### Parabolic growth, exponential decay

The leveraged etfs are fairly complex devices to fully understand. Their short-term behavior is simple enough, but most people don't understand where they will go long-term. I'm going to do my best to analyze them.

The math:

First, the variables have to be defined. I'm going to use x for the change in price of the underlying index each day (a 1% increase would be x=.01), a for the long term ratio of the index level (a 1% increase would be a=1.01), and p for the price of the index. The corresponding values for the leveraged etf will be y, b, and q.

If you have a continuously adjusted leveraged etf, where the leverage is l, then the differential return on the etf would be

dq/q=ldp/p

Integrating both sides (and starting with p=q=1)

ln(q)=l*ln(p)

q=p^l

So a double leveraged etf will have q=p^2, and an inverse double leveraged etf will have q=p^(-2). In particular, at the end of the day the double leveraged etf return is

(1+y)=(1+x)^2=1+2x+x^2

But this is only if you rebalance continuously. If you rebalance once a day, then the daily return is

y=2x

which means that the return is low by a factor of x^2 each day. With the inverse double etf, the continuous return is

(1+y)=(1+x)^(-2)=1-2x+3x^2-4x^3+...

The daily rebalanced return is y=-2x, which is low by 3x^2 every day (the terms beyond x^2 are generally too small to matter, with the exception of single day returns above 7% or so). Note that for typical daily returns of about 1%, the leveraged etfs are underperforming continuous rebalancing by 1-3bp, which doesn't add up very quickly. But when volatility increases and daily returns are typically about 3%-7%, the double etfs will underperform by about 10-50bp/day, and the double short etfs will underperform by 30-150bp/day.

You can try to work out the long term return by multiplying the series together, but it will be much easier to take the logarithms. For the index, the long term logarithm is

ln(a)=sum(log(1+x))=sum(x)-sum(x^2)/2+sum(x^3)/3-...

Again, the x^3 terms can be ignored most of the time. The continuously rebalanced etf returns would be

double long:ln(b)=2ln(a)=2*sum(x)-sum(x^2)

double short:ln(b)=-2ln(a)=-2*sum(x)+sum(x^2)

With daily rebalancing, the returns are instead

double long:ln(b)=sum(y)-sum(y^2)/2=2*sum(x)-2*sum(x^2)

double short:ln(b)=sum(y)-sum(y^2)/2=-2*sum(x)-2*sum(x^2)

So here again, the double etf is underperforming by x^2 each day, while the double short etf is underperforming by 3x^2 each day.

It is useful to call the sum(x^2) term the variance (technically it isn't the variance, but it is pretty close and it is what you can trade as variance futures on SPX), represented by v.

As I said, the double long etf should have a return of b=a^2. But, with daily rebalancing the formula is instead

double long:ln(b)=2*ln(a)-ln(v) or b=a^2*exp(-v)

double short:ln(b)=-2*ln(a)-3*ln(v) or b=(1/a)^2*exp(-3v)

I tested these formulas against simulated double long and double short etfs (which eliminates tracking errors and expenses), and found that they match very well. The only exception is that the '87 crash (20% in a day) caused a jump change of about 2-3% in the value relative to what was expected. The formulas are not valid when there is a move over about 7% because the higher order terms start mattering. I'm surprised that the results matched as closely as they did over the 20% move. If the error from the '87 crash is ignored, the formula has an error on the order of 40-60bp over a few decades. I think that is close enough to call it valid.

The corresponding formula for other leveraged etfs comes out to

1x short:b=(1/a)*exp(-v)

triple long:b=a^3*exp(-3v)

triple short:b=(1/a)^3*exp(-6v)

quadruple long:b=a^4*exp(-6v)

quadruple short:b=(1/a)^4*exp(-10v)

5x long:b=a^5*exp(-10v)

5x short:b=(1/a)^5*exp(-15v)

The coefficient in front of the v is l/2-l^2/2 (and remember that l is negative for short funds).

