Risk frees and currencies: Part 1

Hello friends, it’s been a while. Mostly I’d reckon it has to do with the demands of my work, plus my current efforts to build a decent or crappy systematic trader, which I plan (at least the first iteration) to open source:

https://github.com/thelilypad/systematic-trader

The other facet of this, of course, is that as my following and notoriety grew, my comfort in sharing inane thoughts has overall diminished. I have a ton of drafts saved in Substack at the moment that eventually hopefully and probably I will work through.

Why We Care About Risk

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In my day job right now (Moody’s Analytics) I mostly think about risk. Risk is an interesting concept, and pretty difficult in practice to define as one word. When most retail traders, for example, think about risk, they tend to think in terms of downside risk — what is my worst-case loss if a trade moves against me? In sophisticated cases, we tend to think of risk in terms of some pre-determined limits we will accept, which largely depends on the asset and mandate one has. In terms of options, we usually think about the first-order sensitivities to changes in spot, volatility, and time, which we call the Greeks. For a market maker, risk is often measured in inventory held and variance in spot prices (or its root, volatility). For a fixed income trader, risk might be measured in sensitivity to interest rate changes (one type of risk there, at least), which we call duration. 

In general, people give quite a lot of thought into not only what risk is but how to measure it. This again relates back to the idea of pre-determined limits — to properly constrain outcomes and not pull a Canadian pension fund — we must not only understand what risk is, but figure out:

1. Can we measure what the risk is (which, in most respects, can be restated as sensitivity to X)?

2. Can we figure out if we are overpaying or underpaying for risk?

This is fairly intuitive in trading terms. In the most abstract sense, most forms of trading (or even the big boy cousin, investing) can be understood as a tradeoff between risk and return.If you consistently are overpaid or underpay for risk and stay within your limits (e.g. Kelly sizing), you should in theory make money over time. Conversely, if you consistently overpay or are underpaid for risk, you should in theory lose money over time. Simple.

The crux of this is you need to first understand what we can call a market price for risk. This is because when we define risk, we largely are uncreative about it, and think about risk in terms of additive factors, or in fancy speak, premia. While there are certain financial transactions which net reduce risk exposure (hedging), in general, we can think of risk sort of like entropy - the more counterparties, the more transactions, the higher your net risk. 

This relates to the idea of conditional probabilities - if we imagine a success to be the final outcome of a certain transaction, dependent on a string of intermediate successes each with defined probability, it is intuitive that each successive intermediary reduces the final likelihood of success (increasing the total risk). The simple multiplicative rule works, however, only in the case of independent intermediaries; in the real world, intermediaries tend to show some level of dependency (in hedging, for example, the conditional likelihood of success is higher for A+B than A alone).

Dependency in risk is usually what kills you for two reasons. First and foremost, it often is an unknown unknown, especially before The Really Bad Event happens. An interesting example is the deviation of FX swaps from theoretical pricing after the Global Financial Crisis; once dealers realized that counterparty risk was indeed possible between banks, this meant that charging at the theoretical price (covered interest rate parity) meant underpaying for the true risk. Therefore, a new premium emerged, and several new academic papers along with it.

Secondly, it makes it difficult to model. Much like the n-body problem of physics, in practicality no financial party exists in a vacuum. In the case of equities, for example, it’s well known that in times of crisis, correlations essentially tend to 1 — even unrelated companies will show fairly similar behavior in high volatility regimes. However, the more insidious aspect emerges in the case of hedging or conditional behaviors. As we brutally observed with AIG in 2008, many methods of hedging work well until everyone suddenly needs to be paid out all at once. This can cause essentially massive gaps in risk limits, where slowly but then suddenly, your risk goes from under the limit to much, much higher than expected.

