Several CTAs or fund managers offer a standard version of their fund, along with a **leveraged version** (called enhanced risk, 2x or 3x fund, etc.). However a simple performance comparison of the leveraged option against the standard option usually makes it obvious that the former does not offer a simple performance multiplier of latter.

I was looking at Conquest Capital Group and their MFS fund (based on the Trend Following benchmark used in their paper discussed here), which offers two such options: their standard fund and a 3x option.

The main fund, since inception (2004), returns 56% (net of fees) with a max drawdown of 14%. Their 3x leveraged option, over the same period, returns 85% with a max drawdown of 47%. The *risk* is indeed tripled, but the *reward* falls fairly short of it. When removing the fees on both funds (1% for the standard fund and 3% on the leveraged fund), the performances are 66% vs. 120%. This is just an example. Most CTAs are similar in that respect and it highlights some effects of leverage.

### Leverage is not a magic bullet

A similar concept has become more mainstream with the recent apparition of leveraged ETFs and the realisation by *some* investors of the large volatility decay that they incur (when they are held on a long term-basis, i.e. anything longer than intra-day). Check the long-term charts of FAS and FAZ (3x leveraged bullish and bearish financial ETFs): the only long-term profitable trade would be to short both, to cash in on the volatility decay while being hedged (since Nov 08 they are down respectively 58% and 98%).

This is simply due to the design of these instruments, which aim to replicate and triple the daily returns of the tracked benchmark. This highlights a fact about leverage: **multiplying the arithmetic returns will not result in the same multiplication of the geometric returns**.

### Leverage with Position Sizing

A typical way of leveraging a trading strategy is to increase the position size, and the resulting risk and return for each trade.

Assuming a fractional betting strategy for position sizing, the leverage dictated by the value of the fraction of capital risked on each trade will impact the overall average geometric return. Theories describing this phenomenon are the Kelly criterion and Ralph Vince’s Optimal f.

One of the main implications of this type of Money Management is that, for every strategy, there is an optimal fractional position size to maximize the (past) geometric return (and future if you assume that the past is a good representation of the future). Any fractional position size higher than this optimal value would result in a lower return.

Any strategy (with its resulting stream of returns) has an **embedded maximum leverage**, which can be defined with the optimal f.

### Notional funding: another type of Leverage

Notional funding can be compared to trading with *imaginary money*. Typically, CTAs also offer notional funding: they trade your account based on an agreed notional amount, which is different from the actual funds in the account. For example, you could send 100k to your CTA and instruct them to trade it as if it were a 300k account. Or you could do the same thing with your own trading system.

One could intuitively think that this is exactly the same thing as tripling the position size but there is a difference.

Let’s consider this simple example with a strategy producing two consecutive trades of +10% and -5%:

The notional-funded account would close at 300 x 1.10 x 0.95 = 313.50. This would result in a gain of 13.50 on actual funds of 100: +13.50%, which is exactly triple the return one would get on a fully-funded, unleveraged account (100 x 1.10 x 0.95 = 104.50 for a 4.5% return)

Leveraging by tripling the position size on an account of 100 would result in two consecutive trades of +30% and -15%. Final balance would be: 100 x 1.30 x 0.85 = 110.50 – or a +10.50% return.

Initially, notional funding appears a better solution to the leverage issue as it does not suffer from the *volatility decay* or erosion of returns introduced by increasing position size.

### Issues with Notional Funding: Cost

But funding always comes at a cost, pretty much in the same way as an overdraft gets charged by your bank. Another way to look at notional funding is to compare it to borrowing trading capital. The higher ratio of your notional account you borrow (higher leverage), the more the *cost of borrowing* will impede your trading returns.

In practice, you do not really borrow any funds (only by the power of imagination) but this practice still ends up impacting your return when you start taking into consideration the return on margin.

### Impact of Margin Interest on Trend Following Returns

This paper from EDHEC Risk (PDF) looks at Trend Following/Managed Futures performance and presents an interesting breakdown of the overall return of such strategy. Assuming interest being paid at T-bill rates on the margin used for trading, it represents the bulk of the returns compared to actual Trend Following gains and rebalancing gains.

This is one of the reasons for the *hidden cost* of notional funding: a notional-funded account would enjoy the same Trend Following and rebalancing gains as a fully-funded account of the same size, but only a fraction of the return on margin (which is based on the *actual* account size).

To illustrate that point, below is a chart showing the impact of fees and margin interest on an arbitrary performance curve:

Notice the fairly sizable difference between taking return on margin into account or not (red v. blue curves). The green curve adds fees at 1% annually. The return on margin was assumed to be the T-bill rate for that month.

