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Abstract

Alpha Generation and Risk Smoothing using Managed Volatility

Tony Cooper

Quantitative Analyst

Double-Digit Numerics

August 6, 2010

Abstract

It is difficult to predict stock market returns but relatively easy to predict market

volatility. But volatility predictions don’t easily translate into return predictions since

the two are largely uncorrelated. We put forward a framework that produces a formula

in which returns become a function of volatility and therefore become somewhat more

predictable. We show that this strategy produces excess returns giving us the upside of

leverage without the downside.

As a side-effect the strategy also smoothes out volatility variation over time, reduces

the kurtosis of daily returns, reduces maximum drawdown, and gives us a dynamic timing

signal for tilting asset allocations between conservative and aggressive assets.

It has been said that diversification is the only free lunch in investing. It appears that

once you have diversified away some risk you can get a further free lunch by smoothing

what risk remains.

Keywords and phrases: volatility timing, volatility of volatility, extreme volatility,

volatility drag, managed volatility, continuously dynamic leverage, leveraged exchange

traded funds, alpha generation, kurtosis, maximum drawdown

JEL Classification: C22, C5, G1 www.ddnum.com. This paper differs from the NAAIM Wagner Award (naaim.org/wagneraward.aspx) winning paper in having a title change, minor text improvements, and a new section on maximum drawdown and kurtosis. Thanks to Aaron Brown and Florin Spinu for their related comments.

at: http://ssrn.com/abstract=1664823 https://ssrn.com/abstract=1664823

1 Introduction

Whether or not you believe in the efficient market hypothesis it is difficult to predict market returns (Ferson, Simin, and Sarkissian 2003) whereas market volatility is clearly forecastable (Poon and Granger 2003). Fischer Black (Miller 1999) reasoned that estimating variances isorders of magnitude easier than estimating expected returns. This conclusion does not violate market efficiency since accurate volatility forecasting is not in conflict with underlying assetand option prices being correct. The purpose of this paper is to use the predictability of volatility to generate excess returns (returns over and above the market when adjusted for risk). For such predictions to be useful volatility itself must be volatile. The remainder of this article is structured as follows. The next section presents the math- ematics of compounding daily leveraged returns which gives rise to the formula that motivates the investment strategies. The next two sections discuss the concept of the volatility of volatil- ity and introduce the strategies that exploit the concept. The following section forms themain body of the article where the econometric results are discussed. Finally, the last section summarizes the empirical findings, draws some conclusions, and points out avenues for future research. All index data comes from Yahoo.com except for the US stock market data from 1885 to 1962 which comes from (Schwert 1990) (the capital index is used). We augment this dataset using the S&P 500 index to produce a price series up to 2009.

2 Simplified Mathematics of leverage

To introduce our key investment concept of volatility drag and to see how leverage works westart with an example involving leveraged exchange traded funds (ETFs).

2.1 Definition of Leveraged Exchange Traded Funds

Leveraged ETFs are funds that aim to magnify the daily moves of an index. For example ina double-leveraged fund (a 2x fund), if the index goes up, then the fund goes up twice thatamount. The magnification multiple is specified in the prospectus for the ETF and is a fixed

at: http://ssrn.com/abstract=1664823 https://ssrn.com/abstract=1664823

2.2 Myth about Long Term Holding of Leveraged ETFs

constant. Typical values are 1, 1.5, 2 and 3. Negative multiples such as -3, -2, and -1 are usedas well. For example if an ETF promises a return of 2 times the S&P 500 index then if the S&P 500 index goes up 1.2% in one day the ETF will go up 2 x 1.2% = 2.4%. The salient point about this definition is that the return is marked to the benchmark daily, not annually. This causes confusion for investors because the daily leverage multiple does nottranslate to the same multiple when applied to annual returns of the ETFs or to compound returns. The dynamics of leveraged ETFs are more complicated than just applying a multiple to a return and the correct formula provides the key to our volatility strategies.

2.2 Myth about Long Term Holding of Leveraged ETFs

Leveraged ETFs confuse many investors because of the difference between arithmetic and com- pound returns and because of the effects of volatility drag (explained below). One of the common myths is:

Leveraged ETFs are not suitable for long term buy and hold

This myth is expressed in various ways. Some quotes from the internet about leveraged ETFs:

“unsuitable for buy-and-hold investing,”“leveraged ETFs are bound to deteriorate,” “over time the compounding will kill,”“leveraged ETFs verge on insanity,”“levered ETFs are toxic,” “levered ETFs [are] a horrible idea,” “.. practically guarantees losses,”“in the long run [investors] are almost sure to lose money,”“anyone holding these funds for the long term is an uneducated lame-brain.” “Leveraged ETFs areleaky,”“Warning: Leveraged and Inverse ETFs Kill Portfolios.”

There is even an article comparing these ETFs to swine flu. The explanation popularly given for this myth is that volatility eats away at long termreturns. If this were true then non-leveraged funds would also be unsuitable for buy and holdbecause they too suffer from volatility. We need to more closely examine the effect of volatility.

2.3 Volatility Drag

Daily volatility hurts the returns of leveraged ETFs (including those with leverage 1x). This isdue to the equality

(1 x)(1 + x) = 1 x2

Suppose the market goes down by x and then the next day it goes up by x. For example the market goes down by 5% then up by 5%. Then the net result is that the market has goneto (1-0.05) times (1+0.05) = 0.9975 which is a drop of 0.0025 or 0.25%. That doesn’t seem fair. The market has gone down by 5% then up by 5% but our ETF thathas a leverage of 1x has gone down by 0.25%. This drop always occurs because x2 is always positive and the sign in front is negative. Sowhenever the market has volatility we lose money. We call this loss volatility drag. The larger x is, the larger x2 is, so the larger the volatility drag. For a leveraged ETF theleverage multiplies x and so multiplies the volatility drag. Even an ETF with a leverage of 1xhas volatility drag. The myth has resulted from the belief that volatility drag will drag any leveraged ETFdown to zero given enough time. But we know that leverage of 1x (i.e. no leverage) is safe tohold forever even though leverage 1x still has volatility drag. If 1 times leverage is safe then is 1.01 times leverage safe? Is 1.1 times safe? What’s so special about 2 times? Where are yougoing to draw the line between safe and unsafe?