Crypto especially is interesting because even though it is (at this time) a multi-trillion dollar market, risk has largely escaped traditional modeling methods. While assuredly prop firms and shadowy VCs have their own risk frameworks for analyzing whether they’re overpaying or underpaying for risk, the shadow of regulatory overhang and rapid evolution of the space has given it a Wild West-like reputation, with total loss and horror stories commonplace. It’s undeniable that one of the principal problems of crypto in early 2022 is that the space is still so technical, that without thorough background in computer science and finance, it is difficult to distinguish bad from good actors.

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The Risk Free Rate and the Market Price of Risk

In general, to discuss the market price of risk, we must first work backwards and ask, “what is the risk free rate?” The risk free rate we can further subdivide into a “general” risk free rate, a singular latent rate over the entire universe of available assets, and an asset-specific risk-free rate, which reflects asset-specific considerations like asset carry, hedonic utility, or capital constraints. In the case of crypto, the latter is a more appropriate rate to consider, based on difficulty of converting between crypto and non-crypto capital.

The textbook definition of the general risk-free rate tends to be extremely tautological - it is the rate of return achieved when zero default risk is present. Or in more real world terms - it is the rate a blemish-free borrower, who always pays promptly and on time, can borrow it. However, this tends to be a bit unimaginative to me.

In traditional finance, we view the risk-free rate to be an arm of Pax Americana, a specific grace given to the financial world by the hegemony of America and the assumption that like a Lannister, she will always repay her debts. So far, this has held remarkably true, with some close calls related to political kabuki around the debt ceiling ever so often. This is why we view, depending on your cup of tea, previously LIBOR (the interbank overnight borrow rate), the Fed Funds Rate (the rate the Federal Reserve pays overnight deposits), or the yield of the maturity-matched US Treasury as a ‘true’ general risk free rate.

A more interesting thing to note here is that this risk-free rate has two unique properties:

1. The US Treasury and FFR are risk-free specifically because, among all other sovereigns, US debt is largely denominated in its own currency. This is a very special case, because it implicitly means the US cannot default unless it chooses to — having fiat currency, it can simply inflate the debt away if need be (this of course, would impact future demand for said debt, especially if international relations change).

2. It is only risk-free in its own currency - if I am a Japanese investor, my capital and return is denominated in yen, not US dollars. The cost of a treasury and its yield, however, are in dollars. Therefore regardless of the “risk-free rate” provided by the US Treasury, I incur foreign exchange risk - if the exchange rate between dollars and yen changes during my investment period, I may incur loss relative to my initial capital.

We can abstract this and make three fairly reasonable claims:

1. There is no true way for an investor to get exposure to the universal risk free rate — any investor has returns denominated in some numeraire, or benchmark asset. This asset by its own nature reflects some risk. In the currency case, this risk reflects supply and demand for the currency - if the currency I am holding has zero demand, it is essentially worthless. Therefore I cannot achieve true zero risk, simply because my assets are denominated in something which has risk.

2. An asset may provide a risk free rate in relation to itself, which reflects the universal risk free rate plus exchange risk - A salient example here is share dilution. A company is fully free to provide “risk free return” to its own investors, barring regulatory scrutiny, by simply printing and providing shares to each existing share owner. Much like a pie being cut into progressively smaller slices, this will provide return in relation to the asset itself - you will own more shares. However, in relation to the universe outside the asset (for a company, for example, the market capitalization in USD), this will not change the overall value of all existing shares. This is implicitly exchange risk - an asset has risk in relation to its exchange rate to other assets.

3. The real risk free rate is the unique clock time value of an asset - We can neatly observe this by the “store of value” requirement for money by defining an instantaneous moneyness property - an asset can be considered “money” for some time period if a liquid forward market exists for the time period in observation. This is because the existence of the forward contract allows me to guarantee value preservation over the time period in question, and we can understand (via the basis trade argument next) that the rate of return achieved over this period is compensation for agreeing to that value preservation. The implication here is the real risk-free rate of an asset is the lowest compensation required to believe that asset will have some value in the future, regardless of what happens between then and now.

