If risk is not correlated with returns, then anyone who is mostly concerned with absolute wealth can profit from this by choosing a low beta risk portfolio.
That sounds as though it would only work if risk was negatively-correlated with returns.
Surprisingly not. If returns are uncorrelated with risk, then you choose a low beta portfolio. It will have low overall variance. You can either choose to have a a low variance portfolio with the same expected return as the market portfolio or you can use leverage (doesn’t necessarily require personally taking out loans) to have a higher variance, higher expected return portfolio. Whichever you prefer.
If returns are uncorrelated with risk (average gain in value of an investment is unrelsated to its volatility), why would you choose the low beta portfolio? Couldn’t you increase your personal returns by dollar-cost-averaging into the high-beta portfolio?
Do you have your directions flipped? beta = covariance(portfolio, market)/variance(market) if beta is low for a portfolio, it has low covariance with the market. If not, I don’t understand your logic.
A high beta portfolio will have high variance. If risk is not correlated with return, then it won’t have higher expected return to compensate, which is worse than having the same return and low variance.
I was taking beta to effectively measure average deviation from a trendline exp(k t), k>0, when an investment’s cumulative value is expressed in dollars. In any case, that’s the measure of risk I thought you were using, and by that definition of it, I think I’ve shown it to indicate a way you can make returns greater than k: just buy $X worth of the investment every period (dollar-cost averaging). In that case, you will buy less of the investment when it is above exp(k t) trend and more when it’s below, beating the ROR k.
This would show how investors buying riskier assets would end up with a higher return for the same average return k.
I’d think average deviation of portfolio value from any trendline will be infinite for any risky portfolio, assuming asset prices are a random walk. Your story relies on mean reverting asset returns, and the advantage from that mean reversion rather than Falkenstein’s idea.
Normally, discussions about correlations between assets are actually discussions of correlations between returns of assets.
I’d think average deviation of portfolio value from any trendline will be infinite for any risky portfolio, assuming asset prices are a random walk.
How can that be, when you take the average per unit time.
In any case, I’m confused—I had always taken “risk” in this context to mean the volatility, and the traditional argument to mean that the highest-returning risky assets should have higher returns than the highest-returning less-risky assets. And my point is that the individual’s return will be different from that of the asset’s return.
Do you mean average change in deviation from the trend line? I understood you to mean average absolute deviation from the trend line. Maybe I just misunderstood you. The average change in deviation is non-infiinite, but the average absolute deviation deviation will grow without bound.
Sorry, I meant the signed difference—so that an investment going above and below could count as zero. But I don’t think that’s essential to my point about risk per the standard definition.
I think you’re confused about something, but I’m not sure what it is. I’ve added some clarifying material to the post. Does that clarify our discussion?
Maybe I didn’t make it clear, but difference between investor returns and investment returns is a non-issue, because such models are about instantaneous/incremental returns, not total returns. The simplest way to think about it is to split up time into segments. At the start of each segment you construct your portfolio (which could include cash) and it stays constant over that segment. At the end of the segment you observe the return to your portfolio over that period. The geometric sum of these returns is going to be your total return. Both CAPM and Falkestein’s model are about these period returns, not total returns. They are also only about the change in value beginning to end; the Var() does not describe the back and forth motion during the period, it describes your uncertainty at the beginning of the period about what the end of the period is going to look like.
I think maybe I understand what you were saying in the beginning. I’ll explain that and why I think it’s wrong, and you tell me if I’ve misunderstood something. I’m taking your first post to indicate your core point; if it’s changed, tell me.
OK, so I think what your idea is that your idea was that beta is like the variance of a portfolio price around a path. You can profit from this by buying when it’s below the path since this implies the price will at some point be above the path, giving you a profit.
The problem with is that financial markets are pretty efficient. This means that prices are a random walk, meaning the expected incremental return on an asset is independent of past returns. The price of an asset being below trend does not imply that above average expected incremental returns. You can think of this as saying that any changes in a portfolio price are permanent; if a stock price goes down, on average, it’s going to stay down; it might go up, but it’s just as likely to go down further.