Note that even if stocks fluctuate logarithmically rather than arithmetically (so that changes of (1 x) and (1 + x) are not equally likely) we still get volatility drag because it results fromvolatility per se rather than the nature of the volatility.

2.4 Sample Market Data

Figure 1 shows 135 years worth of daily US index prices going back to 16 February 1885. The construction of this index is described in Schwert (1990) and the index used is the capital index (no dividends reinvested). S&P 500 data from Yahoo.com has been used to augment thedata up to 2009.

2.4 Sample Market Data

The orange circles show popular leverage rates 1, 2, 3, and, where visible, 4. It can be seenthat increasing leverage from zero to 1 increases the annualised return as would be expected. But then, contrary to what the myth propagators say, increasing the leverage even further stillkeeps increasing the returns. There is nothing magic about the leverage value 1. There is no mathematical reason forreturns to suddenly level off at that leverage. We can see that returns do drop off once leverage reaches about 2. That is the effect of volatility drag. What the myth propagators have forgotten is that there are two factors that decide leveraged ETF returns: benchmark returns and benchmark volatility. If the benchmark has a positive return then leveraged exposure to it is good and compensates for volatility drag. Since thereturn is a multiple of leverage and the drag a multiple of the leverage squared then eventually the drag overwhelms the extra return obtained through leverage. So there is a limit to theamount of leverage that can be used. The tradeoff between return acceleration and volatility drag and the management of thetradeoff is the basis for our volatility strategies.

2.5 The Formula for the ETF Long Term Return

The formula for the long term compound annual growth rate of a leveraged ETF cannot bewritten in terms of just the benchmark return and volatility. It also involves terms containing the skewness and kurtosis of the benchmark. It is derived using a Taylor series expansion (details available from the author). It does not assume that benchmark returns are Gaussian or that returns are continuous as do formulae derived using Ito’s lemma. But it turns out that for the world’s stock markets and for low levels of leverage (up toabout 3) the formula can be approximated by this formula:

R=k 0:5k 2 2 =(1 + k ) (1)

where R is the compound daily growth rate of the ETF, k is the ETF leverage (not neces- sarily an integer or positive), is the mean daily return of the benchmark, and is the dailyvolatility (i.e. standard deviation) of the daily return of the benchmark. R is the quantity you use to calculate the long term buy-and-hold return of the ETF. Youcan see from the formula that if the volatility is zero then R = k so that the return of the ETF is k times the return of the benchmark. The 0:5k 2 2 =(1 + k ) term is the volatility drag. Since k 2 2 is always positive and (1 + k ) is always close to 1 then the volatility drag is always positive. R is a quadratic function of k with a negative coefficient for the square term. That meanswe will always get the parabola shape shown in Figure 1 and we will always have a maximum for some value of k. Some algebra shows that the maximum is approximately (for small k ) at

k= = 2 (2)

This clearly shows the return/volatility trade-off that determines the optimal leverage. This formula occurs in an appropriate form in the Kelly Criterion (Thorp 2006) and Merton’s Portfolio Problem (Merton 1969). Its appearance here as the result of an optimisation is nosurprise.

2.6 A Look at Some Stock Markets

Let’s look at some other markets and time frames in Figure 2. The pattern is quite clear. Over various markets over various time periods (mostly the lasttwo or so decades) except for the Nikkei 225 the optimal leverage is about 2.

2. 7 Leveraged ETFs Fees

Unfortunately there may be a reason not to hold leveraged ETFs for the long term but it hasnothing to do with volatility drag. It is because of fees. Most leveraged ETFs where the leverage is greater than one charge an annual fee of aboutone percent. This imposes a “fee drag”on the ETF. Also leveraged ETFs suffer from tracking error. They do not exactly hit their target return for the day every day. From studying a few leveraged ETFs it has been observed that thetracking error is a complicated error and beyond the scope of this document. But tracking error is small and is ignored for this article. An annual fee of 0.95% (a typical value for recent leveraged ETFs) applied daily subtracts 0.95% from the annual return of the ETF. For the 1885 to 2009 data this fee reduces the return of a 2x leveraged ETF to that of a 1x fee-free ETF. So all the benefits of leveraging have beenlost. For other markets over shorter time frames the fees aren’t as destructive. The optimal leverage is still about 2 and even after fees 2x ETFs outperform the benchmark over several decades.

2. 8 Leveraged ETFs Risks

Leveraged ETFs can be held long term provided the market has enough return to overcome volatility drag. It usually does. For most markets in recent times the optimal leverage is about

2. But some markets and time frames will reward a leverage of up to 3. No markets will reward a leverage of 4. So it would seem that using leveraged ETFs is a good idea for enhancing returns. But this is not true for the following reasons:

Leveraged ETFs do not generate alpha. Any leverage that multiplies return also multiplies

2.8 Leveraged ETFs Risks

3 Volatility and Volatility of Volatility

volatility by the same multiple. So risk-adjusted returns are not enhanced.

Nearing the peak of the parabolas from the left we are adding leverage and thus volatility and getting not much extra compounded return for it. And going past the peak we areadding volatility and losing return. This is dangerous and since we do not know in future markets where the peak is exactly we run the risk of negative alpha.

Leveraged ETFs run the risk of ruin— losing all money invested.

So to use leveraged ETFs effectively we want to invest as close to the left as possible andto closely watch volatility to avoid straying over to the right. This is where the volatility ofvolatility comes into play.

3 Volatility and Volatility of Volatility

The optimal leverage k = = 2 gives the optimal value of R. For the charts of the markets in Figure 2 and are the mean values over the lifetime of the ETF. But in reality and arenot regarded as constant but reasonably can be seen to vary over time. So it seems likely thatthe optimal value of k is time varying also. Can we predict and well enough to estimate the optimal value of k for the immediate future?

We cannot easily predict and it is outside the scope of this volatility paper anyway. This is a dynamic asset allocation problem where we want to maximise some measure ofterminal wealth with decisions made at discrete time intervals such as days or months. This isa dynamic stochastic programming problem (Samuelson 1969) but it is outside the scope of thispaper to provide an optimal solution. Instead we provide an indication of the possible gainsfrom solving the problem.

