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.
I think leveraged portfolios have a higher EV according to every theoretical account of finance, and they have had much higher average returns in practice during any reasonably long stretch. I’m not sure what your theorem statement is saying exactly but it shouldn’t end up contradicting that.
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?
NOTE: Don’t believe everything I said in this comment! I elaborate on some of the problems with it in the responses, but I’m leaving this original comment up because I think it’s instructive even though it’s not correct.
There is a theoretical account for why portfolios leveraged beyond a certain point would have poor returns even if prices follow a random process with (almost surely) continuous sample paths: leverage decay. If you could continuously rebalance a leveraged portfolio this would not be an issue, but if you can’t do that then leverage exhibits discontinuous behavior as the frequency of rebalancing goes to infinity.
A simple way to see this is that if the underlying follows Brownian motion dS/S=μdt+σdW and the risk-free return is zero, a portfolio of the underlying leveraged k-fold and rebalanced with a period of T (which has to be small enough for these approximations to be valid) will get a return
r=kS(T)S(0)−k=kexp(ΔlogS)−k
On the other hand, the ideal leveraged portfolio that’s continuously rebalanced would get
ri=(S(T)S(0))k−1=exp(kΔlogS)−1
If we assume the period T is small enough that a second order Taylor approximation is valid, the difference between these two is approximately
E[ri−r]≈k(k−1)2E[(Δlog(S))2]≈k(k−1)2σ2T
In particular, the difference in expected return scales linearly with the period in this regime, which means if we look at returns over the same time interval changing T has no effect on the amount of leverage decay. In particular, we can have a rule of thumb that to find the optimal (from the point of view of maximizing long-term expected return alone) leverage in a market we should maximize an expression of the form
kμ−k(k−1)2σ2
with respect to k, which would have us choose something like k=μ/σ2+1/2. Picking the leverage factor to be any larger than that is not optimal. You can see this effect in practice if you look at how well leveraged ETFs tracking the S&P 500 perform in times of high volatility.
I didn’t follow the math (calculus with stochastic processes is pretty confusing) but something seems obviously wrong here. I think probably your calculation of E[(Δlog(S)2)] is wrong?
Maybe I’m confused, but in addition to common sense and having done the calculation in other ways, the following argument seems pretty solid:
Regardless of k, if you consider a short enough period of time, then with overwhelming probability at all times your total assets will be between 0.999 and 1.001.
So no matter how I choose to rebalance, at all times my total exposure will be between 0.999k and 1.001k.
And if my exposure is between 0.999k and 1.001k, then my expected returns over any time period T are between 0.999kTμ and 1.001kTμ. (Where μ is the expected return of the underlying, maybe that’s different from your μ but it’s definitely just some number.)
So regardless of how I rebalance, doubling k approximately doubles my expected returns.
So clearly for short enough time periods your equation for the optimum can’t be right.
But actually maximizing EV over a long time period is equivalent to maximizing it over each short time period (since final wealth is just linear in your wealth at the end of the initial short period) so the optimum over arbitrary time periods is also to max leverage.
Thanks for the comment—I’m glad people don’t take what I said at face value, since it’s often not correct...
What I actually maximized is (something like, though not quite) the expected value of the logarithm of the return, i.e. what you’d do if you used the Kelly criterion. This is the correct way to maximize long-run expected returns, but it’s not the same thing as maximizing expected returns over any given time horizon.
My computation of E[(Δlog(S))2] is correct, but the problem comes in elsewhere. Obviously if your goal is to just maximize expected return then we have
and to maximize this we would just want to push k as high as possible as long as μ>0, regardless of the horizon at which we would be rebalancing. However, it turns out that this is perfectly consistent with
E[Ik(T)Vk(T)]1/T≈1+k(k−1)2σ2
where Ik is the ideal leveraged portfolio in my comment and Vk is the actual one, both with k-fold leverage. So the leverage decay term is actually correct, the problem is that we actually have
dIkIk=kdSS+k(k−1)2dS2S2=(kμ+k(k−1)2σ2)dt+kσdz
and the leverage decay term is just the second term in the sum multiplying dt. The actual leveraged portfolio we can achieve follows
dVkVk=kμdt+kσdz
which is still good enough for the expected return to be increasing in k. On the other hand, if we look at the logarithm of this, we get
dlog(Vk)=(kμ−k22σ2)dt+kσdz
so now it would be optimal to choose something like k=μ/σ2 if we were interested in maximizing the expected value of the logarithm of the return, i.e. in using Kelly.
The fundamental problem is that Ik is not the good definition of the ideally leveraged portfolio, so trying to minimize the gap between Vk and Ik is not the same thing as maximizing the expected return of Vk. I’m leaving the original comment up anyway because I think it’s instructive and the computation is still useful for other purposes.
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.)
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.
