THIS IS A DRAFT
Beating the market is very difficult, the efficient market hypothesis persists that all information that affects market prices has already been incorporated into stock prices. This is one reason why index investing has seen a surge in popularity over the recent decades - why pay somebody to do an impossible task: pick stocks. Does this mean that for the average person, long term market returns are capped by the broader market? Not necessarily.
Generally, investors can trade some other desirable attribute in exchange a unit of return.
For example:
Liquidity Premium - Investments that are less liquid are less desireable and thus investors are compensated by a higher return.
Value Premium - TODO: Growth stocks are cooler than value stocks so investors are compensated.
This is just like when you walk into a fancy restaurant, you're paying for the staff and th ambiance and whatever and all that is not going to food quality or price.
One of the more common exchanges is trading a unit of risk for a unit of return. In the stock market this comes in many forms:
Equity Risky Premium - Equities are more risky than bonds Small Cap Premium - Smaller company's values are more volatile than larger than bigger companies and thus investors expect larger returns for their investments.
Further risk/return tradeoffs can come in the form of leverage. Leverage can be reckless, but it can also boost returns. The classic example is if you can get a mortgage for 2% and the market returns 9% (before accounting for inflation) shouldn't you invest before paying your mortgage?
So the question is: How much leverage can index fund invertors reasonably apply to their portfolios without being reckless.
We'll explore three different ways of answering: analytical, historical, and monte carlo simulations.
Analytical
Imagine you're in a casino losing all your money. You're down to your last $100 and you decide you're going to blow it all on "black" at the roulette table then hit the road.
Before you get a chance to do so, a casino employee approaches you and says that, since your such a valued customer, you have an opportunity to play a new game: A betting game in which you can wager on the outcome of a coin toss (heads or tails). A bet pays out even money (bet $1 to win $1). And, here's the kicker, the coin is weighted, so it has a 60% chance of coming up heads and a 40% chance of coming out tails. You can play this game for 30 minutes.
You're no dummy and can certainly see that if you guess heads every time you should come out ahead, but how much should you bet?
If you bet your whole $100, you have a decent chance (40%) of losing everything on the first toss. If you bet all your money multiple times in a row, you'll almost certainly walk away empty handed.
On the other hand, if you play things super conservatively and bet $1 each time, even if things really go your way, you wont be up as much as you could have been after half an hour.
Maybe it's best to split the difference and throw down $50?
Discrete Kelly
Turns out, there is mathematical formula for calculating the optimal bet:
fraction_of_bank_role_to_bet = probability of winning / fraction of wager you win in case of a good result - probability of winning / fraction of wager you lose in case of a bad result
This approach is called the Kelly Criterion (https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf).
In this case, the optimal wager calculation would be as follows:
fraction to bet = .6/1-.4/1 = .2 = 20% of your bank roll
As with most math I attempt to read about on Wikipedia (even easy math) my brain shuts down when trying to decipher the meanings of their cryptic mathematical iconography and follow their derivations and proofs. Though my mathematical skills, knowledge, and attention span may lack the ability to verify this result apriori, I can certainly do it a postori via a simulation, and you can too!
Kelly Bet Explorer
Things to try:
- Optimal bet (as a fraction) can go over 1 if the loss is smaller than 1. This implies you should borrow money to wager.
Continuous Kelly
Investing in the market has many more possible outcomes than that of a coin flip, which makes it challenging to apply Kelly to wall street.
However, in 1997 Edward Thorpe published a paper that, among other things, derived a formula for applying the kelly criterion to the stock market.
f* = (μ - r) / σ^2
where,
f* = fraction of your money to apply to the investment
μ = mean return TODO: confirm this
σ = Standard deviation of return
r = The risk free rate of return. In other words, the rate of return of an alternative investment. A common option for this is short term U.S. Treasuries, alternatively holding cash as an alternative makes the r value 0.
For the S&P500 Thorp proposed the following rough values (remember this paper is from 1997) μ = .11, σ = .15, r = .06. This resulted in a leverage value of 2.2! e.g. if you have $100 to invest, this Thorpe derivation of the continuous Kelly Criterion indicates that you should borrow and additional $120 and invest all $220 into the index.