3. 1 EGARCH(1, 1) Volatility Forecasting

, the daily volatility, is a random variable which can vary from day to day. It is defined tobe the instantaneous standard deviation of price fluctuations. Measuring volatility is not asstraightforward as measuring price. It requires measuring price changes over time but during the time interval measured the volatility itself may change. For example, if only daily closing

3.1 EGARCH(1, 1) Volatility Forecasting

prices are observable then to get a reasonable measure of standard deviation 10 prices over 10 days need to be observed. But over those 10 days the volatility itself is sure to have changed. So instantaneous volatility is unobservable. Instantaneous volatility can only be estimated inthe context of a model. We shall use the EGARCH(1, 1) model for volatility forecasting because it is a relatively standard choice and Poon and Granger (2003) report studies where it outperforms other meth- ods, particularly when used on market indexes. It is outside the scope of this expository paperto find the best method for our purposes so an indicative method will suffice. The EGARCH(1,

1. model is

"t 1 "t 1 log( 2t ) = m(1 ) + log( 2t 1 ) + + (3) t 1 t 1

This models the log variance on day t, log( 2t ), as a weighted average of the long run logvariance m, the previous day’s log variance log( 2t 1 ), and the absolute value of the previous day’s return "t 1 (scaled for consistency by the previous day’s volatility t 1 ). The termis a correction1 term that allows for the fact that negative daily returns tend to add positive amounts to the daily volatility. After estimating m,,;and we can estimate volatility onday t from quantities calculated on day t 1. Even though volatility is not a directly observable quantity this method gives us estimates of volatility for days 1;:::; t. But, further, we can plug in values calculated on day t to forecast the volatility for day t + 1. Figure 3 shows the EGARCH estimates for the S&P 500 daily volatility from 1950 to 2009. The left hand charts show the volatility estimates, the right the estimates on a log scale. It isapparent that logs of the volatilities have more symmetry than the raw volatilities; they alsoare less “spiky.”This gives us justification for the use of the log estimator in Equation (3) over 2 an estimator that estimates t directly.

3.2 Volatility of Volatility

3.2 Volatility of Volatility

A feature of the charts in Figure 3 is that the estimates of volatility vary over time. The mostlikely reason is that the actual volatility being estimated varies over time. Let us define thevolatility of volatility (vovo for short) as the standard deviation of the daily volatilities t for

t = 1;:::; n where n represents the number of days of observation. Then for the purposes of this paper we shall estimate vovo by the intuitively reasonable standard deviation of the EGARCH(1, 1) estimates of the daily volatility. It is beyond thescope of this exploratory paper to examine the statistical properties of this estimator. For ourpurposes it suffices to know that volatility does vary and that our EGARCH model gives usestimates of the variable volatility. We don’t take the standard deviation of the daily volatilities, however. We deal with logvolatility where possible and calculate vovo on the log scale.

3. 3 Costs of Volatility of Volatility

In traditional asset allocation vovo has a cost. As an example, consider a balanced-fund manager who uses, say, bonds to reduce the volatility of a fund which contains equities. Or a financial planner who assesses the risk tolerance of a client investor and proposes, say, a 60-40 equities- bonds mix. The traditional way of managing these investments is to look at the long termhistorical volatility of the component asset classes to decide the proportions to invest in. Usually no consideration is given to the vovo of the asset classes and no constant volatility target is set. 1 The term is frequently called the “leverage” term but we won’t use that volatility term here because itcould be confusing.

3.4 Volatility of Volatility Strategies

So the investor or manager with a static or reasonably constant 60-40 mix has to suffer thevarying volatility of the markets— sometimes sleeping well at night, occasionally not. In orderto ensure that the worst volatility is minimised the asset allocation will have erred on the sideof conservative. This will have cost returns. The larger the vovo the more conservative the asset allocation will have to be and therefore the more the cost to returns. In addition to the loss of returns there is a cost to the fund manager due to the extra expected drawdown of the fund arising from vovo. Drawdowns cause worried investors to redeem fundsso funds under management will drop. If we have two fund managers with the same annual returns and same annual volatility the manager with the higher vovo will lose the most fundsin a market drawdown.

3.4 Volatility of Volatility Strategies

We now create three portfolio strategies that use volatility prediction and the = 2 formula ina short-term look-ahead manner. The strategies are:

The Constant Volatility Strategy (CVS)

The Optimal Volatility Strategy (OVS)

Mean Only Strategy (MOS)

The Optimal Volatility Plus Mean Strategy (OVPMS)

The MOS is not a strategy that we consider for investment because it is included only as abenchmark for the the OVPMS strategy. We also construct a data-snooping version of OVPMS called OVPMS+. The purpose ofthis version is to give an indication of how well a calibrated version of OVPMS would havedone. Every day just before the close of trading we take the closing price for the day and put itinto our EGARCH estimator to estimate our volatility for the next day. Call our estimate ofthe ith day’s volatility si. Then we calculate our leverage ki for the ith day using our leverage calculation formula. Then just on the closing bell we execute trades to bring our portfolio to

3.5 Leveraged ETFs and the Probability of Ruin

leverage ki so that it begins the next day at this level of leverage. The fact that the EGARCH estimation and ensuing trading have to be done in the instant just before closing and that thereare transaction costs are implementation inconveniences that we overlook for the purposes ofthis expository study. Also, in practice, we would likely be using intra-day prices for estimates of the volatility rather than end of day prices since they can provide better predictions (Poonand Granger 2003). These strategies are not pattern-detection market timing strategies or technical analysis. The strategies are not obtained by backtesting. They are purely mathematical day-ahead opti- misation strategies that use past data only (OVPMS+ the exception) for volatility prediction.

3. 5 Leveraged ETFs and the Probability of Ruin

The strategies can be implemented as overlays on existing portfolios or as investments in theirown right. They can be implemented by using combinations of leveraged ETFs provided thatsuitable ETFs exist. For example suppose that leverage of 2.4 is required. This can be achieved by investing 0.6 of the asset into a 2x leveraged ETF and 0.4 into a 3x leveraged ETF. A typical range of leverages such as -3, -2, -1, 1, 2, and 3 may exist for any one index. Thatgives a range of possibilities for mixing them. For example 2.4 = 0.6x2 + 0.3x3 but also 2.4 = 0.3x1 + 0.7x3 and also 2.4 = 0.15x(-1) + 0.85x3. More than two ETFs may be used. Which isthe best to choose? One possible answer considers the possibility of the ETF being ruined. An ETF with leverage k will drop to zero if the market drops by 1=k. So in the 2.4 = 0.6x2 + 0.3x3 case the strategy will be ruined if the index drops by 50% or more in one day. But the 2.4 = 0.15x(-1) + 0.85x3 case will still have 0.15x(0.5) left. The probability of ruin when invested in an index is virtually zero. But a leveraged ETFwhere the absolute value of the leverage exceeds one will always have a finite chance of ruin. The 1987 crash would have ruined any 5x leveraged ETF. Fortunately all the strategies reduce leverage when volatility is high so this makes the chances of ruin smaller. We examine thebehaviour of the strategies at the 1987 crash below. Non-market catastrophic events that hitthe market without warning would be the most deadly risk.

