**Summary**

- Potential investors in c-times daily leveraged ETFs might like to know the expected value and variance of monthly gains.
- You can estimate expected value and variance by regressing monthly gains for an ETF, Y, vs. monthly gains for the index, X. In other words, fit the model: E[Y] = alpha + beta X.
- The expected value is alpha + beta E[X], where E[X] is the expected monthly gain of the underlying index. Under certain conditions, this is approximately c E[X].
- The variance is a function of residual variance around the regression line, beta, and the variance of monthly gains for the index, V[X]. Under certain conditions, it is approximately c squared times V[X].
- Leveraged ETFs often have similar risk-adjusted returns as the underlying index, but much greater raw returns. They are attractive funds to include in portfolio optimization problems.

**Monthly Gains for SSO and UPRO vs. the S&P 500**

The figure below (called "security characteristic lines" in the capital assets pricing model framework) shows monthly gains for ProShares Ultra S&P 500 (NYSEARCA:SSO) vs. the S&P 500, and for ProShares UltraPro S&P 500 (NYSEARCA:UPRO) vs. the S&P 500, over each ETF's lifetime. SSO and UPRO seek to multiply daily S&P 500 gains by factors of 2 and 3, respectively.

We see that monthly gains for both SSO and UPRO map almost perfectly to monthly gains for the S&P 500 (which was my main point in this recent article. This is somewhat surprising, because leveraged ETFs work on daily gains, not monthly gains. But it turns out that SSO and UPRO essentially operate as both daily and monthly multipliers of S&P 500 gains.

**Expected Value of Monthly Gains**

The blue lines in the above figures are fitted regression models. Basically, we assumed a linear model relating monthly gains of each ETF to monthly gains of the S&P 500. This model is:

Y = alpha + beta X + e, e ∼ (0, σ^{2})

Let's consider SSO for a moment. The above model says that we assume that monthly growth of SSO is linearly related to monthly growth of the S&P 500. We allow for a non-zero intercept, i.e. that SSO might on average lose or gain in months when the index is flat. In fact, we would expect alpha to be negative, since leveraged ETFs will always deteriorate somewhat when the underlying index bounces around with no net change. Finally, the e term just represents random fluctuations around the regression line.

This is an assumed model, but in the case of SSO and UPRO, it fits the data extremely well.

For SSO, the regression equation was: Y = −0.03 + 2.03X. From this, we can easily calculate conditional expectations. For example, our expected value for SSO monthly gain, given a S&P 500 monthly gain of 3%, is −0.03 + 2.03(3) = 6.06%. For an index gain of -1%, it's −0.03 + 2.03(−1) = −2.06%.

Given that we don't really know what the S&P 500 is going to do next month, potential long-term investors would be more interested in the marginal expected value of Y, rather than the conditional expected value for some predicted S&P 500 gain. Fortunately, we can use properties of conditional expectation from probability theory to figure this out.

The marginal expectation of Y can be written as E[Y] = E[E[Y|X]]. In other words, we can condition on another random variable, and then take the expected value of that conditional expectation. In the current setting, our linear model tells us that E[Y|X] = alpha + beta X. The expectation of that quantity is E[alpha + beta X] = alpha + beta E[X]. For E[X], I would use the S&P 500's mean monthly gain over the past 65 years, which is 0.708.

To illustrate, our estimate for the expected monthly growth of SSO is −0.03 + 2.03(0.708) = 1.41%. For UPRO, it is 0.09 + 3.09(0.708) = 2.28%. Not bad!

To summarize:

*The expected monthly gain for a leveraged ETF is alpha + beta E[X]. With historical data, you can regress monthly gains for the ETF vs. monthly gains for the index to estimate alpha and beta, and use the historical mean monthly gain of the index for E[X]. In the absence of historical data for the ETF, a rough approximation is c E[X] where c is the target daily multiple.*

**Variance of Monthly Gains**

For variance, we take advantage of a similar property from probability theory: V[Y] = E[V[Y|X]] + V[E[Y|X]]. It looks cryptic, but it's really not too hard to apply.

V[Y] = E[V[Y|X]] + V[E[Y|X]] = E[σ^{2}] + V[alpha + beta X] = σ^{2} + beta^{2} V[X].

A natural estimate for σ^{2} is the mean squared error from the linear regression, and we replace beta with our regression estimate. Finally, we plug in the variance of monthly gains for V[X]. For example, the variance of monthly gains for the S&P 500 over the last 65 years is 17.25%.

Our estimate for variance of monthly gains for SSO is 0.5 + 2.0276 2 (17.25) = 71.1%. For UPRO, it's 0.6 + 3.0882 2 (17.25) = 165.1. Wow, that seems pretty high. Then again, I think most of us better sense of standard deviations than variances. The standard deviations for SSO and UPRO are 8.4% and 12.8%, respectively.

For leveraged ETFs that closely track their underlying index, like SSO and UPRO, we could drop the σ^{2} part since it's negligible compared to the second term. It contributed 0.5 to 71.1 for SSO, and 0.6 to 165.1 for UPRO. At the same time, it isn't too hard to include it, so we might as well.

To summarize:

*The variance of monthly gains for a leveraged ETF is σ*^{2}*+ beta*^{2}*V[X]. With historical data, you can regress monthly gains for the ETF vs. monthly gains for the index to estimate beta and σ*^{2}*, and use the variance of historical monthly gains for the index for V[X] . In the absence of historical data for the ETF, a rough approximation for an ETF with very little tracking error on the monthly scale is c*^{2}*V[X], where c is the target daily multiple.*

**Risk-Adjusted Growth (E-SD Ratio)**

With estimates for the expected value and variance of monthly gains for a leveraged ETF, it makes sense to consider the ratio of the two. Ratios like this tend to use standard deviation rather than variance to quantify risk, so we could define the metric as E[Y]/SD[Y], where SD[Y] is just the square root of V[Y]. Let's call it the E-SD ratio for the heck of it.

