The term structure of futures contracts implies a 'commodity interest rate' (comparable to a currency's rate). When long a commodity in contango (and short US dollars) the investor is paying 2 rates...and commodity 'interest rates' can be extremely large numbers.

You need to be "right about oil increasing in price" AND about not paying too much to be long a "currency" with negative interest rates.]]>

The term structure of futures contracts implies a 'commodity interest rate' (comparable to a currency's rate). When long a commodity in contango (and short US dollars) the investor is paying 2 rates...and commodity 'interest rates' can be extremely large numbers.

You need to be "right about oil increasing in price" AND about not paying too much to be long a "currency" with negative interest rates.]]>

Let's take your quote, "...to make up for a 50% loss, you require a 100% gain (not 50%)." Math translation: (1-0.5)*(1+1) = 1. This conclusion is an artifact of the bias inherent in measuring growth/return as change normalized by a value an arbitrarily-chosen time before. This framework also leads to other false intuition--"volatility is a drag on returns"--while LETF decay is very accurately estimated by a function of volatility, the cause is the bias inherent in the metric of percent change over finite periods.

By taking the limit as the 'holding period' tends to zero, the continuously calculated return (log return) model normalizes price changes by the asset's contemporaneous value, not its value at the end of the previous holding period. This allows for an unbiased view of the evolution of a system (here, price): "...to make up for a 50% loss, you require a 50% gain." Math translation: exp(+0.5)*exp(-0.5) = 1.]]>

Let's take your quote, "...to make up for a 50% loss, you require a 100% gain (not 50%)." Math translation: (1-0.5)*(1+1) = 1. This conclusion is an artifact of the bias inherent in measuring growth/return as change normalized by a value an arbitrarily-chosen time before. This framework also leads to other false intuition--"volatility is a drag on returns"--while LETF decay is very accurately estimated by a function of volatility, the cause is the bias inherent in the metric of percent change over finite periods.

By taking the limit as the 'holding period' tends to zero, the continuously calculated return (log return) model normalizes price changes by the asset's contemporaneous value, not its value at the end of the previous holding period. This allows for an unbiased view of the evolution of a system (here, price): "...to make up for a 50% loss, you require a 50% gain." Math translation: exp(+0.5)*exp(-0.5) = 1.]]>

Consider the following characteristic of a log return model: a 1-day negative return, -x%, followed by a 1-day positive return of equal magnitude, +x%, means final value equals initial value...and returns sum to zero.]]>

Consider the following characteristic of a log return model: a 1-day negative return, -x%, followed by a 1-day positive return of equal magnitude, +x%, means final value equals initial value...and returns sum to zero.]]>

Although many academics from the fields of economics & finance measure return by percent change, it is not a mathematically tractable approach...hence the need for ad hoc add-ons like geometric mean (for return data) and convexity adjustment (for economic data). There is even peer-reviewed research published in academic journals suffering from this weakness.]]>

Although many academics from the fields of economics & finance measure return by percent change, it is not a mathematically tractable approach...hence the need for ad hoc add-ons like geometric mean (for return data) and convexity adjustment (for economic data). There is even peer-reviewed research published in academic journals suffering from this weakness.]]>

The magnitude of the effective log leverage is less than nominal when the LETF is gaining value, and the magnitude of the effective log leverage is greater than nominal when the LETF is losing value. For this reason, they are down in flat markets. This is true for long & short LETFs, with the effect being greater in short funds.

In physical systems, path independence implies no loss of potential so path dependence is associated with losses...but that is not its definition. LETFs are path dependent, but that term does not get at the point.

As for nonlinearity, in "Leveraged ETFs" I define a reference leveraged system without drift, and it is also nonlinear. LETFs are nonlinear, but that term does not get at the point, either.]]>

The magnitude of the effective log leverage is less than nominal when the LETF is gaining value, and the magnitude of the effective log leverage is greater than nominal when the LETF is losing value. For this reason, they are down in flat markets. This is true for long & short LETFs, with the effect being greater in short funds.

In physical systems, path independence implies no loss of potential so path dependence is associated with losses...but that is not its definition. LETFs are path dependent, but that term does not get at the point.

As for nonlinearity, in "Leveraged ETFs" I define a reference leveraged system without drift, and it is also nonlinear. LETFs are nonlinear, but that term does not get at the point, either.]]>

The novel log-normal distribution presentation in my book "Leveraged ETFs" clarifies the notion of a leveraged system without decay, which supports a mathematically motivated definition of decay & decay rate.]]>

The novel log-normal distribution presentation in my book "Leveraged ETFs" clarifies the notion of a leveraged system without decay, which supports a mathematically motivated definition of decay & decay rate.]]>