Some results related to logarithmic utility and stock market leverage (I derived these after reading your previous post, but I think it fits better here):
Tl;dr: We can derive the optimal stock market leverage for an agent with utility logarithmic in money. We can also back-derive a utility function from any constant leverage[1], giving us a nice class of utility functions with different levels of risk-aversion. Logarithmic utility is recovered a special case, and has additional nice properties which the others may or may not have.
For an agent investing in a stock whose “instantaneous” price movements are i.i.d. with finite moments:
Suppose, for simplicity, that the agent’s utility function is over the amount of money they have in the next timestep. (As opposed to more realistic cases like “amount they have 20 years from now”.)
If U(x)=ln(x), then:
The optimal leverage for the agent to take is given by the formula L=m/(2s2), where m=E[returnPerTimestep−riskFreeReturnPerTimestep] and s is the standard deviation of the same. Derivation here. By my calculations, this implies a leverage of about 1.8 on the S&P 500.
What if we instead suppose the agent prefers some constant leverage L=m/(2cs2), and try to infer it’s utility function?
The relevant differential equation is xU′′(x)=−cU′(x)
This is solved by U(x)=1−x1−c for c≠1 and U(x)=ln(x) for c=1. You can play with the solutions here.
Now suppose instead that the agent’s utility function is “logarithmic withdrawals, time-discounted exponentially” -- U=∫∞t=0ln(w(t))eγt, where w(t) is the absolute[2] rate of withdrawal at time t. It turns out that optimal leverage is still constant, and is still given by the same formula L=m/(2s2). Furthermore, the optimal rate of withdrawal is a constant w(t)=1−γ, regardless of what happens.
Things probably don’t work out as cleanly for the non-logarithmic case.
1. This assumption of constant leverage is pretty arbitrary, so there’s no normative or descriptive force to the class of utility functions we derive from it
2. We have to make an unrealistic assumption that the utility function is over $$ at the next timestep, rather than further in the future. In the log case, these kind of assumptions tend to not change anything, but I’m not sure whether the general case is as clean.
Some results related to logarithmic utility and stock market leverage (I derived these after reading your previous post, but I think it fits better here):
Tl;dr: We can derive the optimal stock market leverage for an agent with utility logarithmic in money. We can also back-derive a utility function from any constant leverage[1], giving us a nice class of utility functions with different levels of risk-aversion. Logarithmic utility is recovered a special case, and has additional nice properties which the others may or may not have.
For an agent investing in a stock whose “instantaneous” price movements are i.i.d. with finite moments:
Suppose, for simplicity, that the agent’s utility function is over the amount of money they have in the next timestep. (As opposed to more realistic cases like “amount they have 20 years from now”.)
If U(x)=ln(x), then:
The optimal leverage for the agent to take is given by the formula L=m/(2s2), where m=E[returnPerTimestep−riskFreeReturnPerTimestep] and s is the standard deviation of the same. Derivation here. By my calculations, this implies a leverage of about 1.8 on the S&P 500.
What if we instead suppose the agent prefers some constant leverage L=m/(2cs2), and try to infer it’s utility function?
The relevant differential equation is x U′′(x)=−c U′(x)
This is solved by U(x)=1−x1−c for c≠1 and U(x)=ln(x) for c=1. You can play with the solutions here.
Now suppose instead that the agent’s utility function is “logarithmic withdrawals, time-discounted exponentially” -- U=∫∞t=0ln(w(t))eγt, where w(t) is the absolute[2] rate of withdrawal at time t. It turns out that optimal leverage is still constant, and is still given by the same formula L=m/(2s2). Furthermore, the optimal rate of withdrawal is a constant w(t)=1−γ, regardless of what happens.
Things probably don’t work out as cleanly for the non-logarithmic case.
[Disclaimer: This is not investment advice.]
Caveats:
1. This assumption of constant leverage is pretty arbitrary, so there’s no normative or descriptive force to the class of utility functions we derive from it
2. We have to make an unrealistic assumption that the utility function is over $$ at the next timestep, rather than further in the future. In the log case, these kind of assumptions tend to not change anything, but I’m not sure whether the general case is as clean.
i.e. in dollars, not percents