L$-USD exchange rate dynamics -- theory vs. observation

I've just finished a preliminary modelling of the dynamics of the USD-L$ exchange rate over the past year (from November 1, 2004 until GOM closed on October 2, 2005). I'm in the process of writing up the results more formally, but here's the executive overview.

I modelled the evolution of the exchange rate with a Cox-Ingersoll-Ross model. This was chosen because it has some desirable properties that seemed to fit well with observed dynamics. The CIR model requires three parameters -- Long-term average (the exchange rate the market will trend toward), volatility (the amount of randomness in the exchange rate), and mean reversion (the speed with which the rate will trend toward its long-term value).

First of all, the CIR process is mean-reverting, which means it tends toward a long-term average as time passes. I was able to demonstrate that by farming the stipend from premium accounts (and assuming a few things about market price of risk, USD interest rate, etc.), the USD-L$ exchange rate should tend toward approximately $2.88 per block in the long term. This is the value that was used in all simulations.

Secondly, the volatility of the exchange rate depends on the rate itself. As the rate gets smaller, the volatility will decrease. This keeps the exchange rate from becoming negative, and also matches observed market behaviour. A base volatility of 0.5% per day was used in the simulations. Again, this value is consistent with typical intraday movements seen in the market data.

With two of the model's parameters fixed, we were able to tune the mean reversion rate to match observed trends. It is important to note that a single, constant reversion rate was unable to match the market data. This is due to a sudden increase in market liquidity that occured when "partial order filling" was introduced into GOM's trading system. At this point, the mean reversion increased suddenly and significantly, going from a value of ~0.5 to a value of ~2.5 over a timespan of a few days.

Finally, the CIR model was augmented to include jump discontinuities. The jump term is intended to model the effects of Linden Lab actively managing the economy to try to keep the exchange rate near $4/block. Jumps occur at random (Poisson distributed) intervals with an average of two jumps per year. The jump magnitudes have Gaussian distribution with zero mean and a standard deviation proportional to the current exchange rate's distance from Linden Lab's stated target rate of $4.00/block.

Results are summarized in the attached plot. This plot contains the volume-weighted average market exchange rate for the time interval in question, and two simulated exchange rate evolutions. Note that the models are able to faithfully reproduce periodic exchange rate increases (boom periods), sudden rate changes (tweaking of the economy by Linden Lab), and the long term trend downward.

In summary, these results indicate that the exchange rate moves inexorably downward toward its long-term target of $2.88/block. This downward trend was accelerated by the increased market liquidity introduced by the addition of partial order filling into GOM. Significant intervention is required if Linden Lab wants to keep the exchange rate at $4.00/block. This will need to take the form of a reduced stipend or increased premium account fees (back-of-the-envelope calculations indicate an average stipend of ~L$360 per week or an annually-billed premium account cost of ~$100 should stabilize the exchange rate), or a revision of the target exchange rate to ~$2.88/block.


Comments are highly encouraged.

You should try to get that VP os Finance position at LL

Seriously though, thanks for this analysis I find it very useful.

So basically, the sky is falling?

The sky is... trying it's hardest to fall.

On a side-note my estimate for the long-term value of L$ was $2.1... I was only $0.78 off (or 27% off, ouch)

The growth of SL absorbs some L$ and injects a lot of $ back, I think. So it exerces a good upwards force on that "sky"

As long as it's possible to leverage premium accounts to get free money, though, the rate will seek its equilibrium. Jauani has posted a nice summary of the advantage to a consumer-type SL user of buying premium accounts versus paying for L$ through a reseller. It's only a small step from there to farming premium accounts for L$ to exchange directly.

The only reason the target rate isn't zero is that the opportunity cost associated with registering 1,000 free basic accounts to farm for their L$50/week is too high relative to the total payoff.

I wouldn't underestimate LL's anti farming logic.

After all, this problem is fundamental, one would assume they've put significant man hours into coming up with a solution.

