Ultra fund DIY - Constant Leverage Ratio

If we want to create a leveraged portfolio tracking an index, we need to know how to maintain a constant leverage ratio.  The concept is pretty simple.

Say if we want to maintain a leverage ratio, $L = 2$.  If the index $I$ moves $\delta$ percent.  Our portfolio $\pi$ will move $2\delta$ percent.

Assuming the borrowing cost is zero. Our initial capital is $C$.

So at the beginning, our asset is $2C$ and our liability is $C$, the net is $2C-C=C$.

When the index moves $\delta$ percent, then

Asset: $2C*(1+\delta)$

Liability: $C$

Net: $C*(1+2\delta)$

Leverage Ratio: $(2+2\delta)/(1+2\delta) \neq 2$

The leverage ratio is changed due to the index has moved $\delta$ percent.  So we have to adjust our shares to re-balance the leverage ratio to 2.

If our original shares is $S$, the new share price is $2C*(1+\delta)/S$.

Our new net value of the portfolio is $C*(1+2\delta)$, the new shares $T$ is

$T = 2C*(1+2\delta) / (2C*(1+\delta)/S) = (1+2\delta)/(1+\delta)*S = (1+\delta/(1+\delta))*S$

So the percentage of the adjustment is $\delta/(1+\delta)$, when the index moves $\delta$ percent.

If the leverage ratio is $L$, the adjustment is $\dfrac{\delta}{(1+\delta)}(L-1)$.

Let's do some simple calculation for $L=2$

If the index moves up or down 1~5%, the adjustment is listed in the following table.

L=2
1%
2%
3%
4%
5%
down
-1.01%
-2.04%
-3.09%
-4.17%
-5.26%
up
0.99%
1.96%
2.91%
3.85%
4.76%

From the formula $\frac{\delta}{(1+\delta)}(L-1)$, we can also see that when $L>1$, it's similar to a trend following strategy.  Because when the index goes up, we will increase our shares.  When the index goes down, we will decrease our shares.