If I’m understanding correctly, you’re suggesting they should go up at least at the bond market nominal rate…
Yes. It is necessary to disentangle two separate ideas. The first idea is time-discounting. Time-discounting plus arbitrage means that securities grow at no less than the bond market risk-free rate. The price of a stock right now doesn’t reflected future growth. It represents future growth time-discounted by the bond market.
…but they tend to go up much faster than that?
The second idea is risk-adjustment. A 100% chance at $1 million is more valuable than a 1% chance at $100 million. If two securities have equal risk-neural expected value then the security with lower volatility has higher risk-adjusted expected value. If stocks tend to have higher volatility than bonds then an efficient risk-adjusted market ought to price stocks lower than bonds for a given expected (average) growth rate.
The value of capital invested in the bond market goes up via reverse time-discounting. The value of capital invested in the stock market goes up via reverse time-discounting and because you have purchased additional average growth by tolerating higher volatility.
Stocks’ expected values (in terms of time-discounted dividends or similar) have volatility, from things like business decisions, technology development, and capital re-investment.
A stocks expected value serves as an anchor point for its current price (kind of?)
A stock’s price will change around that anchor until it has expected returns that justify the volatility risk.
Thus a stock price will increase when either its expected value goes up, or its volatility goes down (without changing expected value).
In an efficient market, knowledge that changes your expected value of the stock is probably already priced in, but you can still capture the gains due to volatility.
In an efficient market, knowledge that changes your expected value of the stock is probably already priced in, but you can still capture the gains due to volatility.
I’m not certain I understand this bullet point. Can you explain what you mean?
Rephrase attempt: If you were to buy and hold a company’s stock, and you don’t expect to know better than the market, then your anticipated gains over time are independent of what you think the company’s underlying value will do (since the market has already priced that in). But you should still anticipate gains over time due to volatility’s effect on pricing.
Question: say I have a company whose underlying value is volatile, but whose expected underlying value after any time span is the same as today’s value. Both of the above arguments seem to suggest I should expect the company’s price to increase over time, but wouldn’t this unanchor the company’s price from its underlying value?
I’d be interested for both 100% growth and 100% dividend stocks—I’m not sure why they’d behave differently w.r.t. to this.
By underlying value, I’m not sure. Something like the real dollar value of all of its capital—or the real dollar price someone would pay to own it in its entirety?
By price, I mean what you can buy the stock for on the market.
Let me try clarifying: The volatility argument seems formal rather than empirical, so I’m wondering what we formally need to assume to make it go through.
I’d summarize the argument as “since stock prices are volatile, they’re expected to go up over time (more than the risk-free rate)”. But then why are stock prices volatile? I assumed that they’re volatile due to “underlying value” being volatile/hard to predict.
So my hypothetical is a company whose “underlying value” is hard to predict, but where the expectation of its “underlying value” is constant over time. To make it easier, assume the company is a magic vault that currently contains $10 million real dollars, and the money will undergo a stochastic process with the given property, and everyone knows this. Maybe it will disperse it’s full value as a dividend at some random future point.
It seems obvious to me I shouldn’t expect this company’s price to go up faster than the risk free rate, yet the volatility argument seems to apply to it. So I’m trying to identify what I’m missing.
It seems obvious to me I shouldn’t expect this company’s price to go up faster than the risk free rate, yet the volatility argument seems to apply to it.
You should, because the company’s current value will be lower than $10 million due to the risk. Your total return over time will be positive, while the return for a similar company that never varies will be 0 (or the interest rate if nonzero).
I agree the company’s current price should be lower than $10 million. But if it starts at price P and I expect it to go up at the risk free rate r, then at a time T later the company’s price should be PrT in expectation. At some point, that’ll be substantially more than the $10 million I expect it to pay out.
Well there’s some probability of it paying out before then.
If the magic value is a martingale, and the payout timing is given by a poisson process then the stock price should remain a constant discount off of the magic value. You will gain on average by holding the stock until the payout, but won’t gain in expectation by buying and selling the stock.