4 Volatility of Volatility Strategies

Our three strategies differ in the formula used to derive the daily values of the ki.

4.1 The Constant Volatility Strategy (CVS)

This strategy sets the leverage to be ki = c=si for some constant c. This strategy is not optimal in that it uses si in the denominator rather than s2i. But it has a simplicity and appeal that we find irresistible. If tomorrow is predicted to be a high volatility day then it lowers the leverage for that day in proportion to the predicted volatility. This means that it is aiming for a constant volatility of c every day. This has four useful implications for financial planning and risk management:

it provides a target risk level that can be sold as a number (and put into the prospectus)

it reduces the vovo experienced by the strategy.

it increases the portfolio return by reducing the costs of vovo as described in subsection 3.3

it generates alpha.

4. 2 The Optimal Volatility Strategy (OVS)

This strategy sets the leverage to be ki = c=s2i for some constant c. The problem with thisstrategy is that c cannot be easily set in advance to target a specific volatility as it can with CVS. For a given prediction si and leverage c=s2i the predicted strategy volatility of the strategy for day i is c=si rather than the constant c for CVS. So OVS is not targeting constant volatility but targeting extra low volatility the higher predicted volatility is. OVS is not trying to reduce vovo so even though it is optimal it will lose some of itsdesirableness in a management setting when the cost of vovo is taken into account. And afurther implementation problem arises in that in targeting a mean level of volatility the valueof c required is not known in advance. It can be calibrated by examining historical data butthis introduces estimation error and the assumption that the future will behave as the past.

4.3 The Mean Only Strategy (MOS)

A further problem with OVS that we will see below is that vovo introduces extra volatility into the ki values which sometimes get very high. Higher ki values have higher probabilities ofportfolio ruin.

4.3 The Mean Only Strategy (MOS)

This strategy sets the leverage to be ki = cm(si ) for some constant c and some function m(s) which is described more fully in the next strategy. We are not interested in the MOS strategy in its own right— just as a means of looking at the effect of the term contribution in theexpression = 2. That is, this strategy acts as a benchmark.

4. 4 The Optimal Volatility Plus Mean Strategy (OVPMS, OVPMS+)

This strategy sets the leverage to be ki = cm(si )=s2i for some constant c and some function m(s). Here we have some estimate of the mean return as a function of volatility only. Anyother prediction is outside the scope of this volatility paper. The difference between OVPMS and OVPMS+ is that the former only uses past data toestimate the function m(s) and every day when the EGARCH forecast is calculated the estimate of m(s) is updated as well. OVPMS+ uses only one estimate of m(s) and that is calculated from the entire date range. Therefore, for example, it uses in 1951 an estimate of m(s) thatwas calculated from data up to 2009. The reason for doing this is to give us an idea of theperformance of this strategy from 2009 onwards by seeing how well the 2009 parameters didwhen applied from 1950 onwards. This is a valid use of data-snooping in a calibration context but the usual caveats of the future resembling the past apply.

4. 5 Estimating Return as a Function of Volatility

For OVPMS we require an estimate of market return as a function of volatility m(s). Morespecifically, we want a prediction of market return as a function of our prediction of market volatility. The left hand chart of Figure 4 shows for the S&P 500 index from 1950 to 2009 a plot of themarket return versus our predicted EGARCH(1, 1) values for 14823 days. The daily returns di are annualised by scaling to (1 + di )252 and the daily predicted volatility is annualised by

4.5 Estimating Return as a Function of Volatility

pmultiplying by 252. Some extreme points such as the 1987 crash of over 20% have beenomitted to improve readability. At first glance, no obvious relationship between return and volatility can be seen. The cor- relation between the two is 0.005557 (not significantly different from zero, P = 0.50 significance level). Let’s make some changes to the figure to get the right hand chart. Noticing that thespread of points about the horizontal line is expanding as we move left to right we aim for auniform spread by normalising the daily returns by dividing them by the predicted volatility. This produces a uniform spread which allows us to fit curves using unweighted least squares regression. Now we zoom in on the chart into the rectangle shown in the chart. We get Figure

5. We have divided the interval of forecast volatility from 0.05 to 0.25 (which is where 96% of

4.6 Another look at Return as a Function of Volatility

the points lie) into 20 equally spaced bins and calculate the mean normalised return for eachbin. These means are plotted in cyan along with two standard deviation error bars. We seeconvincing evidence that returns decrease with increasing volatility. In an effort to get an idea of the relationship between returns and volatility amongst what isalmost all noise we apply some smoothing techniques to all the points, not just the ones shownin the figure. The gold line is the overall mean. The black line is an application of Friedman and Stuetzle’s (1982) supersmoother which uses local linear fits. This smooths the data without imposing any constraints onto it other than smoothness so is the most truthful smooth. But itdoes not give us a clean functional form. So we try some parametric functions. The green line is a quadratic fit. This imposes a turning point onto the relationship andallows the scaled return to go below zero. The red line a robust fit of a power function f (s) = asb. This imposes positivity onto the relationship and has no requirement for a turning point. The parametric fits are sensitive to the constraints we impose and we have no justification forimposing them. But we have to start somewhere and the final portfolio may not be sensitive to the constraints anyway. So we focus on the power function. This function seems to capture the relationship quite well and it multiplies well with Formula 2 so we use this function for all further work. Table 1 shows the estimates for b and the 95% confidence intervals for these estimates. The power function fit to the scaled returns is m=s = asb or m = asb+1 so our OVPMSstrategy becomes ki = cm(si )=s2i = casbi 1 and we drop the a by absorbing it into the constant c. Thuski = csbi 1

where for OVPMS we update our estimate of b daily and for OVPMS+ we use a single estimate for the whole period.