### Risk and Margin Requirements

Another issue of Notional funding used as leverage are the constraints imposed by the underlying strategy.

Any strategy requires a minimum margin commitment to support the positions. Effectively the **Margin-to-Equity** ratio will dictate the **maximum leverage** one can use to trade a strategy.

The **Drawdown** is also a leverage-limiting factor: the more risk the strategy generates (higher drawdowns), the less leverage can be applied. If a strategy regularly exhibits drawdowns at the 40% level, there is a always the possibility that such drawdown appears at the beginning of trading the strategy. Based on the level of leverage, a notional-funded account might not have the time to build enough equity cushion to withstand the drawdown, which would result in the actual account funds going to 0 or a margin call.

Note that in this case, the sequence of trade returns has an impact on the outcome: if the strategy produces gains of 100% followed by a drawdown of 40% a 3x leverage via notional funding would still result in a gain of 60% (300 x 2 x 0.6 = 360, or a profit of 60 with actual funds of 100). If the 40% drawdown appears first, the notional account would shrink to 180, which would result in a negative actual account balance (trading would have to be stopped before then).

### Leverage Comparisons

With these considerations and the benefit of hindsight, here is a comparison between the same arbitrary strategy as above (fees and interest included) traded with:

- no leverage
- 6x leverage using notional funding
- 6x leverage using position sizing

The notional funding *appears* to be the better option. However the position sizing leverage is independent of the order in the sequence of returns – as opposed to the notional funding leverage. If that nasty current drawdown (which is the largest one historically) had appeared at the beginning of trading and carried on further to reach 17%, the notional-funded account would have been wiped out.

The next chart highlights the impact of leverage on the difference between the two types of leverage for that specific strategy:

The target curve represents a simple multiplier (equal to the leverage factor) of the unleveraged performance (which, as we have shown is not attainable). The position sizing leverage clearly exhibits the behaviour discussed in Optimal f theories with the return breaking down past the optimal value.

Hopefully, this gives you a few ideas about how to work leverage in your trading strategies or when sending funds to CTAs. Taking the time to look into it has cleared up a few points for me.

Matt// Apr 20, 2010 at 9:57 amHonestly, this isn’t surprising. But you really lay things out really well with this post.

Leverage is, indeed, a two-edge sword.

Jez// Apr 20, 2010 at 12:46 pmThanks matt.

Not too surprising indeed, although the level of contribution from the margin interest to the overall return from that EDHEC paper is higher than I would have expected (and i think most people do not realise this). It was quite insightful as well to decompose a real fund’s return and play with its various components.

Adam H// Apr 20, 2010 at 1:24 pmJez,

I thought you might be intersted in this post regarding REVERSE notional position sizing.

If in fact you haven’t come across this guy before on the interwebs, I’d definately recommend learning all you can from him.

link to elitetrader.com post

Thanks for the post, I really enjoy your blog.

-Adam H.

Jez// Apr 20, 2010 at 4:28 pmHi Adam- thanks very much for the comments and suggesting new ideas (I’m not a big fan of EliteTrader but if you fing the odd good thing in there, feel free to share it).

I am confused with this “reverse notional position sizing” and what it is supposed to achieve. For the benefit of other readers, I pasted acrary’s post below:

I’m assuming you’d increase size to take advantage of geometric growth of return. If you risked say 1% to achieve 20% returns you’re saying I could risk 2% and have 50% returns. The same level of reward:risk can be achieved by using reverse notional funding. All you’d do is figure the fully funded account size for the 2% risk per-trade then divide each trade by 1/2 to keep the risk at 1% and then achieve the reward of 25%. The two propositions are mathamatically equal so you would have to have a preference for a higher drawdown to increase size.I believe there are flaws in the reasoning above. A so-called “reverse notional funding” would not increase the return without also increasing the drawdown. To illustrate this, I quickly ran a test using the same equity curve used in the post – returning 66% with a drawdown of 13.4%. When multiplying the position size by 3.5, the return came to +293% but with a drawdown of 41.9%. Applying a “reverse notional funding” to keep the drawdown in the same level as the unleveraged equity curve, the return drops to +18.76%.

Motomoto// Apr 21, 2010 at 7:03 amfantastic coverage – all the way through I was thinking, and what about this, and then you covered it!

