Here I will talk about how longer term P&L (Profit & Loss) volatility affects instantaneous (shorter term) expected returns, and some practical means to apply this information.

In the Black-Scholes framework, the expected return is explained as

E(R) = (u - v^2 / 2) * (T - t)

where

E(R) = Expected return

u = expected drift, usually the risk-free interest rate

v = expected volatility of return

T = time at the end of the trade

t = time at the beginning of the trade

With basic calculus, we can see that the (u - v^2 / 2) portion is derived as an integral of v, so return volatility is important for not just risk management, but also estimating P&L.

So it looks like P&L volatility lowers longer term returns off short term P&L estimates, e.g. if a trading strategy is expected to average 1%/week, and if the P&L volatility, a random component, is expected to be greater than 0, then its expected annual return with weekly compounding would be LESS than (1.01)^52.

Assume that there is a 3% weekly volatility on an asset valued $1. We can visualize that a net loss occurs when when a 3% positive return is followed by a 3% negative move of $1.03.

See Neil A. Chriss' Black-Scholes and Beyond for more details behind the math.

We can now see that applying leverage to a trading strategy does not necessarily increase expected risks more than the expected returns. This is also apparent in the performance of leveraged ETFs.

A theoretical inefficeincy exists if one was to sell short a leveraged ETFs/ETNs, hedged with unleveraged ETFs of identical underlyings. Of course it would still have to overcome transaction costs and product dependent limitations around shortselling.

In the Black-Scholes framework, the expected return is explained as

E(R) = (u - v^2 / 2) * (T - t)

where

E(R) = Expected return

u = expected drift, usually the risk-free interest rate

v = expected volatility of return

T = time at the end of the trade

t = time at the beginning of the trade

With basic calculus, we can see that the (u - v^2 / 2) portion is derived as an integral of v, so return volatility is important for not just risk management, but also estimating P&L.

So it looks like P&L volatility lowers longer term returns off short term P&L estimates, e.g. if a trading strategy is expected to average 1%/week, and if the P&L volatility, a random component, is expected to be greater than 0, then its expected annual return with weekly compounding would be LESS than (1.01)^52.

Assume that there is a 3% weekly volatility on an asset valued $1. We can visualize that a net loss occurs when when a 3% positive return is followed by a 3% negative move of $1.03.

See Neil A. Chriss' Black-Scholes and Beyond for more details behind the math.

**Leverage**We can now see that applying leverage to a trading strategy does not necessarily increase expected risks more than the expected returns. This is also apparent in the performance of leveraged ETFs.

**An example of exploiting this phenomenon**A theoretical inefficeincy exists if one was to sell short a leveraged ETFs/ETNs, hedged with unleveraged ETFs of identical underlyings. Of course it would still have to overcome transaction costs and product dependent limitations around shortselling.

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