4. 6 Another look at Return as a Function of Volatility

Return as a smooth function of volatility may not be the way the market works. It couldbe that there is a threshold of volatility where the returns jump to zero or negative abovethat threshold level. There is evidence that such a threshold exists for all the markets studied and that it has a similar value in all cases. But estimating that value is a statistical exercise

4.7 Backtesting Strategies

Time Annual 95% Confidence Index Period Volatility b Interval for b US Stocks 1885–2009 0.1672 -1.69 (-2.28, -1.09) S&P 500 1950–2009 0.1531 -1.76 (-2.63, -0.90) Dow Jones Industrial 30 1928–2009 0.1846 -1.52 (-2.27, -0.76) NASDAQ 100 1971–2009 0.2020 -1.71 (-2.24, -1.19) Russell 2000 1987–2009 0.2024 -1.82 (-2.60, -1.03) Australian All Ords 1984–2009 0.1584 -1.87 (-2.79, -0.94) Nikkei 225 1984–2009 0.2349 -2.49 (-4.05, -0.92) FTSE 100 1984–2009 0.1833 -3.10 (-4.50, -1.71) NZX All 1986–2009 0.1529 -3.18 (-4.43, -1.93) NZX 10 1988–2009 0.1783 -2.80 (-4.93, -0.66)

Table 1: Indexes, Volatilities, and Power Function Exponents

(estimating peaks in noisy time-series signals) that is outside the scope of this paper.

4. 7 Backtesting Strategies

Backtesting refers to the use of past data to:

test a hypothesis

estimate how a strategy worked in the past as a guide to how it will work in the future.

In this study we use backtesting for both purposes. Our main hypothesis that we are testing is that returns can be increased and risk reduced by using our leverage formulae. This is a validand reasonably safe use of backtesting. The problems come when you use the data to test hypotheses suggested by the data itself. This dual use of the data produces biases which are explained below. The problems with the second use of backtesting are more straightforward. To use the pastas a guide to the future assumes that the future will behave as the past. There is a risk thatit does not. We cannot quantify this risk or correct for it and simply take it as given that anypredictions that our models make in this paper are subject to this risk. For the purposes of this paper we classify the backtesting biases into three categories basedon how dangerous they are when used to predict the future:

Data snooping bias – this occurs when we use data in the past that was obtained fromfuture data. This bias can be small when we use future data to calibrate a model and then

4.7 Backtesting Strategies

use calibrated parameters in the past. An example is when we test an asset allocation strategy that requires a measure of the volatility of the market. We can estimate thevolatility from 50 years worth of data then go back in time and use the strategy asthough we knew that estimate right at the beginning. Provided that the estimate isreasonably close to the actual future volatility this bias may be small. For example, ourestimate of the market volatility may turn out to be too high or too low with roughly equal probability and this may make our estimate of future returns too high or too lowwith equal probability. The equality of the probabilities means that the bias is low. Andin any case it does give us an idea of the potential returns of an accurately calibrated model.

Data mining bias – this occurs when we use data in the past that we obtained fromfuture data where we expect some regression to the mean in future excess returns. Thisbias usually occurs when we optimise some parameter in our strategy. For example, wemight have a strategy based on a moving average of period. We use an optimising algorithm to find the value of that gives the best excess returns. In that case we expect considerable bias in backtesting because the true value of in the future will be different than our estimated value and our actual excess returns will be less than our estimated value no matter whether our estimate was too high or too low. Our actual excess returns will regress towards zero. Hopefully not all the way to zero— our optimal value of willhopefully be in the neighbourhood of the true value— and so data mining can be useful even though biased.

Another situation that creates data mining bias is testing a hypothesis where the dataitself suggested the hypothesis. The correlation between the test and the suggestion means that the significance levels of the hypothesis test are really not as good as the teststatistics suggest. This bias arises frequently when the data is “mined” for ideas whichare then tested using the same data. We avoid this bias in this study by formulating ourequations and strategies before we look at the data and only use the data for calibration not idea generation.

Data torturing bias –this occurs when we “torture” the data into revealing information it does not actually contain. In this case we expect excess returns to regress all the way

5 Volatility of Volatility Strategies Performance

to zero. An example is when we notice that Friday the 13ths have a larger mean excess return than any other day so our strategy is to only invest on Friday the 13. Since theoutperformance of that day is (presumably) just due to chance then the strategy willproduce zero excess returns in future.

The expression comes from the apothegm amongst statisticians that “if you torture adataset long enough it will confess to anything.”

5 Volatility of Volatility Strategies Performance

All the vovo strategies are evaluated in a return-risk framework. We use three different riskformulations for this:

The traditional risk –the standard deviation of daily returns (SDDR)

The extreme risk –vovo-oriented risk –the standard deviation of daily returns plus twotimes the vovo (SDDR+2)

The maximum drawdown –the maximum peak to trough loss experienced by the strategy equity curve

The first risk measure is traditionally the one used in portfolio risk-adjusted measures ofperformance such as the Sharpe Ratio (Sharpe 1966). But in the presence of vovo the standard deviation of daily returns gives a conservative measure of risk in that it averages volatility overthe whole period. For example if volatility moves between 0.10 and 0.20 then SDDR will givea value around 0.15. In contrast SDDR+2 will give a value closer to 0.20 thus reflecting theworst volatility during the investment period. We call the SDDR+2 value the extreme volatility value. It is roughly the volatility valuethat is exceeded only 5% of the time2. We tested all the strategies on the full list of indexes in Table 1 but for brevity we onlyshow here the results for the S&P 500 for 1950–2009. The patterns in the results are similar and consistent across all markets and all indexes except for the magnitudes of the returns. Inparticular the strategies show the same relative rankings. 2 It deals with the volatility of volatility and should not be confused with the concept of value-at-risk whichdeals with the volatility of returns