There is a big different between leverage and exposure that people always seem to miss. Exposure is where the real risk is at, the leverage merely lets you get at that risk….that is why the notional leverage seems the ideal way to go….of course this is still Dependant on the sequence of the trades, the costs of the funding and the drawdowns – and availability of margin calls if you like. Ultimately offering a leveraged version of a fund is really just a marketing ploy, as surely most funds would be offering their clients the best risk reward fund as the original main fund…wouldn’t they? :)

Motomoto// Apr 21, 2010 at 7:10 amside note – I also wonder about the calculations as they are given when it comes to percentages.

Often a straight addition or cumulative approach is incorrect – and is something that needs to be made sure is calculated properly….this might cause confusion in some numbers.

eg; from 100 to 105 is a 5% increase

but for each section 100-101-102-103-104-105 the % returns, gives a cumulative return of only .049029

Jez// Apr 21, 2010 at 3:03 pmThanks Motomoto!

Re: Notional funding, I was thinking a bit more about it and, in summary, if you apply notional funding to a strategy that uses a fixed fractional Money Management strategy, it’s the equivalent of using a martingale-style strategy: upping the stakes in drawdowns and lowering them in winning periods.

Ideal… as long as you dont get hit with a drawdown too big or if your pockets are deep enough to meet the margin call!

Kataphraktos// Apr 21, 2010 at 7:16 pmRegarding the bit about notional funding vs. tripling position size:

If the account is 100k but you are trading it as if it is 300k, how do you do so without tripling position size?

Put another way: in the “notional funding” example, your account after the first trade would be 130 (300 x 1.10). Would the second trade not be calculated using the new account balance of 130 as a base, or 390 x 0.95 = 370.5, leaving the account at 110.5?

So “notional funding” really means, leverage the initial investment, but not any returns on said investment. So if margin was $10/contract on the instrument used in these trades, the first trade and second trade in the “notional funding” account would both be for 30 contracts, whereas the triple-size account would be 30 followed by 39.

These are two black&white examples, one never leveraging the profits, the other doing so in real time. In reality, prudent CTAs – especially those who like to stick to GAAP accounting – set a monthly “reset period” in which they adjust the base account balance and re-calculate the leverage based on this new balance. I prefer quarterly, to match performance fee payout. Others wire profits to clients – and some clients cash them out themselves when they build up.

If a client’s account has shown excellent returns – say that 100k account actually tripled in value to a real 300k – at some point, that client is going to want to see the 300k leveraged to 900k to enhance returns as with the original 100k, if they aren’t pulling the 200k of profits out.

Jez// Apr 22, 2010 at 2:17 amKataphraktos,

The first trade would be an exact tripling of the position size but not after that. As you mention the reinvestment of profits differs. In your example, the second trade would be sized according to the notional balance, which is 330, regardless of the real account size (in practice the position size is less than triple based on the real account balance, which is 130). Note that if the first trade was a loser, the second trade would more than triple the position size (hence my earlier comment about this leverage strategy being a martingale-type money management).

Regarding leveraging/reinvesting the profits: a small correction, in that the notional funding strategy reinvests profit at the same rate as the notional balance growth rate. In your example, the first and second trades would have position size of 30 and 33 (for the notional funding case).

Finally, to reiterate a point made in the post, required margin is a limiting factor. If trading a notional account of 300 with real funds of 100, one could not get a position size of 30 contracts at $10 margin/contract because the margin requirement (300) is higher than the real account equity (100).

Jez// Apr 22, 2010 at 2:28 amNotional funding should really be seen as trading/investing in a fund “on margin”. It can simply be used to avoid sending all available funds to a CTA (whether for risk reasons or optimisation of capital). It can also be used as pure leverage (using the same prudence as one would do when trading on margin).

Campuzan// Jun 17, 2010 at 9:21 pmReducing position size during drawdowns more than proportionaly as compared with the level of drawdown has 2 healthy advantages: avoiding the risk of ruin and logically …. minimizing the level of dd.

Proper use of leverage in any long term trend following strategy is used both to maximise profits during trending periods but also to “recover” from drawdown at a faster pace when the market turns up.

Gdoban// Feb 2, 2011 at 3:27 amAgree with Jez.

The author applied buy-and-hold strategy and did not rebalance between returns. With equity at the end of 1st month = 130 the multiplier is not 3 anymore but 2.5. This preserved profit by applying lower multiplier for the second month. Once you apply proper multiplier (i.e. NF = 390) the final result is exactly the same = 10.5%

Buy and hold strategy has a completely different risk profile. An investor applying buy and hold in highly levered account will go out of busines after a series of negative months.

Gdoban// Feb 2, 2011 at 4:01 amIf returns are -5% and -10% instead of -5% and +10% the buy and hold return on equity is -43.5% instead of +13.5%.