a good job tracking the DAILY movement of the underlying times the leverage factor;

2. over time, it can underperform or outperform depending how the underlying time series were behaving.

 

In order to maintain the constant leverage ratio, the leveraged and inverse funds need to trade in the SAME direction of the intraday movements at closing, ie buy after a run-up and sell after a leg down, around market close at 4 PM EST. So if the market tends to move in the same direction in the following days, ie market is trending/demonstrating momentum, then the rebalancing trades buy at high, but sell higher, or sell at low and cover at even lower price, which leads to a compounding effects and improves performance. However if the market tends to retrace part of the previous movement, ie mean-reverting, then the rebalancing trade is buying at high and selling at low and will hurt the performance of the fund. 
 

Date	SPY	Ret_SPY	Balance	Pre-rebal $SPY	target$SPY	$Cash	Share_SPY	Trade_SPY
0	$  100.00		$  1,000,000	$                       -	$    3,000,000	$  (2,000,000)	30,000			rf	0%
1	$  110.00	10.00%	$  1,300,000	$       3,300,000	$    3,900,000	$  (2,600,000)	35,455	5,455
2	$  100.00	-9.09%	$      945,455	$       3,545,455	$    2,836,364	$  (1,890,909)	28,364	(7,091)
3	$    90.00	-10.00%	$      661,818	$       2,552,727	$    1,985,455	$  (1,323,636)	22,061	(6,303)		W_f	-200%
4	$  100.00	11.11%	$      882,424	$       2,206,061	$    2,647,273	$  (1,764,848)	26,473	4,412		W_s	300%
Date	SPY	Ret_SPY	Balance	Pre-rebal $SPY	target$SPY	$Cash	Share_SPY	Trade_SPY
0	$  100.00		$  1,000,000	$                       -	$    3,000,000	$  (2,000,000)	30,000			rf	0%
1	$  110.00	10.00%	$  1,300,000	$       3,300,000	$    3,900,000	$  (2,600,000)	35,455	5,455
2	$  120.00	9.09%	$  1,654,545	$       4,254,545	$    4,963,636	$  (3,309,091)	41,364	5,909
3	$  130.00	8.33%	$  2,068,182	$       5,377,273	$    6,204,545	$  (4,136,364)	47,727	6,364		W_f	-200%
4	$  140.00	7.69%	$  2,545,455	$       6,681,818	$    7,636,364	$  (5,090,909)	54,545	6,818		W_s	300%
Date	SPY	Ret_SPY	Balance	Pre-rebal $SPY	target$SPY	$Cash	Share_SPY	Trade_SPY
0	$  100.00		$  1,000,000	$                       -	$    3,000,000	$  (2,000,000)	30,000			rf	0%
1	$    90.00	-10.00%	$      700,000	$       2,700,000	$    2,100,000	$  (1,400,000)	23,333	(6,667)
2	$    80.00	-11.11%	$      466,667	$       1,866,667	$    1,400,000	$      (933,333)	17,500	(5,833)
3	$    70.00	-12.50%	$      291,667	$       1,225,000	$       875,000	$      (583,333)	12,500	(5,000)		W_f	-200%
4	$    60.00	-14.29%	$      166,667	$           750,000	$       500,000	$      (333,333)	8,333	(4,167)		W_s	300%