1) I care about the investor return, not the asset return or any of the other metrics you listed, because that’s the return I get.
2) I wasn’t suggesting that an investor could buy on the lows. I was saying that the investor could buy in every period, but only an amount that is constant in dollar terms (or in terms of some other risk-free asset). This would then exceed the asset’s average return, though both might be negative.
1) Obviously you’re concered about invstor return (well utility anyway), I never intended to suggested there was any other metric you’d be interested in. Surely our previous conversations should have convinced you I’m not one to make that kind of mistake frequently.
2) Dollar cost averaging gets you something if there’s mean reversion. Otherwise it gains you nothing (might not cost you anything either) for the simple reason that asset prices are random walks. This is a standard result (http://www.moneychimp.com/features/dollar_cost.htm). If you think Falkenstein’s model implies that dollar cost averaging is a superior strategy, then I think you’ve misunderstood something. I’m not sure what it is though, I tried to reexplain the concepts which I thought you might be misunderstanding. Could you reexplain why you think it would get you superior performance using the kind of multi period model I was talking about?
Okay, I hadn’t actually experimented with the math on that so I can’t defend DCA as amplifying returns under volatility. So I don’t have much more to say in objection to the result you’ve posted either.
1) I’m not sure what different metrics you though I suggested. To clarify, CAPM and Falkenstein’s model are about returns for a constant portfolio over a given period. If you want to talk about a changing portfolio, you’ll have to approximate this as several time periods each with a different constant portolio and geomtrically average the returns.
Actually, IIRC, one of Falkenstein’s findings was that once volatility exceeds a certain level, it actually is negatively correlated with returns. He explains this with the phrase “People pay for hope.”
Not terribly surprising—though you can pretty easily turn a low risk investment into a high-risk one at relatively low cost with a roulette wheel. It is going to be harder to sell high-risk low return bonds when high-risk average return bonds are also available.
That sounds as though it would only work if risk was negatively-correlated with returns.
Surprisingly not. If returns are uncorrelated with risk, then you choose a low beta portfolio. It will have low overall variance. You can either choose to have a a low variance portfolio with the same expected return as the market portfolio or you can use leverage (doesn’t necessarily require personally taking out loans) to have a higher variance, higher expected return portfolio. Whichever you prefer.
If returns are uncorrelated with risk (average gain in value of an investment is unrelsated to its volatility), why would you choose the low beta portfolio? Couldn’t you increase your personal returns by dollar-cost-averaging into the high-beta portfolio?
Do you have your directions flipped? beta = covariance(portfolio, market)/variance(market) if beta is low for a portfolio, it has low covariance with the market. If not, I don’t understand your logic.
A high beta portfolio will have high variance. If risk is not correlated with return, then it won’t have higher expected return to compensate, which is worse than having the same return and low variance.
I was taking beta to effectively measure average deviation from a trendline exp(k t), k>0, when an investment’s cumulative value is expressed in dollars. In any case, that’s the measure of risk I thought you were using, and by that definition of it, I think I’ve shown it to indicate a way you can make returns greater than k: just buy $X worth of the investment every period (dollar-cost averaging). In that case, you will buy less of the investment when it is above exp(k t) trend and more when it’s below, beating the ROR k.
This would show how investors buying riskier assets would end up with a higher return for the same average return k.
I’d think average deviation of portfolio value from any trendline will be infinite for any risky portfolio, assuming asset prices are a random walk. Your story relies on mean reverting asset returns, and the advantage from that mean reversion rather than Falkenstein’s idea.
Normally, discussions about correlations between assets are actually discussions of correlations between returns of assets.
How can that be, when you take the average per unit time.