5.1 Traditional Risk Performance

5.1 Traditional Risk Performance

Figure 6 (right hand chart is a zoomed-in version) shows the returns of the strategies versus theannualised daily standard deviation (SDDR) for the S&P 500 from 1950 to 2009. The curves are traced out as a function of increasing volatility by increasing the constant scaling factor cof Section 4. The vertical grey line is the mean volatility of the index over the time period. Foreach strategy two versions are shown: the solid line is that strategy that invests in the risk freereturn for the balance of any assets not invested in the market when the leverage is less thanone. The dotted line assumes that when leverage is less than one any uninvested funds return zero. (Some dotted lines are hard to see but they all converge onto the same point at zero). The actual value of the risk free rate of return (in this case 4%) can be seen as the intercept of the solid curves when the leverage is zero. We have assumed for simplicity that the risk freerate has been constant over the period. The effect of this investment is to raise the returns alittle when leverage is low— the curve is twisted up near zero. The black curve is a leveraged investment in the index itself as would be produced by aleveraged ETF without fees. The black circles show integer values for the leverage. So theblack circle that intersects the grey vertical line is a 1x leveraged ETF. The next black circle is 2x etc. Up to 4x leverage is shown. The effect of volatility drag is evident on all the strategies including the leveraged ETFs. All the curves have a peak and then drop down to (not shown on these charts) a return of -1. The sharp dropoff down to -1 is a result of the 1987 crash where the S&P 500 had a return of

-20%. Any strategy having a leverage of 5 or more lost all their assets on that day. Due to theone-off nature of the crash and to the high volatility of the strategies on the right hand side ofthe chart we shall confine our interest to the zoomed-in area of the chart near the vertical greyline which is a realistic level of volatility for a fund manager to consider. We choose the value of c that gives the strategy the same volatility as the index volatility. We call this the 1x instance of the strategy. Here 1x refers to a multiple of the index volatility rather than the index return3. The strategy returns can be read off the chart and are given intable 3. All the strategies beat the index but, more importantly, met our expectations. CVSdid well but not as well as OVS and this was expected because OVS used the exponent 2 in theoptimum Formula (2) whereas CVS only used an exponent 1. OVPMS did better than OVSbecause it used the term in the formula whereas OVS set = 1. And OVPMS+ did better than OVPMS because it used data-snooping to optimise the formula m(si ) used to estimate as a function of. OVPMS+ gives us an idea of how well OVPMS will perform in the future with a calibrated m(si ) function. It only provides a small improvement over OVPMS whichsuggests that the daily dynamic updating of the m(si ) worked quite well as an adaptive method of estimating the optimal formula. MOS uses just the part of Formula (2) and did less well than any of the strategies thatused. This indicates that our excess returns over the index did not come just from the factthat returns are high when volatility is low. Further returns came from the denominator of theformula. For all our strategies and for all the markets studied the correlation between dailyreturns and leverages is about 0.005 to 0.02. This is very small— too small to be noticed ifplotted on a chart and it is not statistically different from zero (P = 0.50 to 0.10 typically). But it is large enough to create a significant timing boost to returns. We have charts for all the other markets and for brevity they are omitted here. The results are summarised in Table 2 and are consistent over all the markets and time periods studied. Some special bear, sideways, and bull market time periods are considered below. Figure 7 shows for the strategies CVS 1x, OVS 1x, MOS 1x, and OVPMS+ 1x the dailyleverages used. We see that only CVS keeps maximum leverage down below 3 most of thetime. The other strategies concern us a little because we start getting nervous when volatility 3 For a leveraged ETF the designations 1x, 2x,.. with equal validity could refer to a multiple of the return or the volatility

exceeds that level because of the encroaching risk of ruin. The OVPMS+ strategy actually gets leverage as high as 12.5 which must surely qualify as dangerous because an 8% drop inthe market would ruin the investment. The grey vertical line shows the 1987 crash and wecan easily see that the strategies anticipated the event by reducing leverage. This is discussed further below. Figure 8 shows, on a log scale, the equity curve for the strategies and the equity curves forthe S&P 500 at 1x, 2x, 3x, 4x. The strategies do not do as well at times as the 3x and 4xleveraged ETFs but they all have volatility 1x whereas the ETFs have volatility 3x and 4x. The ETFs also suffer severely at the 1987 crash. Table 2 shows the performance of the strategies for the same range of markets and timeframes as Table 1. As a rough rule of thumb the CVS strategy seems to return about twice asmuch as the market and OVPMS+ three times. All the strategies are the instances whichproduce the same volatility as the market (1x instances). This table provides compelling evidence that the strategies have worked very well in the past. Whether they will continue to do so in the future is an open question.

Annual CVS OVS MOS OVPMS OVPMS+ Index Return Return Return Return Return Return US Stocks 0.0393 0.0647 0.0837 0.0572 0.0953 0.0917 S&P 500 0.0699 0.1012 0.1209 0.0950 0.1260 0.1292 Dow Jones Industrial 30 0.0446 0.0855 0.1046 0.0678 0.0904 0.1071 NASDAQ 100 0.0769 0.1883 0.2540 0.1708 0.2606 0.2681 Russell 2000 0.0640 0.1393 0.1920 0.1296 0.1857 0.2062 Australian All Ords 0.0650 0.1095 0.1371 0.1050 0.1413 0.1516 Nikkei 225 -0.0000 0.0317 0.0920 0.0605 0.1228 0.1552 FTSE 100 0.0370 0.0578 0.0710 0.0720 0.0767 0.0747 NZX All 0.0441 0.1020 0.1565 0.1650 0.1952 0.2060 NZX 10 0.0504 0.0744 0.1030 0.0973 0.1116 0.1394

Table 2: Indexes and Returns for 1x Instances of the Strategies

5.2 Extreme Volatility Risk Performance

5.2 Extreme Volatility Risk Performance

Figure 9 shows the returns of the strategies versus the extreme volatility (SDDR+2). Thisproduces dramatic changes in the rankings of the strategies. CVS with its almost constant volatility and smaller vovo produces returns as good as OVSand both strategies beat the strategies that use the m(si ). It seems that any method thatincorporates m(si ) gets an extra boost in vovo which reduces its effectiveness in this riskframework. How good is the constant volatility of CVS? Since many managers estimate volatility usinghistorical volatility and since investor perception of volatility comes from experience of recent volatility let us estimate the volatility of CVS using 63-day (corresponding to a three monthly reporting period) historical volatility. This is plotted in Figure 10 with the S&P 500 index 63-day volatility in the background. A SDDR+2 horizontal line is drawn for each series. Bothseries have the same mean level of volatility (shown as a grey horizontal line) but CVS is lessextreme both upwards and downwards. CVS is still not as smooth as we might like and this isan area for further research and optimisation. We have looked at extremes in the volatility - what happens to extremes in the returns?