It's also likely that they wouldn't spend a lot of time telling us about the solution.

There's really no evidence they've put in *any* time coming up with a solution. As you say, this is a fundamental economic issue that has existed and been publicly discussed for about a year and a half now.

Now, either they have a solution, or they don't. None of that matters, however. All we can go on is the market data, and that shows a strong downward trend that has been well approximated as a CIR model trending toward $2.88/block. If they come up with a solution that works, then that's fine -- the model will have to be revised and recalibrated. Until then, I'll trust the math.

The high-mean-reversion tail of the process is unambiguous. I've tried to model it with an assumed long-term average of $4/block, and it simply can't be done. Unless we've spent the past five months living 10 standard deviations outside the mean. And I don't buy that.

While the mathematical models are interesting, one has to wonder if there is a quantum effect of sorts going on.

Basically, as volume approaches zero (relative to large forex exchanges), the laws of economics start to change.

ie: since the volume of SL isn't really that high, other factors have come into play which are effecting price rather than simply the laws of supply and demand.

Or, perhaps, the supply and demand is being effected by factors which are more impactful at these levels.

For example:

1. Having to relog every time you sell, buy, or cancel
2. Having all purchases go to the credit card (detering traders)
3. No buy side so it's harder to move the market

etc.

I agree that your "quantum effect" would be a non-negligible component. That's why the marked increase in liquidity when partial fills was introduced is so noteworthy. The ability to instantly convert L$<->USD allows for greater market liquidity and, therefore, greater applicability of the mathematical model. The June-October tail is the region of the model I currently have the most confidence in.

Now, the other factors you quoted are particular to LL's new exchange and should affect the prices listed there. Maybe that's LL's plan to limit stipend farming -- make selling through them a PITA.

The supply of premium accounts for farming purpose is not unlimited though. There is one limit per person. This means everybody can only buy so many accounts.

The only problem with this assumption is the fact tracking someone's identity is so damned hard if they have multiple credit cards, IP logs, and/or mailing addresses. Like the debate on subsidized first land, all bets fly out the window when someone pushes, hard, to cheat the system. But, as noted, there's a limited realistic number of identities one person can have within the system.

That said, this is the first money thread in a hell of a long time that I actually enjoy reading. It's very interesting to me, as a Jr. Mathematician, to see the assumptions at play and simulation data presented in such a professional manner. My hat's off to you for this analysis, Ricky.



I would also be very interested in a GDP value for the fiscal year ending October 2, 2005. That would give us a good shot at speculating the consumer-to-creator ratio present in Second Life, as well as other interesting statistics.

For now, I will withhold judgment on the LindeX's overall effect on the economy, save to say it puts LL in a strong position for managing opportunity cost, similar to the Fed. I speculated on this outcome several months ago, and personally see LL's move as more or less inevitable.



And without rehashing the LL-eats-GOM debate, I do think it puts them in a tenuous position of power to micromanage the economy. Recently select Lindens have even resorted to some fiscal policy within their own economy, for fear adding L$ to the economy to fund private ventures would spark inflation. Very interesting.

Though, honestly, I'm not convinced they know enough of what they're doing to manage it effectively. Yet.

I'm actually impressed with the market's ability to remain near 260 for so long.

I agree with the assessment that too much funny money is flowing into SL. I think the ease with which the rank and file can sell their stipends will tell in the long run.

The new classifieds will introduce a small sink. I would expect LL to introduce even more sinks, but I don't see them stemming the flood of $L they create with stipends and referral bonuses.

Buster

One possibility is that with the switch to lindex, farmers have been pulled from the system.

Every analysis, regardless of its level of sophistication, stands or falls on its initial assumptions. It seems to me that by founding every simulation run on the same assumption that everything is actually trending to a fixed Ricky-decided $2.88/1000 you have prejudged the result and predecided the conclusion.