I think leveraged portfolios have a higher EV according to every theoretical account of finance, and they have had much higher average returns in practice during any reasonably long stretch. I’m not sure what your theorem statement is saying exactly but it shouldn’t end up contradicting that.
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?
NOTE: Don’t believe everything I said in this comment! I elaborate on some of the problems with it in the responses, but I’m leaving this original comment up because I think it’s instructive even though it’s not correct.
There is a theoretical account for why portfolios leveraged beyond a certain point would have poor returns even if prices follow a random process with (almost surely) continuous sample paths: leverage decay. If you could continuously rebalance a leveraged portfolio this would not be an issue, but if you can’t do that then leverage exhibits discontinuous behavior as the frequency of rebalancing goes to infinity.
A simple way to see this is that if the underlying follows Brownian motion dS/S=μdt+σdW and the risk-free return is zero, a portfolio of the underlying leveraged k-fold and rebalanced with a period of T (which has to be small enough for these approximations to be valid) will get a return
r=kS(T)S(0)−k=kexp(ΔlogS)−kOn the other hand, the ideal leveraged portfolio that’s continuously rebalanced would get
ri=(S(T)S(0))k−1=exp(kΔlogS)−1If we assume the period T is small enough that a second order Taylor approximation is valid, the difference between these two is approximately
E[ri−r]≈k(k−1)2E[(Δlog(S))2]≈k(k−1)2σ2TIn particular, the difference in expected return scales linearly with the period in this regime, which means if we look at returns over the same time interval changing T has no effect on the amount of leverage decay. In particular, we can have a rule of thumb that to find the optimal (from the point of view of maximizing long-term expected return alone) leverage in a market we should maximize an expression of the form
kμ−k(k−1)2σ2with respect to k, which would have us choose something like k=μ/σ2+1/2. Picking the leverage factor to be any larger than that is not optimal. You can see this effect in practice if you look at how well leveraged ETFs tracking the S&P 500 perform in times of high volatility.
I didn’t follow the math (calculus with stochastic processes is pretty confusing) but something seems obviously wrong here. I think probably your calculation of E[(Δlog(S)2)] is wrong?
Maybe I’m confused, but in addition to common sense and having done the calculation in other ways, the following argument seems pretty solid:
Regardless of k, if you consider a short enough period of time, then with overwhelming probability at all times your total assets will be between 0.999 and 1.001.
So no matter how I choose to rebalance, at all times my total exposure will be between 0.999k and 1.001k.
And if my exposure is between 0.999k and 1.001k, then my expected returns over any time period T are between 0.999kTμ and 1.001kTμ. (Where μ is the expected return of the underlying, maybe that’s different from your μ but it’s definitely just some number.)
So regardless of how I rebalance, doubling k approximately doubles my expected returns.
So clearly for short enough time periods your equation for the optimum can’t be right.
But actually maximizing EV over a long time period is equivalent to maximizing it over each short time period (since final wealth is just linear in your wealth at the end of the initial short period) so the optimum over arbitrary time periods is also to max leverage.
Thanks for the comment—I’m glad people don’t take what I said at face value, since it’s often not correct...
What I actually maximized is (something like, though not quite) the expected value of the logarithm of the return, i.e. what you’d do if you used the Kelly criterion. This is the correct way to maximize long-run expected returns, but it’s not the same thing as maximizing expected returns over any given time horizon.
My computation of E[(Δlog(S))2] is correct, but the problem comes in elsewhere. Obviously if your goal is to just maximize expected return then we have
E[R(T)]=E[V(T)]V(0)=T−1∏i=0E[V(i+1)V(i)|Fi]=T−1∏i=0E[kS(i+1)S(i)−k|Fi]=kT(exp(μ)−1)Tand to maximize this we would just want to push k as high as possible as long as μ>0, regardless of the horizon at which we would be rebalancing. However, it turns out that this is perfectly consistent with
E[Ik(T)Vk(T)]1/T≈1+k(k−1)2σ2where Ik is the ideal leveraged portfolio in my comment and Vk is the actual one, both with k-fold leverage. So the leverage decay term is actually correct, the problem is that we actually have
dIkIk=kdSS+k(k−1)2dS2S2=(kμ+k(k−1)2σ2)dt+kσdzand the leverage decay term is just the second term in the sum multiplying dt. The actual leveraged portfolio we can achieve follows
dVkVk=kμdt+kσdzwhich is still good enough for the expected return to be increasing in k. On the other hand, if we look at the logarithm of this, we get
dlog(Vk)=(kμ−k22σ2)dt+kσdzso now it would be optimal to choose something like k=μ/σ2 if we were interested in maximizing the expected value of the logarithm of the return, i.e. in using Kelly.
The fundamental problem is that Ik is not the good definition of the ideally leveraged portfolio, so trying to minimize the gap between Vk and Ik is not the same thing as maximizing the expected return of Vk. I’m leaving the original comment up anyway because I think it’s instructive and the computation is still useful for other purposes.
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.)