Leverage and the LTCM model

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Permalink Reply by Sachidanand (Sacha) Singh on December 26, 2009 at 3:32am

Many thanks, Brent, it was very interesting.

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Permalink Reply by Brent Wheeler on December 26, 2009 at 4:19am

Yes Sacha... we all know (more or less) the story but I thought the way it sets out the principles is quite clever. I ran a few numbers through a spreadsheet using his equations and - since you can "see" risk right alongside return and what leverage is doing to both its pretty graphic.

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Permalink Reply by Mukund Bhide on December 26, 2009 at 3:04pm

Hi Brent,is it possible to use your spreadsheet data and fit a statistical distribution to test certain hypothesis pertaining to the high leverage situation to measure the risk of business failure.
The principal policy issue arising out of the events surrounding the collapse of LTCM is how excessive leverage could have been constrained.
Wish your suggestions & advise on carrying out some research study pertaining to leverage constrains.
Thanks
Mukund

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Permalink Reply by Brent Wheeler on December 26, 2009 at 4:03pm

The word "model" is a bit too generous Mukund !!! Its a couple of equations which I have attached.... but it does allow you to see what effect a few changes to the parameters do to the risk one is running to achieve a given level of returns.

The yellow cells are user input. I have set it at "zero debt" in the attached and you can see that return on equity exactly equals risk being taken.... just have a "play" with the spread (equity minus debt) by changing the debt levels and you will see what happens.

The policy question then is how much risk does one want to take!!!.

The troubling assumptions are important though:

- perfect liquidity
- close to zero transaction costs
- no arbitrage... etc.

Attachments:

• LTCM leverage.xls, 17 KB

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Permalink Reply by Sachidanand (Sacha) Singh on December 27, 2009 at 12:24am

Brent,
I could not follow your spreadsheet. I have added one on page 2. I also simulated (with normal distribution - mean = asset return and standard deviation = asset risk) with ~11000 trials - the outputs are as per the formula in the doc you had attached.

Attachments:

• Impact of leverage on risk and return.xls, 1.2 MB

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Permalink Reply by Brent Wheeler on December 27, 2009 at 3:54am

Hmmm.... pretty interesting huh!!! I have plugged in some "roughly normal" numbers for western markets. The equation produces about what I would expect but the simulation? A different story.... when modeled (as it is here for a normal distribution.

My "roughly normal" numbers are from empirical observation and thus incorporate the "fat tail" we see in markets. The simulation is a faithful reproduction of the normal distribution - on which all the models are based.... herin may lie the problem for the LTCM approach!

Attachments:

• LTCM leverage.xls, 1.2 MB

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Permalink Reply by Mukund Bhide on December 27, 2009 at 2:21pm

Hello Brent & Sacha
I appreciate your xls results but they do not substantiate or divulge any commonsense behaviorial patterns of these endogeneous variables.
The sample data you propose to investigate with, prima facie may not follow normal distribution,but you may have to use a logNormal distribution.
The need for a Leverage model is all the more a necessity to comment on ideal mix and match asset returns expectations and business failure analysis.
Your views please.
Mukund

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Permalink Reply by Brent Wheeler on December 27, 2009 at 2:36pm

The simulation produces a normal distribution. The parameters I used are ones commonly observed in the market - a mean return of somewhere round 8% for the S&P 500 with sigma 15% - 17% (the return on assets does reflect some leverage).

You are correct to note that there is not any necessary reason to suppose that what we observe in the market is derived from a normal distribution - in fact it likely comes from a fat tailed distribution (a power distribution as Taleb suggests).

What we can see is the outsize increase in risk which leverage creates... on my original (versus Sachas) you can see that at 0 leverage risk = return according to the LTCM equations.

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Permalink Reply by Mukund Bhide on December 27, 2009 at 3:10pm

Hi Brent,
The S&P 500 charestirustic results would definetly not serve,justify or support the current analysis purpose.Neither could simulation based on random numbers help you out here.I wonder why you are not keen in model building in such an inexact business environment & phenomenon.The risk perceptions are so very individualistic and so do the consequent returns.
Mukund

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Permalink Reply by Brent Wheeler on December 27, 2009 at 4:10pm

Bit of misunderstanding here I think!!! My first spreadsheet simply says "here is one model. It is the one LTCM used" Sacha then said, ok lets look at what happens when we observe a series of possible market parameters drawn, at random, from a normal distribution - that distribution being the one underlying all our financial models"

Naturally parameters for any given case will be individualistic as you put it, but they should fall within the broad range of possibilities which the normal distribution describes.... let's waut for Sacha's comment.

Permalink Reply by Sachidanand (Sacha) Singh on December 28, 2009 at 2:45am

Mukund,

“…they do not substantiate or divulge any commonsense behaviorial patterns of these endogeneous variables.”

In the risk/return models to check the impact of leverage on volatility (Brent’s and mine), return is the exogenous variable and risk (equated with standard deviation of equity returns) is the endogenous variable simply because risk (volatility) is caused by variations in returns. The pattern revealed by the model very much substantiates the analytically established formula that Risk of levered equity = (L+1)* Risk of unlevered equity. I had seen and used this formula earlier without trying to figure out if it is actually observable. The excel page satisfied me that if asset returns are normally distributed, equity risk does indeed multiples by (L+1) after leveraging.

“…sample data … may not follow normal distribution, but you may have to use a logNormal distribution”

Though I do not subscribe to it (that returns are not normally distributed but lognormally distributed, which may be a good approximation for short interval returns) page 3 of the attached Excel sheet investigates with lognormal distribution. The expected returns are as per the formula for lognormal distribution i.e. E(x) = Exp(mean+0.5*standard deviation^2) with impact of leverage. The concomitant risk is much more than the formula – which, it seems to me, is for normal distribution)

Lastly, I have taken actual monthly returns on NIFTY ( I have considered Nifty TRI) – page 4. The returns are unlevered for an investor who does not borrow to buy NIFTY. I then consider that the investor has leveraged her position and see the impact of leverage on return and risk - I find just as per the formula (L+1) times standard deviation.

You have written that “The S&P 500 characteristic results would definitely not serve, justify or support the current analysis purpose.” As the purpose is only to see how volatility increases with leverage, returns from any liquid asset, I suppose, would serve the purpose.

I could not understand “I wonder why you are not keen in model building in such an inexact business environment & phenomenon.” Pl elaborate.

Sacha

(Correction: In the excel sheet uploaded earlier, on page 2 please read norminv instead of normdist)

Attachments:

• Impact of leverage on risk and return.xls, 2.4 MB

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Permalink Reply by Brent Wheeler on December 28, 2009 at 3:00am

Well put Sacha... I would advise careful reading of these comments. They explain very well what the reasoning is here.

Now we have:

- a good explanation of the rationale for what we should expect
- a simple equation to calculate it
- an empirical demonstration of it at work

So back to my simple point which was that one can (as written in the article) see both the power and the peril of leverage.

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