If you had chosen another figure doubtless the simulation parameters relating to other variabilties would have compensated to achieve an equally good fit to the data.

We cannot conclude from this that the $2.88 figure has any reality derived from or reinforced by your simulation runs, since you assumed it before running your simulations at all. It started as a Ricky guess. It ends as a Ricky guess.

What is your purpose in this posting Ricky? I could achieve good data fit with any number of models on a wide variety of assumptions.

It seems to me that the most likely effect of your post on less knowledgeable readers is to give them the impression that some brilliant analytic process has proved that the L$ will trend in future to 2.88, and that this is somehow the "correct" figure.

If anyone comes away with such an impression, and many will, it is entirely erroneous. No such thing has been demonstrated.

If there is an argument in favour of the 2.88 figure it is a very simple one relating to stipends, one which anyone can understand if you care to explain it, and could then also judge for themselves just how valid it is on the basis of how much trouble this "farming" is to do, and whether it is likely on any significant scale.

I judge its current significance as trivial, and the US$2.88/L$1000 figure to be worthless for any practical predictive/analytic purpose.

However, I'm not dogmatic, and I'll go check a few figures in a minute, to estimate what percentage of the market could at most sensibly be involved in this "farming" activity you have assumed to be so very significant as to be the underlying determiner of the market.

For the attention impaired, here is a rather eloquent summary of Ellie's post

An off the wall thought (I'm not good at economics) - maybe LL doesn't really care if our own Lindens are worth $2 per thousand or whatever as much as they care about real dollars being spent by real corporations like Wells Fargo to pay to buy their Sims in real dollars and pay to hire people from SL to design them in real dollars. (And treat and report it all in real IRS wages fashion.)

After all, those of us who don't cash out our Lindens are happy to create things and sell them back and forth in play money.

Course I could be totally off the wall with this. But it's the point of view of someone who doesn't sell Lindens.

coco

Ellie, my goal in performing the analysis was to investigate whether or not a well-established quantitative model could explain the exchange rate movements observed over a year of trading. I've been able to demonstrate that, yes, a modified CIR model with term-structured mean reversion is a good fit to the empirical data.

Your key issue with my analysis seems to be with my choice of $2.88 as the value of the long-term average. I'll explain my reasoning behind that now.

1) I can spend $72 today and receive periodic cash flow of L$500 per week for the next year. Assuming a starting rate of $4.00, I would need to convert L$18,000 into USD to pay for the account.

2) After 1 year, my L$18,000 has turned into L$26,000. That is an annualized rate of return of 44% or a continuously compounded rate of about 37%. We'll call that the base interest rate of a L$ investment.

3) If I were to purchase risk-free USD treasuries, I could realize an interest rate of about 4%.

4) In a multi-currency market, the 1-year forward exchange rate (i.e. the rate we expect to see in a year) is given by S_t = S_o * (1+r_d)/(1+r_f). That is obtained by equating returns in the foreign- and domestic currencies for a unit investment and is a well-known formula.

5) Substituting numbers, we reach a forward rate of approximately $2.88.

Basically, I can either put my $72 into USD treasuries and get 4%, or I can put it into L$ and, if enough people are trading, the market will move to give me the same effective USD return. This is independent of *how much* farming is happening, and simply quantifies the *possibility* of farming.

Next, I assumed an intraday volatility of about 0.5%. That follows from an examination of the market data itself. A volatility coefficient of 0.5% corresponds to a mean intraday standard deviation of rates of roughly $0.04 (95% of the time the rate today would be within 8 cents of yesterday's rate).

Jumps were added by assuming an average of two market corrections per year. Since this is about the frequency with which LL announces a significant change in policy, it seemed reasonable. Since it has been stated in the past that LL likes to see a rate around $4, the magnitude of the jumps is proportional to the distance of the current rate from $4. e.g. a low spot rate will yield a larger jump correction that one that happens when the rate is closer to $4.00.

