Academics have been shouting from the rooftops about risk-efficient portfolios (minimum variance, minimum correlation, minimum expected shortfall etc) and their merits, for some time now. This has led to a suite of indices from EDHEC-Risk, many minimum variance funds, ETFs, risk parity products and the like.

The goal of the experiment here is to see if we can take a minimum variance portfolio, leverage it up to have the same expected volatility as the benchmark and see if it still performs after the additional cost of capital. These are experiments to model Portfolios and not investment advice or a recommendation to buy/sell/hold securities.

The data:

9 SPDR’s from 1998-12-22 to 2016-09-16
SPY as our benchmark from 1998-12-22 to 2016-09-16
Federal Funds Rate Monthly from 1998 to 2016

The process:
First we construct a minimum variance portfolio, that is choose portfolio weights to minimise expected future portfolio risk (volatility). This is repeated every quarter.

We end up with performance that looks like this (which is rather typical of minimum variance strategies):


I’ve also included in that backtest the performance of an equal weight portfolio (called “Naive”) of the same ETF’s as an alternative baseline, as you can see consistent with both the academic research and the products in the market the two outperform the benchmark (S&P500 ETF) over the horizon.

Now when we lever the portfolio up to the same expected volatility as the benchmark we get the following results.

I’ve used a cost of capital of the Federal Funds Rate + 1.5% which is what my broker charges me on overnight margin balances.

Table of Annualised Returns, Annualised Volatility and Sharpe Ratio:

NaiveSPDR MinVOLSPDR SPY MinVolSPDRLeveraged MinVolSPDRLeveragedWCost
Annualized Return 0.0665 0.0641 0.0515 0.1171 0.1097
Annualized Std Dev 0.1876 0.1626 0.1972 0.1934 0.1934
Annualized Sharpe (Rf=0%) 0.3545 0.3945 0.2613 0.6054 0.5672


What is interesting here is that the returns go up by approximately 1.7x that of the unlevered portfolio, with slightly less risk than the benchmark

Let’s look at the leverage ratio over time to see how much leverage was required to generate that:


Authors Note: I use Assets/Equity to define leverage here so a leverage of 1 indicates no borrowing.

The Long term average leverage is 1.2151, which shows some efficiency gains or timing gains here. By varying the level of leverage with each rebalancing period this allows us to capture the efficiency difference between the risk efficient portfolio and the benchmark in a manner that can increase leverage when the difference is large and decrease when it is small (there was even a period in 2009 when the portfolio was not levered at all and held cash).

Let’s look at the rolling volatility and as we can see here in some few periods the portfolio has worse volatility than the benchmark however for most of it it has lower volatility than the benchmark.


Further work can be done in forecasting volatility to improve the calibration of the model.

Let’s look at the worst losses (Maximum Drawdowns):

NaiveSPDR MinVOLSPDR SPY MinVolSPDRLeveraged MinVolSPDRLeveragedWCost
Worst Drawdown 0.5297 0.4321 0.5519 0.4409 0.4453

Even with the leverage the maximum drawdown of the portfolio is far less than the benchmark.

To look at the persistence of outperformance here is the rolling plot of the 1 year annualized return differential between the leveraged portfolio and the SPY.


Finally going back to CAPM and theory, here is a glance at how the Portfolios fare against the benchmark:

NaiveSPDR to SPY MinVOLSPDR to SPY SPY to SPY MinVolSPDRLeveraged to SPY MinVolSPDRLeveragedWCost to SPY
Alpha 1e-04 1e-04 0 3e-04 2e-04
Beta 0.93 0.7794 1 0.9203 0.9202
Beta+ 0.902 0.7425 1 0.8837 0.8837
Beta- 0.9383 0.7895 1 0.8974 0.8974
R-squared 0.9562 0.8939 1 0.8803 0.8803
Annualized Alpha 0.0173 0.0213 0 0.0675 0.0605
Correlation 0.9778 0.9455 1 0.9383 0.9383
Correlation p-value 0 0 0 0 0
Tracking Error 0.2575 0.2393 0.2483 0.261 0.261
Active Premium 0.015 0.0126 0 0.0656 0.0582
Information Ratio 0.0581 0.0527 0 0.2512 0.2229
Treynor Ratio 0.0715 0.0823 0.0515 0.1272 0.1192

In conclusion, adding leverage to a risk-efficient portfolio can generate superior returns, at the same (or slightly less) risk and even generate some alpha. This is NOT investment advice or a recommendation to trade securities.

1 comment

  1. Comment by investor425

    investor425 Reply September 20, 2016 at 6:14 pm

    Event risk is more to be feared and guarded against than volatility risk.

    Increasing sharpe is always nice, but not at the risk of the long term.

    At 1.7% leverage if a major crash event occurs, your portfolio can be decimated.

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