Introduction

In this small document I want to examine the so called “Volatility Drag”. The hypothesis is that when we have normal distributed returns (not log normal) that the geometric return is lower the higher the variance of the distribution is.

Setup

Let’s first create some random series. With 10’000 data points and 3% annual mean and 20% annual standard deviation

Analyzing

Now let’s see how the numbers look like.

##                           arithmetic results geometric results
## Annualized Return                     0.0118           -0.0085
## Annualized Std Dev                    0.2019            0.2019
## Annualized Sharpe (Rf=0%)             0.0586           -0.0422

Here we already see quite a difference. Now let’s see what happens when we increase/decrease volatility.

##                           simple_1X compund_1x simple_2X compound_2X
## Annualized Return            0.0118    -0.0085    0.0237     -0.0563
## Annualized Std Dev           0.2019     0.2019    0.4039      0.4039
## Annualized Sharpe (Rf=0%)    0.0586    -0.0422    0.0586     -0.1393
##                           simple_0.5X compound_0.5X
## Annualized Return              0.0059        0.0008
## Annualized Std Dev             0.1010        0.1010
## Annualized Sharpe (Rf=0%)      0.0586        0.0081

Eurostoxx and S&P500

So is leverage hurting? So to say do I get more risk for less return if I use excessive leverage? If I am a passive investor do I get arithmetic returns or geometric returns? http://www.investopedia.com/articles/investing/041114/most-accurate-way-gauge-returns-compound-annual-growth-rate.asp
Probably the compounded return is more accurate from an investors point of view. Therefore let’s have a look at some real data.

Now let’s play with the vola and see how the sharpe changes.

##                               ES  ES_2X ES_0.5X
## Annualized Return         0.0678 0.0960  0.0201
## Annualized Std Dev        0.1925 0.3849  0.0481
## Annualized Sharpe (Rf=0%) 0.3523 0.2493  0.4184
##                            Stoxx Stoxx_2X Stoxx_0.5X
## Annualized Return         0.0068  -0.0484     0.0076
## Annualized Std Dev        0.2510   0.5020     0.0627
## Annualized Sharpe (Rf=0%) 0.0270  -0.0964     0.1214

Summary

It is true that if I increase the leverage I take on more risk for less reward which might not be obvious in the first place. https://cssanalytics.wordpress.com/2012/03/12/understanding-the-link-between-volatility-and-compound-returns/ On this side they provide a proxy formula for the Volatility Drag which is

Volatility Drag= 0.5*(Volatility)2