Doesn’t it reduce your returns though? It increases transaction costs, and it increases variance drag. If a highly-leveraged stock is more volatile then it will perform worse under daily rebalancing than a stock with equal returns but less volatility.
Under the assumption that each day’s expected return is a constant that doesn’t depend on the previous days, the average growth rate of your investment is basically the same whether you rebalance daily or weekly or monthly. It decreases slightly as you go to longer time scales (e.g. at 2x leverage annually you will get wiped out pretty often, so your average growth rate is negative).
(Of course if days are correlated with each other, then different rebalancing periods will have different values. E.g. if there is short-term momentum then rebalancing frequently will be good and if there is short-term reversal then it will be bad.)
If you care about expected returns rather than growth rates, then rebalancing daily is a good idea iff you are using >1x leverage. Well, hard to say what it means for it to be a “good idea,” since your optimal strategy is infinitely much leverage, which yields infinite expected returns. But having Nx leverage rebalanced daily will lead to higher expected returns than rebalancing monthly.
Yes, more volatile stocks with the same expected (i.e. arithmetic average) returns have lower growth rates. That’s true whether or not you use leverage though. More volatile stocks with the same growth rate will have lower growth rates when leveraged, whether you rebalance daily or annually.
Thanks, Paul. To further address Michael’s question, I think the reason why leverage gives returns raised to a power, rather than multiplying by a constant factor is the rebalancing. Let’s say we have $100 and we take out a $100 loan and invest those $200. This is 2x leverage. However, if the fund increases 10%, we would have $220 and still a $100 loan, which is less than 2x leverage. In order to maintain the leverage, you should take out $20 more in loan and then have $120 in loan and $240 in the market. Then you can see why as the stock goes up, a given percent increase will give you a greater dollar increase if you rebalance. The converse is true if the stock goes down. And this is how you protect yourself from going to zero (a given percent decrease in the market means a smaller dollar value loss if you reduce your loan).
The problem comes if gains and losses alternate each rebalancing period. One day it goes down and you sell some, and the next day goes up and you buy some. Since you don’t want to sell low and buy high, I believe this is the volatility drag.
Doesn’t it reduce your returns though? It increases transaction costs, and it increases variance drag. If a highly-leveraged stock is more volatile then it will perform worse under daily rebalancing than a stock with equal returns but less volatility.
Under the assumption that each day’s expected return is a constant that doesn’t depend on the previous days, the average growth rate of your investment is basically the same whether you rebalance daily or weekly or monthly. It decreases slightly as you go to longer time scales (e.g. at 2x leverage annually you will get wiped out pretty often, so your average growth rate is negative).
(Of course if days are correlated with each other, then different rebalancing periods will have different values. E.g. if there is short-term momentum then rebalancing frequently will be good and if there is short-term reversal then it will be bad.)
If you care about expected returns rather than growth rates, then rebalancing daily is a good idea iff you are using >1x leverage. Well, hard to say what it means for it to be a “good idea,” since your optimal strategy is infinitely much leverage, which yields infinite expected returns. But having Nx leverage rebalanced daily will lead to higher expected returns than rebalancing monthly.
Yes, more volatile stocks with the same expected (i.e. arithmetic average) returns have lower growth rates. That’s true whether or not you use leverage though. More volatile stocks with the same growth rate will have lower growth rates when leveraged, whether you rebalance daily or annually.
Transaction costs are not zero. If you assume they are you can make an easier model but you fail to capture the reality of how RUSS/RUSL act.
Thanks, Paul. To further address Michael’s question, I think the reason why leverage gives returns raised to a power, rather than multiplying by a constant factor is the rebalancing. Let’s say we have $100 and we take out a $100 loan and invest those $200. This is 2x leverage. However, if the fund increases 10%, we would have $220 and still a $100 loan, which is less than 2x leverage. In order to maintain the leverage, you should take out $20 more in loan and then have $120 in loan and $240 in the market. Then you can see why as the stock goes up, a given percent increase will give you a greater dollar increase if you rebalance. The converse is true if the stock goes down. And this is how you protect yourself from going to zero (a given percent decrease in the market means a smaller dollar value loss if you reduce your loan). The problem comes if gains and losses alternate each rebalancing period. One day it goes down and you sell some, and the next day goes up and you buy some. Since you don’t want to sell low and buy high, I believe this is the volatility drag.