Conclusions:

The double leveraged etfs will have parabolic price curves relative to the underlying indices. But those parabolas decay at an exponential rate based on the volatility. Which provides for some interesting possibilities when trading combinations of them, or options based on them. The major indices aren't terribly volatile, but some of the smaller indices are. The financial and semiconductor indices are probably sufficiently volatile that both the double long and double short etfs will eventually be pushed to 0. But remember that there will be wild swings on the way, so you can only short them if you have a lot of capital.

You can use combinations of the double etfs to trade consolidation and breakout patterns. If you are certain that the market will be consolidating, then you can short both etfs. The volatility decline will puch their prices down, and small movements will be basically offset. If you are expecting a breakout (perhaps at the point of a triangle) you can buy both. The parabolic return profile will mean that you make money as long as the market actually makes a big move, regardless of the direction. But remember that you have a volatility decay eating away at the prices every day.

A quick way to estimate variance is that the variance is equal to the square of the volatility. So when VIX is at 20, the expected variance for the year is .04. With VIX at about 70, the expected variance for the year is .49 (or .04/month). The exponential decay will cut the value in half whenever the exponent goes up by about .7, so SSO is expected to drop by 50% (relative to the parabolic curve from SPX returns) every 17 years normally, or every 17 months if VIX stays at 70. SDS is expected to drop by 50% every 6 years normally, or every 6 months right now. Which is why SDS is only up a little bit despite the huge drop in SPX. And SSO is plummeting due to the parabolic returns compounded by exponential decay.

Another point is that the equations are valid for other rebalancing periods as well (as long as the returns are small enough for the x^3 terms to be ignored). So you can change the rate of decay by changing the rebalancing frequency. The formulas are probably good up to about monthly rebalancing (and will certainly work for intraday rebalancing). According to EMH and random-walk theories, the variance will add up at the same rate regardless of the rebalancing frequency. But I don't think many people take the random-walk theory seriously anymore.

At some point I will look at how these formulas compare to the actual etfs, which will show the effects of tracking error and costs.

The math:

First, the variables have to be defined. I'm going to use x for the change in price of the underlying index each day (a 1% increase would be x=.01), a for the long term ratio of the index level (a 1% increase would be a=1.01), and p for the price of the index. The corresponding values for the leveraged etf will be y, b, and q.

If you have a continuously adjusted leveraged etf, where the leverage is l, then the differential return on the etf would be

dq/q=ldp/p

Integrating both sides (and starting with p=q=1)

ln(q)=l*ln(p)

q=p^l

So a double leveraged etf will have q=p^2, and an inverse double leveraged etf will have q=p^(-2). In particular, at the end of the day the double leveraged etf return is

(1+y)=(1+x)^2=1+2x+x^2

But this is only if you rebalance continuously. If you rebalance once a day, then the daily return is

y=2x

which means that the return is low by a factor of x^2 each day. With the inverse double etf, the continuous return is

(1+y)=(1+x)^(-2)=1-2x+3x^2-4x^3+...

The daily rebalanced return is y=-2x, which is low by 3x^2 every day (the terms beyond x^2 are generally too small to matter, with the exception of single day returns above 7% or so). Note that for typical daily returns of about 1%, the leveraged etfs are underperforming continuous rebalancing by 1-3bp, which doesn't add up very quickly. But when volatility increases and daily returns are typically about 3%-7%, the double etfs will underperform by about 10-50bp/day, and the double short etfs will underperform by 30-150bp/day.

You can try to work out the long term return by multiplying the series together, but it will be much easier to take the logarithms. For the index, the long term logarithm is

ln(a)=sum(log(1+x))=sum(x)-sum(x^2)/2+sum(x^3)/3-...

