Kelly Criterion: Why You SHOULD Use Leverage — And By How MuchS&P 500SP:SPXHenriqueCentieiroHow much should you allocate to a certain investment? How much should you leverage? If you have $10,000 available to invest, how much should you put into a stock like Tesla, and how much should go into an index fund like the S&P 500? Most investors have no idea what the optimal allocations are to maximize their returns. "Tesla stock? I'll put $5,000. S&P? Let's do $2,000." Don't ask them why. They wouldn't know how to answer. And that's exactly why they're leaving huge profits on the table. In this educational TradingView post, I will show you how much you should allocate to that Tesla stock and why you should use leverage (yes, leverage) to invest in the S&P 500 index fund. The Expected Value Equation That Most Investors Ignore There are two types of investors: those who calculate their EV and make money, and those who ignore it and lose money. You probably have that friend who said they were going to buy a meme coin or a stock because it could go up 100x. The problem? They didn't calculate the probabilities. There's less than a 0.1% chance that a meme coin/stock delivers 100x returns. So investing in a meme coin is more like a gamble with a very negative Expected Value. Here's how you calculate the EV of this bet: EV = (prob. success x exp. returns) + (prob. loss x exp. returns) EV = (0.001 × 10,000%) + (0.999 × -100%) = 10% - 99.9% = -89.9% In other words, you would lose an average of 89.9% of your capital by making this bet repeatedly over time. But there are also good bets — ones with a positive EV. For example, the annual EV of an S&P 500 ETF like SPY is around 11%: EV = (0.75 × +19%) + (0.25 × -11%) = 11% Now that you have a positive EV bet in front of you, the question becomes: how much should you leverage it? If an asset has a positive expected value, you're leaving money on the table by not leveraging it. But there's an extremely fine balance: Too much leverage will destroy you. Too little, and you're missing out on gains. The Kelly Criterion: Your Leverage GPS Harry Markowitz, the father of Modern Portfolio Theory, once said: "The only free lunch in investing is diversification." He was half right. There's a second free lunch he missed: optimal leverage. Diversification reduces risk, but Kelly leverage maximizes growth. The Kelly Criterion, originally formulated by John Kelly Jr., answers an extremely important question that 99% of investors never ask: "What leverage maximizes the long-term growth of my portfolio?" Here's the formula: Optimal Leverage (f*) = Expected Return (μ) / Volatility² (σ²) In simple terms: Expected Return (μ) = How much you expect to gain per year. The more you can gain, the higher the leverage potential. SPY: 10% annual return → Higher leverage potential Bonds: 4% annual return → Lower leverage potential Volatility² (σ²) = How much the asset's price swings — squared. The higher the volatility, the lower the leverage you can safely use. SPY: 18% volatility → (0.18)² = 0.0324 Bitcoin: 90% volatility → (0.90)² = 0.81 The ratio between these two gives you the optimal leverage: High return + Low volatility = Use MORE leverage High return + High volatility = Use LESS leverage Low return + High volatility = Use NO leverage Example: SPY: 10% / (18%)² = 10% / 0.0324 = 3.1x optimal leverage Bitcoin: 80% / (90%)² = 80% / 0.81 = 0.99x optimal leverage (barely 1x!) What You See on the Chart Applied to SPY (S&P 500 ETF), the leverage/growth curve passes through different zones: underinvesting, optimal sizing, high risk, never logical, and suicidal. Underinvesting means you're being too conservative and leaving money on the table — like holding SPY at 1x when the optimal leverage is 3x. Optimal Sizing is the sweet spot where you use some leverage, but not too much. This is usually where half-Kelly falls — and what most experienced investors use. High Risk involves higher leverage and sits close to what's mathematically optimal. But this could be too high if there's a crash tomorrow — use with extreme caution. Never Logical means risk far outweighs reward, and expected growth actually declines as you take on more leverage. Suicidal means guaranteed capital destruction over time. Even coming close to these levels is almost certain to cost you money. How to Use the Kelly Criterion Curve Step 1: Load the Indicator Load the indicator on an index ETF, stock, or crypto. Set the lookback period to a realistic value — I like to use 2,000 days for long-term investing on a daily chart. Step 2: Find Your Leverage Sweet Spot Optimal Kelly: Maximum long-term growth (but very aggressive) ½ Kelly: Gives you 75% of the max growth with only 50% of the volatility (this is what I usually target) Step 3: Implement with Leveraged ETFs Say QQQ shows Optimal Kelly = 3x and 1/2 Kelly at 1.5x. What comes next depends on your risk appetite: If you're very adventurous, use TQQQ (3x leveraged QQQ ETF). Or go with QLD (2x leveraged QQQ ETF). Or use a combination of cash + QQQ + QLD or TQQQ to reach your desired leverage. You could also use a margin account or futures, but these are riskier and harder to manage. The Truth About 1x Investing Here's what 99% of investors don't know: if an asset has a positive expected value, 1x (no leverage) is mathematically suboptimal. For SPY: At 1x leverage, your expected growth rate is far from optimal. At 2x leverage, you fall into the optimal sizing zone. At 3x leverage, you're close to Kelly's optimal. Here's what would have happened if you'd applied this over the last 15 years with S&P 500 leveraged ETFs: 1x leverage with SPY: 765% return 2x leverage with SSO: 2,973% return 3x leverage with UPRO: 7,200% return 😲 Leverage does increase volatility — but it also increases returns. And these excess returns outweigh the added volatility. If you can stomach that volatility over time, it's a win. But better returns are not guaranteed: Drawdowns are psychologically brutal There's path dependency: if you start using leverage now and the market crashes tomorrow, you'll be in a tough spot We can't predict black swans That's why I like to combine leveraged ETFs with DCA. Using the Kelly Criterion Across Different Assets I got hooked on Kelly Criterion math after reading the paper Alpha Generation and Risk Smoothing using Managed Volatility by Tony Cooper. Cooper shows that over very long periods, applying leverage would have improved returns. According to the paper, these were the optimal leverages: S&P 500 (SPY): 3x leverage Dow Jones: 2x leverage Nasdaq-100 (QQQ): 2x leverage Russell 2000 (IWM): 2x leverage My indicator shows similar results: S&P 500 (SPY): Optimal Kelly of 3.3x — pretty close to the 3x in the paper. Nasdaq-100 (QQQ): Optimal Kelly of 2.99x — higher than the 2x in Cooper's paper. I'd still go for something closer to half Kelly. Russell 2000 (IWM): Optimal Kelly at 1.8x, very close to the 2x in Cooper's paper. You can also apply it to individual stocks and crypto: Tesla: Despite good performance, Tesla is fairly volatile. The indicator shows an Optimal Kelly of 0.94x — basically no leverage. Bitcoin: Full Kelly at 0.9x — also no leverage. This makes sense — many people who leveraged Bitcoin at "only" 2x lost everything in November 2025. Most investors use 1x leverage simply because it's a round number and because it's what's readily available. That's not how hedge funds invest. This piece of math puts you closer to how they invest: with mathematical precision and in a systematic manner. But most investors will read this, nod along, and do nothing. You now have the math. Use it to improve your returns.