Let’s ignore risk. Suppose the company has a market value of P right now at time T0=0, you expect the vault to open at time ΔT, and the bond rate (which equals the equities rate because this is a risk-free thought experiment) equals r. Then the value of the company at time ΔT is 10 million dollars. If the value of the company at time ΔT is 10 million dollars then the market value (price) of the company right now is 10 million dollars times r−ΔT.
Suppose vaults are a fungible liquid securitized asset and that you can buy fractions of them. Suppose you invest in these vaults. Whenever a vault opens, you immediately invest your 10 million dollars cash in more vaults. Your investment grows at a rate r, exactly equal to the bond rate.
I think what you’re missing is that whenever a vault opens you immediately reinvest the cash. The vault has different time-adjusted value depending on when it opens. On a long enough time horizon, $10 million now is worth more than $10 billion later.
Yes, the volatility argument is formal rather than empirical. Whether it actually exists in practice is dubious. Fortunately, this does not affect the core issue of the subject at hand. This discussion concerns the theoretical market. For the volatility argument to apply, all we have to assume is an efficient market, rational actors and the Law of Diminishing Returns.
By “real” do you mean physical dollars or “real value”? In this answer, I ignore inflation and treat the vault as if it contains physical dollars.
We can build a physical system that replicates the effect of your magic vault. Suppose there is a vault tied whose lock mechanism is connected to a radioactive isotope. Each second there is a small chance the isotope will decay and the vault will open and the owner will receive $10 million cash. Each second, there is a large chance the isotope will decay and the vault will remain shut. Radioactive decay is a stocastic process. Therefore if the vault remains shut then the price of the vault remains at a constant price less than $10 million.
At every instant there is a small chance Schrödinger’s vault will open and a large chance the vault will stay shut. In the quantum future where the vault stays shut you are correct and the vault’s nominal market value stays constant. The time-discounted price of a closed vault actually goes down it it stays shut.
It’s not the price of the closed vault that goes up faster than time-discounted non-risk-adjusted value. It’s the average time-discounted risk-adjusted probability-weighted price of all possible future vault states (open and closed) that goes up faster than the time-discounted non-risk-adjusted value of the initial closed vault.
Yes. It is necessary to disentangle two separate ideas. The first idea is time-discounting. Time-discounting plus arbitrage means that securities grow at no less than the bond market risk-free rate. The price of a stock right now doesn’t reflected future growth. It represents future growth time-discounted by the bond market.
The second idea is risk-adjustment. A 100% chance at $1 million is more valuable than a 1% chance at $100 million. If two securities have equal risk-neural expected value then the security with lower volatility has higher risk-adjusted expected value. If stocks tend to have higher volatility than bonds then an efficient risk-adjusted market ought to price stocks lower than bonds for a given expected (average) growth rate.
The value of capital invested in the bond market goes up via reverse time-discounting. The value of capital invested in the stock market goes up via reverse time-discounting and because you have purchased additional average growth by tolerating higher volatility.
The opposite, right?
Fixed. Thanks.
Ok, so the argument would go:
Stocks’ expected values (in terms of time-discounted dividends or similar) have volatility, from things like business decisions, technology development, and capital re-investment.
A stocks expected value serves as an anchor point for its current price (kind of?)
A stock’s price will change around that anchor until it has expected returns that justify the volatility risk.
Thus a stock price will increase when either its expected value goes up, or its volatility goes down (without changing expected value).
In an efficient market, knowledge that changes your expected value of the stock is probably already priced in, but you can still capture the gains due to volatility.
Does that seem in line with what you’re saying?
I’m not certain I understand this bullet point. Can you explain what you mean?
[Fixed.]
Typo: “volatility goes up” → “volatility goes down”Fixed.
Rephrase attempt: If you were to buy and hold a company’s stock, and you don’t expect to know better than the market, then your anticipated gains over time are independent of what you think the company’s underlying value will do (since the market has already priced that in). But you should still anticipate gains over time due to volatility’s effect on pricing.
Yes. Your understanding is in line with the idea of risk-adjustment.
Question: say I have a company whose underlying value is volatile, but whose expected underlying value after any time span is the same as today’s value. Both of the above arguments seem to suggest I should expect the company’s price to increase over time, but wouldn’t this unanchor the company’s price from its underlying value?