The leaves only a single parameter that can't be pinned down with the data I have, and must be tuned by hand -- the mean reversion rate. This is the only parameter that deals with the amount of farming going on, and was adjusted to provide a reasonable fit to the market data. Now if, as you say, the degree of farming is small then the mean reversion parameter should also be small, and the market should just bounce around stochasically near the starting rate of $4.00. That is clearly not happening, which implies that there is enough stipend sell-off (perhaps not premeditated farming, but simply people who have no need for their stockpiles of currency) to drive down the rate.

In fact, I would say it is a credit to the model that it effectively discovered the date that partial fills were introduced. At that point the mean reversion increased markedly, and the initial model was no longer a good fit to the data.

Is the long-term rate definitely going to $2.88? No, not for sure. It is, however, dropping in a manner consistent with a trend toward that value.

I'm just not even sure what that means or how it at all relates to this discussion.....

I don't know about using your CIR model but I highly doubt 2.88 would be long run. Instead of complicating things by taking all these variables here is how I would simply calculate it.

{[$72(1 year subscription) + $5(transfer SL funds into paypal)]/.965(L sell transaction fee)/26(year worth of stipend)}

{[(72+5)/.965]/26} = $3.069

Then factor in the rate of return on your investment. You suggested 4% which is for riskless investments. I wouldn't consider investing in SL riskless, in fact I would consider it very risky. Exchange rate fluctuations, LL could go out of business, SL can change stipends, can take your chars, close your accts etc. Factor in all of these risks you should be expecting a return well into the double digits.

RAndon number, say you expect a 10% return at least for the risks(I personally would expect much more for the risks involved) So 3.069*1.1= 3.376.

So about 3.38 which is 50 cents greater then the number you got.

You don't know the Flying Spaghetti Monster? Heresy!
What I meant was that Ellie seemed to be accusing you of trying to torture data into fitting your model, as the people behind "Intelligent Design" do. The FSM is a parody of ID: http://www.venganza.org

That graph has become a bit of a symbol for flawed logic, silly or biased statistical studies, and the media's lack of understanding and subsequently poor coverage of real studies.
But hey I don't want to start any drama. I don't care about the stupid linden dollar... it was an inside joke.

Could anyone tell me why parity wouldn't be desirable as a long-term target? (Or even stronger ratios?) So, 1$L to 1$USD/EUR, etc.. Or is the supply just too large to even contemplate that happening?

Two of the more straight forward answers:
We'd like to maintain the L$ as a currency with micropayment-like value.
We also don't want to start supporting fractional amounts of L$s, as it would require quite a bit of technical rework.

Cheers,
Lawrence

Technical issues aside, from a fundamental economic point of view 1:1 FX parity would be undesirable/infeasible for a variety of reasons.

Firstly, the exchange rate is *so* far from there, it would require one of: (1) an active peg from LL to get there, (2) a "new L$" revaluation, or (3) extreme growing pains and rampant deflation of in-world prices. I can all but guarantee you that LL won't put up their own cash to make (1) happen. (2) is unlikely because of the technical and aesthetic reasons Lawrence pointed out. (3) is possible if LL adjusts sources and sinks in a sufficiently radical manner that SL becomes a sort of Mad Max-esque hunt for L$ (a market-induced change of 30,000% in the exchange rate would be reflective of the devaluation of the Mark in post-WWI-Weimar Germany) . I suspect another barter currency would arise before the L$ reached that point.

Secondly, if you consider SL as its own "nation", then it would be essentially an exporting nation. The product it exports is entertainment. As with other net exporting countries, all else being equal, a weaker currency has the effect of boosting investment in the exporter's economy. If the L$ strengthens against the dollar, it discourages foreign investors (i.e. all of us) from investing in the exported goods and services.

Those are the fundamental economic reasons why a strong L$ is undesirable.

Sounds like very good news. I will be able to buy more lindens for less money.