# Conceptualizing The Volatility Decay Of Leveraged Funds

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by: Michael Z. Yu
Summary

In this article I will walk you through the commonly used but wrong way to think about volatility decay.

We'll then walk through an example of volatility decay with realistic numbers.

Afterwards, we'll build a simple mathematical model to calculate volatility decay from first principles.

We'll compare the model with back-test simulations done on portfoliovisualizer.com.

## The Wrong Story About Volatility Drag

On SeekingAlpha and various other financial forums, whenever leveraged funds are mentioned, the concept of "daily decay" of the funds is sure to be mentioned. Just what do we mean by "decay"? Let's walk through the classic example:

(For the sake of simplicity let's just assume that the margin debt is interest-free, and the stock is dividend-free. Given current market conditions it's a reasonable assumption.)

A fund holds \$200 of stock and \$100 of margin loan. During Day 1, the value of the underlying stock falls by 5%, so at the end of the day the fund has \$190 of stock and \$100 of debt, netting to \$90 of equity.

The fund must rebalance its assets to maintain its leverage ratio. So, it pays down \$10 of its excess debt by selling \$10 of stock, and now its balance sheet includes \$180 of stock, and \$90 of debt, netting to \$90 of equity.

During Day 2, the stock rises by 5.263%, which nearly exactly cancels out its previous day loss of 5%. The fund's \$180 of stock is now worth \$189.47. To maintain the leverage ratio, the amount of debt in the fund is raised from \$90 to \$94.74. The fund has \$189.47 in stock and \$94.74 in debt, netting to \$94.73 in equity.

What was the net result? An investor who owned \$100 of stock would still have \$100 of stock at the end of Day 2, but an investor who owned \$100 of equity in the 2x levered fund would only own \$94.73 in equity. Sounds like a terrible deal - over 2 days, the leveraged investor lost 5.27% of their equity!

What's the problem with this storyline? Does the stock market typically go down 5% on one day, and then go up 5.26% on the next day? Absolutely not. Just how absurd was this example? Let's walk through some numbers.

The long-term volatility of the S&P 500 is 15% per year. This is the standard deviation of the S&P 500 annual returns. But, we need to scale this to daily returns. We will divide 15% by the square root of 365 to get the daily volatility, which is 0.7851%.

So, a 5% move in a single day is a 6.37 standard deviation event, which is a near zero-probability event. The more mathematically astute reader might object: but stock returns are heavy-tailed, how about we see some real data?

This link (Average Daily Percent Move Of The Stock Market: S&P Volatility Returns) nicely shows the distribution of S&P 500 moves in a 10-year period. Just based on the helpful graphic, we can conclude that a 5% move (either up or down) in a single day happens with a less than 1% probability.

Should we judge leveraged funds, based on a single example, based on an event with a less than 1% chance of happening, as though it happens on a daily basis? Absolutely not.

## The Correct Story Explained With Realistic Numbers

Since we want to know how a leverage fund behaves on a typical day, let's assume that the S&P 500 moves up or down each day by exactly 0.7851%, which is exactly 1 standard deviation of volatility.

On Day 1, the levered fund holds \$200 of stock and \$100 of debt. During Day 1, the stock goes down by 0.7851%, with final value of \$198.4298. So, with just \$98.4298 of equity, the fund must sell \$1.5702 of stock to pay down that much debt, so the fund ends Day 1 with \$98.4298 of equity and \$98.4298 of debt.

On Day 2, the remaining \$196.8596 of stock rises by 0.7913% (just enough to cancel out a 0.7851% drop), ending at \$198.4174 of value. Since the fund had \$98.4298 of debt, it now owns \$99.9876 of equity. And at the end of the day it must rebalance so that it has \$99.9876 of debt too.

How did our leveraged investor do? Over 2 days, \$100 of equity turned into \$99.9876 of equity: just a 0.0124% loss over those two days.

A far cry from the fabled 5.27% told with the wrong story...