Estimates for μ, σ, and r
Taking this all together, and applying nominal annualized return (CAGR 1871 - 2021) and standard deviation from Shiller's data as well as the loan rate you can get from a box spread gives us:
Thorpe Paper ("rough estimates"): μ = .11, σ = .15
Shiller (1871-2020): μ = .092 σ = .20
Ibbotson SBBI 2014 (1825-1925): μ = .073 σ = .163
Ibbotson SBBI 2014 (1926-2014): μ = .101 σ = .201
Ibbotson SBBI 2014 (1825-2014): μ = .086 σ = .182
These are total return values (include capital appreciation and re-invested dividends)
The risk free rate changes frequently, but so it's likely best to pick it from treasuries when re-balancing leverage. Just to give an idea of what is, here are a few examples:
Risk Free Rate:
Thorpe Paper ("rough estimates"): r = .06
Ibbotson SBBI 2014 = .035 (long term geometric mean of treasuries)
TODO: WHY ARE THORPS ESTIMATED THE ONLY ONES THAT ARE GOOD???
Try it with your own numbers:
What about f* < 1?
A leverage fraction less than one implies that the investor should invest less than he or she has in the market and invest any remainder at the risk free rate - r (e.g. short term U.S. treasuries).
e.g. for a $10k investment and f* = .8
- $8k in the the S&P500
- $2k in Treasury Bonds
What about f* > 1?
As previously mentioned, this implies the a Kelly investor should borrow money to increase their investment until meeting the formula's fractional amount.
e.g. for a $10k investment and f* = 1.2
- all $10k in the the S&P500
- borrow an additional $2k to invest in the S&P500
However, this presumes we can borrow money for free, which is almost certainly not an option.
Thorpe provides an example for how to account for the interest rates on the loan, he re-calculates f* using an r value equal to the rate of the loan (which he calls r_b). This effectively provides us with two versions of the same equation, one in which we are lending (aka investing) at the rate 'r' and one in which we are borrowing the the rate 'r':
f*_lend = (μ - risk_free_rate) / σ^2
then if f*_lend > 1,
f*_borrow = (μ - loan_interest_rate) / σ^2
Since you probably can't borrow at the same rate the government can, the loan_interest_rate is likely to be higher than the risk_free_rate in the above equations.
Example:
What about f* < 0?
This means that you should short the securities in question, we are going to largly skip over this because it's not likely to happen in this context and therefore not worth figuring out the costs of shorting.
f* Summary (assuming f* > 0)
In Summary (Using S&P500 as the investment, and T-Bills as the alternate):
f_lend* > 1 - Calculate f*_borrow
f*_borrow > 1 - Proceed with borrowing up to f*_borrow
f*_borrow = 1 - Hold 100% S&P500
f*_borrow <= 1 - Borrowing costs make leverage prohibitive, hold 100% S&P500
f*_lend = 1 - Hold 100% S&P500
f_lend < 1 - Hold both S&P500 and T-Bills according in proportion to f_lend's value
TODO: fractional kelly
f* = μ / σ^2
Disadvantages
The definition of ruin is zero money, however for me, I think I'd like it to be getting a margin call.
Normal distribution 20 standard deviations.
Continuous rebalancing.
Sell into losses and buy into gains
This comes up with a slew of assumptions used in the derivation. Thorpe says:
Assume that prices change “continuously” (no “jumps”), that portfolios may be revised “continuously”, and that there are no transactions costs (market impact, commissions, “overhead”), or taxes (Federal, State, city, exchange, etc.).
Historical
//μ = .11, σ = .15, r = .06
(0.11 - r) / (.15*.15)
(0.092 - r) / (.2*.2)
(0.101 - r) / (.201*.201)
(0.121 - r) / (.2*.2)
(0.121 - r) / (.165*.165)
// Yearly
Return pyramid
Constant
Re-ballancing
// Daily
Monte Carlo
https://edoc.hu-berlin.de/bitstream/handle/18452/14923/wesselhoefft.pdf?sequence=1
For a final bet fraction:
f* = 1.925
Notes
- How does re-balancing frequency affect returns?
- How does borrowing rate affect returns?
- Fractional kelly preforms badly when f* < 1. Is there a better stratagy?
- Getting a hight alternative rate seems like it can decrease your long term returns. Is this right or an error?