One measure of extreme returns is the kurtosis of the returns which measure how ‘fat’ thetails are. It is known (Poon and Granger 2003) that vovo increases kurtosis in the returns

5.3 Maximum Drawdown

so a reasonable question to ask is whether smoothing the volatility reduces vovo. Figure 11 shows the kurtosis of the returns for the strategies for various indexes (the lines connecting thepoints have no meaning and are there to guide the eye). We see that the CVS, OVS, and MOSstrategies produce less kurtosis but the other strategies increase the kurtosis to exceed that forthe index.

5. 3 Maximum Drawdown

Kurtosis measures the tail risk of returns and SDDR+2 measure the tail risk of volatility. Butinvestors are not concerned with these risks per se because volatility brings both rewards alongwith losses. What really keeps investors awake at night is a sustained sequence of losses thatproduces drawdown in the investment. So we look at maximum drawdown which is defined asthe greatest peak-to-trough loss in the investment in any given time period. We use a timeperiod of one year for our investigation so that we get more observations although in practice longer periods such as a decade might be more useful. A sustained sequences of losses can occur with both high and low accompanying volatility so it is not obvious that reducing volatility reduces drawdowns. Magdon-Ismail, Atiya, Pratap, and Abu-Mostafa (2004) find that in a simplified random walk model expected maximum draw- 2 down approximately increases with and decreases with where these quantities refer to the

daily returns4. Even though stockmarkets follow paths more complicated than simple random walks— particularly in regards to the interaction between returns and changes in volatility— itseems reasonable to expect drawdowns to reduce with reduced volatility. It is even less clear what the effect of vovo is on maximum drawdown. Vovo both reduces and increases volatility so may reduce or increase drawdown which may reduce or increase expected maximum drawdown. In the absence of a formula for expected maximum drawdown for processes with vovo we did some simulations of EGARCH models. We defined vovo to bethe standard deviation of the log daily EGARCH(1, 1) volatilities and allowed this to rangefrom 0 to 3. We fixed the mean daily volatility and the mean daily return to be the sameas those for the S&P 500. Then we ran enough simulations to get a smooth curve of meanmaximum drawdown versus vovo and plot the curve in Figure 12. We see that mean maximum drawdown does increase with increasing vovo. This is a useful result because it suggests thateven if risk smoothing does not produce benefits in the tradition risk framework (i.e. produces alpha) it can still reduce maximum drawdowns. Let’s look at some real data. In Figure 13 the period from 1950 to the start of 2010 has been divided up into individual years and the maximum drawdown has been calculated for each year for the 1x version ofeach strategy. Since the strategies apply leverage inversely proportional to volatility then theyincrease maximum drawdowns in years with low volatility and decrease maximum drawdowns inyears with high volatility. This makes it difficult to get an overall impression of which strategy works best. But it is evident that the market (black) has the highest maximum drawdowns sothe strategies improve on the market. Comparing the CVS (cyan) and OVS (green) we see that OVS almost always has less maximum drawdown than CVS even though CVS has less vovo. This is because OVS has a higher mean return than CVS and return is more important thanvovo in reducing drawdown.

4 In fact they show that expected maximum drawdown over an extended period of time is approximately proportional to 2 = which, interestingly, is inversely proportional to our optimum leverage. It appears thatmanagement of the optimum leverage may be equivalent to management of expected maximum drawdown.

5.4 Alpha Beta charts

Annualised Annualised Mean Daily Strategy Volatility Return SDDR+2 Leverage Alpha Beta S&P 500 0.153 0.070 0.283 1.000 0.000000 1.000 CVS 1x 0.153 0.101 0.191 1.303 0.000147 0.896 OVS 1x 0.153 0.121 0.222 1.380 0.000270 0.731 MOS 1x 0.153 0.095 0.205 1.257 0.000113 0.932 OVPMS 1x 0.153 0.126 0.291 1.310 0.000330 0.594 OVPMS+ 1x 0.153 0.129 0.264 1.352 0.000336 0.612

Table 3: Statistics of the Strategies

5.4 Alpha Beta charts

Now we get to a paradox about leverage and beta. Traditionally they are thought of as measures of the same thing. But here we find it is not so. Figure 14 shows plots of the strategy daily returns versus the S&P 500 index daily returns for the instances of the strategies that have the same volatility as the S&P 500 index (the 1xversions, OVPMS 1x omitted to save space since it is similar to OVPMS+ 1x). Alpha and Beta are the intercept and slope of the regression lines— the least squares fit to the equation StrategyReturn = + IndexReturn. measures the amount the strategy returns for eachunit of return in the index. It is thus a measure of leverage. Table 3 shows that all the strategies have a less than one. On average, the strategies move less than the index moves. The table also shows the mean daily leverage that the strategies applied to the S&P 500 index over the duration of the study. The figure shows the mean leverages as a blue line. Allthe strategies have a mean leverage greater than one. This explains why the strategies beatthe index— the index has a long term trend that is upwards and the strategies have a leveraged exposure to this trend. The vovo strategies in a sense apportion out this leverage so that theoverall volatility is not increased yet the overall leverage is increased. So why is telling us the exposure to the index decreased overall? The reason for theparadox is that in the regression of daily returns the points with the most leverage have thelowest volatility and so tend to be near the origin of the charts and have less influence on theregression line5. 5 Statisticians would say that those points have less “leverage” on the regression line but we won’t use thatstatistical term here because it could be confusing.

5.5 Sideways and Bear Market Performance

5.5 Sideways and Bear Market Performance

The Nikkei 225 index from 1984 to 2009 is an interesting case of a sideways market. This indexwas 10149.00 on 14 June 1984 and 10135.82 on 12 June 2009. So the annualised rate of return was -0.0053% over this period which is practically zero. This lets us see how the strategies performed in a market that would not benefit from leverage under a buy-and-hold strategy. The strategies that use mean prediction (Table 2 OVPMS and OVPMS+) did very well inthis market. It appears that using volatility to predict returns is an effective strategy for the Nikkei 225— we got 12.5% or 16% annual returns out of a market that went nowhere. The next index we look at (Figure 15) is the S&P 500 from 24 March 2000 to 9 March 2009 when the index went from 1527.46 to 676.53— a sustained bear market where the annualised return was -8.72%. All the strategies beat or equalled the index but they still lost money overthe period. So it appears that in a bear market when leverage is expected to amplify losses thevovo strategies do no worse than the index. But they do not turn a bear market into a bull.