## Estimating Annual Returns Of A Leveraged Fund

For the sake of simplicity, let's assume that the S&P 500 still moves in 1-standard deviation increments (0.7851% down or 0.7913% up) on a daily basis, up or down.

Just how many more up than down motions are needed to deliver the average annual return of approximately 8.00% on the SPX?

We'll need to set up an ugly equation (where n is the number of up days) & solve it. Fortunately we can use computers, so I'll just show you the equation and the final result: Solving the equation by computer, we get n = 187.509. So this means that we need 187.509 up-days and 177.5491 down days, or almost exactly 10 more up days than down days to get the year's return.

Let's move on to the leveraged fund. Since the 2x levered fund's daily movements are by definition twice as extreme, we simply need to include a factor of two into our last equation to be able to use it to calculate the expected returns on a leveraged fund after volatility decay:

Our annual expected leveraged return after volatility decay is 14.37%/year! If in theory you could earn 16%/year on a 2x levered investment, this means that 2x leveraged funds suffer from about 1.63%/year of lost returns because of volatility drag.

## What If We Put Dividends & Margin Interest Back In?

The SPX's historical return has been around 8.00%. If we add in an estimated dividend yield of 2%, we get an expected return of 10%/year on an unleveraged S&P500 fund.

Our 2x levered fund would be a bit messier. Let's assume for the sake of simplicity that the dividends received from the entire stock holding and the cost of carrying the margin loan cancel each other out. Given current conditions, the dividend yield on the SPY is about 1.9%. This would suggest that the interest rate on the margin loan is about 3.8% - I think reasonable numbers. This would mean that our previously calculated number of 14.37%/year should hold.

## How About 3x Leveraged Funds?

Since the daily movements of a 3x levered fund are by definition 3x those of the underlying asset, we can still use our old equation, swapping all the 2's with 3's:

This suggests that a 3x leveraged fund can achieve an expected annual return of 18.224%. Since without volatility drag, simply tripling the 8.00%/yr return on the SPX yields 24%, we can suggest that the volatility drag on a 3x portfolio is about 5.78%/year of lost return.

Again, we'll assume that margin interest & dividends from the underlying roughly cancel each other out, just to get a first estimate. (VFINX) is a Vanguard S&P 500 mutual fund, while (VFISX) is a Vanguard short-term Treasury fund.

## Theory vs. Reality

Below is a portfoliovisualizer.com run on each of the following simulated strategies starting from 1992 and running up to the present:

100% VFINX - simulating just the S&P 500 - [blue]

200% VFINX & -100% VFISX - simulating the 2x S&P 500 - [red]

300% VFINX & -200% VFISX - simulating the 3x S&P 500 - [yellow]

Let's compare theory estimated CAGR with backtest CAGR:

S&P 500: 10.00% (theory) vs. 9.79% (backtest)

2x S&P 500: 14.37% (theory) vs. 13.39% (backtest)

3x S&P 500: 18.22% (theory) vs. 14.06% (backtest)

Clearly, something is wrong with the pure theory - somewhere along the way, an assumption was wrong. Which one? Hmm...

Our calculations assumed that the only possible daily moves were exactly 1 standard deviation. It simplified the math, but of course there definitely exist 2, 3, and 4 and beyond standard deviation moves in the market. The bigger the move, the bigger the contribution of the move to volatility decay.

Circled in light green in the backtest are periods during which the 3x and 2x levered funds suffered the most from volatility decay: exactly when the underlying asset, the S&P 500, suffered the most volatility.

Hence, our model based on 1-standard deviation moves actually underestimated volatility decay. If you thought that a modeled decay of 5.78%/year on a 3x leverage was bad enough, reality is actually even worse.

## Final Takeaways

• The higher the leverage multiple present in a portfolio, the more volatility decay it will suffer.
• Volatility decay scales far more than proportionally with the leverage multiple.
• 2x leveraged funds, with an estimated volatility decay of about 1.63%/year, are probably the highest amount of leverage any buy & hold strategy should employ.

Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.