The existence of the risk free rate allows us in general to make reasoned claims about the nature of risk in assets. If we lend out money in an efficient market, all other things held equal, we can understand the return on that loan to be roughly equal to the risk free rate plus our counterparty’s credit risk. This is additive — if this loan can be traded openly, there may be a premium in relation to how liquid the loan is (as I’ve talked about before in liquidity versus fundamental valuation). If the loan is between multiple currencies, there will be foreign exchange risk. And so forth.

So, trivially, we can start our understanding of quantitative risk modeling in crypto and hunt for the crypto Market Price of Risk by pinning down an uncontroversial risk free rate. Traditionally, many in and outside of the crypto community have viewed the basis trade as the fundamental risk-free rate in crypto.

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The Basis Trade

For those less deep in the lore, the basis trade is a specialized carry trade involving purchase of a currency and shorting of the associated currency forward. There is a fairly important mathematical relationship here.

For any given asset with static interest rates and no other gotchas (dividends, cost of carry, Trump tweets, declarations of war from various prior superpowers), the forward value of an asset is simple to calculate:

S = P*e^rt.

This is identical to the idea of compounded interest, because it is compounded interest, or the formula you learn in Finance 101.

The intuitive basis here is that the forward price of an asset - the price of the contract if you agree to buy or sell an asset at a certain date in the future - reflects the opportunity cost of otherwise investing the money now. I can simply do this by buying the underlying asset, shorting the forward, and taking the money gained from the forward and investing it in some sort of bank account (good luck nowadays). This, if interest rates don’t change, is a type of static replication - theoretically, the cash flows from both side should exactly balance at expiry.

However this gets much more complicated when we consider the behavior across currencies.

In the Treasury yield case, the yield of the bond moves inversely to the price of the bond asset (this is true in well, all bond cases I personally know of).

This is a simple case of supply and demand - the bond represents a fixed payout at some point in the future, and when demand is high, I can charge more now for that payout in the future, meaning the overall amount gained owning the bond will be less. Conversely, when demand is low, I must charge less for the same payout, meaning the overall amount gained will be higher. Notably, in the real world, most “risk-free” rates from sovereign bonds represent a fixed rate - when I purchase the bond asset, I am locked into receiving that rate of return for the lifetime of my ownership of the bond. Therefore, marked to market, if rates move higher in the future while I still own the bond, I have recorded a loss. Alternatively, if rates move lower, I will record a gain.

In the absence of a fundamental demand pull (or as nerds call it, the balance of payments model - demand for a currency to buy actual things with said currency), we’re left with two simple variables that should determine the value of a currency in relation to other things: its market price and its return (the interest rate). 

If I am a young, spry currency arbitrageur, I can put on the basis trade (let’s say EURUSD to keep it simple) by purchasing euros through loaned (shorted) US dollars.

If I take the spot EUR (that I bought by “shorting” USD, let’s say at a 1.1 USD to 1 EUR ratio) and invest it at a risk-free rate of 10% APY for 6 months (the ECB has gotten really into crypto in this example), paid out in EUR, I should end the 6 month period with roughly 5.1% more EUR than when I started. Assuming the conversion between USD and EUR does not change, when I sell the EUR to turn back into freedom bucks, this means I’ve similarly made 5.1% on my USD. 

Let’s make the example more interesting and pretend that Jerome Powell decided at the same time to jack up the matched-duration Treasury to 20% APY for 6 months, paid out in USD. If I had instead bought Treasuries like a true patriot, I would’ve achieved 10.51% return in the same time period. Assuming again fixed exchange rates, I effectively lost about 5.4% due to the difference in interest rates!

If we flip this around, we can observe that due to this, if I was a European, I would be crazy, all other factors held equal, to buy European sovereign debt over Treasuries (I’d be losing in opportunity cost). This, by the basic laws of the market, should increase demand for USD and decrease demand for EUR, reflecting in a relative decrease of EURUSD spot prices.

