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To maintain leverage of L you need the stock you hold to be worth L(1+LR) so you need to buy/sell X = L(1+LR) - L(1+R) = L + L^2R - L - LR = L(L-1)R Remember we are doing this per $1 of equity, so multiply by your initial account size to get a $ trade size.
Couple of interesting things: 1. If L = 0 (you hold cash) or L = 1 (unlevered) then you never need to trade, your portfolio automatically tracks. 2. If 0 < L < 1 then L(L-1) is negative so you sell on price increases and buy on decreases, which is stabilising
3. If L < 0 or L > 1 then you need to buy when the stock is up and sell when it is down, which is trend-following/destabilising behaviour. A recent L < 0 case is the XIV blowup and a recent L > 1 case is the Archegos blowup
4. L(L-1) is quadratic, so when L gets big, X gets really big. For Archegos if we assume L=5 then L(L-1)=20 so whenever the stock went up 5% they would need to buy 100% of their initial position size to maintain leverage -- in retrospect not surprising that VIAC was up so much.
Either big leverage (high L) or moderate leverage and volatile stocks (high R) is enough to create interesting feedback effects that can lead to blowups in less liquid markets like VIX futures or mid cap stocks. Big leverage *and* volatile stocks is spectacular :)
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