I think the two mutual funds theorem roughly holds as long as asset prices change continuously. I agree that if asset prices can make large jumps (or if you are a large fraction of the market) such that you aren’t able to liquidate positions at intermediate prices, then the statement is only true approximately.
That’s not the key assumption for purposes of this discussion. The key assumption is that you can short arbitrary amounts of either fund, and hold those short positions even if the portfolio value dips close to zero or even negative from time to time.
Leveraged portfolios will of course have a higher EV if they don’t get margin called. But in an efficient market, the probability of a margin call (and the loss taken when the call hits) offsets the higher EV—otherwise the lender would have a below-market expected return on their loan. Unfortunately most theoretical accounts assume you can get arbitrary amounts of leverage without ever having to worry about margin calls—a lesson I learned the hard way, back in the day.
In general, if leveraged portfolios have higher EV, then we need to have some explanation of why someone is making the loan.
Maybe the weirdness is that you are assuming all investors have linear utility? (I don’t know what the symbol V_t means, and generally don’t quite understand the theorem statement.) Or maybe in your theorem “E” is an expectation that weights worlds based on the marginal value of $ in those worlds, whereas in other places we are using “probability” to refer to the fraction of worlds in which an event occurs?
Nope, not linear utility, unless we’re using the risk-free rate for r, which is not the case in general. The symbol V_t is a lazy shorthand for the price of the asset, plus the value of any dividends paid up to time t. The weighting by marginal value of $ in different worlds comes from the discount rate r; E is just a plain old expectation.
Leveraged portfolios will of course have a higher EV if they don’t get margin called. But in an efficient market, the probability of a margin call (and the loss taken when the call hits) offsets the higher EV—otherwise the lender would have a below-market expected return on their loan. Unfortunately most theoretical accounts assume you can get arbitrary amounts of leverage without ever having to worry about margin calls—a lesson I learned the hard way, back in the day.
In general, if leveraged portfolios have higher EV, then we need to have some explanation of why someone is making the loan.
Imagine someone who is extremely risk-averse. They aren’t willing to invest much in equities, but they are willing to make a margin loan with ~0 risk. They will get lower return than if they had invested in equities, and they’ll have less risk. I don’t see what’s contradictory about this.
I was being a bit lazy earlier—when I said “EV”, I was using that as a shorthand for “expected discounted value”, which in hindsight I probably should have made explicit. The discount factor is crucial, because it’s the discount factor which makes risk aversion a thing: marginal dollars are worth more to me in worlds where I have fewer dollars, therefore my discount factor is smaller in those worlds.
The person making the margin loan does accept a lower expected return in exchange for lower risk, but their expected discounted return should be the same as equities—otherwise they’d invest in equities.
(In practice the Volker rule and similar rules can break this argument: if banks aren’t allowed to hold stock, then in-principle there can be arbitrage opportunities which involve borrowing margin from a bank to buy stock. But that is itself an exploitation of an inefficiency, insofar as there aren’t enough people already doing it to wipe out the excess expected discounted returns.)
That’s not the key assumption for purposes of this discussion. The key assumption is that you can short arbitrary amounts of either fund, and hold those short positions even if the portfolio value dips close to zero or even negative from time to time.
Leveraged portfolios will of course have a higher EV if they don’t get margin called. But in an efficient market, the probability of a margin call (and the loss taken when the call hits) offsets the higher EV—otherwise the lender would have a below-market expected return on their loan. Unfortunately most theoretical accounts assume you can get arbitrary amounts of leverage without ever having to worry about margin calls—a lesson I learned the hard way, back in the day.
In general, if leveraged portfolios have higher EV, then we need to have some explanation of why someone is making the loan.
Nope, not linear utility, unless we’re using the risk-free rate for r, which is not the case in general. The symbol V_t is a lazy shorthand for the price of the asset, plus the value of any dividends paid up to time t. The weighting by marginal value of $ in different worlds comes from the discount rate r; E is just a plain old expectation.
Imagine someone who is extremely risk-averse. They aren’t willing to invest much in equities, but they are willing to make a margin loan with ~0 risk. They will get lower return than if they had invested in equities, and they’ll have less risk. I don’t see what’s contradictory about this.
Indeed there is nothing contradictory about that.
I was being a bit lazy earlier—when I said “EV”, I was using that as a shorthand for “expected discounted value”, which in hindsight I probably should have made explicit. The discount factor is crucial, because it’s the discount factor which makes risk aversion a thing: marginal dollars are worth more to me in worlds where I have fewer dollars, therefore my discount factor is smaller in those worlds.
The person making the margin loan does accept a lower expected return in exchange for lower risk, but their expected discounted return should be the same as equities—otherwise they’d invest in equities.
(In practice the Volker rule and similar rules can break this argument: if banks aren’t allowed to hold stock, then in-principle there can be arbitrage opportunities which involve borrowing margin from a bank to buy stock. But that is itself an exploitation of an inefficiency, insofar as there aren’t enough people already doing it to wipe out the excess expected discounted returns.)