5.6 Bull Market Performance

5.6 Bull Market Performance

9 Dec 1994 to 24 March 2000 was a bull market where the S&P 500 index went from 446.96 to 1527.46— an annualised rate of 26.11%. Figure 16 shows that the vovo strategies outperformed the index but not by much. The reason is possibly because this was a period of low volatility and low vovo. The vovo strategies did not have much vovo to work on.

5. 7 1987 Crash Protection

We are interested in two aspects of the 1987 crash — did the strategies protect the portfolio from the downside of the crash and did the crash come close to ruining the strategies?

Table 4 shows the leverage that the strategies determined the day before the crash couldhave been used, successfully, the day of the crash. It can be seen that the strategies greatly reduced their leverage for the day after the crashso to a certain extent they reacted as panicked investors and missed some of the next day’s

5.8 The Long-Only Maximum Leverage 1 Portfolio

Strategy Leverage Leverage 19 October 1987 20 October 1987 CVS 0.455 0.160 OVS 0.152 0.019 MOS 0.568 0.256 OVPMS 0.034 0.002 OVPMS+ 0.058 0.003

Table 4: Special Leverage Values of the Strategies

bounce. But the important point here is that all the strategies saw the crash coming and had agreatly reduced exposure to it thereby gaining overall (relative to the market) from the crash. Any conclusions we make here constitute data torturing because the crash was a one-time-only event and is (hopefully) unlikely to ever be repeated due to measures put into place since then (such as closing the markets when they move too far).

5.8 The Long-Only Maximum Leverage 1 Portfolio

Our vovo strategies had no limit to the amount of leverage applied but we need to look at howwell they perform when they are restricted to a maximum leverage of one. Figure 17 shows theresults for the Dow Jones Industrial Average for 1984 to 2009. Setting the maximum leverage to one means that the volatility of the strategy cannot exceed the volatility of the index. Sowe only show the region of the chart to the left of the index volatility.

6 Conclusion and Directions for Further Research

The most interesting result is that our returns are reduced to about the index return. Thisis perhaps to be expected since our excess returns come from having mean leverage greater than one in favourable markets so the maximum leverage of one will reduce the mean leverage to below one. But this is not a law, however. By reducing leverage when the index is doingpoorly means that the mean return can exceed the return of the index. For some indexes suchas the NASDAQ 100 (omitted for brevity) the returns do exceed the index returns. But not bymuch. A limit to the actual and mean leverage does impact strategy returns. The chart shows that we can approximately halve the annualised volatility down to about 0.1 while slightly increasing returns. So the effect of the strategies on the traditional portfolio is likely to be a reduction in volatility rather than an increase in returns.

6 Conclusion and Directions for Further Research

Returns are not easily predictable but volatility is. By allowing for leveraged investing we haveintroduced a formula for compound returns that depends on volatility thereby introducing an element of predictability into returns. We have identified three sources of alpha from thevolatility predictability:

The extra overall leverage allowed by the apportionment of leverage without increasing overall portfolio volatility

The relationship between returns and volatility and the use of extra leverage when returns are higher

The deleveraging prior to the 1987 crash

We have introduced three simple intuitive investment strategies that use these sources togenerate alpha in the various markets that we have tested it in. The three strategies are: the Constant Volatility Strategy (CVS) which aims to achieve a constant daily volatility by predicting the volatility and setting a leverage value to aim atthe target; the Optimal Volatility Strategy (OVS) which sets the daily leverage to be inversely proportional to the predicted volatility squared; and the Optimal Volatility Plus Mean Strategy (OVPMS) which estimates the expected return as a power function of volatility and sets thedaily leverage to be the expected return divided by the predicted volatility squared.

6.1 Multiple Assets Model

All three strategies manage to get the upsides of leverage without the downsides— theygenerate excess returns in bull and sideways markets and do no worse than the market in bearmarkets. If they are restricted to leverage no more than one then they manage to greatly reduce the volatility of a portfolio without significantly reducing returns. And all three strate- gies reduce the volatility of volatility of the portfolio which allows the portfolio manager todeliberately seek more volatility than normal. The order of performance of the strategies in increasing order when measured using risk- adjusted returns is CVS, OVS, OVPMS. But of the three strategies the one that we prefer isthe worst performing one CVS. We prefer this because:

It allows us to seek a specified constant target volatility that can be put into a prospectus and in practice the realised volatility is much less variable than the market and otherstrategies

It usually has leverage less than 3 so has the least maximum risk of all the strategies (andthe least probability of ruin) and can be implemented using existing 3x leveraged ETFs

It has the best return profile when measured against the maximum lifetime volatility ofthe portfolio (as measured by mean volatility plus two times the standard deviation ofvolatility)

It produces daily returns with the lowest kurtosis of all the strategies

It has been said that diversification is the only free lunch in investing. It appears that onceyou have diversified away some risk you can get a further free lunch by smoothing what riskremains. The following subsections discuss a few directions for further research currently under in- vestigation.

6.1 Multiple Assets Model

We only considered the volatility of stock market indexes in any detail which is a univariate situation. When we consider multiple assets we get not only volatilities but correlations between 1 returns. Then Formula (2) becomes k = where is a variance-covariance matrix and kand are vectors. We might consider a multivariate EGARCH model for estimating.

6.2 Interaction of Volatility Changes and Return

We have ignored the relationship between volatility changes and return. We have only estimated return in as much as it relates to absolute levels of volatility rather than to changes in volatility. But there is evidence that returns and volatility changes are highly negatively correlated and inmore complicated ways than given by in Equation (3). For example Bouchaud, Matacz, and Potters (2001) find in their “Retarded Volatility Model”that the correlation effect for individual stocks is moderate and decays over 50 days, while for stock indices it is much stronger but decays faster.

6.3 Implications for Technical Trading Rules

The above analysis investigated dynamic leverage as an overlay for buy-and-hold strategies. But some investment strategies (such as those using technical trading rules) that use leverage of either zero or one cannot use a dynamic leverage overlay. An example is a strategy that saysto be in the market (leverage one) when the market is above its 200 day moving average elseit is out of the market (leverage zero). For those strategies a volatility overlay has to either beconverted into modifications of the existing timing signals or modifications of the zero and oneleverage values (position sizing). To achieve this the technical rules have to be represented in a framework where the pre- dictability of volatility can be used. This is an interesting area for research.

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