In any case, I’m confused—I had always taken “risk” in this context to mean the volatility, and the traditional argument to mean that the highest-returning risky assets should have higher returns than the highest-returning less-risky assets. And my point is that the individual’s return will be different from that of the asset’s return.
Do you mean average change in deviation from the trend line? I understood you to mean average absolute deviation from the trend line. Maybe I just misunderstood you. The average change in deviation is non-infiinite, but the average absolute deviation deviation will grow without bound.
Sorry, I meant the signed difference—so that an investment going above and below could count as zero. But I don’t think that’s essential to my point about risk per the standard definition.
I think you’re confused about something, but I’m not sure what it is. I’ve added some clarifying material to the post. Does that clarify our discussion?
Given as it doesn’t address any of the issues I raised (like the difference between investor returns and investment returns), no.
Maybe I didn’t make it clear, but difference between investor returns and investment returns is a non-issue, because such models are about instantaneous/incremental returns, not total returns. The simplest way to think about it is to split up time into segments. At the start of each segment you construct your portfolio (which could include cash) and it stays constant over that segment. At the end of the segment you observe the return to your portfolio over that period. The geometric sum of these returns is going to be your total return. Both CAPM and Falkestein’s model are about these period returns, not total returns. They are also only about the change in value beginning to end; the Var() does not describe the back and forth motion during the period, it describes your uncertainty at the beginning of the period about what the end of the period is going to look like.
I think maybe I understand what you were saying in the beginning. I’ll explain that and why I think it’s wrong, and you tell me if I’ve misunderstood something. I’m taking your first post to indicate your core point; if it’s changed, tell me.
OK, so I think what your idea is that your idea was that beta is like the variance of a portfolio price around a path. You can profit from this by buying when it’s below the path since this implies the price will at some point be above the path, giving you a profit.
The problem with is that financial markets are pretty efficient. This means that prices are a random walk, meaning the expected incremental return on an asset is independent of past returns. The price of an asset being below trend does not imply that above average expected incremental returns. You can think of this as saying that any changes in a portfolio price are permanent; if a stock price goes down, on average, it’s going to stay down; it might go up, but it’s just as likely to go down further.
1) I care about the investor return, not the asset return or any of the other metrics you listed, because that’s the return I get.
2) I wasn’t suggesting that an investor could buy on the lows. I was saying that the investor could buy in every period, but only an amount that is constant in dollar terms (or in terms of some other risk-free asset). This would then exceed the asset’s average return, though both might be negative.
1) Obviously you’re concered about invstor return (well utility anyway), I never intended to suggested there was any other metric you’d be interested in. Surely our previous conversations should have convinced you I’m not one to make that kind of mistake frequently.
2) Dollar cost averaging gets you something if there’s mean reversion. Otherwise it gains you nothing (might not cost you anything either) for the simple reason that asset prices are random walks. This is a standard result (http://www.moneychimp.com/features/dollar_cost.htm). If you think Falkenstein’s model implies that dollar cost averaging is a superior strategy, then I think you’ve misunderstood something. I’m not sure what it is though, I tried to reexplain the concepts which I thought you might be misunderstanding. Could you reexplain why you think it would get you superior performance using the kind of multi period model I was talking about?
Okay, I hadn’t actually experimented with the math on that so I can’t defend DCA as amplifying returns under volatility. So I don’t have much more to say in objection to the result you’ve posted either.
1) I’m not sure what different metrics you though I suggested. To clarify, CAPM and Falkenstein’s model are about returns for a constant portfolio over a given period. If you want to talk about a changing portfolio, you’ll have to approximate this as several time periods each with a different constant portolio and geomtrically average the returns.
Actually, IIRC, one of Falkenstein’s findings was that once volatility exceeds a certain level, it actually is negatively correlated with returns. He explains this with the phrase “People pay for hope.”
Not terribly surprising—though you can pretty easily turn a low risk investment into a high-risk one at relatively low cost with a roulette wheel. It is going to be harder to sell high-risk low return bonds when high-risk average return bonds are also available.