However, as we previously identified, the yield from bond assets is also sensitive to supply and demand. As demand increases for this interest-based trade from foreign investors, assuming supply (Treasury issuance) doesn’t change, I would anticipate that the yield would decrease. This, similarly, would decrease demand for USD (and relatively increase demand for EUR), increasing EURUSD spot prices.

That said, there is plain risk here in this carry change - if exchange rates settle unfavorably, you can record (especially with the use of leverage) quite a massive loss.

An even more clever arbitrageur could try to take advantage of the very liquid currency forward markets. Simply, a forward is an agreement to purchase (or in this case, repurchase) an asset at a fixed time (unlike an option, which is, well, an option to do so).

If the forward was priced in accordance to the home currency (in this case USD) interest rate, then we would be able to conduct a true arbitrage - we would be able to simply long EURUSD spot, invest in euros for the duration of the forward agreement, and sell it back to USD at the USD interest rate-implied forward price. This would give us the differential of interest rates (assuming they don’t change over the time period) as profit.

But, the forward markets are smart, and their traders have just as good information about the world as you and I. It would be crazy if you know there’s an interest rate disparity (in the absence of other factors like liquidity needs) to lose out on that free money. As we discussed above, there are two ways this adjustment will occur:

(1) If the interest rate in EUR < USD, I would expect a forward premium to occur - I would expect the USD to weaken relative to the EUR in the future (which would mean, in quote terms, that the price goes up). This would, in equilibrium, balance out the gains of the carry trade.

(2) If the interest rate in EUR > USD, I would expect a forward discount to occur - this would be the reverse of the above (I would expect traders to sell EUR to buy USD to invest).

This, in theory, should exactly counterbalance the expected arbitrage (this isn’t true in practice due to liquidity, credit risk, and other constraints, but holds mostly true). More interestingly, the flip side is also true — the forward premium or discount represents the market’s ‘best guess’ of the evolution of currency spot prices. In the absence of factors (like, sanctioning the central bank for example), this contains embedded information about the relative interest rates possible to achieve in both currencies — the forward premium or discount should balance out this differential.

In the multi-currency world, this equilibrium gives us a property called Covered Interest Rate Parity. A weaker form, called Uncovered Interest Rate Parity, is similarly implied by the argument - at every point in time during the lifetime of the forward, we expect the change in the two currency prices (above - EURUSD) to equal the interest achievable by investing in the higher interest-bearing (EUR). This is intuitive - to remove arbitrage opportunities, we would anticipate at every time period the currency exchange rate is slightly ticking against us, or we could sell out and achieve risk-free profit.

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What is Risk Free in Crypto?

As mentioned, most literature considers the basis trade to be the final arbiter of the risk free rate in crypto. For a quick primer, the basis trade in crypto is a bit different than what was previously discussed.

A basis in crypto can occur in two separate but implicitly related ways:

(1) Via the use of traditional futures or OTC forward contracts — this is surprisingly less common, and largely is constrained to major coins like Bitcoin or Ether.

(2) Via the use of perpetual swaps - these instruments essentially act as a contract for differences between assets paid continuously. These can be levered (often up to 100x on some exchanges) or unlevered, with the simplified basis being the difference between the spot value and the swap value at each funding time (accounting for interest and magic, often grifty pixie dust).

The basis trade can occur on any exchange that hosts both futures and spot-type instruments, or more sketchily, cross-exchange as well (this however, becomes often inefficient in terms of cross-margining, and has other risk factors). It is an incredibly popular trade, given the high and nearly “risk-free” yields.

Many people claim it is the true risk-free rate in crypto, owing to lack of, well, other obvious risk-free claimants. This has many problems.

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Since this post is already incredibly long, I will save it for the next article. Sorry, but not really, to end on a cliffhanger. In the next post in the series, I’ll discuss what the basis trade looks like in crypto, what a true risk-free rate may look like, and what implications (potentially tradeable, not just pseudo-philosophical navel gazing!) it may have.

Ciao for now, and happy to return.

• Lily