We see that for a leveraged ETF, the ratio of expected value to standard deviation is (alpha + beta E[X])/ sqrt(σ^{2} + beta^{2} V[X]).

If alpha = 0 (i.e. no deterioration due to volatility decay), beta = c (daily multiple maps to same monthly multiple), and σ^{2} = 0 (zero tracking error on monthly scale), then this ratio reduces to c E[X] / sqrt(c^{2} V[X]) = E[X]/SD[X]. In other words, the ratio is the same as for the underlying index. The extra return comes with a perfectly proportionate extra risk.

In practice, we expect a small negative alpha due to volatility decay, and a small σ^{2} since monthly gains will not map perfectly to the index. The negative alpha decreases the numerator, and the positive σ^{2}increases the denominator, both of which have the effect of reducing the ratio.

Our estimates for E-SD are 1.41/sqrt(71.1) = 0.167 for SSO, (2.28/sqrt(165.1) = 0.177 for UPRO, and 0.708/sqrt(17.25) = 0.170 for the S&P 500.

It can be hard to keep track of the various metrics used to quantify risk-adjusted return. The ratio described here is different from what we usually call the "risk-return ratio," which is the ratio of total growth over a period of time to maximum drawdown during that period. The term "risk-reward ratio" can refer to a lot of different things, e.g. (1) the ratio of your maximum potential loss to your expected return; (2) the ratio of your maximum potential loss to your maximum potential gain; and (3) the standard deviation of gains divided by the mean of gains.

**How it Relates to Portfolio Optimization**

If you're like me, you're probably excited by the expected returns of leveraged ETFs, but wary of the high level of risk (i.e. variance). I think these funds have great potential in portfolio optimization for this very reason.

In portfolio optimization, we usually aim to maximize some measure of risk-adjusted return. The E-SD ratio is a great metric because it uses the marginal expectation and variance of the portfolio, as opposed to the sample mean and sample variance over some particular time window.

Suppose you want to find the best possible weighted combination of UPRO and some other fund, let's call it FUNDZ. We define the weights as a 2-element column vector **w**, the expected returns for UPRO and FUNDZ as a 2-element column vector **µ**, and the variance-covariance matrix for UPRO and FUNDZ as a 2x2 matrix **V**.

**w** = (w_{1}, w_{2})'

**µ** = (µ_{1}, µ_{2})'

**V** = (σ_{1}^{2}, σ_{12}; σ_{12}, σ_{2}^{2})

We can estimate σ_{12} as either the sample covariance between monthly gains for UPRO and FUNDZ over some time period, or better yet using the formula:

Cov(alpha_{1} + beta_{1} X + e_{1}, alpha_{2} + beta_{2} X + e_{2}) = beta_{1} beta_{2} V[X] + Cov(e_{1}, e_{2}).

where e_{1} and e_{2} are residuals from the CAPM regressions for UPRO and FUNDZ, respectively.

With estimates for **µ** and **V**, we can use optimization routines to figure out the values of w_{1} and w_{2} that maximize the E-SD ratio of the portfolio, which is given by the matrix equation: (**w'** **µ**) / sqrt(**w'** **V** **w**). This optimization can be done in R, for example, using the nlminb function.

I used two funds here for illustration, but you can throw UPRO in with any number of other funds and use this method to calculate optimal weights to maximize E-SD.

**Sensitivity to Market Conditions**

I'd like to point out that E[Y], V[Y], and E-SD outlined here are metrics of fund performance in general, not in a particular market. For example, the characteristic line for UPRO shown earlier came from a strong bull market. However, by nature of the fact that leveraged ETFs directly target the underlying index, I would expect the line to look very similar if the data came from a strong bear market instead. If estimates of alpha, beta, and σ^{2} are not sensitive to market conditions, then our resulting estimates for E[Y], V[Y], and E-SD are also not sensitive to market conditions. This is in contrast to most measures of fund performance, which are strongly dependent on market conditions (e.g. Sharpe ratios are always high during a bull market).

**Generalizability**

Nothing here is specific to leveraged ETFs. For any fund, we can plot gains vs. S&P 500 gains (or any other benchmark), and then estimate the expected value and variance of the fund's gains based on the fitted linear regression model.

**Conclusions**

If you've read my other articles, you know that I like the idea of long-term investments in leveraged ETFs. One reason is that long-term gains on average beat the market. In this article, I show that the expected value for monthly gains is approximately c E[X]. E[X] is positive (otherwise we wouldn't invest in the stock market), so c E[X] amplifies returns favorably for any c > 1. At the same time, the variance of monthly gains is pretty high, which speaks to the riskiness of these funds.

From a risk-adjusted return perspective, leveraged ETFs take on a little bit more risk than is warranted by their extra returns, but only because of technical/practical issues (slightly negative alpha; not quite perfect tracking of monthly gains to index). For SSO and UPRO, these effects seem to be negligible.

**R** **Code**

The R code that I used to fit the regressions for SSO and UPRO, generate the figures, and calculate the historical mean and variance of S&P 500 monthly gains is available on my website.

**Disclosure:** The author is long UPRO.

**Additional disclosure:** The author used Yahoo! Finance to obtain historical prices for SSO, UPRO, and the S&P 500, and used R to analyze the data and generate figures. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.