Again, the x^3 terms can be ignored most of the time. The continuously rebalanced etf returns would be

double long:ln(b)=2ln(a)=2*sum(x)-sum(x^2)

double short:ln(b)=-2ln(a)=-2*sum(x)+sum(x^2)

With daily rebalancing, the returns are instead

double long:ln(b)=sum(y)-sum(y^2)/2=2*sum(x)-2*sum(x^2)

double short:ln(b)=sum(y)-sum(y^2)/2=-2*sum(x)-2*sum(x^2)

So here again, the double etf is underperforming by x^2 each day, while the double short etf is underperforming by 3x^2 each day.

It is useful to call the sum(x^2) term the variance (technically it isn't the variance, but it is pretty close and it is what you can trade as variance futures on SPX), represented by v.

As I said, the double long etf should have a return of b=a^2. But, with daily rebalancing the formula is instead

double long:ln(b)=2*ln(a)-ln(v) or b=a^2*exp(-v)

double short:ln(b)=-2*ln(a)-3*ln(v) or b=(1/a)^2*exp(-3v)

I tested these formulas against simulated double long and double short etfs (which eliminates tracking errors and expenses), and found that they match very well. The only exception is that the '87 crash (20% in a day) caused a jump change of about 2-3% in the value relative to what was expected. The formulas are not valid when there is a move over about 7% because the higher order terms start mattering. I'm surprised that the results matched as closely as they did over the 20% move. If the error from the '87 crash is ignored, the formula has an error on the order of 40-60bp over a few decades. I think that is close enough to call it valid.

The corresponding formula for other leveraged etfs comes out to

1x short:b=(1/a)*exp(-v)

triple long:b=a^3*exp(-3v)

triple short:b=(1/a)^3*exp(-6v)

quadruple long:b=a^4*exp(-6v)

quadruple short:b=(1/a)^4*exp(-10v)

5x long:b=a^5*exp(-10v)

5x short:b=(1/a)^5*exp(-15v)

The coefficient in front of the v is l/2-l^2/2 (and remember that l is negative for short funds).

Conclusions:

The double leveraged etfs will have parabolic price curves relative to the underlying indices. But those parabolas decay at an exponential rate based on the volatility. Which provides for some interesting possibilities when trading combinations of them, or options based on them. The major indices aren't terribly volatile, but some of the smaller indices are. The financial and semiconductor indices are probably sufficiently volatile that both the double long and double short etfs will eventually be pushed to 0. But remember that there will be wild swings on the way, so you can only short them if you have a lot of capital.

You can use combinations of the double etfs to trade consolidation and breakout patterns. If you are certain that the market will be consolidating, then you can short both etfs. The volatility decline will puch their prices down, and small movements will be basically offset. If you are expecting a breakout (perhaps at the point of a triangle) you can buy both. The parabolic return profile will mean that you make money as long as the market actually makes a big move, regardless of the direction. But remember that you have a volatility decay eating away at the prices every day.

A quick way to estimate variance is that the variance is equal to the square of the volatility. So when VIX is at 20, the expected variance for the year is .04. With VIX at about 70, the expected variance for the year is .49 (or .04/month). The exponential decay will cut the value in half whenever the exponent goes up by about .7, so SSO is expected to drop by 50% (relative to the parabolic curve from SPX returns) every 17 years normally, or every 17 months if VIX stays at 70. SDS is expected to drop by 50% every 6 years normally, or every 6 months right now. Which is why SDS is only up a little bit despite the huge drop in SPX. And SSO is plummeting due to the parabolic returns compounded by exponential decay.

Another point is that the equations are valid for other rebalancing periods as well (as long as the returns are small enough for the x^3 terms to be ignored). So you can change the rate of decay by changing the rebalancing frequency. The formulas are probably good up to about monthly rebalancing (and will certainly work for intraday rebalancing). According to EMH and random-walk theories, the variance will add up at the same rate regardless of the rebalancing frequency. But I don't think many people take the random-walk theory seriously anymore.

At some point I will look at how these formulas compare to the actual etfs, which will show the effects of tracking error and costs.

Labels: leveraged etfs

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