Is the company a growth stock or a dividends stock?
What do you mean by “underlying value”?
Does this question concern risk-adjusted price or non-risk-adjusted price? Does it concern time-discounted price or non-time-discounted-price?
I’d be interested for both 100% growth and 100% dividend stocks—I’m not sure why they’d behave differently w.r.t. to this.
By underlying value, I’m not sure. Something like the real dollar value of all of its capital—or the real dollar price someone would pay to own it in its entirety?
By price, I mean what you can buy the stock for on the market.
Let me try clarifying: The volatility argument seems formal rather than empirical, so I’m wondering what we formally need to assume to make it go through.
I’d summarize the argument as “since stock prices are volatile, they’re expected to go up over time (more than the risk-free rate)”. But then why are stock prices volatile? I assumed that they’re volatile due to “underlying value” being volatile/hard to predict.
So my hypothetical is a company whose “underlying value” is hard to predict, but where the expectation of its “underlying value” is constant over time. To make it easier, assume the company is a magic vault that currently contains $10 million real dollars, and the money will undergo a stochastic process with the given property, and everyone knows this. Maybe it will disperse it’s full value as a dividend at some random future point.
It seems obvious to me I shouldn’t expect this company’s price to go up faster than the risk free rate, yet the volatility argument seems to apply to it. So I’m trying to identify what I’m missing.
You should, because the company’s current value will be lower than $10 million due to the risk. Your total return over time will be positive, while the return for a similar company that never varies will be 0 (or the interest rate if nonzero).
I agree the company’s current price should be lower than $10 million. But if it starts at price P and I expect it to go up at the risk free rate r, then at a time T later the company’s price should be PrT in expectation. At some point, that’ll be substantially more than the $10 million I expect it to pay out.
Well there’s some probability of it paying out before then.
If the magic value is a martingale, and the payout timing is given by a poisson process then the stock price should remain a constant discount off of the magic value. You will gain on average by holding the stock until the payout, but won’t gain in expectation by buying and selling the stock.
Let’s ignore risk. Suppose the company has a market value of P right now at time T0=0, you expect the vault to open at time ΔT, and the bond rate (which equals the equities rate because this is a risk-free thought experiment) equals r. Then the value of the company at time ΔT is 10 million dollars. If the value of the company at time ΔT is 10 million dollars then the market value (price) of the company right now is 10 million dollars times r−ΔT.
Suppose vaults are a fungible liquid securitized asset and that you can buy fractions of them. Suppose you invest in these vaults. Whenever a vault opens, you immediately invest your 10 million dollars cash in more vaults. Your investment grows at a rate r, exactly equal to the bond rate.
I think what you’re missing is that whenever a vault opens you immediately reinvest the cash. The vault has different time-adjusted value depending on when it opens. On a long enough time horizon, $10 million now is worth more than $10 billion later.
Yes, the volatility argument is formal rather than empirical. Whether it actually exists in practice is dubious. Fortunately, this does not affect the core issue of the subject at hand. This discussion concerns the theoretical market. For the volatility argument to apply, all we have to assume is an efficient market, rational actors and the Law of Diminishing Returns.
By “real” do you mean physical dollars or “real value”? In this answer, I ignore inflation and treat the vault as if it contains physical dollars.
We can build a physical system that replicates the effect of your magic vault. Suppose there is a vault tied whose lock mechanism is connected to a radioactive isotope. Each second there is a small chance the isotope will decay and the vault will open and the owner will receive $10 million cash. Each second, there is a large chance the isotope will decay and the vault will remain shut. Radioactive decay is a stocastic process. Therefore if the vault remains shut then the price of the vault remains at a constant price less than $10 million.
At every instant there is a small chance Schrödinger’s vault will open and a large chance the vault will stay shut. In the quantum future where the vault stays shut you are correct and the vault’s nominal market value stays constant. The time-discounted price of a closed vault actually goes down it it stays shut.
It’s not the price of the closed vault that goes up faster than time-discounted non-risk-adjusted value. It’s the average time-discounted risk-adjusted probability-weighted price of all possible future vault states (open and closed) that goes up faster than the time-discounted non-risk-adjusted value of the initial closed vault.